Section 1 Theory of Polynomials

Chapter 1

Abstract

1. Introduction

translation τ in the derived category D<sup>b</sup>

the algebra A have been initiated.

3

Algebras

José-Antonio de la Peña

Cyclotomic and Littlewood

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕAð Þ T as the automorphism of the Grothendieck group K0ð Þ A induced by the Auslander-Reiten translation τ in the derived category D<sup>b</sup>ð Þ mod<sup>A</sup> of the module category mod<sup>A</sup> of finite dimensional left A-modules. In this paper we study the Mahler measure Mð Þ χ<sup>A</sup> of the Coxeter polynomial χ<sup>A</sup> of certain algebras A. We consider in more detail two cases: (a) A is said to be cyclotomic if all eigenvalues of χ<sup>A</sup> are roots of unity; (b) A is said to be of Littlewood type if all coefficients of χ<sup>A</sup> are �1, 0 or 1. We find criteria in order that A is

of one of those types. In particular, we establish new records according to

as possible, ordered by their number of roots outside the unit circle.

Mossingshoff's list of Record Mahler measures of polynomials q with 1< Mð Þq as small

Keywords: finite dimensional algebra, coxeter transformation, derived category, accessible algebra, characteristic polynomial, cyclotomic polynomial, littlewod type

Assume throughout the paper that K is an algebraically closed field. We assume that A is a triangular finite dimensional basic K-algebra, that is, of the form A ¼ KQ=I, where I is an ideal of the path algebra KQ for Q a quiver without oriented cycles. In particular, A has finite global dimension. The Coxeter transformation ϕ<sup>A</sup> is the automorphism of the Grothendieck group K0ð Þ A induced by the Auslander-Reiten

of ϕ<sup>A</sup> is called the Coxeter polynomial χ<sup>A</sup> of A, or simply χ<sup>A</sup> see [15, 17]. It is a monic self-reciprocal polynomial, therefore it is <sup>χ</sup><sup>A</sup> <sup>¼</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>T</sup> <sup>þ</sup> <sup>a</sup>2T<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> an�<sup>2</sup>T<sup>n</sup>�<sup>2</sup>

Consider the roots λ1, …, λ<sup>n</sup> of χA, the so called spectrum of A. There is a number of measures associated to the absolute values ∣λ∣ for λ in the spectrum Specð Þ ϕ<sup>A</sup> of A. For instance, the spectral radius of A is defined as ρ<sup>A</sup> ¼ maxf g jλj : λ∈Specð Þ ϕ<sup>A</sup>

some explorations on the relations of the Mahler measure Mð Þ χ<sup>A</sup> and properties of

For a one-point extension A ¼ B N½ �, we show that M χ<sup>B</sup> ð Þ≤ Mð Þ χ<sup>A</sup> . The strongest statements and examples will be given for the class of accessible algebras. We say

an�<sup>1</sup>T<sup>n</sup>�<sup>1</sup> <sup>þ</sup> anT<sup>n</sup> <sup>∈</sup>Z½ � <sup>T</sup> , with ai <sup>¼</sup> an�<sup>i</sup> for 0<sup>≤</sup> <sup>i</sup><sup>≤</sup> <sup>n</sup>, and <sup>a</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> an.

and the Mahler measure of χ<sup>A</sup> defined as Mð Þ¼ χ<sup>A</sup> max 1;

ð Þ A see [1]. The characteristic polynomial χ<sup>A</sup>

Q <sup>∣</sup>λ∣>1jλj n o. Recently,

þ

Polynomials Associated to

#### Chapter 1

## Cyclotomic and Littlewood Polynomials Associated to Algebras

José-Antonio de la Peña

#### Abstract

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕAð Þ T as the automorphism of the Grothendieck group K0ð Þ A induced by the Auslander-Reiten translation τ in the derived category D<sup>b</sup>ð Þ mod<sup>A</sup> of the module category mod<sup>A</sup> of finite dimensional left A-modules. In this paper we study the Mahler measure Mð Þ χ<sup>A</sup> of the Coxeter polynomial χ<sup>A</sup> of certain algebras A. We consider in more detail two cases: (a) A is said to be cyclotomic if all eigenvalues of χ<sup>A</sup> are roots of unity; (b) A is said to be of Littlewood type if all coefficients of χ<sup>A</sup> are �1, 0 or 1. We find criteria in order that A is of one of those types. In particular, we establish new records according to Mossingshoff's list of Record Mahler measures of polynomials q with 1< Mð Þq as small as possible, ordered by their number of roots outside the unit circle.

Keywords: finite dimensional algebra, coxeter transformation, derived category, accessible algebra, characteristic polynomial, cyclotomic polynomial, littlewod type

#### 1. Introduction

Assume throughout the paper that K is an algebraically closed field. We assume that A is a triangular finite dimensional basic K-algebra, that is, of the form A ¼ KQ=I, where I is an ideal of the path algebra KQ for Q a quiver without oriented cycles. In particular, A has finite global dimension. The Coxeter transformation ϕ<sup>A</sup> is the automorphism of the Grothendieck group K0ð Þ A induced by the Auslander-Reiten translation τ in the derived category D<sup>b</sup> ð Þ A see [1]. The characteristic polynomial χ<sup>A</sup> of ϕ<sup>A</sup> is called the Coxeter polynomial χ<sup>A</sup> of A, or simply χ<sup>A</sup> see [15, 17]. It is a monic self-reciprocal polynomial, therefore it is <sup>χ</sup><sup>A</sup> <sup>¼</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>T</sup> <sup>þ</sup> <sup>a</sup>2T<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> an�<sup>2</sup>T<sup>n</sup>�<sup>2</sup> þ an�<sup>1</sup>T<sup>n</sup>�<sup>1</sup> <sup>þ</sup> anT<sup>n</sup> <sup>∈</sup>Z½ � <sup>T</sup> , with ai <sup>¼</sup> an�<sup>i</sup> for 0<sup>≤</sup> <sup>i</sup><sup>≤</sup> <sup>n</sup>, and <sup>a</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> an.

Consider the roots λ1, …, λ<sup>n</sup> of χA, the so called spectrum of A. There is a number of measures associated to the absolute values ∣λ∣ for λ in the spectrum Specð Þ ϕ<sup>A</sup> of A. For instance, the spectral radius of A is defined as ρ<sup>A</sup> ¼ maxf g jλj : λ∈Specð Þ ϕ<sup>A</sup> and the Mahler measure of χ<sup>A</sup> defined as Mð Þ¼ χ<sup>A</sup> max 1; Q <sup>∣</sup>λ∣>1jλj n o. Recently, some explorations on the relations of the Mahler measure Mð Þ χ<sup>A</sup> and properties of the algebra A have been initiated.

For a one-point extension A ¼ B N½ �, we show that M χ<sup>B</sup> ð Þ≤ Mð Þ χ<sup>A</sup> . The strongest statements and examples will be given for the class of accessible algebras. We say

that an algebra A is accessible from B if there is a sequence B ¼ B1, B2, …, Bs ¼ A of algebras such that each Biþ<sup>1</sup> is a one-point extension (resp. coextension) of Bi for some exceptional Bi-module Mi. As a special case, a K-algebra A is called accessible if A is accessible from the one vertex algebra K.

Indeed, there is an efficient algorithm to determine such polynomials of given

Cyclotomic polynomials Φ<sup>n</sup> and their products are a natural source for self-

It is not a coincidence that in the above tables we have b nð Þ¼ c nð Þ þ 1 for n even

<sup>n</sup> <sup>≥</sup>2Φen n

degree <sup>n</sup>, based on a quadratic bound for <sup>n</sup>≤4f nð Þ<sup>2</sup> in terms of Euler totient

Cyclotomic and Littlewood Polynomials Associated to Algebras

DOI: http://dx.doi.org/10.5772/intechopen.82309

reciprocal polynomials. Clearly, Φ1ð Þ¼ z z � 1 is not self-reciprocal, but all remaining <sup>Φ</sup><sup>n</sup> (with <sup>n</sup> <sup>≥</sup>2) are. Hence, exactly the polynomials ð Þ <sup>z</sup> � <sup>1</sup> <sup>2</sup>k<sup>Q</sup>

with natural numbers k and en are self-reciprocal with spectral radio one and

and b nð Þ¼ c nð Þ for n odd. Indeed, if p is self-reciprocal of odd degree then

Let A be a finite dimensional K-algebra with finite global dimension. The Grothendieck group K0ð Þ A of the category mod<sup>A</sup> of finite dimensional (right) A-modules, formed with respect to short exact sequences, is naturally isomorphic to the Grothendieck group of the derived category, formed with respect to exact

self-reciprocal polynomial, therefore it is <sup>χ</sup>Að Þ¼ <sup>T</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>T</sup> <sup>þ</sup> <sup>a</sup>2T<sup>2</sup> <sup>þ</sup> …<sup>þ</sup>

an�<sup>2</sup>T<sup>n</sup>�<sup>2</sup> <sup>þ</sup> an�<sup>1</sup>T<sup>n</sup>�<sup>1</sup> <sup>þ</sup> anT<sup>n</sup> <sup>∈</sup>Z½ � <sup>T</sup> , with ai <sup>¼</sup> an�<sup>i</sup> for 0 <sup>≤</sup><sup>i</sup> <sup>≤</sup>n, and <sup>a</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> an. Consider the roots λ1ð Þ A , …, λnð Þ A of χA, the so called spectrum of A. In [15], a measure for polynomials was introduced. Namely, the Mahler measure of χ<sup>A</sup> is

Prop. 1.2.1], a monic integral polynomial p, with pð Þ 0 6¼ 0, has Mð Þ¼ p 1 if and only if p factorizes as product of cyclotomic polynomials. As observed in [18], A is of cyclotomic type if and only if Mð Þ¼ χ<sup>A</sup> 1, that is, χAð Þ T factorizes as product of

If the spectrum of A lies in the unit disk, then all roots of χ<sup>A</sup> lie on the unit circle, hence A has spectral radius ρ<sup>A</sup> ¼ 1. Clearly, for fixed degree there are only finitely

4.generalizing (2), (some) algebras which are derived equivalent to categories of

We put vn <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> xn�1. Note that vn has degree <sup>n</sup> � 1. There are several reasons for this choice: first of all vnð Þ¼ 1 n, second this normalization yields

convincing formulas for the Coxeter polynomials of canonical algebras and

The following finite dimensional algebras are known to produce Coxeter

The Coxeter transformation ϕ<sup>A</sup> is the automorphism of the Grothendieck group K0ð Þ A induced by the Auslander-Reiten translation τ. The characteristic polynomial χAð Þ T of ϕ<sup>A</sup> is called the Coxeter polynomial χAð Þ T of A, or simply χA. It is a monic

<sup>i</sup>¼<sup>1</sup> <sup>j</sup>λi<sup>j</sup> � � . By a celebrated result of Kronecker [9], see also [7,

pð Þ¼ �1 0, hence p zð Þ¼ ð Þ z þ 1 q zð Þ where q is also self-reciprocal.

function, f nð Þ.

without eigenvalue zero.

2.2 Mahler measure

triangles.

Mð Þ¼ χ<sup>A</sup> max 1;

cyclotomic polynomials.

Q<sup>n</sup>

2.3 Spectral radius one, periodicity

polynomials of spectral radius one:

3. (some) extended canonical algebras;

2. all canonical algebras;

coherent sheaves.

5

many monic integral polynomials with this property.

1. hereditary algebras of finite or tame representation type;

We say that A is of cyclotomic type if the eigenvalues of ϕ<sup>A</sup> lie on the unit circle. Many important finite dimensional algebras are known to be of cyclotomic type: hereditary algebras of finite or tame representation type, canonical algebras, some extended canonical algebras and many others. On the other hand, there are wellknown classes of algebras with a mixed behavior with respect to cyclotomicity. For instance, in Section 6 below we consider the class of Nakayama algebras. Let N nð Þ ;r be the quotient obtained from the linear quiver with n vertices

> •! x • ! x ⋯•! x •

with relations xr <sup>¼</sup> 0. The Nakayama algebras N nð Þ ; <sup>2</sup> are easily proven to be of cyclotomic type, while those of the form N nð Þ ; 3 are of cyclotomic type as consequence of lengthly considerations in [18]. The case r ¼ 4 is more representative: N nð Þ ; 4 is of cyclotomic type for all 0≤n ≤100 except for n ¼ 10; 22; 30; 42; 50; 62; 70; 82 and 90. Clearly, if A is of cyclotomic type then ∣Tr ð Þ ϕ<sup>A</sup> k ∣ ≤n, for k≥0. We show the following theorem.

Theorem 1: Let M be an unimodular n � n-matrix. The following are equivalent:

a. M is of cyclotomic type;

b.for every positive integer 0≤k≤n, we have ∣Tr M<sup>k</sup> ∣ ≤n.

We also consider algebras A of Littlewood type where χ<sup>A</sup> has all its coefficients in the set f g �1; 0; 1 . Among other structure results, we prove.

Proposition. The closure P of the set P of roots of Littlewood polynomials, equals the set R of roots of Littlewood series.

Our results make use of well established techniques in the representation theory of algebras, as well as results from the theory of polynomials and transcendental number theory, where Mahler measure has its usual habitat. We stress here the natural context of these investigations on the largely unexplored overlapping area of these important subjects. Hence, rather than a comprehensive study we understand our work as a preliminary exploration where examples are most valuable.

#### 2. Measures for polynomials

#### 2.1 Self-reciprocal polynomials

A polynomial p zð Þ of degree <sup>n</sup> is said to be self-reciprocal if p zð Þ¼ znpð Þ <sup>1</sup>=<sup>z</sup> . The following table displays the number a nð Þ of polynomials p of degree n (for small n) with pð Þ 0 non-zero, b nð Þ is the number of such polynomials which are additionally self-reciprocal, and c nð Þ is the number of those which are self-reciprocal and where pð Þ �1 is the square of an integer.


Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

Indeed, there is an efficient algorithm to determine such polynomials of given degree <sup>n</sup>, based on a quadratic bound for <sup>n</sup>≤4f nð Þ<sup>2</sup> in terms of Euler totient function, f nð Þ.

Cyclotomic polynomials Φ<sup>n</sup> and their products are a natural source for selfreciprocal polynomials. Clearly, Φ1ð Þ¼ z z � 1 is not self-reciprocal, but all remaining <sup>Φ</sup><sup>n</sup> (with <sup>n</sup> <sup>≥</sup>2) are. Hence, exactly the polynomials ð Þ <sup>z</sup> � <sup>1</sup> <sup>2</sup>k<sup>Q</sup> <sup>n</sup> <sup>≥</sup>2Φen n with natural numbers k and en are self-reciprocal with spectral radio one and without eigenvalue zero.

It is not a coincidence that in the above tables we have b nð Þ¼ c nð Þ þ 1 for n even and b nð Þ¼ c nð Þ for n odd. Indeed, if p is self-reciprocal of odd degree then pð Þ¼ �1 0, hence p zð Þ¼ ð Þ z þ 1 q zð Þ where q is also self-reciprocal.

#### 2.2 Mahler measure

that an algebra A is accessible from B if there is a sequence B ¼ B1, B2, …, Bs ¼ A of algebras such that each Biþ<sup>1</sup> is a one-point extension (resp. coextension) of Bi for some exceptional Bi-module Mi. As a special case, a K-algebra A is called accessible if

We say that A is of cyclotomic type if the eigenvalues of ϕ<sup>A</sup> lie on the unit circle. Many important finite dimensional algebras are known to be of cyclotomic type: hereditary algebras of finite or tame representation type, canonical algebras, some extended canonical algebras and many others. On the other hand, there are wellknown classes of algebras with a mixed behavior with respect to cyclotomicity. For instance, in Section 6 below we consider the class of Nakayama algebras. Let N nð Þ ;r

> ⋯•! x •

> > k

∣ ≤n, for k≥0.

with relations xr <sup>¼</sup> 0. The Nakayama algebras N nð Þ ; <sup>2</sup> are easily proven to be of cyclotomic type, while those of the form N nð Þ ; 3 are of cyclotomic type as consequence of lengthly considerations in [18]. The case r ¼ 4 is more representative: N nð Þ ; 4 is of cyclotomic type for all 0≤n ≤100 except for n ¼ 10; 22; 30; 42; 50;

Theorem 1: Let M be an unimodular n � n-matrix. The following are equivalent:

We also consider algebras A of Littlewood type where χ<sup>A</sup> has all its coefficients in

Proposition. The closure P of the set P of roots of Littlewood polynomials, equals the

Our results make use of well established techniques in the representation theory of algebras, as well as results from the theory of polynomials and transcendental number theory, where Mahler measure has its usual habitat. We stress here the natural context of these investigations on the largely unexplored overlapping area of these important subjects. Hence, rather than a comprehensive study we understand our

A polynomial p zð Þ of degree <sup>n</sup> is said to be self-reciprocal if p zð Þ¼ znpð Þ <sup>1</sup>=<sup>z</sup> . The following table displays the number a nð Þ of polynomials p of degree n (for small n) with pð Þ 0 non-zero, b nð Þ is the number of such polynomials which are additionally self-reciprocal, and c nð Þ is the number of those which are self-reciprocal and where

n 1 2 3 4 5 6 7 8 9 10 11 12 15 20 25 a nð Þ 2 6 10 24 38 78 118 224 330 584 838 1420 4514 30,532 152,170 b nð Þ 1 5 5 19 19 59 59 165 165 419 419 1001 2257 20,399 76,085 c nð Þ 1 3 5 12 19 34 59 99 165 244 419 598 2257 12,526 76,085

A is accessible from the one vertex algebra K.

Polynomials - Theory and Application

We show the following theorem.

a. M is of cyclotomic type;

set R of roots of Littlewood series.

2. Measures for polynomials

2.1 Self-reciprocal polynomials

pð Þ �1 is the square of an integer.

4

be the quotient obtained from the linear quiver with n vertices

62; 70; 82 and 90. Clearly, if A is of cyclotomic type then ∣Tr ð Þ ϕ<sup>A</sup>

b.for every positive integer 0≤k≤n, we have ∣Tr M<sup>k</sup> ∣ ≤n.

work as a preliminary exploration where examples are most valuable.

the set f g �1; 0; 1 . Among other structure results, we prove.

•! x • ! x

> Let A be a finite dimensional K-algebra with finite global dimension. The Grothendieck group K0ð Þ A of the category mod<sup>A</sup> of finite dimensional (right) A-modules, formed with respect to short exact sequences, is naturally isomorphic to the Grothendieck group of the derived category, formed with respect to exact triangles.

> The Coxeter transformation ϕ<sup>A</sup> is the automorphism of the Grothendieck group K0ð Þ A induced by the Auslander-Reiten translation τ. The characteristic polynomial χAð Þ T of ϕ<sup>A</sup> is called the Coxeter polynomial χAð Þ T of A, or simply χA. It is a monic self-reciprocal polynomial, therefore it is <sup>χ</sup>Að Þ¼ <sup>T</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>T</sup> <sup>þ</sup> <sup>a</sup>2T<sup>2</sup> <sup>þ</sup> …<sup>þ</sup> an�<sup>2</sup>T<sup>n</sup>�<sup>2</sup> <sup>þ</sup> an�<sup>1</sup>T<sup>n</sup>�<sup>1</sup> <sup>þ</sup> anT<sup>n</sup> <sup>∈</sup>Z½ � <sup>T</sup> , with ai <sup>¼</sup> an�<sup>i</sup> for 0 <sup>≤</sup><sup>i</sup> <sup>≤</sup>n, and <sup>a</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> an.

> Consider the roots λ1ð Þ A , …, λnð Þ A of χA, the so called spectrum of A. In [15], a measure for polynomials was introduced. Namely, the Mahler measure of χ<sup>A</sup> is Mð Þ¼ χ<sup>A</sup> max 1; Q<sup>n</sup> <sup>i</sup>¼<sup>1</sup> <sup>j</sup>λi<sup>j</sup> � � . By a celebrated result of Kronecker [9], see also [7, Prop. 1.2.1], a monic integral polynomial p, with pð Þ 0 6¼ 0, has Mð Þ¼ p 1 if and only if p factorizes as product of cyclotomic polynomials. As observed in [18], A is of cyclotomic type if and only if Mð Þ¼ χ<sup>A</sup> 1, that is, χAð Þ T factorizes as product of cyclotomic polynomials.

#### 2.3 Spectral radius one, periodicity

If the spectrum of A lies in the unit disk, then all roots of χ<sup>A</sup> lie on the unit circle, hence A has spectral radius ρ<sup>A</sup> ¼ 1. Clearly, for fixed degree there are only finitely many monic integral polynomials with this property.

The following finite dimensional algebras are known to produce Coxeter polynomials of spectral radius one:

1. hereditary algebras of finite or tame representation type;


We put vn <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> xn�1. Note that vn has degree <sup>n</sup> � 1. There are several reasons for this choice: first of all vnð Þ¼ 1 n, second this normalization yields convincing formulas for the Coxeter polynomials of canonical algebras and


hereditary stars, third representing a Coxeter polynomial — for spectral radius one — as a rational function in the vn's relates to a Poincaré series, naturally attached to the setting.

algebras. Restricting to algebraically closed fields, already the request that χAð Þ �1

1 a b 0 1 c 001

yields the Coxeter polynomial <sup>f</sup> <sup>¼</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>α</sup>x<sup>2</sup> <sup>þ</sup> <sup>α</sup><sup>x</sup> <sup>þ</sup> 1, where <sup>α</sup> <sup>¼</sup> abc � <sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup>

<sup>c</sup><sup>2</sup> <sup>þ</sup> 3. The equation <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> � abc <sup>¼</sup> 3 of Hurwitz-Markov type does not have an integral solution. (Use that reduction modulo 3 only yields the trivial solution in F3.)

Given a (non-oriented) graph Δ, its characteristic polynomial κ<sup>Δ</sup> is defined as the characteristic polynomial of the adjacency matrix M<sup>Δ</sup> of Δ. Observe that, since M<sup>Δ</sup> is symmetric, all its eigenvalues are real numbers. For general results on graph

There are important interactions between the theory of graph spectra and the representation theory of algebras, due to the fact that if C is the Cartan matrix of

, then <sup>M</sup><sup>Δ</sup> is determined by the symmetrization <sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>t</sup> of <sup>C</sup>, since

h i !

be a hereditary algebra with Δ

is bipartite, we may assume that the first m vertices are sources

, when Δ !

!

In <sup>þ</sup> ð Þ In � <sup>N</sup> ð Þ In <sup>þ</sup> <sup>N</sup> <sup>t</sup> � �det In � <sup>N</sup><sup>t</sup> ð Þ

N � �

is a bipartite

a bipartite quiver

! .

∪R<sup>þ</sup>. This was

,

□

<sup>M</sup><sup>Δ</sup> <sup>¼</sup> <sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>t</sup> � <sup>2</sup>I. We shall see that information on the spectra of <sup>M</sup><sup>Δ</sup> provides fundamental insights into the spectral analysis of the Coxeter matrix Φ<sup>A</sup> and the

without oriented cycles. Then <sup>χ</sup><sup>A</sup> <sup>x</sup><sup>2</sup> ð Þ¼ <sup>x</sup><sup>n</sup>κ<sup>Δ</sup> <sup>x</sup> <sup>þ</sup> <sup>x</sup>�<sup>1</sup> ð Þ, where n is the number of vertices

and κ<sup>Δ</sup> is the characteristic polynomial of the underlying graph Δ of Δ

and the last n � m vertices are sinks. Then the adjacency matrix A of Δ and the Cartan matrix <sup>C</sup> of <sup>A</sup>, in the basis of simple modules, take the form: <sup>A</sup> <sup>¼</sup> <sup>N</sup> <sup>þ</sup> <sup>N</sup><sup>t</sup>

> <sup>N</sup> <sup>¼</sup> <sup>0</sup> <sup>D</sup> 0 0 � �

for certain <sup>m</sup> � <sup>m</sup>-matrix <sup>D</sup>. Since <sup>N</sup><sup>2</sup> <sup>¼</sup> 0, then <sup>C</sup>�<sup>1</sup> <sup>¼</sup> In <sup>þ</sup> <sup>N</sup>. Therefore

<sup>¼</sup> det <sup>x</sup><sup>2</sup>

In � <sup>x</sup><sup>2</sup>

<sup>¼</sup> <sup>x</sup><sup>n</sup>det <sup>x</sup> <sup>þ</sup> <sup>x</sup>�<sup>1</sup> � �In � <sup>A</sup> � �:

<sup>N</sup><sup>t</sup> <sup>þ</sup> ð Þ In � <sup>N</sup> � �

<sup>¼</sup> <sup>x</sup><sup>n</sup>det <sup>x</sup> <sup>þ</sup> <sup>x</sup>�<sup>1</sup> � �In � xN<sup>t</sup> � <sup>x</sup>�<sup>1</sup>

A fundamental fact for a hereditary algebra A ¼ K Δ

shown as a consequence of the following important identity.

quiver, that is, every vertex is a sink or source, is that Specð Þ <sup>Φ</sup><sup>A</sup> <sup>⊂</sup>S<sup>1</sup>

h i !

3 7 5

�

cyclotomic polynomials <sup>Φ</sup>4, <sup>Φ</sup>6, <sup>Φ</sup>8, <sup>Φ</sup>10. Moreover, the polynomial <sup>f</sup> <sup>¼</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> 1, which is the Coxeter polynomial of the non simply-laced Dynkin diagram B3, does not appear as the Coxeter polynomial of a triangular algebra over an algebraically closed field, despite of the fact that fð Þ¼ �1 0 is a square. Indeed,

is a square discards many self-reciprocal polynomials, for instance the

Cyclotomic and Littlewood Polynomials Associated to Algebras

DOI: http://dx.doi.org/10.5772/intechopen.82309

2 6 4

the Cartan matrix

A ¼ K Δ h i !

of Δ !

7

2.5 Relationship with graph theory

theory and spectra of graphs see [4].

Proposition. [2] Let A ¼ K Δ

!

det x<sup>2</sup>

In � Φ<sup>A</sup> � � <sup>¼</sup> det <sup>x</sup><sup>2</sup>

structure of the algebra A.

Proof. Since Δ

C ¼ In � N, where

In the column 'v-factorization', we have added some extra terms in the nominator and denominator which obviously cancel.

Inspection of the table shows the following result:

Proposition. Let k be an algebraically closed field and A be a connected, hereditary k-algebra which is representation-finite. Then the Coxeter polynomial χ<sup>A</sup> determines A up to derived equivalence. □

#### 2.4 Triangular algebras

Nearly all algebras considered in this survey are triangular. By definition, a finite dimensional algebra is called triangular if it has triangular matrix shape


where the diagonal entries Ai are skew-fields and the off-diagonal entries Mij, j>i, are Ai, Aj-bimodules. Each triangular algebra has finite global dimension.

Proposition. Let A be a triangular algebra over an algebraically closed field K. Then χAð Þ �1 is the square of an integer.

Proof. Let C be the Cartan matrix of A with respect to the basis of indecomposable projectives. Since A is triangular and K is algebraically closed, we get detC ¼ 1, yielding

$$\mathcal{X}\_A = \left| \mathbf{x}I + \mathbf{C}^{-1}\mathbf{C}^t \right| = \left| \mathbf{C}^{-1} \right| \cdot \left| \mathbf{x}\mathbf{C} + \mathbf{C}^t \right| = \left| \mathbf{C}^t + \mathbf{x}\mathbf{C} \right|.$$

Hence <sup>χ</sup>Að Þ �<sup>1</sup> is the determinant of the skew-symmetric matrix <sup>S</sup> <sup>¼</sup> <sup>C</sup><sup>t</sup> � <sup>C</sup>. Using the skew-normal form of <sup>S</sup>, see [16, Theorem IV.1], we obtain <sup>S</sup><sup>0</sup> <sup>¼</sup> <sup>U</sup><sup>t</sup> SU for some U ∈ GLnð Þ Z , where S<sup>0</sup> is a block-diagonal matrix whose first block is the zero matrix of a certain size and where the remaining blocks have the shape <sup>0</sup> mi �mi 0 � � with integers mi. The claim follows. □

Which self-reciprocal polynomials of spectral radius one are Coxeter polynomials? The answer is not known. If arbitrary base fields are allowed, we conjecture that all self-reciprocal polynomials are realizable as Coxeter polynomials of triangular

algebras. Restricting to algebraically closed fields, already the request that χAð Þ �1 is a square discards many self-reciprocal polynomials, for instance the cyclotomic polynomials <sup>Φ</sup>4, <sup>Φ</sup>6, <sup>Φ</sup>8, <sup>Φ</sup>10. Moreover, the polynomial <sup>f</sup> <sup>¼</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> 1, which is the Coxeter polynomial of the non simply-laced Dynkin diagram B3, does not appear as the Coxeter polynomial of a triangular algebra over an algebraically closed field, despite of the fact that fð Þ¼ �1 0 is a square. Indeed, the Cartan matrix

$$
\begin{bmatrix}
\mathbf{1} & a & b \\
\mathbf{0} & \mathbf{1} & c \\
\mathbf{0} & \mathbf{0} & \mathbf{1}
\end{bmatrix}
$$

yields the Coxeter polynomial <sup>f</sup> <sup>¼</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>α</sup>x<sup>2</sup> <sup>þ</sup> <sup>α</sup><sup>x</sup> <sup>þ</sup> 1, where <sup>α</sup> <sup>¼</sup> abc � <sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � <sup>c</sup><sup>2</sup> <sup>þ</sup> 3. The equation <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> � abc <sup>¼</sup> 3 of Hurwitz-Markov type does not have an integral solution. (Use that reduction modulo 3 only yields the trivial solution in F3.)

#### 2.5 Relationship with graph theory

hereditary stars, third representing a Coxeter polynomial — for spectral radius one — as a rational function in the vn's relates to a Poincaré series, naturally

Dynkin type Star symbol v-factorization Cyclotomic factorization Coxeter number

In the column 'v-factorization', we have added some extra terms in the nomina-

Proposition. Let k be an algebraically closed field and A be a connected, hereditary k-algebra which is representation-finite. Then the Coxeter polynomial χ<sup>A</sup> determines A up to derived equivalence. □

Nearly all algebras considered in this survey are triangular. By definition, a finite

⋱ ⋮

A<sup>1</sup> M<sup>12</sup> ⋯ M1<sup>n</sup> 0 A<sup>2</sup> ⋯ M2<sup>n</sup>

0 0 ⋯ An

where the diagonal entries Ai are skew-fields and the off-diagonal entries Mij, j>i, are Ai, Aj-bimodules. Each triangular algebra has finite global dimension.

Proposition. Let A be a triangular algebra over an algebraically closed field K. Then

Proof. Let C be the Cartan matrix of A with respect to the basis of indecomposable projectives. Since A is triangular and K is algebraically closed, we get detC ¼ 1,

> � <sup>¼</sup> <sup>C</sup>�<sup>1</sup> � � �

Hence <sup>χ</sup>Að Þ �<sup>1</sup> is the determinant of the skew-symmetric matrix <sup>S</sup> <sup>¼</sup> <sup>C</sup><sup>t</sup> � <sup>C</sup>. Using the skew-normal form of <sup>S</sup>, see [16, Theorem IV.1], we obtain <sup>S</sup><sup>0</sup> <sup>¼</sup> <sup>U</sup><sup>t</sup>

some U ∈ GLnð Þ Z , where S<sup>0</sup> is a block-diagonal matrix whose first block is the zero matrix of a certain size and where the remaining blocks have the shape <sup>0</sup> mi

with integers mi. The claim follows. □ Which self-reciprocal polynomials of spectral radius one are Coxeter polynomials? The answer is not known. If arbitrary base fields are allowed, we conjecture that all self-reciprocal polynomials are realizable as Coxeter polynomials of triangular

dimensional algebra is called triangular if it has triangular matrix shape

ð Þ <sup>v</sup><sup>2</sup> vn�<sup>2</sup> vn�<sup>1</sup> <sup>v</sup><sup>2</sup>ð Þ <sup>n</sup>�<sup>1</sup> <sup>Φ</sup><sup>2</sup>

Q

Y d∣2ð Þ n � 1 d 6¼ 1, d 6¼ n � 1

ð Þ <sup>v</sup><sup>3</sup> <sup>v</sup>4v<sup>6</sup> <sup>v</sup><sup>12</sup> <sup>Φ</sup>3Φ<sup>12</sup> <sup>12</sup>

ð Þ <sup>v</sup><sup>4</sup> <sup>v</sup><sup>6</sup> <sup>v</sup><sup>9</sup> <sup>v</sup><sup>18</sup> <sup>Φ</sup>2Φ<sup>18</sup> <sup>18</sup>

<sup>v</sup><sup>6</sup> <sup>v</sup>10v<sup>15</sup> v<sup>30</sup> Φ<sup>30</sup> 30

� � xC <sup>þ</sup> <sup>C</sup><sup>t</sup> j j <sup>¼</sup> <sup>C</sup><sup>t</sup> j j <sup>þ</sup> xC :

SU for

�mi 0 � �

<sup>d</sup>∣n, <sup>d</sup>><sup>1</sup> Φ<sup>d</sup> n þ 1

Φ<sup>d</sup> 2ð Þ n � 1

attached to the setting.

Polynomials - Theory and Application

A<sup>n</sup> ½ � n vnþ<sup>1</sup>

D<sup>n</sup> ½ � 2; 2; n � 2 <sup>v</sup><sup>2</sup> ð Þ <sup>v</sup><sup>2</sup> vn�<sup>2</sup>

E<sup>6</sup> ½ � 2; 3; 3 <sup>v</sup><sup>2</sup> <sup>v</sup><sup>3</sup> ð Þ <sup>v</sup><sup>3</sup>

E<sup>7</sup> ½ � 2; 3; 4 <sup>v</sup><sup>2</sup> <sup>v</sup><sup>3</sup> ð Þ <sup>v</sup><sup>4</sup>

<sup>E</sup><sup>8</sup> ½ � <sup>2</sup>; <sup>3</sup>; <sup>5</sup> <sup>v</sup><sup>2</sup> <sup>v</sup><sup>3</sup> <sup>v</sup><sup>5</sup>

2.4 Triangular algebras

χAð Þ �1 is the square of an integer.

<sup>χ</sup><sup>A</sup> <sup>¼</sup> xI <sup>þ</sup> <sup>C</sup>�<sup>1</sup> C<sup>t</sup> � � �

yielding

6

tor and denominator which obviously cancel.

Inspection of the table shows the following result:

Given a (non-oriented) graph Δ, its characteristic polynomial κ<sup>Δ</sup> is defined as the characteristic polynomial of the adjacency matrix M<sup>Δ</sup> of Δ. Observe that, since M<sup>Δ</sup> is symmetric, all its eigenvalues are real numbers. For general results on graph theory and spectra of graphs see [4].

There are important interactions between the theory of graph spectra and the representation theory of algebras, due to the fact that if C is the Cartan matrix of A ¼ K Δ h i ! , then <sup>M</sup><sup>Δ</sup> is determined by the symmetrization <sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>t</sup> of <sup>C</sup>, since <sup>M</sup><sup>Δ</sup> <sup>¼</sup> <sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>t</sup> � <sup>2</sup>I. We shall see that information on the spectra of <sup>M</sup><sup>Δ</sup> provides fundamental insights into the spectral analysis of the Coxeter matrix Φ<sup>A</sup> and the structure of the algebra A.

A fundamental fact for a hereditary algebra A ¼ K Δ h i ! , when Δ ! is a bipartite quiver, that is, every vertex is a sink or source, is that Specð Þ <sup>Φ</sup><sup>A</sup> <sup>⊂</sup>S<sup>1</sup> ∪R<sup>þ</sup>. This was shown as a consequence of the following important identity.

Proposition. [2] Let A ¼ K Δ h i ! be a hereditary algebra with Δ ! a bipartite quiver without oriented cycles. Then <sup>χ</sup><sup>A</sup> <sup>x</sup><sup>2</sup> ð Þ¼ <sup>x</sup><sup>n</sup>κ<sup>Δ</sup> <sup>x</sup> <sup>þ</sup> <sup>x</sup>�<sup>1</sup> ð Þ, where n is the number of vertices of Δ ! and κ<sup>Δ</sup> is the characteristic polynomial of the underlying graph Δ of Δ ! .

Proof. Since Δ ! is bipartite, we may assume that the first m vertices are sources and the last n � m vertices are sinks. Then the adjacency matrix A of Δ and the Cartan matrix <sup>C</sup> of <sup>A</sup>, in the basis of simple modules, take the form: <sup>A</sup> <sup>¼</sup> <sup>N</sup> <sup>þ</sup> <sup>N</sup><sup>t</sup> , C ¼ In � N, where

$$N = \begin{pmatrix} \mathbf{0} & D \\ \mathbf{0} & \mathbf{0} \end{pmatrix}.$$

for certain <sup>m</sup> � <sup>m</sup>-matrix <sup>D</sup>. Since <sup>N</sup><sup>2</sup> <sup>¼</sup> 0, then <sup>C</sup>�<sup>1</sup> <sup>¼</sup> In <sup>þ</sup> <sup>N</sup>. Therefore

$$\det\left(\mathbf{x}^2 I\_n - \Phi\_A\right) = \det\left(\mathbf{x}^2 I\_n + (I\_n - N)(I\_n + N)^t\right) \det(I\_n - N^t)$$

$$= \det\left(\mathbf{x}^2 I\_n - \mathbf{x}^2 N^t + (I\_n - N)\right)$$

$$= \mathbf{x}^n \det\left((\mathbf{x} + \mathbf{x}^{-1})I\_n - \mathbf{x}N^t - \mathbf{x}^{-1}N\right)$$

$$= \mathbf{x}^n \det\left((\mathbf{x} + \mathbf{x}^{-1})I\_n - A\right).$$


The above result is important since it makes the spectral analysis of bipartite quivers and their underlying graphs almost equivalent. Note, however, that the representation theoretic context is much richer, given the categorical context behind the spectral analysis of quivers. The representation theory of bipartite quivers may thus be seen as a categorification of the class of graphs, allowing a bipartite structure.

i. Let S1, …, Sn be a complete system of pairwise non-isomorphic simple Amodules, P1, …, Pn the corresponding projective covers and I1, …, In the injective envelopes. Then ϕ<sup>A</sup> is the automorphism of K0ð Þ A defined by Φ<sup>A</sup> Pi ½ �¼� Ii ½ �, where ½ � X denotes the class of a module X in K0ð Þ A .

h i !

a. A is representation-finite if 1 ¼ ρ<sup>A</sup> is not a root of the Coxeter

c. A is wild if 1< ρA. Moreover, if A is wild connected, Ringel [20] shows that the spectral radius ρ<sup>A</sup> is a simple root of χA. Then Perron-Frobenius theory yields a vector y<sup>þ</sup> ∈K0ð Þ A ⊗ ZR with positive coordinates such

vectors yþ, y� play an important role in the representation theory of

Explicit formulas, special values. We are discussing various instances where an

Since the Coxeter polynomial χ<sup>A</sup> does not depend on the orientation of A we will

vpi ð Þ� x þ 1 x ∑

t i¼1

!

<sup>1</sup> � <sup>1</sup> pi vpi �1 vpi

� � � � : (2)

� � is of Dynkin type, correspondingly the

star quivers. Let A be the path algebra of a hereditary star p1; …; pt

that ΦAy<sup>þ</sup> ¼ ρ<sup>A</sup> yþ. Since χ<sup>A</sup> is self reciprocal, there is a vector <sup>y</sup>� <sup>∈</sup>K0ð Þ <sup>A</sup> <sup>⊗</sup> ZR with positive coordinates such that <sup>Φ</sup>Ay� <sup>¼</sup> <sup>ρ</sup>�<sup>1</sup>

the representation type of A in the following manner:

, the spectral radius ρ<sup>A</sup> ¼ ρ<sup>Φ</sup><sup>A</sup> determines

<sup>A</sup> y�. The

� � with respect

: (1)

ii. For a hereditary algebra A ¼ K Δ

DOI: http://dx.doi.org/10.5772/intechopen.82309

b.A is tame if 1 ¼ ρ<sup>A</sup> ∈Rootsð Þ χ<sup>A</sup> .

Cyclotomic and Littlewood Polynomials Associated to Algebras

, see [5, 17].

explicit formula for the Coxeter polynomial is known.

denote it by <sup>χ</sup> <sup>p</sup>1;…;pt ½ �. It follows from [11, prop. 9.1] or [2] that

<sup>χ</sup>ð Þ¼ <sup>1</sup> <sup>Y</sup><sup>t</sup>

� �. Moreover,

1. χð Þ1 >0 if and only if the star p1; …; pt

algebra A is representation-finite.

i¼1

i¼1

In particular, we have an explicit formula for the sum of coefficients of

pi 2 � ∑ t i¼1

This special value of χ has a specific mathematical meaning: up to the factor

<sup>i</sup>¼<sup>1</sup> pi this is just the orbifold-Euler characteristic of a weighted projective line <sup>X</sup> of

<sup>χ</sup> <sup>p</sup>1;…; pt ½ � <sup>¼</sup> <sup>Y</sup><sup>t</sup>

polynomial χA.

A ¼ K Δ h i !

to the standard orientation, see

<sup>χ</sup> <sup>¼</sup> <sup>χ</sup> <sup>p</sup>1;…;pt ½ � as follows:

weight type p1; …; pt

Q<sup>t</sup>

9

Constructions in graph theory. Several simple constructions in graph theory provide tools to obtain in practice the characteristic polynomial of a graph. We recall two of them (see [4] for related results):

a. Assume that a is a vertex in the graph Δ with a unique neighbor b and Δ0 (resp. Δ00) is the full subgraph of Δ with vertices Δ0\f ga (resp. Δ0\f g a; b ), then

$$
\kappa\_{\Delta} = \mathfrak{x}\kappa\_{\Delta'} - \kappa\_{\Delta''}
$$

b.Let Δ<sup>i</sup> be the graph obtained by deleting the vertex i in Δ. Then the first derivative of κ<sup>Δ</sup> is given by

$$\kappa\_{\Delta}' = \sum\_{i} \kappa\_{\Delta\_{i}}$$

The above formulas can be used inductively to calculate the characteristic polynomial of trees and other graphs. They immediately imply the following result that will be used often to calculate Coxeter polynomials of algebras.

Proposition. Let A ¼ K Δ h i ! be a bipartite hereditary algebra. The following holds:

i. Let a be a vertex in the graph Δ with a unique neighbor b. Consider the algebras B and C obtained as quotients of A modulo the ideal generated by the vertices a and a, b, respectively. Then

$$\chi\_A = (\mathfrak{x} + \mathfrak{1})\chi\_B - \mathfrak{x}\chi\_C$$

ii. The first derivative of the Coxeter polynomial satisfies:

$$2\chi\_A^{'} = n\chi\_A + (\mathbf{x} - \mathbf{1})\sum\_i \chi\_{A^{(i)}}$$

where <sup>A</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>K</sup> <sup>Δ</sup> ! \f gi h i is an algebra obtained from <sup>A</sup> by 'killing' a vertex <sup>i</sup>.

Proof. Use the corresponding results for graphs and A'Campo's formula for the algebras <sup>A</sup> and its quotients <sup>A</sup>ð Þ<sup>i</sup> . □

#### 3. Important classes of algebras

In this section we give the definitions and main properties of such classes of finite dimensional algebras where information on their spectral properties is available.

#### 3.1 Hereditary algebras

Let A be a finite dimensional K-algebra. For simplicity we assume A ¼ K Δ h i ! =I

for a quiver Δ ! without oriented cycles and I an ideal of the path algebra. The following facts about the Coxeter transformation Φ<sup>A</sup> of A are fundamental:

The above result is important since it makes the spectral analysis of bipartite quivers and their underlying graphs almost equivalent. Note, however, that the representation theoretic context is much richer, given the categorical context behind the spectral analysis of quivers. The representation theory of bipartite quivers may thus be seen as a categorification of the class of graphs, allowing

Constructions in graph theory. Several simple constructions in graph theory provide tools to obtain in practice the characteristic polynomial of a graph. We

κ<sup>Δ</sup> ¼ xκ<sup>Δ</sup><sup>0</sup> � κ<sup>Δ</sup><sup>00</sup>

b.Let Δ<sup>i</sup> be the graph obtained by deleting the vertex i in Δ. Then the first

The above formulas can be used inductively to calculate the characteristic polynomial of trees and other graphs. They immediately imply the following result that

i. Let a be a vertex in the graph Δ with a unique neighbor b. Consider the algebras B and C obtained as quotients of A modulo the ideal generated by the vertices a and

χ<sup>A</sup> ¼ ð Þ x þ 1 χ<sup>B</sup> � xχ<sup>C</sup>

¼ nχ<sup>A</sup> þ ð Þ x � 1 ∑

Proof. Use the corresponding results for graphs and A'Campo's formula for the algebras <sup>A</sup> and its quotients <sup>A</sup>ð Þ<sup>i</sup> . □

In this section we give the definitions and main properties of such classes of finite

dimensional algebras where information on their spectral properties is available.

Let A be a finite dimensional K-algebra. For simplicity we assume A ¼ K Δ

following facts about the Coxeter transformation Φ<sup>A</sup> of A are fundamental:

without oriented cycles and I an ideal of the path algebra. The

i χ<sup>A</sup>ð Þ<sup>i</sup>

is an algebra obtained from A by 'killing' a vertex i.

h i ! =I

be a bipartite hereditary algebra. The following holds:

κ0 <sup>Δ</sup> ¼ ∑ i κΔi

will be used often to calculate Coxeter polynomials of algebras.

ii. The first derivative of the Coxeter polynomial satisfies: 2xχ<sup>A</sup> 0

h i !

(resp. Δ00) is the full subgraph of Δ with vertices Δ0\f ga (resp. Δ0\f g a; b ), then

a. Assume that a is a vertex in the graph Δ with a unique neighbor b and

a bipartite structure.

Polynomials - Theory and Application

Δ0

recall two of them (see [4] for related results):

derivative of κ<sup>Δ</sup> is given by

Proposition. Let A ¼ K Δ

a, b, respectively. Then

! \f gi h i

3. Important classes of algebras

where <sup>A</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>K</sup> <sup>Δ</sup>

3.1 Hereditary algebras

!

for a quiver Δ

8

	- a. A is representation-finite if 1 ¼ ρ<sup>A</sup> is not a root of the Coxeter polynomial χA.
	- b.A is tame if 1 ¼ ρ<sup>A</sup> ∈Rootsð Þ χ<sup>A</sup> .
	- c. A is wild if 1< ρA. Moreover, if A is wild connected, Ringel [20] shows that the spectral radius ρ<sup>A</sup> is a simple root of χA. Then Perron-Frobenius theory yields a vector y<sup>þ</sup> ∈K0ð Þ A ⊗ ZR with positive coordinates such that ΦAy<sup>þ</sup> ¼ ρ<sup>A</sup> yþ. Since χ<sup>A</sup> is self reciprocal, there is a vector <sup>y</sup>� <sup>∈</sup>K0ð Þ <sup>A</sup> <sup>⊗</sup> ZR with positive coordinates such that <sup>Φ</sup>Ay� <sup>¼</sup> <sup>ρ</sup>�<sup>1</sup> <sup>A</sup> y�. The vectors yþ, y� play an important role in the representation theory of A ¼ K Δ h i ! , see [5, 17].

Explicit formulas, special values. We are discussing various instances where an explicit formula for the Coxeter polynomial is known.

star quivers. Let A be the path algebra of a hereditary star p1; …; pt � � with respect to the standard orientation, see

Since the Coxeter polynomial χ<sup>A</sup> does not depend on the orientation of A we will denote it by <sup>χ</sup> <sup>p</sup>1;…;pt ½ �. It follows from [11, prop. 9.1] or [2] that

$$\mathcal{X}\_{[p\_1,\dots,p\_t]} = \prod\_{i=1}^t \nu\_{p\_i} \left( (\varkappa + 1) - \varkappa \sum\_{i=1}^t \frac{\nu\_{p\_i - 1}}{\nu\_{p\_i}} \right). \tag{1}$$

In particular, we have an explicit formula for the sum of coefficients of <sup>χ</sup> <sup>¼</sup> <sup>χ</sup> <sup>p</sup>1;…;pt ½ � as follows:

$$\chi(\mathbf{1}) = \prod\_{i=1}^{t} p\_i \left( 2 - \sum\_{i=1}^{t} \left( 1 - \frac{\mathbf{1}}{p\_i} \right) \right). \tag{2}$$

This special value of χ has a specific mathematical meaning: up to the factor Q<sup>t</sup> <sup>i</sup>¼<sup>1</sup> pi this is just the orbifold-Euler characteristic of a weighted projective line <sup>X</sup> of weight type p1; …; pt � �. Moreover,

1. χð Þ1 >0 if and only if the star p1; …; pt � � is of Dynkin type, correspondingly the algebra A is representation-finite.


The above deals with all the Dynkin types and with the extended Dynkin diagrams of type <sup>D</sup><sup>~</sup> <sup>n</sup>, <sup>n</sup> <sup>≥</sup>4, and <sup>E</sup>~n, <sup>n</sup> <sup>¼</sup> <sup>6</sup>; <sup>7</sup>; 8. To complete the picture, we also consider the extended Dynkin quivers of type <sup>A</sup><sup>~</sup> <sup>n</sup> (<sup>n</sup> <sup>≥</sup>2) restricting, of course, to quivers without oriented cycles. Here, the Coxeter polynomial depends on the orientation: If p (resp. q) denotes the number of arrows in clockwise (resp. anticlockwise) orientation (p, q≥ 1, p þ q ¼ n þ 1), that is, the quiver has type Að Þ p; q , the Coxeter polynomial χ is given by

$$
\chi\_{(p,q)} = \left(\boldsymbol{\omega} - \mathbf{1}\right)^2 \boldsymbol{v}\_p \boldsymbol{v}\_q. \tag{3}
$$

The Coxeter polynomial therefore only depends on the weight sequence p. Conversely, the Coxeter polynomial determines the weight sequence — up to

Let X be a finite partially ordered set (poset). The incidence algebra KX is

multiplication defined by exyezw ¼ δyzexw. Finite dimensional right modules over KX can be identified with commutative diagrams of finite dimensional K-vector spaces over the Hasse diagram of X, which is the directed graph whose vertices are the points of X, with an arrow from x to y if x<y and there is no z∈ X

We recollect the basic facts on the Euler form of posets and refer the reader to [6] for details. The algebra KX is of finite global dimension, hence its Euler form is well-defined and non-degenerate. Denote by CX, Φ<sup>X</sup> the matrices of the bilinear form and the corresponding Coxeter transformation with respect to the basis of

The incidence matrix of X, denoted 1X, is the X � X matrix defined by 1ð Þ <sup>X</sup> xy ¼ 1

Proposition. If X and Y are posets, then CX�<sup>Y</sup> ¼ CX ⊗ CY and Φ<sup>X</sup>�<sup>Y</sup> ¼ �Φ<sup>X</sup> ⊗ ΦY.

xy .

if x≤y and otherwise 1ð Þ <sup>X</sup> xy ¼ 0. By extending the partial order on X to a linear order, we can always arrange the elements of X such that the incidence matrix is uni-triangular. In particular, 1<sup>X</sup> is invertible over Z. Recall that the Möbius function

b. Let x, y ∈X. Then ð Þ Φ<sup>X</sup> xy ¼ �∑<sup>z</sup>:z≥<sup>x</sup>μXð Þ y; z .

4. Cyclotomic polynomials and polynomials of Littlewood type

The n-cyclotomic polynomial Φnð Þ T is inductively defined by the formula

d∣n

ð Þ �<sup>1</sup> <sup>r</sup> if <sup>n</sup> <sup>¼</sup> <sup>p</sup>1, … pr is a factorization into distinct primes:

.

Φdð Þ T : (5)

vn=<sup>d</sup>ð Þ <sup>T</sup> <sup>μ</sup>ð Þ <sup>d</sup> (6)

<sup>T</sup><sup>n</sup> � <sup>1</sup> <sup>¼</sup> <sup>Y</sup>

A more explicit expression for the cyclotomic polynomials is given by

1≤ d< n d∣n

<sup>Φ</sup>nð Þ¼ <sup>T</sup> <sup>Y</sup>

0 if n is divisible by a square

We recall some facts about cyclotomic polynomials.

The Möbius function is defined as follows:

for <sup>n</sup><sup>≥</sup> 2, where vn <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup>

the K-algebra spanned by elements exy for the pairs x≤y in X, with

Cyclotomic and Littlewood Polynomials Associated to Algebras

DOI: http://dx.doi.org/10.5772/intechopen.82309

ordering.

with x< z<y.

the simple KX-modules.

Lemma. a. CX <sup>¼</sup> <sup>1</sup>�<sup>1</sup>

4.1 Cyclotomic polynomials

μð Þ¼ n

11

�

<sup>μ</sup><sup>X</sup> : <sup>X</sup> � <sup>X</sup> ! <sup>Z</sup> is defined by <sup>μ</sup>Xð Þ¼ <sup>x</sup>; <sup>y</sup> ð Þ <sup>1</sup><sup>X</sup> �<sup>1</sup>

X .

3.3 Incidence algebras of posets

Hence χð Þ¼ 1 0, fitting into the above picture.

The next table displays the v-factorization of extended Dynkin quivers.


Remark: As is shown by the above table, proposition 2.3 extends to the tame hereditary case. That is, the Coxeter polynomial of a connected, tame hereditary K-algebra A (remember, K is algebraically closed) determines the algebra A up to derived equivalence. This is no longer true for wild hereditary algebras, not even for trees.

#### 3.2 Canonical algebras

Canonical algebras were introduced by Ringel [19]. They form a key class to study important features of representation theory. In the form of tubular canonical algebras they provide the standard examples of tame algebras of linear growth. Up to tilting canonical algebras are characterized as the connected K-algebras with a separating exact subcategory or a separating tubular one-parameter family (see [12]). That is, the module category mod � Λ accepts a separating tubular family T ¼ ð Þ T<sup>λ</sup> <sup>λ</sup> <sup>∈</sup>P1<sup>K</sup>, where T<sup>λ</sup> is a homogeneous tube for all λ with the exception of t tubes T<sup>λ</sup><sup>1</sup> , …, T<sup>λ</sup><sup>t</sup> with T<sup>λ</sup><sup>i</sup> of rank pi (1≤i ≤t).

Canonical algebras constitute an instance, where the explicit form of the Coxeter polynomial is known, see [11] or [10].

Proposition. Let Λ be a canonical algebra with weight and parameter data (p,λ). Then the Coxeter polynomial of Λ is given by

$$\chi\_{\Lambda} = \left(\mathbf{x} - \mathbf{1}\right)^{2} \prod\_{i=1}^{t} v\_{p\_{i}}.\tag{4}$$

The Coxeter polynomial therefore only depends on the weight sequence p. Conversely, the Coxeter polynomial determines the weight sequence — up to ordering.

#### 3.3 Incidence algebras of posets

2. χð Þ¼ 1 0 if and only if the star p1; …; pt

3. χð Þ1 <0 if and only if p1; …; pt

Polynomials - Theory and Application

the Coxeter polynomial χ is given by

for trees.

10

3.2 Canonical algebras

t tubes T<sup>λ</sup><sup>1</sup> , …, T<sup>λ</sup><sup>t</sup> with T<sup>λ</sup><sup>i</sup> of rank pi (1≤i ≤t).

polynomial is known, see [11] or [10].

Then the Coxeter polynomial of Λ is given by

Hence χð Þ¼ 1 0, fitting into the above picture.

correspondingly the algebra A is of tame (domestic) type.

correspondingly the algebra A is of wild representation type.

The above deals with all the Dynkin types and with the extended Dynkin dia-

grams of type <sup>D</sup><sup>~</sup> <sup>n</sup>, <sup>n</sup> <sup>≥</sup>4, and <sup>E</sup>~n, <sup>n</sup> <sup>¼</sup> <sup>6</sup>; <sup>7</sup>; 8. To complete the picture, we also consider the extended Dynkin quivers of type <sup>A</sup><sup>~</sup> <sup>n</sup> (<sup>n</sup> <sup>≥</sup>2) restricting, of course, to quivers without oriented cycles. Here, the Coxeter polynomial depends on the orientation: If p (resp. q) denotes the number of arrows in clockwise (resp. anticlockwise) orientation (p, q≥ 1, p þ q ¼ n þ 1), that is, the quiver has type Að Þ p; q ,

<sup>χ</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup>

The next table displays the v-factorization of extended Dynkin quivers.

<sup>A</sup><sup>~</sup> p,q — ð Þ <sup>p</sup>; <sup>q</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup>

<sup>D</sup><sup>~</sup> <sup>n</sup>, <sup>n</sup> <sup>≥</sup><sup>4</sup> [2,2,n-2] ð Þ <sup>2</sup>; <sup>2</sup>; <sup>n</sup> � <sup>2</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup>

<sup>E</sup>~<sup>6</sup> ½ � <sup>3</sup>; <sup>3</sup>; <sup>3</sup> ð Þ <sup>2</sup>; <sup>3</sup>; <sup>3</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup>

<sup>E</sup>~<sup>7</sup> ½ � <sup>2</sup>; <sup>4</sup>; <sup>4</sup> ð Þ <sup>2</sup>; <sup>3</sup>; <sup>4</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup>

<sup>E</sup>~<sup>8</sup> ½ � <sup>2</sup>; <sup>3</sup>; <sup>6</sup> ð Þ <sup>2</sup>; <sup>3</sup>; <sup>5</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup>

Remark: As is shown by the above table, proposition 2.3 extends to the tame hereditary case. That is, the Coxeter polynomial of a connected, tame hereditary K-algebra A (remember, K is algebraically closed) determines the algebra A up to derived equivalence. This is no longer true for wild hereditary algebras, not even

Canonical algebras were introduced by Ringel [19]. They form a key class to study important features of representation theory. In the form of tubular canonical algebras they provide the standard examples of tame algebras of linear growth. Up to tilting canonical algebras are characterized as the connected K-algebras with a separating exact subcategory or a separating tubular one-parameter family

(see [12]). That is, the module category mod � Λ accepts a separating tubular family T ¼ ð Þ T<sup>λ</sup> <sup>λ</sup> <sup>∈</sup>P1<sup>K</sup>, where T<sup>λ</sup> is a homogeneous tube for all λ with the exception of

Canonical algebras constitute an instance, where the explicit form of the Coxeter

Proposition. Let Λ be a canonical algebra with weight and parameter data (p,λ).

i¼1 vpi

<sup>χ</sup><sup>Λ</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup> □ <sup>Y</sup><sup>t</sup>

: (4)

Extended Dynkin type Star symbol Weight symbol Coxeter polynomial

� � is of extended Dynkin type,

vpvq: (3)

vp vq

v2v<sup>2</sup> 3

v2v3v<sup>4</sup>

v2v3v<sup>5</sup>

v2 <sup>2</sup>vn�<sup>2</sup>

� � is not Dynkin or extended Dynkin,

Let X be a finite partially ordered set (poset). The incidence algebra KX is the K-algebra spanned by elements exy for the pairs x≤y in X, with multiplication defined by exyezw ¼ δyzexw. Finite dimensional right modules over KX can be identified with commutative diagrams of finite dimensional K-vector spaces over the Hasse diagram of X, which is the directed graph whose vertices are the points of X, with an arrow from x to y if x<y and there is no z∈ X with x< z<y.

We recollect the basic facts on the Euler form of posets and refer the reader to [6] for details. The algebra KX is of finite global dimension, hence its Euler form is well-defined and non-degenerate. Denote by CX, Φ<sup>X</sup> the matrices of the bilinear form and the corresponding Coxeter transformation with respect to the basis of the simple KX-modules.

The incidence matrix of X, denoted 1X, is the X � X matrix defined by 1ð Þ <sup>X</sup> xy ¼ 1 if x≤y and otherwise 1ð Þ <sup>X</sup> xy ¼ 0. By extending the partial order on X to a linear order, we can always arrange the elements of X such that the incidence matrix is uni-triangular. In particular, 1<sup>X</sup> is invertible over Z. Recall that the Möbius function <sup>μ</sup><sup>X</sup> : <sup>X</sup> � <sup>X</sup> ! <sup>Z</sup> is defined by <sup>μ</sup>Xð Þ¼ <sup>x</sup>; <sup>y</sup> ð Þ <sup>1</sup><sup>X</sup> �<sup>1</sup> xy .

Lemma. a. CX <sup>¼</sup> <sup>1</sup>�<sup>1</sup> X .

$$\text{b. Let } \mathfrak{x}, \mathfrak{y} \in \mathcal{X}. \text{ Then } (\Phi\_{\mathcal{X}})\_{\mathfrak{x}\mathfrak{y}} = -\sum\_{x:x \geq \mathfrak{x}} \mu\_{\mathcal{X}}(\mathfrak{y}, x).$$

Proposition. If X and Y are posets, then CX�<sup>Y</sup> ¼ CX ⊗ CY and Φ<sup>X</sup>�<sup>Y</sup> ¼ �Φ<sup>X</sup> ⊗ ΦY.

### 4. Cyclotomic polynomials and polynomials of Littlewood type

#### 4.1 Cyclotomic polynomials

We recall some facts about cyclotomic polynomials. The n-cyclotomic polynomial Φnð Þ T is inductively defined by the formula

$$T^n - 1 = \prod\_{d|n} \Phi\_d(T). \tag{5}$$

The Möbius function is defined as follows:

$$
\mu(n) = \begin{pmatrix} 0 & \text{if } n \text{ is divisible by a square} \\ \left(-1\right)^r & \text{if } n = p\_1, \dots \ p\_r \text{ is a factorization into distinct primes.} \end{pmatrix}
$$

A more explicit expression for the cyclotomic polynomials is given by

$$\Phi\_n(T) = \prod\_{\substack{1 \le d < n \\ d \mid n}} \nu\_{n/d}(T)^{\mu(d)} \tag{6}$$

for <sup>n</sup><sup>≥</sup> 2, where vn <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> .

#### 4.2 Hereditary stars

A path algebra KΔ is said to be of Dynkin type if the underlying graph ∣Δ∣ of Δ is one of the ADE-series, that is, of type, An, Dn, for some n ≥1 or Ek, for k ¼ 6; 7; 8.

has Mahler measure μ<sup>0</sup> ¼ 1:176280…, and he asked if there exist any smaller values exceeding 1. In fact, the polynomial above is the Coxeter polynomial of the

We say that a matrix M is of Mahler type (resp. strictly Mahler type) if either Mð Þ¼ M 1 or Mð Þ M ≥μ<sup>0</sup> (resp. Mð Þ M >μ0). Earlier this year, Jean-Louis Verger-Gaugry announced a proof of Lehmer's conjecture, see https://arxiv.org/pdf/ 1709.03771.pdf. The key result (Theorem 5.28, p. 122) is a Dobrowolski type minoration of the Mahler Measure Mð Þ β . Experts are still reading the arguments,

In [8], Happel shows that the trace of the Coxeter matrix can be expressed as

ð Þ �<sup>1</sup> <sup>k</sup>

where <sup>H</sup><sup>k</sup>ð Þ <sup>A</sup> denotes the <sup>k</sup>-th Hochschild cohomology group. In particular, if

A N½ �¼ <sup>A</sup> <sup>0</sup>

Let B be an algebra and M a B-module. Consider the one-point extension A ¼ B N½ �. In [19] it is shown the Coxeter transformations of A and B are related by

<sup>ϕ</sup><sup>A</sup> <sup>¼</sup> <sup>ϕ</sup><sup>B</sup> �C<sup>T</sup>

where CB is the Cartan matrix of <sup>B</sup> which satisfies <sup>ϕ</sup><sup>B</sup> ¼ �C�<sup>T</sup>

We recall that the Euler quadratic form is defined as qAð Þ¼ <sup>x</sup> xCt

�nϕ<sup>B</sup> nC<sup>T</sup>

class of N in the Grothendieck group K0ð Þ B . In case A ¼ B N½ � with N an exceptional

Trð Þ¼ ϕ<sup>A</sup> Tr ϕ<sup>B</sup> ð Þ

A ¼ B M½ � for an algebra B and an indecomposable module M. In many cases, we get

!

N K � �

<sup>B</sup> n<sup>T</sup>

<sup>B</sup> nT � <sup>1</sup>

the one-point extension of A by N. This construction provides an order of vertices to deal with triangular algebras, that is, algebras KQ=I, where I is an ideal of the path

dimKH<sup>k</sup>

ð Þ A (7)

ð Þ¼ A 0 for i >0 and

(8)

. Assume that

<sup>B</sup> CB and n is the

Axt

∞ k¼0

�Trð Þ¼ ϕ<sup>A</sup> ∑

the Hochschild cohomology ring <sup>H</sup><sup>∗</sup> ð Þ <sup>A</sup> is trivial, that is, <sup>H</sup><sup>i</sup>

For an algebra A and a left A-module N we call

algebra KQ for Q a quiver without oriented cycles.

hereditary algebra whose underlying graph 2½ � ; 3; 7 is depicted below.

Cyclotomic and Littlewood Polynomials Associated to Algebras

DOI: http://dx.doi.org/10.5772/intechopen.82309

but there is no conclusive opinion.

4.5 Happel's trace formula

<sup>H</sup><sup>0</sup>ð Þ¼ <sup>A</sup> <sup>K</sup>, then Trð Þ¼� <sup>ϕ</sup><sup>A</sup> 1.

4.6 One-point extensions

module, it follows that

13

follows:

There are various instances where an explicit formula for the Coxeter polynomial is known.

Let A be the path algebra of a hereditary star p1; …; pt � � with respect to the standard orientation, see [13].

Since the Coxeter polynomial χ<sup>A</sup> does not depend on the orientation of A we will denote it by <sup>χ</sup> <sup>p</sup>1;…;pt ½ �. It follows that

$$\chi\_{\left[p\_1,\dots,p\_t\right]} = \prod\_{i=1}^t \nu\_{p\_i} \left( (T+1) - T \sum\_{j=1}^t \frac{\nu\_{p\_j-1}}{\nu\_{p\_j}} \right).$$

In particular, we have an explicit formula for the sum of coefficients of <sup>χ</sup> <sup>p</sup>1;…;pt ½ � as follows:

$$\sum\_{i=0}^{n} a\_i = \chi\_{\left[p\_1,\dots,p\_t\right]}(\mathbf{1}) = \prod\_{i=1}^{t} p\_i \left( 2 - \sum\_{i=1}^{t} \left( 1 - \frac{\mathbf{1}}{p\_i} \right) \right).$$

#### 4.3 Wild algebras

Let <sup>c</sup> be the real root of the polynomial <sup>T</sup><sup>3</sup> � <sup>T</sup> � 1, approximately <sup>c</sup> <sup>¼</sup> <sup>1</sup>:325. As observed in [21], a wild hereditary algebra A associated to a graph Δ without multiple arrows has spectral radius ρ<sup>A</sup> > c unless Δ is one of the following graphs:

In these cases, for m ≥ 8

$$c > \rho\_{[2,4,5]} > \rho\_{[2,3,m]} > \rho\_{[2,3,\overline{\mathbb{Z}}]} = \mu\_0$$

where μ<sup>0</sup> ¼ 1:176280… is the real root of the Coxeter polynomial

$$T^{10} + T^9 - T^7 - T^6 - T^5 - T^4 - T^3 + T + 1$$

associated to any hereditary algebra whose underlying graph is 2½ � ; 3; 7 . Observe that in these cases, the Mahler measure of the algebra equals the spectral radius.

#### 4.4 Lehmer polynomial

In 1933, D. H. Lehmer found that the polynomial

$$T^{10} + T^9 - T^7 - T^6 - T^5 - T^4 - T^3 + T + \mathbf{1}$$

has Mahler measure μ<sup>0</sup> ¼ 1:176280…, and he asked if there exist any smaller values exceeding 1. In fact, the polynomial above is the Coxeter polynomial of the hereditary algebra whose underlying graph 2½ � ; 3; 7 is depicted below.

We say that a matrix M is of Mahler type (resp. strictly Mahler type) if either Mð Þ¼ M 1 or Mð Þ M ≥μ<sup>0</sup> (resp. Mð Þ M >μ0). Earlier this year, Jean-Louis Verger-Gaugry announced a proof of Lehmer's conjecture, see https://arxiv.org/pdf/ 1709.03771.pdf. The key result (Theorem 5.28, p. 122) is a Dobrowolski type minoration of the Mahler Measure Mð Þ β . Experts are still reading the arguments, but there is no conclusive opinion.

#### 4.5 Happel's trace formula

4.2 Hereditary stars

standard orientation, see [13].

Polynomials - Theory and Application

denote it by <sup>χ</sup> <sup>p</sup>1;…;pt ½ �. It follows that

∑ n i¼0

mial is known.

as follows:

4.3 Wild algebras

In these cases, for m ≥ 8

4.4 Lehmer polynomial

12

A path algebra KΔ is said to be of Dynkin type if the underlying graph ∣Δ∣ of Δ is one of the ADE-series, that is, of type, An, Dn, for some n ≥1 or Ek, for k ¼ 6; 7; 8. There are various instances where an explicit formula for the Coxeter polyno-

Since the Coxeter polynomial χ<sup>A</sup> does not depend on the orientation of A we will

vpi ð Þ� T þ 1 T ∑

In particular, we have an explicit formula for the sum of coefficients of <sup>χ</sup> <sup>p</sup>1;…;pt ½ �

i¼1

Let <sup>c</sup> be the real root of the polynomial <sup>T</sup><sup>3</sup> � <sup>T</sup> � 1, approximately <sup>c</sup> <sup>¼</sup> <sup>1</sup>:325. As

<sup>c</sup>>ρ½ � <sup>2</sup>;4;<sup>5</sup> <sup>&</sup>gt;ρ½ � <sup>2</sup>;3;<sup>m</sup> <sup>&</sup>gt; <sup>ρ</sup>½ � <sup>2</sup>;3;<sup>7</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup>

<sup>T</sup><sup>10</sup> <sup>þ</sup> <sup>T</sup><sup>9</sup> � <sup>T</sup><sup>7</sup> � <sup>T</sup><sup>6</sup> � <sup>T</sup><sup>5</sup> � <sup>T</sup><sup>4</sup> � <sup>T</sup><sup>3</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup>

<sup>T</sup><sup>10</sup> <sup>þ</sup> <sup>T</sup><sup>9</sup> � <sup>T</sup><sup>7</sup> � <sup>T</sup><sup>6</sup> � <sup>T</sup><sup>5</sup> � <sup>T</sup><sup>4</sup> � <sup>T</sup><sup>3</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup>

associated to any hereditary algebra whose underlying graph is 2½ � ; 3; 7 . Observe that in these cases, the Mahler measure of the algebra equals the spectral radius.

where μ<sup>0</sup> ¼ 1:176280… is the real root of the Coxeter polynomial

In 1933, D. H. Lehmer found that the polynomial

observed in [21], a wild hereditary algebra A associated to a graph Δ without multiple arrows has spectral radius ρ<sup>A</sup> > c unless Δ is one of the following graphs:

t j¼1

!

pi 2 � ∑ t i¼1 vpj �1 vpj

<sup>1</sup> � <sup>1</sup> pi

� � � �

:

:

� � with respect to the

Let A be the path algebra of a hereditary star p1; …; pt

<sup>χ</sup> <sup>p</sup>1;…;pt ½ � <sup>¼</sup> <sup>Y</sup><sup>t</sup>

i¼1

ai <sup>¼</sup> <sup>χ</sup> <sup>p</sup>1;…;pt ½ �ð Þ¼ <sup>1</sup> <sup>Y</sup><sup>t</sup>

In [8], Happel shows that the trace of the Coxeter matrix can be expressed as follows:

$$-\text{Tr}\,(\phi\_A) = \sum\_{k=0}^{\infty} (-1)^k \text{dim}\_K H^k(A) \tag{7}$$

where <sup>H</sup><sup>k</sup>ð Þ <sup>A</sup> denotes the <sup>k</sup>-th Hochschild cohomology group. In particular, if the Hochschild cohomology ring <sup>H</sup><sup>∗</sup> ð Þ <sup>A</sup> is trivial, that is, <sup>H</sup><sup>i</sup> ð Þ¼ A 0 for i >0 and <sup>H</sup><sup>0</sup>ð Þ¼ <sup>A</sup> <sup>K</sup>, then Trð Þ¼� <sup>ϕ</sup><sup>A</sup> 1.

For an algebra A and a left A-module N we call

$$\mathcal{A}[\mathbf{N}] = \begin{bmatrix} \mathbf{A} & \mathbf{0} \\ \mathbf{N} & \mathbf{K} \end{bmatrix}$$

the one-point extension of A by N. This construction provides an order of vertices to deal with triangular algebras, that is, algebras KQ=I, where I is an ideal of the path algebra KQ for Q a quiver without oriented cycles.

#### 4.6 One-point extensions

Let B be an algebra and M a B-module. Consider the one-point extension A ¼ B N½ �. In [19] it is shown the Coxeter transformations of A and B are related by

$$\phi\_A = \begin{pmatrix} \phi\_B & -\mathbf{C}\_B^T \boldsymbol{n}^T \\ -\boldsymbol{n}\phi\_B & \boldsymbol{n}\mathbf{C}\_B^T \boldsymbol{n}^T - \mathbf{1} \end{pmatrix} \tag{8}$$

where CB is the Cartan matrix of <sup>B</sup> which satisfies <sup>ϕ</sup><sup>B</sup> ¼ �C�<sup>T</sup> <sup>B</sup> CB and n is the class of N in the Grothendieck group K0ð Þ B . In case A ¼ B N½ � with N an exceptional module, it follows that

$$\operatorname{Tr}\left(\phi\_A\right) = \operatorname{Tr}\left(\phi\_B\right).$$

We recall that the Euler quadratic form is defined as qAð Þ¼ <sup>x</sup> xCt Axt . Assume that A ¼ B M½ � for an algebra B and an indecomposable module M. In many cases, we get that qAð Þ m >0, for m the dimension vector of M (for instance, if M is preprojective, or if qA coincides with the Tits form of A...)

Proposition. Let A be an accessible algebra, such that qAð Þ m >0 for m the dimension vector of M, where A ¼ B M½ � for certain algebra B and an indecomposable module M. Then the following happens:

a. Trð Þ ϕ<sup>A</sup> ≥ � 1;

$$b. \text{ if } \operatorname{Tr}(\phi\_B) = -\mathbf{1} \text{ and } q\_B(m) = \mathbf{1}, \text{ then } \operatorname{Tr}(\phi\_A) = -\mathbf{1}.$$

Proof. Assume that A ¼ B M½ � for an algebra B and an indecomposable module M such that qAð Þ m >0 for m the dimension vector of M. Then B is also accessible. By induction hypothesis, Tr ϕ<sup>B</sup> ð Þ≥ � 1. Then

$$\operatorname{Tr}(\phi\_A) = \operatorname{Tr}(\phi\_B) + \left(m\mathbf{C}\_B^T m^T - \mathbf{1}\right) \ge -\mathbf{1} + \left(m\mathbf{C}\_B^T m^T - \mathbf{1}\right) = -\mathbf{1} + \left(q\_B(m) - \mathbf{1}\right) \ge -\mathbf{1}$$

This shows (a).

For (b) assume that Tr ϕ<sup>B</sup> ð Þ¼�1 and qBð Þ¼ m 1, then

$$\operatorname{Tr}\left(\phi\_A\right) = \operatorname{Tr}\left(\phi\_B\right) + \left(m\mathbf{C}\_B^T m^T - \mathbf{1}\right) = -\mathbf{1} + \left(m\mathbf{C}\_B^T m^T - \mathbf{1}\right) = -\mathbf{1} + \left(q\_B(m) - \mathbf{1}\right) = -\mathbf{1}$$

$$\Box$$

□

<sup>0</sup> <sup>¼</sup> <sup>c</sup>01<sup>n</sup> <sup>þ</sup> <sup>c</sup>1<sup>M</sup> <sup>þ</sup> <sup>c</sup>2M<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> cn�1Mn�<sup>1</sup> <sup>þ</sup> cnMn

<sup>c</sup>01<sup>n</sup> <sup>þ</sup> <sup>c</sup>2M<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>c</sup>2mM<sup>2</sup><sup>m</sup> <sup>¼</sup> <sup>c</sup>1<sup>M</sup> <sup>þ</sup> <sup>c</sup>3M<sup>3</sup> <sup>þ</sup> … <sup>þ</sup> <sup>c</sup>2m�1M2m�<sup>1</sup> <sup>þ</sup> <sup>c</sup>2ð Þ� <sup>m</sup>þ<sup>r</sup> <sup>1</sup>M2ð Þ� <sup>m</sup>þ<sup>r</sup> <sup>1</sup>

<sup>c</sup>01<sup>n</sup> <sup>þ</sup> <sup>c</sup>2M<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>c</sup>2mM<sup>2</sup><sup>m</sup>

so that we get two expressions of P as positive linear combinations of powers of M. Suppose that n ¼ 2m þ 1. By hypothesis we have Tr ð Þ P ≤n. Moreover, since

Tr <sup>M</sup><sup>n</sup> ð Þ<sup>≤</sup> Trð Þ¼ <sup>Q</sup> Trð Þ <sup>P</sup> <sup>≤</sup> <sup>n</sup>

Otherwise, <sup>n</sup> <sup>¼</sup> <sup>2</sup>m. The claim follows similarly. □

Proof of Theorem 1. Observe that M ¼ ϕ<sup>A</sup> is a real unimodular matrix. One implication of the Theorem was shown before. Suppose that ∣Tr M<sup>k</sup> ∣ ≤n or equiv-

cyclotomic. □

of Littlewood type if every coefficient non-zero pi has modulus 1. A polynomial p tð Þ

1=2 <∣z∣<2

<sup>1</sup> <sup>¼</sup> <sup>ϵ</sup>1<sup>z</sup> <sup>þ</sup> <sup>ϵ</sup>2z<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>ϵ</sup>nz<sup>n</sup>

Moreover, if <sup>∣</sup>z∣>1, then 1=∣z∣<1 and 1=<sup>2</sup> <sup>&</sup>lt;1=∣z∣<2. Hence 1=<sup>2</sup> <sup>&</sup>lt;∣z∣<2. □

Definition. A Littlewood series is a power series all of whose coefficients are

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> j j <sup>z</sup> <sup>n</sup> <sup>&</sup>lt;∣z∣=ð Þ <sup>1</sup> � jz<sup>j</sup> so <sup>∣</sup>z∣>1=2. Since <sup>z</sup> is the root

<sup>n</sup>�<sup>1</sup> <sup>þ</sup> pnt

<sup>n</sup> is

alently, �n<sup>≤</sup> Tr <sup>M</sup><sup>k</sup> <sup>≤</sup><sup>n</sup> for 0 <sup>≤</sup>k<sup>≤</sup> <sup>n</sup>. The Proposition above yields that <sup>M</sup> is

An integral self-reciprocal polynomial p tðÞ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>p</sup>1<sup>t</sup> <sup>þ</sup> … <sup>þ</sup> pn�<sup>1</sup><sup>t</sup>

Lemma. If z is a root of a polynomial of Littlewood type, then

of Littlewood type with all pi 6¼ 0, for i ¼ 0, 1, …, n, is said to be Littlewood.

Proof. Suppose z is a root of a polynomial of Littlewood type. Then

of a polynomial of Littlewood type if and only if z�<sup>1</sup> is, then 1=2< ∣z∣<2.

Let P ¼ fz∈ C : z is the root of some Littlewood polynomial g.

<sup>c</sup>1<sup>M</sup> <sup>þ</sup> <sup>c</sup>3M<sup>3</sup> <sup>þ</sup> … <sup>þ</sup> <sup>c</sup>2m�1M2m�<sup>1</sup> <sup>þ</sup> <sup>c</sup>2ð Þ� <sup>m</sup>þ<sup>r</sup> <sup>1</sup>M2ð Þ� <sup>m</sup>þ<sup>r</sup> <sup>1</sup>

Let c>0 be the common value of the trace of this matrix. Write n ¼ 2m þ r for r ¼ 0 or 1. Consider the matrices

> <sup>P</sup> <sup>¼</sup> <sup>1</sup> c

Cyclotomic and Littlewood Polynomials Associated to Algebras

DOI: http://dx.doi.org/10.5772/intechopen.82309

Then

cn ¼ 1 then

4.9 Theorem 1

<sup>Q</sup> ¼ � <sup>1</sup> c

The claim follows by induction.

4.10 Polynomials of Littlewood type

for some ϵ<sup>i</sup> ∈f g �1; 0; 1 . If ∣z∣<1 then 1≤ ∣z∣ þ j j z

4.11 Littlewood series

1, 0 or �1.

15

#### 4.7 Strongly accessible algebras

Theorem: A finite dimensional accessible algebra A then it is strongly accessible if and only if Trð Þ¼� ϕ<sup>A</sup> 1.

Proof. Assume A is strongly accessible from A0. Since qAð Þ m ≥1, for A ¼ B M½ � a one-point extension of the subcategory B of A by the exceptional module M (since then qAð Þ¼ m dimKEndAð Þ M ). By the Proposition above

$$\operatorname{Tr}\left(\phi\_A\right) = \operatorname{Tr}\left(\phi\_{A\_{n-1}}\right) = \dots = \operatorname{Tr}\left(\phi\_{A\_0}\right) = -\mathbf{1}$$

Conversely, assume that Trð Þ¼� ϕ<sup>A</sup> 1 and write A ¼ B M½ � as a one-point extension of the subcategory B of A by the module M. We shall prove that M is exceptional.

$$-\mathbf{1} = \operatorname{Tr}\left(\phi\_A\right) = \operatorname{Tr}\left(\phi\_B\right) + \left(m\mathbf{C}\_B^T m^T - \mathbf{1}\right) \ge -\mathbf{1} + \left(m\mathbf{C}\_B^T m^T - \mathbf{1}\right) = -\mathbf{1} + \left(q\_B(m) - \mathbf{1}\right) \ge -\mathbf{1}$$

Equality holds and qBð Þ¼ m 1, since M is indecomposable, it follows that the extension ring of <sup>M</sup> is trivial. □

#### 4.8 Stable matrices

The following statement is Theorem 1 for stable matrices.

Proposition. Suppose M is a stable unimodular n � n-matrix. Let χ<sup>M</sup> ¼ c<sup>0</sup> þ c1Tþ <sup>c</sup>2T<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> cn�<sup>2</sup>T<sup>n</sup>�<sup>2</sup> <sup>þ</sup> cn�<sup>1</sup>T<sup>n</sup>�<sup>1</sup> <sup>þ</sup> cnT<sup>n</sup> be its characteristic polynomial. Suppose that <sup>0</sup><sup>&</sup>lt; TrMk <sup>≤</sup> m for p <sup>≤</sup>k≤<sup>p</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup> and certain integers <sup>1</sup><sup>≤</sup> p and m. Then 0< TrM<sup>k</sup> ≤ m for all integers p≤ k. In particular, M is of cyclotomic type. Proof. Consider the coefficients c0, c1, …cn of χM. Since M is stable then cn ¼ 1, cn�<sup>1</sup> <0, cn�<sup>2</sup> > 0 and the signs alternate until we meet a j with cjc<sup>0</sup> <0. Cayley-Hamilton theorem states that χMð Þ¼ M 0. Then

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

$$\mathbf{0} = \mathbf{c}\_0 \mathbf{1}\_n + \mathbf{c}\_1 \mathbf{M} + \mathbf{c}\_2 \mathbf{M}^2 + \dots + \mathbf{c}\_{n-1} \mathbf{M}^{n-1} + \mathbf{c}\_n \mathbf{M}^n$$

Then

that qAð Þ m >0, for m the dimension vector of M (for instance, if M is preprojective,

Proposition. Let A be an accessible algebra, such that qAð Þ m >0 for m the dimension vector of M, where A ¼ B M½ � for certain algebra B and an indecomposable module M.

Proof. Assume that A ¼ B M½ � for an algebra B and an indecomposable module M such that qAð Þ m >0 for m the dimension vector of M. Then B is also accessible. By

Theorem: A finite dimensional accessible algebra A then it is strongly accessible if

Proof. Assume A is strongly accessible from A0. Since qAð Þ m ≥1, for A ¼ B M½ � a one-point extension of the subcategory B of A by the exceptional module M (since

<sup>¼</sup> … <sup>¼</sup> Tr <sup>ϕ</sup><sup>A</sup><sup>0</sup>

Conversely, assume that Trð Þ¼� ϕ<sup>A</sup> 1 and write A ¼ B M½ � as a one-point extension of the subcategory B of A by the module M. We shall prove that M is

Equality holds and qBð Þ¼ m 1, since M is indecomposable, it follows that the extension ring of <sup>M</sup> is trivial. □

Proposition. Suppose M is a stable unimodular n � n-matrix. Let χ<sup>M</sup> ¼ c<sup>0</sup> þ c1Tþ

Suppose that <sup>0</sup><sup>&</sup>lt; TrMk <sup>≤</sup> m for p <sup>≤</sup>k≤<sup>p</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup> and certain integers <sup>1</sup><sup>≤</sup> p and m.

Proof. Consider the coefficients c0, c1, …cn of χM. Since M is stable then cn ¼ 1, cn�<sup>1</sup> <0, cn�<sup>2</sup> > 0 and the signs alternate until we meet a j with cjc<sup>0</sup> <0.

BmT � <sup>1</sup> <sup>≥</sup> � <sup>1</sup> <sup>þ</sup> mCT

¼ �<sup>1</sup>

Bm<sup>T</sup> � <sup>1</sup> ¼ �<sup>1</sup> <sup>þ</sup> qBð Þ� <sup>m</sup> <sup>1</sup> <sup>≥</sup> � <sup>1</sup>

Bm<sup>T</sup> � <sup>1</sup> ¼ �<sup>1</sup> <sup>þ</sup> qBð Þ� <sup>m</sup> <sup>1</sup> <sup>≥</sup> � <sup>1</sup>

Bm<sup>T</sup> � <sup>1</sup> ¼ �<sup>1</sup> <sup>þ</sup> qBð Þ� <sup>m</sup> <sup>1</sup> ¼ �<sup>1</sup>

□

or if qA coincides with the Tits form of A...)

induction hypothesis, Tr ϕ<sup>B</sup> ð Þ≥ � 1. Then

b. if Tr ϕ<sup>B</sup> ð Þ¼�1 and qBð Þ¼ m 1, then Trð Þ¼� ϕ<sup>A</sup> 1.

Bm<sup>T</sup> � <sup>1</sup> <sup>≥</sup> � <sup>1</sup> <sup>þ</sup> mC<sup>T</sup>

Bm<sup>T</sup> � <sup>1</sup> ¼ �<sup>1</sup> <sup>þ</sup> mC<sup>T</sup>

For (b) assume that Tr ϕ<sup>B</sup> ð Þ¼�1 and qBð Þ¼ m 1, then

then qAð Þ¼ m dimKEndAð Þ M ). By the Proposition above

Trð Þ¼ ϕ<sup>A</sup> Tr ϕAn�<sup>1</sup>

The following statement is Theorem 1 for stable matrices.

Then 0< TrM<sup>k</sup> ≤ m for all integers p≤ k. In particular, M is of cyclotomic type.

Cayley-Hamilton theorem states that χMð Þ¼ M 0. Then

<sup>c</sup>2T<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> cn�<sup>2</sup>T<sup>n</sup>�<sup>2</sup> <sup>þ</sup> cn�<sup>1</sup>T<sup>n</sup>�<sup>1</sup> <sup>þ</sup> cnT<sup>n</sup> be its characteristic polynomial.

Then the following happens:

Polynomials - Theory and Application

a. Trð Þ ϕ<sup>A</sup> ≥ � 1;

Trð Þ¼ <sup>ϕ</sup><sup>A</sup> Tr <sup>ϕ</sup><sup>B</sup> ð Þþ mC<sup>T</sup>

<sup>T</sup>rð Þ¼ <sup>ϕ</sup><sup>A</sup> <sup>T</sup><sup>r</sup> <sup>ϕ</sup><sup>B</sup> ð Þþ mC<sup>T</sup>

and only if Trð Þ¼� ϕ<sup>A</sup> 1.

exceptional.

14

4.8 Stable matrices

4.7 Strongly accessible algebras

�<sup>1</sup> <sup>¼</sup> Trð Þ¼ <sup>ϕ</sup><sup>A</sup> Tr <sup>ϕ</sup><sup>B</sup> ð Þþ mC<sup>T</sup>

This shows (a).

$$c\_0 \mathbf{1}\_n + c\_2 \mathbf{M}^2 + \dots + c\_{2m} \mathbf{M}^{2m} = c\_1 \mathbf{M} + c\_3 \mathbf{M}^3 + \dots + c\_{2m-1} \mathbf{M}^{2m-1} + c\_{2(m+r)-1} \mathbf{M}^{2(m+r)-1}$$

Let c>0 be the common value of the trace of this matrix. Write n ¼ 2m þ r for r ¼ 0 or 1. Consider the matrices

$$P = \frac{1}{\mathcal{c}} \left( c\_0 \mathbf{1}\_n + c\_2 \mathbf{M}^2 + \dots + c\_{2m} \mathbf{M}^{2m} \right)$$

$$\mathcal{Q} = -\frac{1}{\mathcal{c}} \left( \left( c\_1 \mathbf{M} + c\_3 \mathbf{M}^3 + \dots + c\_{2m-1} \mathbf{M}^{2m-1} + c\_{2(m+r)-1} \mathbf{M}^{2(m+r)-1} \right) \right)$$

so that we get two expressions of P as positive linear combinations of powers of M. Suppose that n ¼ 2m þ 1. By hypothesis we have Tr ð Þ P ≤n. Moreover, since cn ¼ 1 then

$$\operatorname{Tr}\left(\mathcal{M}^n\right) \le \operatorname{Tr}\left(Q\right) = \operatorname{Tr}\left(P\right) \le n$$

The claim follows by induction.

Otherwise, <sup>n</sup> <sup>¼</sup> <sup>2</sup>m. The claim follows similarly. □

#### 4.9 Theorem 1

Proof of Theorem 1. Observe that M ¼ ϕ<sup>A</sup> is a real unimodular matrix. One implication of the Theorem was shown before. Suppose that ∣Tr M<sup>k</sup> ∣ ≤n or equivalently, �n<sup>≤</sup> Tr <sup>M</sup><sup>k</sup> <sup>≤</sup><sup>n</sup> for 0 <sup>≤</sup>k<sup>≤</sup> <sup>n</sup>. The Proposition above yields that <sup>M</sup> is cyclotomic. □

#### 4.10 Polynomials of Littlewood type

An integral self-reciprocal polynomial p tðÞ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>p</sup>1<sup>t</sup> <sup>þ</sup> … <sup>þ</sup> pn�<sup>1</sup><sup>t</sup> <sup>n</sup>�<sup>1</sup> <sup>þ</sup> pnt <sup>n</sup> is of Littlewood type if every coefficient non-zero pi has modulus 1. A polynomial p tð Þ of Littlewood type with all pi 6¼ 0, for i ¼ 0, 1, …, n, is said to be Littlewood.

Lemma. If z is a root of a polynomial of Littlewood type, then

$$1/2 < |z| < 2$$

Proof. Suppose z is a root of a polynomial of Littlewood type. Then

$$\mathbf{1} = \epsilon\_1 \mathbf{z} + \epsilon\_2 \mathbf{z}^2 + \dots + \epsilon\_n \mathbf{z}^n$$

for some ϵ<sup>i</sup> ∈f g �1; 0; 1 .

If ∣z∣<1 then 1≤ ∣z∣ þ j j z <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> j j <sup>z</sup> <sup>n</sup> <sup>&</sup>lt;∣z∣=ð Þ <sup>1</sup> � jz<sup>j</sup> so <sup>∣</sup>z∣>1=2. Since <sup>z</sup> is the root of a polynomial of Littlewood type if and only if z�<sup>1</sup> is, then 1=2< ∣z∣<2.

Moreover, if <sup>∣</sup>z∣>1, then 1=∣z∣<1 and 1=<sup>2</sup> <sup>&</sup>lt;1=∣z∣<2. Hence 1=<sup>2</sup> <sup>&</sup>lt;∣z∣<2. □

#### 4.11 Littlewood series

Definition. A Littlewood series is a power series all of whose coefficients are 1, 0 or �1.

Let P ¼ fz∈ C : z is the root of some Littlewood polynomial g.

#### Remarks:

a. Littlewood series converge for ∣z∣<1.

b. A point z∈ C with ∣z∣<1 lies in P if and only if some Littlewood series vanishes at this point.

Proof. (a): Consider m ≥1, n ¼ 3 þ 6m and the algebra Bn ¼ R3þ6<sup>m</sup> such that

<sup>¼</sup> ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>T</sup>nþ<sup>6</sup> <sup>þ</sup> <sup>T</sup>nþ<sup>5</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup> � � � <sup>T</sup><sup>3</sup> ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>χ</sup>Rn � <sup>T</sup>χCnþ<sup>5</sup>

modð Þ Cn�<sup>1</sup> where Cn�<sup>1</sup> is a proper quotient of an algebra derived equivalent to

We shall calculate χC2þ6<sup>m</sup> . Observe that C2þ6<sup>m</sup> is tilting equivalent to the

<sup>χ</sup>C2þ6<sup>m</sup> <sup>¼</sup> ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>χ</sup>R1þ6<sup>m</sup> � <sup>T</sup>χR6<sup>m</sup> <sup>¼</sup> <sup>T</sup>2þ6<sup>m</sup> <sup>þ</sup> <sup>T</sup>1þ6<sup>m</sup> � <sup>T</sup><sup>3</sup> ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>χ</sup>R1þ6ð Þ <sup>m</sup>�<sup>1</sup> � <sup>T</sup>χR6ð Þ <sup>m</sup>�<sup>1</sup>

<sup>χ</sup>Amþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>T</sup><sup>n</sup>þ<sup>6</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>þ<sup>5</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup> � � � <sup>T</sup><sup>3</sup> ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>χ</sup>Rn � T T<sup>n</sup>þ<sup>5</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>þ<sup>4</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup> � �

(b) By induction, we shall construct polynomials rm representing χAm .

Observe that <sup>T</sup><sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>1</sup> � � <sup>¼</sup> vn � Tvn�<sup>2</sup> then <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>¼</sup>

ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>1</sup> � � is represented by wn <sup>¼</sup> T uð Þ <sup>n</sup>�<sup>1</sup> � un�<sup>3</sup> .

<sup>χ</sup>Am <sup>T</sup><sup>2</sup> � � <sup>¼</sup> <sup>T</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> � � <sup>T</sup><sup>2</sup>n�<sup>2</sup> <sup>þ</sup> <sup>1</sup> � � � <sup>T</sup><sup>6</sup> <sup>χ</sup>Am�<sup>1</sup> <sup>T</sup><sup>2</sup> � �

<sup>¼</sup> <sup>T</sup><sup>10</sup> � <sup>9</sup>T<sup>8</sup> � <sup>T</sup><sup>7</sup> <sup>þ</sup> <sup>27</sup>T<sup>6</sup> <sup>þ</sup> <sup>3</sup>T<sup>5</sup> � <sup>30</sup>T<sup>4</sup> � <sup>T</sup><sup>3</sup> <sup>þ</sup> <sup>9</sup>T<sup>2</sup>

rm <sup>¼</sup> <sup>T</sup><sup>n</sup> � ð Þ <sup>n</sup> � <sup>1</sup> <sup>T</sup><sup>n</sup>�<sup>2</sup> � <sup>T</sup><sup>3</sup>

For <sup>n</sup> <sup>¼</sup> <sup>4</sup> <sup>þ</sup> <sup>6</sup>m, we define rm <sup>¼</sup> wn � <sup>T</sup><sup>3</sup>

For <sup>m</sup> <sup>¼</sup> 0, we have <sup>χ</sup><sup>A</sup><sup>0</sup> <sup>¼</sup> <sup>T</sup><sup>4</sup> <sup>þ</sup> <sup>T</sup><sup>3</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> 1, which is represented by the

<sup>¼</sup> <sup>T</sup><sup>n</sup> wn <sup>T</sup> <sup>þ</sup> <sup>T</sup>�<sup>1</sup> � � � <sup>T</sup><sup>6</sup>T<sup>n</sup>�<sup>6</sup> rm�<sup>1</sup> <sup>T</sup> <sup>þ</sup> <sup>T</sup>�<sup>1</sup> � � <sup>¼</sup> <sup>T</sup><sup>n</sup> rm <sup>T</sup> <sup>þ</sup> <sup>T</sup>�<sup>1</sup> � �

which has ξð Þ¼ r<sup>1</sup> 4 changes of sign in the sequence of coefficients. According to Descartes rule of signs, r<sup>1</sup> has at most ξð Þ¼ r<sup>1</sup> 4 positive real roots. Since r<sup>1</sup> represents

, then χ<sup>A</sup><sup>1</sup> has at most 2ξð Þ¼ r<sup>1</sup> 8 roots in the unit circle. That is, χ<sup>A</sup><sup>1</sup> has at least 2

We shall prove, by induction, that rm has at most ξð Þ¼ rm 2ð Þ m þ 1 positive real

<sup>r</sup><sup>0</sup> <sup>¼</sup> T T<sup>9</sup> � <sup>8</sup>T<sup>7</sup> <sup>þ</sup> <sup>21</sup>T<sup>5</sup> � <sup>20</sup>T<sup>3</sup> <sup>þ</sup> <sup>5</sup><sup>T</sup> � � � <sup>T</sup><sup>7</sup> � <sup>6</sup>T<sup>5</sup> <sup>þ</sup> <sup>10</sup>T<sup>3</sup> � <sup>4</sup><sup>T</sup> � � � �

qm <sup>þ</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>T</sup><sup>2</sup>

<sup>χ</sup><sup>C</sup>2þ6ð Þ <sup>m</sup>�<sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup>

χAm þ T þ 1

<sup>n</sup> in <sup>D</sup>bð Þ Bn is derived equivalent to

n o

rm�1. We verify by induction on m that

Am <sup>¼</sup> Bn½ � Pn and the perpendicular category <sup>P</sup><sup>⊥</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82309

Cyclotomic and Littlewood Polynomials Associated to Algebras

χAmþ<sup>1</sup> ¼ ð Þ T þ 1 χRnþ<sup>6</sup> � TχCnþ<sup>5</sup>

<sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>¼</sup> <sup>T</sup>2þ6<sup>m</sup> <sup>þ</sup> <sup>T</sup>1þ6<sup>m</sup> � <sup>T</sup><sup>3</sup>

� <sup>T</sup><sup>3</sup>TχCn�<sup>1</sup> <sup>¼</sup> <sup>T</sup><sup>n</sup>þ<sup>7</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>þ<sup>6</sup> � <sup>T</sup><sup>3</sup>

As consequence of formula (a) we observe the following:

one-point extension R1þ6m½ � P<sup>1</sup> . Hence

R2þ6m. Therefore

which implies

as claimed.

) L χAm

rm represents χAm :

For instance.

roots z with ∣z∣ 6¼ 1.

roots. Indeed, write

<sup>r</sup><sup>1</sup> <sup>¼</sup> <sup>w</sup><sup>10</sup> � <sup>T</sup><sup>3</sup>

χA1

17

� � <sup>¼</sup> <sup>4</sup><sup>m</sup> <sup>þ</sup> 5.

polynomial <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>T</sup><sup>4</sup> � <sup>3</sup>T<sup>2</sup> <sup>þ</sup> 1.

� <sup>T</sup><sup>3</sup> <sup>T</sup><sup>4</sup> � <sup>3</sup>T<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �

(a<sup>0</sup>

c. A Littlewood polynomial is not a Littlewood series. But any Littlewood polynomial, say p zð Þ¼ <sup>a</sup><sup>0</sup> <sup>þ</sup> … <sup>þ</sup> adzd yields a Littlewood series having the same roots <sup>z</sup> with ∣z∣<1: indeed, consider the series

P zð Þ¼ p zð Þ<sup>=</sup> <sup>1</sup> � <sup>z</sup>dþ<sup>1</sup> <sup>¼</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup> … <sup>þ</sup> adzd <sup>þ</sup> <sup>a</sup>0zdþ<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> adz<sup>2</sup>dþ<sup>1</sup> <sup>þ</sup> <sup>a</sup>0z2dþ<sup>2</sup> <sup>þ</sup> …

Thus P⊂ R, where R is the set of roots of Littlewood series. We shall show the Proposition at the Introduction.

Proof. Let <sup>L</sup> be the set of Littlewood series. Then <sup>L</sup> ¼ �f g <sup>1</sup>; <sup>0</sup>; <sup>1</sup> <sup>ℕ</sup>, so with the product topology it is homeomorphic to the Cantor set. Choose 0< r< 1. Let F be the space of finite multisets of points z with ∣z∣<r, modulo the equivalence relation generated by S ffi S∪X when ∣X∣ ¼ r .

Claim. Any Littlewood series has finitely many roots in the disc ∣z∣ ≤r. The map f : L ! F sending a Littlewood series to its multiset of roots in this disc is continuous.

Since L is compact, the image of f is closed. From this we can show that R, the set of roots of Littlewood series, is closed. Since Littlewood polynomials are densely included in L and f is continuous, we get that P, the set of roots of Littlewood polynomials, is dense in <sup>R</sup>. It follows that <sup>P</sup> <sup>¼</sup> <sup>R</sup>, as we wanted to show. □

#### 5. An example

#### 5.1 Construction

For m a natural number and let n ¼ 3 þ 6m. Let Rn be an algebra formed by n commutative squares. Consider the one-point extension Am ¼ Rn½ � Pn with Pn the unique indecomposable projective Rn-module of K-dimension 2. Observe that Am (resp. Cn�1) is given by the following quiver with n þ 1 vertices and commutative relations (resp. n � 1 vertices and relations):

We claim:

a. <sup>χ</sup>Am <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> � <sup>T</sup><sup>3</sup> <sup>χ</sup>Am�<sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> 1, for all <sup>n</sup> <sup>≥</sup>1. As consequence, the algebras Am and Cn are of Littlewood type;

b. the number of eigenvalues of ϕAm not lying in the unit disk is at least m;

$$\text{c. } \mathsf{M}(\mathsf{X}\_{A\_{\mathsf{m}}}) \leq \mathsf{8}.$$

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

Proof. (a): Consider m ≥1, n ¼ 3 þ 6m and the algebra Bn ¼ R3þ6<sup>m</sup> such that Am <sup>¼</sup> Bn½ � Pn and the perpendicular category <sup>P</sup><sup>⊥</sup> <sup>n</sup> in <sup>D</sup>bð Þ Bn is derived equivalent to modð Þ Cn�<sup>1</sup> where Cn�<sup>1</sup> is a proper quotient of an algebra derived equivalent to R2þ6m. Therefore

$$\begin{aligned} \chi\_{A\_{m+1}} &= (T+\mathbf{1})\chi\_{R\_{n+6}} - T\chi\_{C\_{n+5}} \\ &= (T+\mathbf{1}) \left( T^{n+6} + T^{n+5} + T + \mathbf{1} \right) - T^3 (T+\mathbf{1})\chi\_{R\_n} - T\chi\_{C\_{n+5}} \end{aligned}$$

We shall calculate χC2þ6<sup>m</sup> . Observe that C2þ6<sup>m</sup> is tilting equivalent to the one-point extension R1þ6m½ � P<sup>1</sup> . Hence

$$\begin{aligned} \chi\_{C\_{2+\delta m}} &= (T+\mathbf{1})\chi\_{R\_{1+\delta m}} - T\chi\_{R\_{\delta m}} = T^{2+\delta m} + T^{1+\delta m} - T^3 \left\{ (T+\mathbf{1})\chi\_{R\_{1+\delta(m-1)}} - T\chi\_{R\_{\delta(m-1)}} \right\} \\ &+ T + \mathbf{1} = T^{2+\delta m} + T^{1+\delta m} - T^3 \chi\_{C\_{2+\delta(m-1)}} + T + \mathbf{1} \end{aligned}$$

which implies

Remarks:

at this point.

is continuous.

5. An example

5.1 Construction

We claim:

c. M χAm

16

≤8.

a. Littlewood series converge for ∣z∣<1.

Polynomials - Theory and Application

with ∣z∣<1: indeed, consider the series

Proposition at the Introduction.

generated by S ffi S∪X when ∣X∣ ¼ r .

relations (resp. n � 1 vertices and relations):

Am and Cn are of Littlewood type;

b. A point z∈ C with ∣z∣<1 lies in P if and only if some Littlewood series vanishes

c. A Littlewood polynomial is not a Littlewood series. But any Littlewood polynomial, say p zð Þ¼ <sup>a</sup><sup>0</sup> <sup>þ</sup> … <sup>þ</sup> adzd yields a Littlewood series having the same roots <sup>z</sup>

P zð Þ¼ p zð Þ<sup>=</sup> <sup>1</sup> � <sup>z</sup>dþ<sup>1</sup> <sup>¼</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup> … <sup>þ</sup> adzd <sup>þ</sup> <sup>a</sup>0zdþ<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> adz<sup>2</sup>dþ<sup>1</sup> <sup>þ</sup> <sup>a</sup>0z2dþ<sup>2</sup> <sup>þ</sup> …

Thus P⊂ R, where R is the set of roots of Littlewood series. We shall show the

Proof. Let <sup>L</sup> be the set of Littlewood series. Then <sup>L</sup> ¼ �f g <sup>1</sup>; <sup>0</sup>; <sup>1</sup> <sup>ℕ</sup>, so with the product topology it is homeomorphic to the Cantor set. Choose 0< r< 1. Let F be the space of finite multisets of points z with ∣z∣<r, modulo the equivalence relation

Claim. Any Littlewood series has finitely many roots in the disc ∣z∣ ≤r. The map f : L ! F sending a Littlewood series to its multiset of roots in this disc

Since L is compact, the image of f is closed. From this we can show that R, the set of roots of Littlewood series, is closed. Since Littlewood polynomials are densely included in L and f is continuous, we get that P, the set of roots of Littlewood

polynomials, is dense in <sup>R</sup>. It follows that <sup>P</sup> <sup>¼</sup> <sup>R</sup>, as we wanted to show. □

For m a natural number and let n ¼ 3 þ 6m. Let Rn be an algebra formed by n commutative squares. Consider the one-point extension Am ¼ Rn½ � Pn with Pn the unique indecomposable projective Rn-module of K-dimension 2. Observe that Am (resp. Cn�1) is given by the following quiver with n þ 1 vertices and commutative

a. <sup>χ</sup>Am <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> � <sup>T</sup><sup>3</sup> <sup>χ</sup>Am�<sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> 1, for all <sup>n</sup> <sup>≥</sup>1. As consequence, the algebras

b. the number of eigenvalues of ϕAm not lying in the unit disk is at least m;

$$\begin{aligned} \chi\_{A\_{n+1}} &= (T+\mathbf{1}) \left( T^{n+6} + T^{n+5} + T + \mathbf{1} \right) - T^3 (T+\mathbf{1}) \chi\_{R\_n} - T \left( T^{n+5} + T^{n+4} + T + \mathbf{1} \right) \\ &- T^3 T \chi\_{C\_{n-1}} = T^{n+7} + T^{n+6} - T^3 \chi\_{A\_n} + T + \mathbf{1} \end{aligned}$$

as claimed.

As consequence of formula (a) we observe the following:

(a<sup>0</sup> ) L χAm � � <sup>¼</sup> <sup>4</sup><sup>m</sup> <sup>þ</sup> 5.

(b) By induction, we shall construct polynomials rm representing χAm .

For <sup>m</sup> <sup>¼</sup> 0, we have <sup>χ</sup><sup>A</sup><sup>0</sup> <sup>¼</sup> <sup>T</sup><sup>4</sup> <sup>þ</sup> <sup>T</sup><sup>3</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> 1, which is represented by the polynomial <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>T</sup><sup>4</sup> � <sup>3</sup>T<sup>2</sup> <sup>þ</sup> 1.

Observe that <sup>T</sup><sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>1</sup> � � <sup>¼</sup> vn � Tvn�<sup>2</sup> then <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>¼</sup>

ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>1</sup> � � is represented by wn <sup>¼</sup> T uð Þ <sup>n</sup>�<sup>1</sup> � un�<sup>3</sup> .

For <sup>n</sup> <sup>¼</sup> <sup>4</sup> <sup>þ</sup> <sup>6</sup>m, we define rm <sup>¼</sup> wn � <sup>T</sup><sup>3</sup> rm�1. We verify by induction on m that rm represents χAm :

$$\begin{aligned} \chi\_{A\_{\mathfrak{m}}} \left( T^2 \right) &= \left( T^2 + \mathbf{1} \right) \left( T^{2\mathfrak{n}-2} + \mathbf{1} \right) - T^6 \chi\_{A\_{\mathfrak{m}-1}} \left( T^2 \right) \\ &= T^{\mathfrak{n}} w\_{\mathfrak{n}} \left( T + T^{-1} \right) - T^6 T^{\mathfrak{n}-6} r\_{\mathfrak{m}-1} \left( T + T^{-1} \right) = T^{\mathfrak{n}} r\_{\mathfrak{m}} \left( T + T^{-1} \right) \end{aligned}$$

For instance.

$$\begin{aligned} r\_1 &= w\_{10} - T^3 r\_0 = T \left\{ \left( T^9 - 8T^7 + 21T^5 - 20T^3 + 5T \right) - \left( T^7 - 6T^6 + 10T^3 - 4T \right) \right\} \\ &- T^3 \left\{ T^4 - 3T^2 + 1 \right\} \\ &= T^{10} - 9T^8 - T^7 + 27T^6 + 3T^5 - 30T^4 - T^3 + 9T^2 \end{aligned}$$

which has ξð Þ¼ r<sup>1</sup> 4 changes of sign in the sequence of coefficients. According to Descartes rule of signs, r<sup>1</sup> has at most ξð Þ¼ r<sup>1</sup> 4 positive real roots. Since r<sup>1</sup> represents χA1 , then χ<sup>A</sup><sup>1</sup> has at most 2ξð Þ¼ r<sup>1</sup> 8 roots in the unit circle. That is, χ<sup>A</sup><sup>1</sup> has at least 2 roots z with ∣z∣ 6¼ 1.

We shall prove, by induction, that rm has at most ξð Þ¼ rm 2ð Þ m þ 1 positive real roots. Indeed, write

$$r\_m = T^n - (n-1)T^{n-2} - T^3 q\_m + (n-1)T^2 q\_n$$

for some polynomial qm of degree n � 6 with signs of its coefficients þ��þþ��⋯� so that ξ qm � � <sup>¼</sup> <sup>2</sup>m. Then

$$r\_{m+1} = w\_{n+6} - T^3 r\_m = T u\_{n+5} - T u\_{n+3} - T^3 r\_m$$

M χAm

Mahler measure of some of the above examples:

DOI: http://dx.doi.org/10.5772/intechopen.82309

Cyclotomic and Littlewood Polynomials Associated to Algebras

Mossinghoff's web page we see:

new records;

calculated.

6.2 Strong towers

(resp. <sup>⊥</sup>Mi

19

6.1 Derived tubular algebras

6. Coefficients of Coxeter polynomials

nents of the algebra Ci; in particular, s<sup>1</sup> ¼ 1.

<sup>≤</sup> <sup>M</sup> <sup>f</sup> <sup>m</sup>

L gm

With the help of computer programs we calculate more accurate values of the

No. vertices No. roots outside unit disk Mahler measure 29 1:28368024451292 30 1:28327850483340 31 1:28386917621114 32 1:28395305512596

Comparing with the list of Record Mahler measures by roots outside the unit circle in

ii. the entries 30 and 31 have a smaller Mahler measure in our table, establishing

There are interesting invariants associated to the Coxeter polynomial of a triangular algebra A ¼ k½ � Δ =I. For instance, the evaluation of the Coxeter polynomial <sup>χ</sup>Að Þ¼ �<sup>1</sup> <sup>m</sup><sup>2</sup> for some integer <sup>m</sup>. Clearly, this number is a derived invariant. A simple argument yields that m ¼ 0 in case Δ has an odd number of vertices. In [14], it was shown that for a representation-finite accessible algebra A with gl.dim A ≤2 the invariant χAð Þ �1 equals zero or one. The criterion was applied to show that a canonical algebra is derived equivalent to a representation-finite algebra if and only if it has weight type 2ð Þ ; p; p þ k , where p≥2 and k≥0. In particular, the tubular canonical algebra of type 3ð Þ ; 3; 3 is not derived equivalent to a representation-finite

Recall from [14] that a strong tower T ¼ ð Þ A<sup>0</sup> ¼ k; A1; …; An ¼ A of access to A satisfies that Aiþ<sup>1</sup> ¼ Ai Mi ½ � or Mi ½ �Ai for some exceptional module Mi in such a way that, in case Aiþ<sup>1</sup> <sup>¼</sup> Ai Mi ½ � (resp. Aiþ<sup>1</sup> <sup>¼</sup> Mi ½ �Ai), the perpendicular category <sup>M</sup><sup>⊥</sup>

) of Mi in modAi is equivalent to modCi�<sup>1</sup> for some accessible algebra Ci�1,

i

<sup>i</sup> (resp. <sup>⊥</sup>Mi

i. for the entry 29 the Mahler measure is the same in both tables;

iii. the entry 32 of our table seems to be new. Further entries could be

algebra, while the tubular algebras of type 2ð Þ ; 4; 4 or 2ð Þ ; 3; 6 are.

<sup>i</sup> <sup>¼</sup> <sup>1</sup>, …, n � 1. In the extension situation the perpendicular category <sup>M</sup><sup>⊥</sup>

in the coextension situation) in D<sup>b</sup> modAi ð Þ is equivalent to D<sup>b</sup> modCi�<sup>1</sup> ð Þ and Biis derived equivalent to a one-point (co-)extension of Ci�1. An algebra Ci as above is called an i-th perpendicular restriction of the tower T, observe that it is well-defined only up to derived equivalence. We denote by si the number of connected compo-

<sup>¼</sup> <sup>8</sup>

an addition of three polynomials with signs of coefficients given as follows:


Hence rmþ<sup>1</sup> <sup>¼</sup> <sup>T</sup>nþ<sup>6</sup> � ð Þ <sup>n</sup> <sup>þ</sup> <sup>5</sup> <sup>T</sup>nþ<sup>4</sup> � <sup>T</sup><sup>3</sup> qmþ<sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>5</sup> <sup>T</sup><sup>2</sup> where the polynomial qmþ<sup>1</sup> of degree <sup>n</sup> has signs of its coefficients þ��þþ��⋯� so that <sup>ξ</sup> qmþ<sup>1</sup> � � <sup>¼</sup> <sup>ξ</sup> qm � � <sup>þ</sup> <sup>2</sup> <sup>¼</sup> <sup>2</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> . Hence <sup>ξ</sup>ð Þ¼ rm <sup>2</sup> <sup>þ</sup> <sup>ξ</sup> qm � � <sup>¼</sup> <sup>2</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> .

By the Lemma below, χAm has at most 4ð Þ m þ 1 roots in the unit circle. Equivalently, χAm has at least 4 þ 6m � 4ð Þ¼ m þ 1 2m roots outside the unit circle. Hence χAm has at least m roots z satisfying ∣z∣>1.

Lemma. Let q be a polynomial representing the polynomial p. Assume q accepts at most s positive real roots, then p has at most 2 s roots in the unit circle.

Proof. Let μ1, …, μ<sup>s</sup> be the positive real roots of q. Let z ¼ a þ ib be a root of p with <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>¼</sup> 1. Consider <sup>w</sup> <sup>¼</sup> <sup>c</sup> <sup>þ</sup> id a complex number with <sup>w</sup><sup>2</sup> <sup>¼</sup> <sup>z</sup>. Then <sup>0</sup> <sup>¼</sup> p zð Þ¼ wnq w <sup>þ</sup> <sup>w</sup>�<sup>1</sup> ð Þ where <sup>w</sup> <sup>þ</sup> <sup>w</sup>�<sup>1</sup> <sup>¼</sup> ð Þþ <sup>c</sup> <sup>þ</sup> id ð Þ¼ <sup>c</sup> � id <sup>2</sup>c. Then 2<sup>c</sup> <sup>¼</sup> <sup>ϵ</sup>λ<sup>j</sup>

$$z = w^2 = \left(\frac{1}{2}\lambda\_j^2 - \mathbf{1}\right) + i\left(2\epsilon\lambda\_j\sqrt{1 - \lambda\_j^2}\right)$$

for some ϵ∈f g 1; �1 and 1≤j≤s. Hence

can be selected in two different ways. □ (c) For <sup>n</sup> <sup>¼</sup> <sup>6</sup><sup>m</sup> <sup>þ</sup> 4 we have <sup>χ</sup>Am <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> � <sup>T</sup><sup>3</sup> <sup>χ</sup>Am�<sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> 1. Then

$$\chi\_{A\_m} = \xi\_m + (-1)^{m-1} T^{2m+4} \chi\_{10}, \\
\text{where } \xi\_m = T^n + T^{n-1} - T^3 \xi\_{m-1} + T + \mathbf{1}.$$

for m ≥2 and ξ<sup>1</sup> ¼ 0.

We observe that ξ<sup>m</sup> is a product of cyclotomic polynomials. Indeed, since ξmð Þ¼ �1 0 we can write

$$\xi\_m = (T+1)\sigma\_m \text{ and } \sigma\_m = T^{n-1} - T^3 \sigma\_{m-1} + 1$$

for m ≥2 and σ<sup>1</sup> ¼ 0.

Recall Φ2<sup>s</sup> �<sup>1</sup> <sup>¼</sup> <sup>T</sup><sup>s</sup>�<sup>1</sup> <sup>þ</sup> <sup>T</sup><sup>s</sup>�<sup>2</sup> <sup>þ</sup> …<sup>T</sup> <sup>þ</sup> 1 and <sup>Φ</sup>2<sup>s</sup>ð Þ¼ <sup>T</sup> <sup>Φ</sup>sð Þ �<sup>T</sup> . Moreover, <sup>Φ</sup>3<sup>p</sup>ð Þ¼ <sup>T</sup> <sup>Φ</sup><sup>p</sup> <sup>T</sup><sup>3</sup> � �, if <sup>p</sup> is a power of 2. Altogether this yields

$$\begin{aligned} \Phi\_{6\binom{2^{(m+1)}-1}{}}(T) &= \Phi\_{2\binom{2^{(m+1)}-1}{}}(T^3) = \Phi\_{2^{2(m+1)}-1}(-T^3) \\ &= T^{6m+3} - T^{6m} + \dots - T^3 + \mathbf{1} = \sigma\_m \end{aligned}$$

hence

$$\xi\_{\mathfrak{m}} = \Phi\_2 \Phi\_{\mathfrak{G}\left(2^{2(m+1)} - 1\right)}.$$

confirming the claim.

We estimate the Mahler measure of <sup>χ</sup>Am <sup>¼</sup> <sup>ξ</sup><sup>m</sup> þ �ð Þ<sup>1</sup> <sup>m</sup>�<sup>1</sup> T<sup>2</sup>mþ<sup>4</sup>χ<sup>A</sup><sup>10</sup> . Write χAm ¼ f <sup>m</sup> þ gm, where f <sup>m</sup> is the cyclotomic summand. Observe that L gm � � <sup>¼</sup> <sup>L</sup> <sup>χ</sup><sup>A</sup><sup>10</sup> � � <sup>¼</sup> 8 and apply Lemma (3.4) with <sup>M</sup> <sup>f</sup> <sup>m</sup> � � <sup>¼</sup> 1 to get

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

for some polynomial qm of degree n � 6 with signs of its coefficients

� � <sup>¼</sup> <sup>2</sup>m. Then

qmþ<sup>1</sup> of degree <sup>n</sup> has signs of its coefficients þ��þþ��⋯� so that

� � <sup>þ</sup> <sup>2</sup> <sup>¼</sup> <sup>2</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> . Hence <sup>ξ</sup>ð Þ¼ rm <sup>2</sup> <sup>þ</sup> <sup>ξ</sup> qm

most s positive real roots, then p has at most 2 s roots in the unit circle.

<sup>z</sup> <sup>¼</sup> <sup>w</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>

<sup>Φ</sup>3<sup>p</sup>ð Þ¼ <sup>T</sup> <sup>Φ</sup><sup>p</sup> <sup>T</sup><sup>3</sup> � �, if <sup>p</sup> is a power of 2. Altogether this yields

We estimate the Mahler measure of <sup>χ</sup>Am <sup>¼</sup> <sup>ξ</sup><sup>m</sup> þ �ð Þ<sup>1</sup> <sup>m</sup>�<sup>1</sup>

χAm ¼ f <sup>m</sup> þ gm, where f <sup>m</sup> is the cyclotomic summand. Observe that

� � <sup>¼</sup> 8 and apply Lemma (3.4) with <sup>M</sup> <sup>f</sup> <sup>m</sup>

<sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>¼</sup> 1. Consider <sup>w</sup> <sup>¼</sup> <sup>c</sup> <sup>þ</sup> id a complex number with <sup>w</sup><sup>2</sup> <sup>¼</sup> <sup>z</sup>. Then

2 λ2 <sup>j</sup> � 1 � �

an addition of three polynomials with signs of coefficients given as follows:

þ 0 � 0 þ 0 � 0 ⋯ þ 0 0

By the Lemma below, χAm has at most 4ð Þ m þ 1 roots in the unit circle. Equivalently, χAm has at least 4 þ 6m � 4ð Þ¼ m þ 1 2m roots outside the unit circle. Hence

Lemma. Let q be a polynomial representing the polynomial p. Assume q accepts at

Proof. Let μ1, …, μ<sup>s</sup> be the positive real roots of q. Let z ¼ a þ ib be a root of p with

can be selected in two different ways. □

(c) For <sup>n</sup> <sup>¼</sup> <sup>6</sup><sup>m</sup> <sup>þ</sup> 4 we have <sup>χ</sup>Am <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> � <sup>T</sup><sup>3</sup> <sup>χ</sup>Am�<sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> 1. Then

We observe that ξ<sup>m</sup> is a product of cyclotomic polynomials. Indeed, since

�<sup>1</sup> <sup>¼</sup> <sup>T</sup><sup>s</sup>�<sup>1</sup> <sup>þ</sup> <sup>T</sup><sup>s</sup>�<sup>2</sup> <sup>þ</sup> …<sup>T</sup> <sup>þ</sup> 1 and <sup>Φ</sup>2<sup>s</sup>ð Þ¼ <sup>T</sup> <sup>Φ</sup>sð Þ �<sup>T</sup> . Moreover,

<sup>Φ</sup>6 22ð Þ <sup>m</sup>þ<sup>1</sup> ð Þ �<sup>1</sup> ð Þ¼ <sup>T</sup> <sup>Φ</sup>2 22ð Þ <sup>m</sup>þ<sup>1</sup> ð Þ �<sup>1</sup> <sup>T</sup><sup>3</sup> � � <sup>¼</sup> <sup>Φ</sup>22ð Þ <sup>m</sup>þ<sup>1</sup> �<sup>1</sup> �T<sup>3</sup> � �

<sup>ξ</sup><sup>m</sup> <sup>¼</sup> <sup>Φ</sup>2Φ6 22ð Þ <sup>m</sup>þ<sup>1</sup> ð Þ �<sup>1</sup>

<sup>¼</sup> <sup>T</sup><sup>6</sup>mþ<sup>3</sup> � <sup>T</sup><sup>6</sup><sup>m</sup> <sup>þ</sup> … � <sup>T</sup><sup>3</sup> <sup>þ</sup> <sup>1</sup> <sup>¼</sup> <sup>σ</sup><sup>m</sup>

<sup>ξ</sup><sup>m</sup> <sup>¼</sup> ð Þ <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>σ</sup><sup>m</sup> and <sup>σ</sup><sup>m</sup> <sup>¼</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> � <sup>T</sup><sup>3</sup>

þ i 2ϵλ<sup>j</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>λ</sup><sup>2</sup> j

σ<sup>m</sup>�<sup>1</sup> þ 1

T<sup>2</sup>mþ<sup>4</sup>χ<sup>A</sup><sup>10</sup> . Write

� � <sup>¼</sup> 1 to get

� � q

<sup>T</sup><sup>2</sup>mþ<sup>4</sup>χ10, where <sup>ξ</sup><sup>m</sup> <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup><sup>n</sup>�<sup>1</sup> � <sup>T</sup><sup>3</sup> <sup>ξ</sup><sup>m</sup>�<sup>1</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup>

<sup>0</sup> <sup>¼</sup> p zð Þ¼ wnq w <sup>þ</sup> <sup>w</sup>�<sup>1</sup> ð Þ where <sup>w</sup> <sup>þ</sup> <sup>w</sup>�<sup>1</sup> <sup>¼</sup> ð Þþ <sup>c</sup> <sup>þ</sup> id ð Þ¼ <sup>c</sup> � id <sup>2</sup>c. Then 2<sup>c</sup> <sup>¼</sup> <sup>ϵ</sup>λ<sup>j</sup>

� 0 þ 0 � 0 ⋯ þ 0 0 �þþ�� ⋯ 0 00

rm <sup>¼</sup> Tunþ<sup>5</sup> � Tunþ<sup>3</sup> � <sup>T</sup><sup>3</sup>

rm

qmþ<sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>5</sup> <sup>T</sup><sup>2</sup> where the polynomial

� � <sup>¼</sup> <sup>2</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> .

rmþ<sup>1</sup> <sup>¼</sup> wnþ<sup>6</sup> � <sup>T</sup><sup>3</sup>

Hence rmþ<sup>1</sup> <sup>¼</sup> <sup>T</sup>nþ<sup>6</sup> � ð Þ <sup>n</sup> <sup>þ</sup> <sup>5</sup> <sup>T</sup>nþ<sup>4</sup> � <sup>T</sup><sup>3</sup>

χAm has at least m roots z satisfying ∣z∣>1.

for some ϵ∈f g 1; �1 and 1≤j≤s. Hence

<sup>χ</sup>Am <sup>¼</sup> <sup>ξ</sup><sup>m</sup> þ �ð Þ<sup>1</sup> <sup>m</sup>�<sup>1</sup>

for m ≥2 and ξ<sup>1</sup> ¼ 0.

for m ≥2 and σ<sup>1</sup> ¼ 0.

confirming the claim.

Recall Φ2<sup>s</sup>

hence

� � <sup>¼</sup> <sup>L</sup> <sup>χ</sup><sup>A</sup><sup>10</sup>

L gm

18

ξmð Þ¼ �1 0 we can write

þ��þþ��⋯� so that ξ qm

Polynomials - Theory and Application

<sup>ξ</sup> qmþ<sup>1</sup>

� � <sup>¼</sup> <sup>ξ</sup> qm

$$\mathbb{M}(\mathbb{X}\_{A\_m}) \le \mathbb{M}(\mathcal{f}\_m) L(\mathfrak{g}\_m) = \mathbf{8}$$

With the help of computer programs we calculate more accurate values of the Mahler measure of some of the above examples:


Comparing with the list of Record Mahler measures by roots outside the unit circle in Mossinghoff's web page we see:


#### 6. Coefficients of Coxeter polynomials

#### 6.1 Derived tubular algebras

There are interesting invariants associated to the Coxeter polynomial of a triangular algebra A ¼ k½ � Δ =I. For instance, the evaluation of the Coxeter polynomial <sup>χ</sup>Að Þ¼ �<sup>1</sup> <sup>m</sup><sup>2</sup> for some integer <sup>m</sup>. Clearly, this number is a derived invariant. A simple argument yields that m ¼ 0 in case Δ has an odd number of vertices. In [14], it was shown that for a representation-finite accessible algebra A with gl.dim A ≤2 the invariant χAð Þ �1 equals zero or one. The criterion was applied to show that a canonical algebra is derived equivalent to a representation-finite algebra if and only if it has weight type 2ð Þ ; p; p þ k , where p≥2 and k≥0. In particular, the tubular canonical algebra of type 3ð Þ ; 3; 3 is not derived equivalent to a representation-finite algebra, while the tubular algebras of type 2ð Þ ; 4; 4 or 2ð Þ ; 3; 6 are.

#### 6.2 Strong towers

Recall from [14] that a strong tower T ¼ ð Þ A<sup>0</sup> ¼ k; A1; …; An ¼ A of access to A satisfies that Aiþ<sup>1</sup> ¼ Ai Mi ½ � or Mi ½ �Ai for some exceptional module Mi in such a way that, in case Aiþ<sup>1</sup> <sup>¼</sup> Ai Mi ½ � (resp. Aiþ<sup>1</sup> <sup>¼</sup> Mi ½ �Ai), the perpendicular category <sup>M</sup><sup>⊥</sup> i (resp. <sup>⊥</sup>Mi ) of Mi in modAi is equivalent to modCi�<sup>1</sup> for some accessible algebra Ci�1, <sup>i</sup> <sup>¼</sup> <sup>1</sup>, …, n � 1. In the extension situation the perpendicular category <sup>M</sup><sup>⊥</sup> <sup>i</sup> (resp. <sup>⊥</sup>Mi in the coextension situation) in D<sup>b</sup> modAi ð Þ is equivalent to D<sup>b</sup> modCi�<sup>1</sup> ð Þ and Biis derived equivalent to a one-point (co-)extension of Ci�1. An algebra Ci as above is called an i-th perpendicular restriction of the tower T, observe that it is well-defined only up to derived equivalence. We denote by si the number of connected components of the algebra Ci; in particular, s<sup>1</sup> ¼ 1.

There are many examples of strongly accessible algebras, that is, algebras derived equivalent to algebras with a strong tower of access. The following are some instances:

which is derived equivalent to the quiver algebra B with the zero relation as

type A<sup>4</sup> and M is an indecomposable module with M<sup>⊥</sup> the category of modules of the disconnected quiver • ! • •, that is s3ð Þ¼ A 2. Moreover s2ð Þ¼ A s<sup>2</sup> A<sup>0</sup> ð Þ¼ 1 and s Að Þ¼ 4. On the other hand B ¼ ½ � N B<sup>0</sup> such that B<sup>0</sup> is not hereditary. A calculation

yields s3ð Þ¼ B 1 and s2ð Þ¼ B s<sup>2</sup> B<sup>0</sup> ð Þ¼ 2, obviously implying that s Bð Þ¼ 4.

s Bð Þ≤s Að Þ. Equality holds exactly when A ¼ B.

, Cð Þ <sup>n</sup>�<sup>2</sup> , Cð Þ <sup>n</sup>�<sup>4</sup> , …, Cð Þ <sup>r</sup><sup>0</sup> satisfying:

i. Let A and B be accessible algebras and A be accessible from B, then

ii. Let A be an accessible schurian algebra (that is for every couple of vertices i, j, dim<sup>k</sup> A ið Þ ; j ≤1), then for every convex subcategory B we have s Bð Þ≤s Að Þ.

An accessible algebra A with n ¼ 2r þ r<sup>0</sup> vertices, and r<sup>0</sup> ∈f g 0; 1 , is said to be

b.for each 0<sup>≤</sup> <sup>i</sup> <sup>¼</sup> <sup>n</sup> � <sup>2</sup>j≤n, there is a strong tower <sup>T</sup>ð Þ<sup>j</sup> <sup>¼</sup> <sup>C</sup>ð Þ <sup>j</sup>;<sup>1</sup> <sup>¼</sup> <sup>k</sup>; …;Cð Þ <sup>j</sup>;<sup>i</sup> <sup>¼</sup>

c. <sup>C</sup>ð Þ <sup>i</sup>�<sup>2</sup> is an <sup>i</sup> � 1-th perpendicular restriction of <sup>T</sup>ð Þ<sup>j</sup> , that is, <sup>C</sup>ð Þ<sup>i</sup> is a one-point (co)extension of <sup>C</sup>ð Þ <sup>j</sup>;i�<sup>1</sup> by a module Ni�<sup>1</sup> and <sup>C</sup>ð Þ <sup>i</sup>�<sup>2</sup> is a perpendicular

i. Hereditary tree algebras: since for any conneceted hereditary tree algebra A with at least 3 vertices, there is an arrow a ! b with a a source (or dually a sink) and A ¼ B P½ � <sup>b</sup> such that the perpendicular restriction of B via Pb is the algebra hereditary tree algebra C obtained from A by deleting the vertices a, b.

ii. Accessible representation-finite algebras A with gl.dim A ≤2, since then the perpendicular restrictions of any strong tower (which exists by [14]) satisfy

iii. Certain canonical algebras: for instance the tame canonical algebra A of weight type 2ð Þ ; 4; 4 is an extension A ¼ B M½ � of a hereditary algebra B of extended Dynkin type 2½ � ; 4; 4 by a module M in a tube of rank 4, then the perpendicular restriction of B via M is the hereditary algebra C of extended Dynkin type 3½ � ; 3; 3 , see for example [?](10.1). Since C is totally accessible, so

iv. Let A be an accessible algebra of the form A ¼ B M½ � for an algebra B and an exceptional module M and let C the perpendicular restriction of B via M. If A

is totally accessible, then B and C are totally accessible.

totally accessible if there is a family of (not necessarily connected) algebras

;

The tower Tð Þ<sup>j</sup> is said to be a j-th derivative of the tower Tð Þ <sup>0</sup> .

Examples that we have encountered of totally accessible algebras are:

½ � M , where A<sup>0</sup> is a quiver algebra of

depicted in the second diagram. Clearly, A ¼ A<sup>0</sup>

Cyclotomic and Littlewood Polynomials Associated to Algebras

Some properties of the invariant s:

DOI: http://dx.doi.org/10.5772/intechopen.82309

6.4 Totally accessible algebras

a. A is derived equivalent to A<sup>0</sup>

restriction of <sup>C</sup>ð Þ <sup>j</sup>;i�<sup>1</sup> via Ni�1.

the same set of conditions.

A is. Moreover s Að Þ¼ 8.

21

<sup>C</sup>ð Þ<sup>i</sup> <sup>Þ</sup> of access to <sup>C</sup>ð Þ<sup>i</sup> ;

<sup>C</sup>ð Þ <sup>n</sup> <sup>¼</sup> <sup>A</sup><sup>0</sup>


#### 6.3 Towering numbers

Consider a strong tower T ¼ ð Þ A<sup>0</sup> ¼ k; A1; …; An ¼ A of access to A such that Aiþ<sup>1</sup> is an one-point (co)extension of Ai by Mi and Ci�<sup>1</sup> the corresponding i-th perpendicular restriction of T. Let Ci�<sup>1</sup> have si�<sup>1</sup> connected components, i ¼ 2, …, n � 1. Define the first towering number of T as the sum <sup>s</sup>Tð Þ¼ <sup>A</sup> <sup>∑</sup><sup>n</sup>�<sup>2</sup> <sup>i</sup>¼<sup>1</sup> si.

Theorem. Let A be a strongly accessible algebra with n vertices, then the first towering number sTð Þ¼ <sup>A</sup> <sup>∑</sup><sup>n</sup>�<sup>2</sup> <sup>i</sup>¼<sup>1</sup> si of <sup>T</sup> is a derived invariant, that is, depends only on the derived class of A. It is sTð Þ¼ A n � 1 � a2, where a<sup>2</sup> is the coefficient of the quadratic term in the Coxeter polynomial of A.

Proof. Assume A ¼ An and B ¼ An�<sup>1</sup> such that A ¼ B M½ � for M an exceptional B-module and let C ¼ Cn�<sup>2</sup> be the algebra such that mod<sup>C</sup> is derived equivalent to the perpendicular category <sup>M</sup><sup>⊥</sup> formed in D<sup>b</sup>ð Þ mod<sup>B</sup> . Then

<sup>χ</sup>AðÞ¼ <sup>t</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>χ</sup>Bð Þ�<sup>t</sup> <sup>t</sup>χCð Þ<sup>t</sup> . Write <sup>χ</sup>BðÞ¼ <sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>þ</sup> <sup>∑</sup><sup>n</sup>�<sup>3</sup> <sup>i</sup>¼<sup>2</sup> bit <sup>i</sup> <sup>þ</sup> <sup>t</sup> <sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>t</sup> <sup>n</sup>�<sup>1</sup> and <sup>χ</sup>CðÞ¼ <sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>∑</sup><sup>n</sup>�<sup>3</sup> <sup>i</sup>¼<sup>1</sup> cit <sup>i</sup> <sup>þ</sup> <sup>t</sup> <sup>n</sup>�2. By induction hypothesis we may assume that s Bð Þ¼ n � 2 � b2. Then a<sup>2</sup> ¼ b<sup>2</sup> þ 1 � c1. Moreover, since C is a direct sum accessible algebras, then <sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>∑</sup><sup>n</sup>�<sup>2</sup> <sup>i</sup>¼<sup>0</sup> ð Þ �<sup>1</sup> <sup>i</sup> dimkH<sup>i</sup> ð Þ¼ <sup>C</sup> dimkH<sup>0</sup>ð Þ¼ <sup>C</sup> sn�2. Hence <sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup> � <sup>1</sup> � s Bð Þ� sn�<sup>2</sup> <sup>¼</sup> <sup>n</sup> � <sup>1</sup> � s Að Þ. □

Corollary. Let T ¼ ð Þ A<sup>1</sup> ¼ k; …; An ¼ A be a strong tower of access to A. Let A ¼ B M½ � for B ¼ An�<sup>1</sup> with M exceptional and C a perpendicular restriction of B via M. Consider the Coxeter polynomials χAðÞ¼ t 1 þ t þ a2t <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> an�<sup>2</sup><sup>t</sup> <sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>t</sup> <sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>t</sup> n and χBðÞ¼ t 1 þ t þ b2t <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> bn�<sup>3</sup><sup>t</sup> <sup>n</sup>�<sup>3</sup> <sup>þ</sup> <sup>t</sup> <sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>t</sup> <sup>n</sup>�<sup>1</sup>, then a<sup>2</sup> ≤b2, with equality if and only if C is connected. In particular, a<sup>2</sup> ≤1.

Proof. First recall that for a connected accessible algebra the linear term of the Coxeter polynomial has coefficient 1. Let

χCðÞ¼ t 1 þ c1t þ c2t <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> cn�<sup>4</sup><sup>t</sup> <sup>n</sup>�<sup>4</sup> <sup>þ</sup> cn�<sup>3</sup><sup>t</sup> <sup>n</sup>�<sup>3</sup> <sup>þ</sup> <sup>t</sup> <sup>n</sup>�<sup>2</sup> be the Coxeter polynomial of C. If C is the direct sum of connected accessible algebras C1, …, Cs, then c<sup>1</sup> ¼ s. Therefore, a<sup>2</sup> ¼ b<sup>2</sup> þ b<sup>1</sup> � c<sup>1</sup> ¼ b<sup>2</sup> � ð Þ s � 1 ≤ b2. By induction hypothesis, we get <sup>a</sup><sup>2</sup> <sup>≤</sup>1. □

Let A be the algebra given by the following quiver with relation γβα ¼ 0:

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

which is derived equivalent to the quiver algebra B with the zero relation as depicted in the second diagram. Clearly, A ¼ A<sup>0</sup> ½ � M , where A<sup>0</sup> is a quiver algebra of type A<sup>4</sup> and M is an indecomposable module with M<sup>⊥</sup> the category of modules of the disconnected quiver • ! • •, that is s3ð Þ¼ A 2. Moreover s2ð Þ¼ A s<sup>2</sup> A<sup>0</sup> ð Þ¼ 1 and s Að Þ¼ 4. On the other hand B ¼ ½ � N B<sup>0</sup> such that B<sup>0</sup> is not hereditary. A calculation yields s3ð Þ¼ B 1 and s2ð Þ¼ B s<sup>2</sup> B<sup>0</sup> ð Þ¼ 2, obviously implying that s Bð Þ¼ 4.

Some properties of the invariant s:


#### 6.4 Totally accessible algebras

There are many examples of strongly accessible algebras, that is, algebras derived

t ¼ 3, in that case, C is derived-equivalent to a representation-finite algebra if

b.The following sequence of poset algebras defines strong towers of access:

Consider a strong tower T ¼ ð Þ A<sup>0</sup> ¼ k; A1; …; An ¼ A of access to A such

corresponding i-th perpendicular restriction of T. Let Ci�<sup>1</sup> have si�<sup>1</sup> connected components, i ¼ 2, …, n � 1. Define the first towering number of T as the sum

Theorem. Let A be a strongly accessible algebra with n vertices, then the first

derived class of A. It is sTð Þ¼ A n � 1 � a2, where a<sup>2</sup> is the coefficient of the quadratic

Proof. Assume A ¼ An and B ¼ An�<sup>1</sup> such that A ¼ B M½ � for M an exceptional B-module and let C ¼ Cn�<sup>2</sup> be the algebra such that mod<sup>C</sup> is derived equivalent to

s Bð Þ¼ n � 2 � b2. Then a<sup>2</sup> ¼ b<sup>2</sup> þ 1 � c1. Moreover, since C is a direct sum accessi-

<sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup> � <sup>1</sup> � s Bð Þ� sn�<sup>2</sup> <sup>¼</sup> <sup>n</sup> � <sup>1</sup> � s Að Þ. □ Corollary. Let T ¼ ð Þ A<sup>1</sup> ¼ k; …; An ¼ A be a strong tower of access to A. Let A ¼ B M½ � for B ¼ An�<sup>1</sup> with M exceptional and C a perpendicular restriction of B via

Proof. First recall that for a connected accessible algebra the linear term of the

<sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>t</sup>

<sup>n</sup>�<sup>3</sup> <sup>þ</sup> <sup>t</sup>

dimkH<sup>i</sup>

<sup>n</sup>�<sup>3</sup> <sup>þ</sup> <sup>t</sup>

<sup>n</sup>�<sup>4</sup> <sup>þ</sup> cn�<sup>3</sup><sup>t</sup>

C. If C is the direct sum of connected accessible algebras C1, …, Cs, then c<sup>1</sup> ¼ s. Therefore, a<sup>2</sup> ¼ b<sup>2</sup> þ b<sup>1</sup> � c<sup>1</sup> ¼ b<sup>2</sup> � ð Þ s � 1 ≤ b2. By induction hypothesis, we get <sup>a</sup><sup>2</sup> <sup>≤</sup>1. □ Let A be the algebra given by the following quiver with relation γβα ¼ 0:

is strongly accessible if and only if

<sup>i</sup>¼<sup>1</sup> si of <sup>T</sup> is a derived invariant, that is, depends only on the

<sup>i</sup>¼<sup>2</sup> bit

ð Þ¼ <sup>C</sup> dimkH<sup>0</sup>ð Þ¼ <sup>C</sup> sn�2. Hence

<sup>n</sup>�2. By induction hypothesis we may assume that

<sup>i</sup> <sup>þ</sup> <sup>t</sup>

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> an�<sup>2</sup><sup>t</sup>

<sup>n</sup>�<sup>1</sup>, then a<sup>2</sup> ≤b2, with equality if

<sup>n</sup>�<sup>2</sup> be the Coxeter polynomial of

<sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>t</sup>

<sup>n</sup>�<sup>1</sup> and

<sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>t</sup>

<sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>t</sup> n

equivalent to algebras with a strong tower of access. The following are some

and only if the weight type does not dominate 3ð Þ ; 3; 3 .

that Aiþ<sup>1</sup> is an one-point (co)extension of Ai by Mi and Ci�<sup>1</sup> the

the perpendicular category <sup>M</sup><sup>⊥</sup> formed in D<sup>b</sup>ð Þ mod<sup>B</sup> . Then <sup>χ</sup>AðÞ¼ <sup>t</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>χ</sup>Bð Þ�<sup>t</sup> <sup>t</sup>χCð Þ<sup>t</sup> . Write <sup>χ</sup>BðÞ¼ <sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>þ</sup> <sup>∑</sup><sup>n</sup>�<sup>3</sup>

<sup>i</sup>¼<sup>0</sup> ð Þ �<sup>1</sup> <sup>i</sup>

M. Consider the Coxeter polynomials χAðÞ¼ t 1 þ t þ a2t

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> cn�<sup>4</sup><sup>t</sup>

and only if C is connected. In particular, a<sup>2</sup> ≤1.

Coxeter polynomial has coefficient 1. Let

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> bn�<sup>3</sup><sup>t</sup>

a. A canonical algebra C of weight p1; …; pt

Polynomials - Theory and Application

instances:

6.3 Towering numbers

<sup>i</sup>¼<sup>1</sup> si.

towering number sTð Þ¼ <sup>A</sup> <sup>∑</sup><sup>n</sup>�<sup>2</sup>

term in the Coxeter polynomial of A.

<sup>i</sup>¼<sup>1</sup> cit

ble algebras, then <sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>∑</sup><sup>n</sup>�<sup>2</sup>

and χBðÞ¼ t 1 þ t þ b2t

χCðÞ¼ t 1 þ c1t þ c2t

20

<sup>i</sup> <sup>þ</sup> <sup>t</sup>

<sup>s</sup>Tð Þ¼ <sup>A</sup> <sup>∑</sup><sup>n</sup>�<sup>2</sup>

<sup>χ</sup>CðÞ¼ <sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>∑</sup><sup>n</sup>�<sup>3</sup>

An accessible algebra A with n ¼ 2r þ r<sup>0</sup> vertices, and r<sup>0</sup> ∈f g 0; 1 , is said to be totally accessible if there is a family of (not necessarily connected) algebras <sup>C</sup>ð Þ <sup>n</sup> <sup>¼</sup> <sup>A</sup><sup>0</sup> , Cð Þ <sup>n</sup>�<sup>2</sup> , Cð Þ <sup>n</sup>�<sup>4</sup> , …, Cð Þ <sup>r</sup><sup>0</sup> satisfying:

a. A is derived equivalent to A<sup>0</sup> ;


The tower Tð Þ<sup>j</sup> is said to be a j-th derivative of the tower Tð Þ <sup>0</sup> . Examples that we have encountered of totally accessible algebras are:


The following results extend some of the features observed in the examples above.

Proposition. a. Assume that A is a totally accessible algebra, then χAð Þ �1 ∈f g 0; 1 .

Proof. (i) ⇔ (ii): Let T ¼ ð Þ A<sup>1</sup> ¼ k; …; An ¼ A be a strong tower of access to A. In

<sup>i</sup>¼<sup>1</sup> si with each si <sup>≥</sup>1. (i) <sup>⇔</sup> (iii): We know that an algebra <sup>A</sup>

<sup>i</sup>¼0<sup>t</sup> i

, hence <sup>A</sup> is derived of type <sup>A</sup>n. □

<sup>i</sup>�<sup>1</sup> <sup>þ</sup> <sup>t</sup> i ,

> ni�<sup>1</sup> <sup>þ</sup> <sup>t</sup> ni ,

ni,j�<sup>1</sup> <sup>þ</sup> <sup>t</sup>

<sup>j</sup> ≤1.

ð Þ i�1 <sup>j</sup>þ<sup>1</sup> <sup>≤</sup> … <sup>≤</sup> <sup>a</sup>

<sup>0</sup> ¼ 1 ¼ ci,<sup>0</sup> and a

<sup>j</sup> <sup>≤</sup>ci,j and að Þ<sup>i</sup>

ni,j ,

<sup>j</sup> <sup>≤</sup>að Þ <sup>i</sup>�<sup>1</sup> <sup>j</sup> .

<sup>j</sup> <sup>≤</sup> ci,j and <sup>a</sup>ð Þ<sup>i</sup>

ð Þ jþ1 <sup>j</sup>þ<sup>1</sup> <sup>¼</sup> <sup>1</sup>:

ð Þi

<sup>j</sup> ≤ a

<sup>1</sup> ≤ s1ð Þ¼ A ci, 1. Moreover

ð Þ i�1 <sup>j</sup> .

<sup>i</sup> , for 1≤j≤si, are the connected compo-

, in particular,

<sup>i</sup> is derived

case each Ci is connected, then s Að Þ¼ n � 2, that is a<sup>2</sup> ¼ 1. If a<sup>2</sup> ¼ 1, then

a<sup>2</sup> ¼ 1. Assume that an accessible algebra A has the quadratic coefficient of its Coxeter polynomial a<sup>2</sup> ¼ 1. Let A ¼ B M½ � for an accessible algebra B ¼ An�<sup>1</sup> and an

¼ ð Þ A<sup>1</sup> ¼ k; …; An�<sup>1</sup> ¼ B satisfying (ii), then the quadratic coefficient of the Coxeter polynomial of B is b<sup>2</sup> ¼ 1 and we may assume that B is derived equivalent to

Consider a tower A1, …, An ¼ A of accessible algebras where Aiþ<sup>1</sup> is a one-point

i�2 j¼2 að Þi j t <sup>j</sup> <sup>þ</sup> <sup>t</sup>

ni�2 r¼2 ci,rt <sup>r</sup> <sup>þ</sup> sit

ni,j�2 s¼2 c ð Þj i,s t <sup>s</sup> <sup>þ</sup> <sup>t</sup>

Proof. We shall check that (α) implies (αα), then we show that (a') holds by

Indeed, assume that (α) holds and proceed to show (αα) by induction on j. If

ð Þi

. Assume that 2≤j<sup>≤</sup> <sup>i</sup> � 2 and <sup>a</sup>ð Þ<sup>i</sup>

(co)extension of Ai by the indecomposable Mi and Ci is such that M<sup>⊥</sup>

ðÞ¼ t 1 þ t þ ∑

ðÞ¼ t 1 þ sit þ ∑

ðÞ¼ t 1 þ t þ ∑

ð Þ i�1 <sup>j</sup> � cj,i�<sup>1</sup> <sup>≤</sup><sup>a</sup>

nents of the category Ci. Consider the corresponding Coxeter polynomials:

a quiver algebra of type An�1. In particular, B is representation-finite with a preprojective component <sup>P</sup> such that the orbit graph O Pð Þ<sup>τ</sup> is of type <sup>A</sup>n�<sup>1</sup> (recall that the orbit graph has vertices the τ-orbits in the quiver P with Auslander-Reiten translation τ and there is an edge between the orbit of X and the orbit of Y if there is some numbers a, b and an irreducible morphism <sup>τ</sup>aX ! <sup>τ</sup>bY). Observe that for any <sup>X</sup> in D<sup>b</sup>ð Þ mod<sup>A</sup> not in the orbit of <sup>M</sup>, there is some translation <sup>τ</sup>aX belonging to <sup>M</sup><sup>⊥</sup>, implying that in case M<sup>τ</sup> has two neighbors in the orbit graph then M<sup>⊥</sup> is not connected, that is sn�<sup>2</sup> <sup>&</sup>gt;1 and <sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup> � <sup>1</sup> � s Að Þ≤0, a contradiction. Therefore, <sup>M</sup><sup>τ</sup>

derived equivalent to a quiver algebra of type <sup>A</sup><sup>n</sup> has <sup>χ</sup>AðÞ¼ <sup>t</sup> <sup>∑</sup><sup>n</sup>

Cyclotomic and Littlewood Polynomials Associated to Algebras

DOI: http://dx.doi.org/10.5772/intechopen.82309

exceptional module M. Since B is also totally accessible with a tower

<sup>n</sup> � <sup>2</sup> <sup>¼</sup> <sup>s</sup>Tð Þ¼ <sup>A</sup> <sup>∑</sup><sup>n</sup>�<sup>2</sup>

has just one neighbor in O Pð Þ<sup>τ</sup>

equivalent to D<sup>b</sup> modCi ð Þ. Assume that <sup>C</sup>ð Þ<sup>j</sup>

χAi

χCi

χCð Þ<sup>j</sup> i

(αα) For every <sup>1</sup>≤j≤<sup>i</sup> � <sup>2</sup>, we have að Þ<sup>i</sup>

<sup>j</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> <sup>a</sup>ð Þ<sup>i</sup>

Let 0 ≤j≤ i � 2. If j ¼ 0, 1 we have a

að Þi <sup>j</sup>þ<sup>1</sup> <sup>¼</sup> <sup>a</sup>

<sup>j</sup>¼<sup>1</sup>ni,j <sup>¼</sup> ni. Lemma. (α) For every <sup>1</sup>≤j<sup>≤</sup> <sup>i</sup> � <sup>2</sup>, we have að Þ<sup>i</sup>

i�j

ð Þ i�1 <sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>a</sup>

<sup>1</sup> . Assume (α) holds for j≥2, then.

where clearly, ∑si

induction on j.

<sup>j</sup> <sup>¼</sup> <sup>0</sup>, 1, then <sup>a</sup>ð Þ<sup>i</sup>

Then

að Þi

23

<sup>1</sup> <sup>¼</sup> <sup>a</sup>ð Þ <sup>i</sup>�<sup>1</sup>

7.2 Theorem 2

T0

b. Assume that A is an accessible but not totally accessible algebra with gl.dim A ≤2, then one of the following conditions hold:

i. for every exceptional B-module such that A ¼ B M½ � and any perpendicular restriction C of B via M, then C is not accessible;

ii. there exists a homological epimorphism ϕ : A ! B such that χBð Þ �1 > 1.

Proof. (a): Consider the perpendicular restriction C of B via M, such that χAðÞ¼ t ð Þ 1 þ t χBð Þ�t tχCð Þt . Therefore χAð Þ¼ �1 χCð Þ �1 and moreover, C is totally accessible. Then by induction hypothesis, χAð Þ¼ �1 χCð Þ <sup>m</sup> ð Þ �1 for a totally accessible algebra <sup>C</sup>ð Þ <sup>m</sup> with number of vertices <sup>m</sup> <sup>¼</sup> 1 or <sup>m</sup> <sup>¼</sup> 2. Clearly, <sup>C</sup>ð Þ <sup>m</sup> is either <sup>k</sup>, <sup>k</sup> <sup>⊕</sup> <sup>k</sup> or hereditary of type A2, which yields the desired result.

(b): Assume A is an accessible algebra with gl.dim A ≤2 and such that for every homological epimorphism ϕ : A ! B we have χBð Þ �1 ∈ f g 0; 1 . Let A ¼ B M½ � for an accessible algebra B and an exceptional B-module M such that C is a perpendicular restriction of B via M. Since gl.dim A ≤ 2 then there is a homological epimorphism A ! C and gl.dim C≤2. Observe that for every homological epimorphism ψ : B ! B<sup>0</sup> (resp. ψ : C ! C<sup>0</sup> ) there is a homological epimorphism ϕ : A ! B<sup>0</sup> (resp. ϕ : A ! C<sup>0</sup> ), hence χB0ð Þ �1 (resp. χC0ð Þ �1 ) is 0 or 1. By induction hypothesis, B is totally accessible. Moreover if C is accessible, then the induction hypothesis yields that C is totally accessible and also A is totally accessible, a contradiction. Therefore <sup>C</sup> is not accessible. □

#### 7. On the quadratic coefficient of the Coxeter polynomial of a totally accessible algebra

#### 7.1 Derived algebras of linear type

Recall that an extended canonical algebra of weight type p1; …; pt is a one-point extension of the canonical algebra of weight type p1; …; pt by an indecomposable projective module. As in (1.3), the extended canonical algebras of type p1; p2; p<sup>3</sup> is strongly accessible. Moreover, the extended canonical algebra A of type 3h i ; 4; 5 (with 12 points) has Coxeter polynomial 1 þ t þ t <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>t</sup> <sup>12</sup> which is also the Coxeter polynomial of a linear hereditary algebra H with 12 vertices. Clearly A and H are not derived equivalent.

The following generalizes a result of Happel who considers the case of Coxeter polynomials associated to hereditary algebras [8].

Theorem 1. Let A be a totally accessible algebra with n vertices and let <sup>χ</sup>AðÞ¼ <sup>t</sup> <sup>∑</sup><sup>n</sup> <sup>i</sup>¼<sup>0</sup>ait <sup>i</sup> be the Coxeter polynomial of A. The following are equivalent:

i. a<sup>2</sup> ¼ 1;

ii. let T ¼ ð Þ A<sup>1</sup> ¼ k; …; An�<sup>1</sup>; An ¼ A be a strong tower of access to A and Ci the i-th perpendicular restriction of T, for all 1≤ i≤n � 2, then the algebras Ci are connected;

iii. A is derived equivalent to a quiver algebra of type An.

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

Proof. (i) ⇔ (ii): Let T ¼ ð Þ A<sup>1</sup> ¼ k; …; An ¼ A be a strong tower of access to A. In case each Ci is connected, then s Að Þ¼ n � 2, that is a<sup>2</sup> ¼ 1. If a<sup>2</sup> ¼ 1, then <sup>n</sup> � <sup>2</sup> <sup>¼</sup> <sup>s</sup>Tð Þ¼ <sup>A</sup> <sup>∑</sup><sup>n</sup>�<sup>2</sup> <sup>i</sup>¼<sup>1</sup> si with each si <sup>≥</sup>1. (i) <sup>⇔</sup> (iii): We know that an algebra <sup>A</sup> derived equivalent to a quiver algebra of type <sup>A</sup><sup>n</sup> has <sup>χ</sup>AðÞ¼ <sup>t</sup> <sup>∑</sup><sup>n</sup> <sup>i</sup>¼0<sup>t</sup> i , in particular, a<sup>2</sup> ¼ 1. Assume that an accessible algebra A has the quadratic coefficient of its Coxeter polynomial a<sup>2</sup> ¼ 1. Let A ¼ B M½ � for an accessible algebra B ¼ An�<sup>1</sup> and an exceptional module M. Since B is also totally accessible with a tower T0 ¼ ð Þ A<sup>1</sup> ¼ k; …; An�<sup>1</sup> ¼ B satisfying (ii), then the quadratic coefficient of the Coxeter polynomial of B is b<sup>2</sup> ¼ 1 and we may assume that B is derived equivalent to a quiver algebra of type An�1. In particular, B is representation-finite with a preprojective component <sup>P</sup> such that the orbit graph O Pð Þ<sup>τ</sup> is of type <sup>A</sup>n�<sup>1</sup> (recall that the orbit graph has vertices the τ-orbits in the quiver P with Auslander-Reiten translation τ and there is an edge between the orbit of X and the orbit of Y if there is some numbers a, b and an irreducible morphism <sup>τ</sup>aX ! <sup>τ</sup>bY). Observe that for any <sup>X</sup> in D<sup>b</sup>ð Þ mod<sup>A</sup> not in the orbit of <sup>M</sup>, there is some translation <sup>τ</sup>aX belonging to <sup>M</sup><sup>⊥</sup>, implying that in case M<sup>τ</sup> has two neighbors in the orbit graph then M<sup>⊥</sup> is not connected, that is sn�<sup>2</sup> <sup>&</sup>gt;1 and <sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup> � <sup>1</sup> � s Að Þ≤0, a contradiction. Therefore, <sup>M</sup><sup>τ</sup> has just one neighbor in O Pð Þ<sup>τ</sup> , hence <sup>A</sup> is derived of type <sup>A</sup>n. □

#### 7.2 Theorem 2

The following results extend some of the features observed in the examples

with gl.dim A ≤2, then one of the following conditions hold:

or hereditary of type A2, which yields the desired result.

ψ : B ! B<sup>0</sup> (resp. ψ : C ! C<sup>0</sup>

accessible algebra

H are not derived equivalent.

<sup>i</sup>¼<sup>0</sup>ait

<sup>χ</sup>AðÞ¼ <sup>t</sup> <sup>∑</sup><sup>n</sup>

22

i. a<sup>2</sup> ¼ 1;

connected;

7.1 Derived algebras of linear type

ϕ : A ! C<sup>0</sup>

restriction C of B via M, then C is not accessible;

Polynomials - Theory and Application

Proposition. a. Assume that A is a totally accessible algebra, then χAð Þ �1 ∈f g 0; 1 .

i. for every exceptional B-module such that A ¼ B M½ � and any perpendicular

ii. there exists a homological epimorphism ϕ : A ! B such that χBð Þ �1 > 1.

Proof. (a): Consider the perpendicular restriction C of B via M, such that χAðÞ¼ t ð Þ 1 þ t χBð Þ�t tχCð Þt . Therefore χAð Þ¼ �1 χCð Þ �1 and moreover, C is totally accessible. Then by induction hypothesis, χAð Þ¼ �1 χCð Þ <sup>m</sup> ð Þ �1 for a totally accessible algebra <sup>C</sup>ð Þ <sup>m</sup> with number of vertices <sup>m</sup> <sup>¼</sup> 1 or <sup>m</sup> <sup>¼</sup> 2. Clearly, <sup>C</sup>ð Þ <sup>m</sup> is either <sup>k</sup>, <sup>k</sup> <sup>⊕</sup> <sup>k</sup>

A ! C and gl.dim C≤2. Observe that for every homological epimorphism

(b): Assume A is an accessible algebra with gl.dim A ≤2 and such that for every homological epimorphism ϕ : A ! B we have χBð Þ �1 ∈ f g 0; 1 . Let A ¼ B M½ � for an accessible algebra B and an exceptional B-module M such that C is a perpendicular restriction of B via M. Since gl.dim A ≤ 2 then there is a homological epimorphism

), hence χB0ð Þ �1 (resp. χC0ð Þ �1 ) is 0 or 1. By induction hypothesis, B is

totally accessible. Moreover if C is accessible, then the induction hypothesis yields that C is totally accessible and also A is totally accessible, a contradiction. Therefore <sup>C</sup> is not accessible. □

7. On the quadratic coefficient of the Coxeter polynomial of a totally

projective module. As in (1.3), the extended canonical algebras of type p1; p2; p<sup>3</sup>

strongly accessible. Moreover, the extended canonical algebra A of type 3h i ; 4; 5

Theorem 1. Let A be a totally accessible algebra with n vertices and let

Coxeter polynomial of a linear hereditary algebra H with 12 vertices. Clearly A and

The following generalizes a result of Happel who considers the case of Coxeter

<sup>i</sup> be the Coxeter polynomial of A. The following are equivalent:

ii. let T ¼ ð Þ A<sup>1</sup> ¼ k; …; An�<sup>1</sup>; An ¼ A be a strong tower of access to A and Ci the i-th perpendicular restriction of T, for all 1≤ i≤n � 2, then the algebras Ci are

Recall that an extended canonical algebra of weight type p1; …; pt

extension of the canonical algebra of weight type p1; …; pt

iii. A is derived equivalent to a quiver algebra of type An.

(with 12 points) has Coxeter polynomial 1 þ t þ t

polynomials associated to hereditary algebras [8].

) there is a homological epimorphism ϕ : A ! B<sup>0</sup> (resp.

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>t</sup>

is a one-point

is

by an indecomposable

<sup>12</sup> which is also the

b. Assume that A is an accessible but not totally accessible algebra

above.

Consider a tower A1, …, An ¼ A of accessible algebras where Aiþ<sup>1</sup> is a one-point (co)extension of Ai by the indecomposable Mi and Ci is such that M<sup>⊥</sup> <sup>i</sup> is derived equivalent to D<sup>b</sup> modCi ð Þ. Assume that <sup>C</sup>ð Þ<sup>j</sup> <sup>i</sup> , for 1≤j≤si, are the connected components of the category Ci. Consider the corresponding Coxeter polynomials:

$$\begin{aligned} \chi\_{A\_i}(t) &= \mathbf{1} + t + \sum\_{j=2}^{i-2} a\_j^{(i)} t^j + t^{i-1} + t^i, \\ \chi\_{C\_i}(t) &= \mathbf{1} + s\_i t + \sum\_{r=2}^{n\_i - 2} c\_{i,r} t^r + s\_i t^{n\_i - 1} + t^{n\_i}, \\ \chi\_{C\_i^{(j)}}(t) &= \mathbf{1} + t + \sum\_{s=2}^{n\_{i,j} - 2} c\_{i,s}^{(j)} t^s + t^{n\_{i,j} - 1} + t^{n\_{i,j}}, \end{aligned}$$

where clearly, ∑si <sup>j</sup>¼<sup>1</sup>ni,j <sup>¼</sup> ni.

Lemma. (α) For every <sup>1</sup>≤j<sup>≤</sup> <sup>i</sup> � <sup>2</sup>, we have að Þ<sup>i</sup> <sup>j</sup> ≤1.

(αα) For every <sup>1</sup>≤j≤<sup>i</sup> � <sup>2</sup>, we have að Þ<sup>i</sup> <sup>j</sup> <sup>≤</sup>ci,j and að Þ<sup>i</sup> <sup>j</sup> <sup>≤</sup>að Þ <sup>i</sup>�<sup>1</sup> <sup>j</sup> .

Proof. We shall check that (α) implies (αα), then we show that (a') holds by induction on j.

Indeed, assume that (α) holds and proceed to show (αα) by induction on j. If <sup>j</sup> <sup>¼</sup> <sup>0</sup>, 1, then <sup>a</sup>ð Þ<sup>i</sup> <sup>j</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> <sup>a</sup>ð Þ<sup>i</sup> i�j . Assume that 2≤j<sup>≤</sup> <sup>i</sup> � 2 and <sup>a</sup>ð Þ<sup>i</sup> <sup>j</sup> <sup>≤</sup> ci,j and <sup>a</sup>ð Þ<sup>i</sup> <sup>j</sup> ≤ a ð Þ i�1 <sup>j</sup> . Then

$$a\_{j+1}^{(i)} = a\_{j+1}^{(i-1)} + \left(a\_{j}^{(i-1)} - c\_{j,i-1}\right) \le a\_{j+1}^{(i-1)} \le \dots \le a\_{j+1}^{(j+1)} = \mathbf{1}.$$

Let 0 ≤j≤ i � 2. If j ¼ 0, 1 we have a ð Þi <sup>0</sup> ¼ 1 ¼ ci,<sup>0</sup> and a ð Þi <sup>1</sup> ≤ s1ð Þ¼ A ci, 1. Moreover að Þi <sup>1</sup> <sup>¼</sup> <sup>a</sup>ð Þ <sup>i</sup>�<sup>1</sup> <sup>1</sup> . Assume (α) holds for j≥2, then.

Polynomials - Theory and Application

$$a\_{j+1}^{(i)} = a\_{j+1}^{(i-1)} + \left(a\_j^{(i-1)} - c\_{j,i-1}\right) \le a\_{j+1}^{(i-1)},$$

$$a\_{j+1}^{(i)} - c\_{i,j+1} = a\_{j+2}^{(i)} - a\_{j+2}^{(i-1)} \le \mathbf{0}.\tag{7}$$

∣detϕA∣ ¼ r

We define the critical power κð Þ A as the minimal k such that

to the existence of k satisfying the following chain of inequalities:

0 1 <sup>ϕ</sup><sup>A</sup> ¼ �C�<sup>1</sup>

Let <sup>M</sup> be a real invertible <sup>n</sup> � <sup>n</sup>-matrix with eigenvalues <sup>λ</sup><sup>j</sup> <sup>¼</sup> rjei <sup>θ</sup><sup>j</sup>

numbers θ<sup>j</sup> ∈½ Þ 0; 2π and j ¼ 1, …, n. We will say that M is stable (resp. semi-stable) if

non-negative), for every j ¼ 1, …, n. The following is well-known, we sketch a proof

Proposition. Let M be a stable (resp. semi-stable) n � n-matrix. Then the charac-

and ð Þ <sup>T</sup> � ð Þ <sup>α</sup> <sup>þ</sup> <sup>i</sup><sup>β</sup> ð Þ¼ <sup>T</sup> � ð Þ <sup>α</sup> � <sup>i</sup><sup>β</sup> <sup>T</sup><sup>2</sup> � <sup>2</sup>α<sup>T</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> with 0 6¼ <sup>β</sup>, <sup>α</sup> <sup>∈</sup> <sup>R</sup>. Stability (resp. semi-stability) implies that <sup>α</sup><0 (resp. <sup>α</sup><sup>≤</sup> 0) above. Therefore, ð Þ �<sup>1</sup> <sup>n</sup>

is product of polynomials with positive coefficients. □ Remark: In most of the literature the stability concept we use goes by the name

ðÞ¼� t My tð Þ

of positive stability, while the stability name is used also as Hurwitz stability, or

y0

teristic polynomial <sup>χ</sup><sup>M</sup> <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> an�<sup>1</sup>T<sup>n</sup>�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> <sup>a</sup>1<sup>T</sup> <sup>þ</sup> <sup>a</sup><sup>0</sup> has coefficients satisfying

<sup>¼</sup> cos <sup>θ</sup><sup>j</sup> of the argument of the eigenvalue <sup>λ</sup><sup>j</sup> is positive (resp.

pð Þ �T is the product of polynomials T � α with α ∈ R

n j¼1 μk <sup>j</sup> ∣ ≥r km<sup>1</sup> <sup>1</sup> � ∑ s j¼2 r kmj <sup>j</sup> ≥r k

The following is a reformulation of Theorem 2.

Cyclotomic and Littlewood Polynomials Associated to Algebras

DOI: http://dx.doi.org/10.5772/intechopen.82309

∣Tr ϕ<sup>k</sup> A <sup>∣</sup> <sup>¼</sup> ∣ ∑

ously, the Cartan matrix of A is of the form

<sup>C</sup> <sup>¼</sup> <sup>1</sup> <sup>a</sup>

9.1 Stability of matrices and the Lyapunov criterion

aj > 0 (resp. ≥ 0), for j ¼ 0, 1, …, n;

The system of differential equations

<sup>¼</sup> <sup>a</sup>ð Þ <sup>2</sup> � <sup>2</sup> <sup>2</sup> � <sup>2</sup>>2.

9. Stability of a real matrix

for the sake of completeness.

Proof. Observe that ð Þ �<sup>1</sup> <sup>n</sup>

have κð Þ A ≤ n:

∣Tr ϕ<sup>k</sup> A

and Tr ϕ<sup>2</sup>

A

the real part Re e<sup>i</sup> <sup>θ</sup><sup>j</sup>

Lyapunov stability.

ð Þ �<sup>1</sup> <sup>n</sup>�<sup>j</sup>

25

m<sup>1</sup> <sup>1</sup> r m<sup>2</sup> <sup>2</sup> …r ms <sup>s</sup> ¼ 1:

∣Tr ϕ<sup>k</sup> A ∣> n

Since r<sup>1</sup> is a simple eigenvalue of ϕA, then it follows that κð Þ A is well defined due

Theorem. Let A be an algebra such that not all roots of χ<sup>A</sup> are roots of unity. We

Proof. Indeed, suppose that A is not of cyclotomic type and κð Þ A > n, that is,

 ∣ ≤<sup>n</sup> for all 0 <sup>≤</sup>k≤n. Observe that <sup>M</sup> <sup>¼</sup> <sup>ϕ</sup><sup>A</sup> is a unimodular matrix and therefore, Theorem 2 implies that M is of cyclotomic type, which yields a contradiction. □ Remark: We consider explicitly the case n ¼ 2 in the above Theorem. Obvi-

for some a≥1. Then ϕ<sup>A</sup> has the indicated shape. If A is not cyclotomic, then a≥3

<sup>1</sup> � ð Þ n � 1 r

<sup>C</sup><sup>T</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> � <sup>1</sup> <sup>a</sup>

�a �1 

, for some

pð Þ �T

k <sup>2</sup> >n:

Theorem 2. Let A be a totally accessible algebra with Coxeter polynomial χAðÞ¼ t 1 þ t þ a2t <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> an�2<sup>t</sup> <sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>t</sup> <sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>t</sup> n, then:

a. aj ≤1, for every 2 ≤j≤n � 2;

b.if for some 2 ≤j≤n � 2, we have aj ¼ 1 then A is derived equivalent to a hereditary algebra of type An.

Proof. Keep the notation as in (4.1). Then (a) is the case i ¼ n of the Lemma above.

We shall prove (b) by induction on n the number of vertices of A. Let j ¼ 2 and assume a<sup>2</sup> ¼ 1, then (3.1) implies that A is derived equivalent to An. Consider now 2< j<n � 2 and assume that aj ¼ 1, we get:

$$\mathbf{1} = \mathbf{a}\_j^{(n)} = \mathbf{a}\_j^{(n-1)} + \left(\mathbf{a}\_{j-1}^{(n-1)} - \mathbf{c}\_{n-1,j-1}\right) \le \mathbf{a}\_j^{(n-1)} \le \mathbf{1}$$

The last inequality due to (a), hence að Þ <sup>n</sup>�<sup>1</sup> <sup>j</sup> ¼ 1. Induction hypothesis yields that An�<sup>1</sup> is derived equivalent to A<sup>n</sup>�<sup>1</sup> and its Auslander-Reiten quiver consists of a preprojective component P. In particular, að Þ <sup>n</sup>�<sup>1</sup> <sup>2</sup> ¼ 1, which implies that sn�<sup>3</sup>ð Þ¼ An�<sup>1</sup> 1, that is, <sup>A</sup> <sup>¼</sup> An�<sup>1</sup>½ � <sup>M</sup> for some exceptional module <sup>M</sup> such that <sup>M</sup><sup>⊥</sup> is derived equivalent to mod<sup>C</sup> for a connected algebra C, that is, s Að Þ¼ n � 2 and by (3.1), <sup>A</sup> <sup>¼</sup> B M½ � is derived equivalent to a hereditary algebra of type <sup>A</sup>n. □

#### 7.3 Examples

If A is a representation-finite accessible algebra with gl.dim A ≤2, then A is totally accessible. On the other hand the algebra B with quiver:

$$\mathbf{1} \xrightarrow{\mathbf{x}} \mathbf{2} \xrightarrow{\mathbf{x}} \mathbf{3} \xrightarrow{\mathbf{x}} \mathbf{4} \dots \xrightarrow{\mathbf{x}} \mathbf{11} \xrightarrow{\mathbf{x}} \mathbf{12}$$

and <sup>x</sup><sup>3</sup> <sup>¼</sup> 0 is representation-finite and accessible (but not gl.dim <sup>B</sup>≤2). The Coxeter polynomial of B is:

$$\chi\_B(t) = \mathbf{1} + \mathbf{t} - \mathbf{t}^3 - \mathbf{t}^4 + \mathbf{t}^6 - \mathbf{t}^8 - \mathbf{t}^9 + \mathbf{t}^{11} + \mathbf{t}^{12}.$$

Then observe that the 6-th coefficient is 1 but the algebra B is not derived equivalent to Dynkin type A12.

#### 8. On the traces of Coxeter matrices

Let A be an algebra such that not all roots of χ<sup>A</sup> are roots of unity. By the result of Kronecker [36], not all of the spectrum of A lies in the unit disk. Equivalently, the spectral radius <sup>ρ</sup><sup>A</sup> <sup>¼</sup> maxf g <sup>j</sup>λ<sup>j</sup> : <sup>λ</sup> eigenvalue  of <sup>ϕ</sup><sup>A</sup> <sup>&</sup>gt;1. Arrange the eigenvalues of ϕ<sup>A</sup> so that μ1, μ2, …, μ<sup>n</sup> have absolute values ρ<sup>A</sup> ¼ r<sup>1</sup> > r<sup>2</sup> > … > rs and multiplicities m1, …, ms, respectively. Therefore s≥2 and

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

að Þi

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> an�2<sup>t</sup>

a. aj ≤1, for every 2 ≤j≤n � 2;

Polynomials - Theory and Application

2< j<n � 2 and assume that aj ¼ 1, we get:

The last inequality due to (a), hence að Þ <sup>n</sup>�<sup>1</sup>

preprojective component P. In particular, að Þ <sup>n</sup>�<sup>1</sup>

1 ¼ a ð Þ n <sup>j</sup> ¼ a

algebra of type An.

χAðÞ¼ t 1 þ t þ a2t

above.

7.3 Examples

24

Coxeter polynomial of B is:

equivalent to Dynkin type A12.

8. On the traces of Coxeter matrices

m1, …, ms, respectively. Therefore s≥2 and

<sup>j</sup>þ<sup>1</sup> <sup>¼</sup> <sup>a</sup>ð Þ <sup>i</sup>�<sup>1</sup>

a ð Þi <sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>a</sup>

<sup>j</sup>þ<sup>1</sup> � ci,jþ<sup>1</sup> <sup>¼</sup> <sup>a</sup>ð Þ<sup>i</sup>

<sup>n</sup>�<sup>2</sup> <sup>þ</sup> <sup>t</sup>

ð Þ n�1

totally accessible. On the other hand the algebra B with quiver:

1 ! <sup>x</sup> <sup>2</sup>! <sup>x</sup> <sup>3</sup> !

χBðÞ¼ t 1 þ t � t

<sup>j</sup> <sup>þ</sup> <sup>a</sup>ð Þ <sup>n</sup>�<sup>1</sup>

An�<sup>1</sup> is derived equivalent to A<sup>n</sup>�<sup>1</sup> and its Auslander-Reiten quiver consists of a

sn�<sup>3</sup>ð Þ¼ An�<sup>1</sup> 1, that is, <sup>A</sup> <sup>¼</sup> An�<sup>1</sup>½ � <sup>M</sup> for some exceptional module <sup>M</sup> such that <sup>M</sup><sup>⊥</sup> is derived equivalent to mod<sup>C</sup> for a connected algebra C, that is, s Að Þ¼ n � 2 and by (3.1), <sup>A</sup> <sup>¼</sup> B M½ � is derived equivalent to a hereditary algebra of type <sup>A</sup>n. □

If A is a representation-finite accessible algebra with gl.dim A ≤2, then A is

<sup>x</sup> <sup>4</sup>… !

and <sup>x</sup><sup>3</sup> <sup>¼</sup> 0 is representation-finite and accessible (but not gl.dim <sup>B</sup>≤2). The

Then observe that the 6-th coefficient is 1 but the algebra B is not derived

Let A be an algebra such that not all roots of χ<sup>A</sup> are roots of unity. By the result of Kronecker [36], not all of the spectrum of A lies in the unit disk. Equivalently, the spectral radius <sup>ρ</sup><sup>A</sup> <sup>¼</sup> maxf g <sup>j</sup>λ<sup>j</sup> : <sup>λ</sup> eigenvalue  of <sup>ϕ</sup><sup>A</sup> <sup>&</sup>gt;1. Arrange the eigenvalues of ϕ<sup>A</sup> so that μ1, μ2, …, μ<sup>n</sup> have absolute values ρ<sup>A</sup> ¼ r<sup>1</sup> > r<sup>2</sup> > … > rs and multiplicities

<sup>3</sup> � <sup>t</sup> <sup>4</sup> <sup>þ</sup> <sup>t</sup> <sup>6</sup> � <sup>t</sup> <sup>8</sup> � <sup>t</sup> <sup>9</sup> <sup>þ</sup> <sup>t</sup>

<sup>x</sup> <sup>11</sup> ! <sup>x</sup> 12

Theorem 2. Let A be a totally accessible algebra with Coxeter polynomial

ð Þ i�1 <sup>j</sup> � cj,i�<sup>1</sup> 

<sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>t</sup>

b.if for some 2 ≤j≤n � 2, we have aj ¼ 1 then A is derived equivalent to a hereditary

We shall prove (b) by induction on n the number of vertices of A. Let j ¼ 2 and assume a<sup>2</sup> ¼ 1, then (3.1) implies that A is derived equivalent to An. Consider now

> <sup>j</sup>�<sup>1</sup> � cn�1,j�<sup>1</sup>

≤að Þ <sup>n</sup>�<sup>1</sup> <sup>j</sup> ≤ 1

<sup>2</sup> ¼ 1, which implies that

<sup>11</sup> <sup>þ</sup> <sup>t</sup> 12:

<sup>j</sup> ¼ 1. Induction hypothesis yields that

Proof. Keep the notation as in (4.1). Then (a) is the case i ¼ n of the Lemma

<sup>j</sup>þ<sup>2</sup> � <sup>a</sup>

n, then:

≤ a ð Þ i�1 <sup>j</sup>þ<sup>1</sup> ,

<sup>j</sup>þ<sup>2</sup> <sup>≤</sup> <sup>0</sup>: □

ð Þ i�1

$$|\det \phi\_A| = r\_1^{m\_1} r\_2^{m\_2} \dots r\_s^{m\_s} = \mathbf{1}.$$

We define the critical power κð Þ A as the minimal k such that

∣Tr ϕ<sup>k</sup> A ∣> n

Since r<sup>1</sup> is a simple eigenvalue of ϕA, then it follows that κð Þ A is well defined due to the existence of k satisfying the following chain of inequalities:

$$|\mathrm{Tr}\left(\phi\_A^k\right)| = |\sum\_{j=1}^n \mu\_j^k| \ge r\_1^{km\_1} - \sum\_{j=2}^s r\_j^{km\_j} \ge r\_1^k - (n-1)r\_2^k > n.$$

The following is a reformulation of Theorem 2.

Theorem. Let A be an algebra such that not all roots of χ<sup>A</sup> are roots of unity. We have κð Þ A ≤ n:

Proof. Indeed, suppose that A is not of cyclotomic type and κð Þ A > n, that is, ∣Tr ϕ<sup>k</sup> A ∣ ≤<sup>n</sup> for all 0 <sup>≤</sup>k≤n. Observe that <sup>M</sup> <sup>¼</sup> <sup>ϕ</sup><sup>A</sup> is a unimodular matrix and therefore, Theorem 2 implies that M is of cyclotomic type, which yields a contradiction. □

Remark: We consider explicitly the case n ¼ 2 in the above Theorem. Obviously, the Cartan matrix of A is of the form

$$\mathbf{C} = \begin{pmatrix} \mathbf{1} & a \\ \mathbf{0} & \mathbf{1} \end{pmatrix} \quad \phi\_A = -\mathbf{C}^{-1}\mathbf{C}^T = \begin{pmatrix} a^2 - \mathbf{1} & a \\ -a & -\mathbf{1} \end{pmatrix}.$$

for some a≥1. Then ϕ<sup>A</sup> has the indicated shape. If A is not cyclotomic, then a≥3 and Tr ϕ<sup>2</sup> A <sup>¼</sup> <sup>a</sup>ð Þ <sup>2</sup> � <sup>2</sup> <sup>2</sup> � <sup>2</sup>>2.

#### 9. Stability of a real matrix

#### 9.1 Stability of matrices and the Lyapunov criterion

Let <sup>M</sup> be a real invertible <sup>n</sup> � <sup>n</sup>-matrix with eigenvalues <sup>λ</sup><sup>j</sup> <sup>¼</sup> rjei <sup>θ</sup><sup>j</sup> , for some numbers θ<sup>j</sup> ∈½ Þ 0; 2π and j ¼ 1, …, n. We will say that M is stable (resp. semi-stable) if the real part Re e<sup>i</sup> <sup>θ</sup><sup>j</sup> <sup>¼</sup> cos <sup>θ</sup><sup>j</sup> of the argument of the eigenvalue <sup>λ</sup><sup>j</sup> is positive (resp. non-negative), for every j ¼ 1, …, n. The following is well-known, we sketch a proof for the sake of completeness.

Proposition. Let M be a stable (resp. semi-stable) n � n-matrix. Then the characteristic polynomial <sup>χ</sup><sup>M</sup> <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> an�<sup>1</sup>T<sup>n</sup>�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> <sup>a</sup>1<sup>T</sup> <sup>þ</sup> <sup>a</sup><sup>0</sup> has coefficients satisfying ð Þ �<sup>1</sup> <sup>n</sup>�<sup>j</sup> aj > 0 (resp. ≥ 0), for j ¼ 0, 1, …, n;

Proof. Observe that ð Þ �<sup>1</sup> <sup>n</sup> pð Þ �T is the product of polynomials T � α with α ∈ R and ð Þ <sup>T</sup> � ð Þ <sup>α</sup> <sup>þ</sup> <sup>i</sup><sup>β</sup> ð Þ¼ <sup>T</sup> � ð Þ <sup>α</sup> � <sup>i</sup><sup>β</sup> <sup>T</sup><sup>2</sup> � <sup>2</sup>α<sup>T</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> with 0 6¼ <sup>β</sup>, <sup>α</sup> <sup>∈</sup> <sup>R</sup>. Stability (resp. semi-stability) implies that <sup>α</sup><0 (resp. <sup>α</sup><sup>≤</sup> 0) above. Therefore, ð Þ �<sup>1</sup> <sup>n</sup> pð Þ �T is product of polynomials with positive coefficients. □

Remark: In most of the literature the stability concept we use goes by the name of positive stability, while the stability name is used also as Hurwitz stability, or Lyapunov stability.

The system of differential equations

$$\mathbf{y}'(t) = -\mathbf{M}\mathbf{y}(t)$$

is said to be stable if for every vector <sup>d</sup> <sup>¼</sup> ð Þ <sup>d</sup>1; …; dn , the solution v tðÞ¼ <sup>e</sup>�tMd of the above system has the property that limt!<sup>∞</sup> v tðÞ¼ 0.

C ¼

0

BBBBBBBB@

verified calculating Tr ϕ<sup>k</sup>

k ¼ Trχ<sup>k</sup>

11 �1 1; 2; 5; 7; 9; 10; 13; 14; 17 ¼ 1 3; 6; 15 ¼ 2 4; 8; 16 ¼ 3 12 ¼ 6

1. Indeed, for.

Tr χ<sup>k</sup> A

χRn :

27

10.2 An example

N nð Þ ; 3 , for all n ≥1.

linear quiver 1 ! 2 ! ⋯ ! s. For 2m þ 1 odd, we consider.

a. Rn is derived equivalent to N nð Þ ; 3 .

100000 1 10000 1 1 1000 01 1 100 001 1 10 0001 1 1

DOI: http://dx.doi.org/10.5772/intechopen.82309

Cyclotomic and Littlewood Polynomials Associated to Algebras

B

<sup>A</sup> ¼

1

CCCCCCCCA

and ϕ ¼

whose characteristic polynomial is cyclotomic as we know from [18] or might be

Starting with k ¼ 17 the sequence of traces repeats cyclically. Therefore,

We recall in some length the argument given in [18] for the cyclotomicity of

Consider the algebra R2<sup>n</sup> with 2n vertices and whose quiver is given as

with all commutative relations. The corresponding Coxeter polynomial

χ<sup>R</sup>2<sup>n</sup> ¼ χ<sup>A</sup><sup>n</sup> ⊗ χ<sup>A</sup><sup>2</sup> ¼ vnþ<sup>1</sup> ⊗ v<sup>3</sup>

is a product of cyclotomic polynomials, therefore χ<sup>R</sup>2<sup>n</sup> is a cyclotomic polynomial. In fact R2<sup>n</sup> ¼ A<sup>n</sup> ⊗ A2, where A<sup>s</sup> is the hereditary algebra associated to the

The following holds for the sequence of algebras Rn and its Coxeter polynomials

� �≤6 for all 0 <sup>≤</sup>k. Then <sup>N</sup>ð Þ <sup>6</sup>; <sup>3</sup> is of cyclotomic type.

0

BBBBBBBB@

� �≤n, for 1≤ k≤72 and applying the criterion of Theorem

�11 0 �111 �10 1 �101 �1 0 0 0 00 0 �1 0 0 00 0 0 �1 0 00 000 �110

1

CCCCCCCCA

We recall here the celebrated.

Lyapunov criterion: The system y<sup>0</sup> ðÞ¼� t My tð Þ is stable if and only if M is a stable matrix, equivalently there is a real positive definite matrix P such that

$$\boldsymbol{M}^T \boldsymbol{P} + \boldsymbol{P} \boldsymbol{M} = \boldsymbol{I}\_n \dots$$

It is not hard to see that given M, the corresponding P is unique. A proof of the criterion and its equivalence to other stability conditions are considered in [13].

#### 9.2 Semi-stable powers

Let <sup>μ</sup>1, …, <sup>μ</sup><sup>n</sup> be the eigenvalues of the real matrix <sup>M</sup> with <sup>μ</sup><sup>j</sup> <sup>¼</sup> <sup>ρ</sup>je<sup>2</sup> <sup>π</sup><sup>i</sup> <sup>θ</sup><sup>j</sup> in polar form. Observe that μ<sup>k</sup> <sup>j</sup> , for <sup>j</sup> <sup>¼</sup> <sup>1</sup>, …, n, are the eigenvalues of Mk and

$$\operatorname{Tr} \mathbf{M}^k = \sum\_{j=1}^n \rho\_j^k \cos \left(k \theta\_j \right) \le \sum\_{j=1}^n |\mu\_j^k| |\cos \left(k \theta\_j \right)|.$$

Lemma. For a positive integer k≥1 the following assertions are equivalent: a. M<sup>k</sup> is a semi-stable matrix;

b. Tr <sup>M</sup><sup>k</sup> <sup>¼</sup> <sup>∑</sup><sup>n</sup> <sup>j</sup>¼<sup>1</sup> <sup>μ</sup><sup>j</sup> k ∣ cos kθ<sup>j</sup> ∣.

Proof. If Mk is a semi-stable matrix, then <sup>μ</sup><sup>k</sup> <sup>¼</sup> <sup>ρ</sup><sup>k</sup> <sup>j</sup> cos kθ<sup>j</sup> <sup>þ</sup> <sup>i</sup>sin <sup>k</sup>θ<sup>j</sup> has cos kθ<sup>j</sup> <sup>≥</sup>0. Since <sup>M</sup> is a real matrix then Tr Mk <sup>¼</sup> <sup>∑</sup><sup>n</sup> <sup>j</sup>¼<sup>1</sup> <sup>ρ</sup><sup>k</sup> <sup>j</sup> cos kθ<sup>j</sup> ≥ 0. Therefore

$$\operatorname{Tr}\left(\mathcal{M}^k\right) = \sum\_{j=1}^n \rho\_j^k |\cos\left(k\theta\_j\right)|.$$

Assume that Tr Mk <sup>¼</sup> <sup>∑</sup><sup>n</sup> <sup>j</sup>¼<sup>1</sup> λ<sup>j</sup> k ∣ cos kθ<sup>j</sup> ∣. Since ∣λ<sup>k</sup> <sup>j</sup> ∣ ≥ρ<sup>k</sup> <sup>j</sup> cos kθ<sup>j</sup> for j ¼ 1, …, n, adding up, we get

$$\operatorname{Tr}\left(\mathcal{M}^k\right) \ge \sum\_{j=1}^n \rho\_j^k \cos\left(k\theta\_j\right) = \operatorname{Tr}\left(\mathcal{M}^k\right),$$

Hence we have equalities ∣λ<sup>k</sup> <sup>j</sup> k cos kθ<sup>j</sup> <sup>∣</sup> <sup>¼</sup> <sup>ρ</sup><sup>k</sup> <sup>j</sup> cos kθ<sup>j</sup> for <sup>j</sup> <sup>¼</sup> <sup>1</sup>, …, n. Then Mk is semi-stable. □

We say that k is a stable power (resp. semi-stable power) of M if Mk is a stable (resp. semi-stable) matrix.

#### 10. Nakayama algebras

#### 10.1 Cyclotomic Nakayama algebras

As a well-understood example the representation theory of the Nakayama algebras stands appart. Let N nð Þ ;r be the quotient obtained from the linear quiver with n vertices with radical rad<sup>A</sup> of nilpotency index r.

For instance, for A ¼ Nð Þ 6; 3 the Cartan matrix C and Coxeter matrix ϕ are:

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

$$C = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \end{pmatrix} \text{ and } \phi = \begin{pmatrix} -1 & 1 & 0 & -1 & 1 & 1 \\ -1 & 0 & 1 & -1 & 0 & 1 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 0 \end{pmatrix}.$$

whose characteristic polynomial is cyclotomic as we know from [18] or might be verified calculating Tr ϕ<sup>k</sup> B � �≤n, for 1≤ k≤72 and applying the criterion of Theorem 1. Indeed, for.


is said to be stable if for every vector <sup>d</sup> <sup>¼</sup> ð Þ <sup>d</sup>1; …; dn , the solution v tðÞ¼ <sup>e</sup>�tMd of

stable matrix, equivalently there is a real positive definite matrix P such that

It is not hard to see that given M, the corresponding P is unique. A proof of the criterion and its equivalence to other stability conditions are considered in [13].

<sup>M</sup>TP <sup>þ</sup> PM <sup>¼</sup> In:

Let <sup>μ</sup>1, …, <sup>μ</sup><sup>n</sup> be the eigenvalues of the real matrix <sup>M</sup> with <sup>μ</sup><sup>j</sup> <sup>¼</sup> <sup>ρ</sup>je<sup>2</sup> <sup>π</sup><sup>i</sup> <sup>θ</sup><sup>j</sup> in polar

<sup>j</sup> , for <sup>j</sup> <sup>¼</sup> <sup>1</sup>, …, n, are the eigenvalues of Mk and

≤ ∑<sup>n</sup>

Lemma. For a positive integer k≥1 the following assertions are equivalent:

n j¼1 ρk

∣ cos kθ<sup>j</sup>

<sup>∣</sup> <sup>¼</sup> <sup>ρ</sup><sup>k</sup>

semi-stable. □ We say that k is a stable power (resp. semi-stable power) of M if Mk is a stable

As a well-understood example the representation theory of the Nakayama algebras stands appart. Let N nð Þ ;r be the quotient obtained from the linear quiver with

For instance, for A ¼ Nð Þ 6; 3 the Cartan matrix C and Coxeter matrix ϕ are:

j¼1 ∣μk

<sup>j</sup> ∣ cos kθ<sup>j</sup> ∣:

∣. Since ∣λ<sup>k</sup>

<sup>¼</sup> Tr <sup>M</sup><sup>k</sup>

<sup>j</sup> cos kθ<sup>j</sup>

<sup>j</sup> k cos kθ<sup>j</sup> ∣

<sup>j</sup> cos kθ<sup>j</sup>

<sup>j</sup>¼<sup>1</sup> <sup>ρ</sup><sup>k</sup>

<sup>j</sup> ∣ ≥ρ<sup>k</sup>

 <sup>þ</sup> <sup>i</sup>sin <sup>k</sup>θ<sup>j</sup> has

≥ 0. Therefore

<sup>j</sup> cos kθ<sup>j</sup>

<sup>j</sup> cos kθ<sup>j</sup>

for

for <sup>j</sup> <sup>¼</sup> <sup>1</sup>, …, n. Then Mk is

ðÞ¼� t My tð Þ is stable if and only if M is a

the above system has the property that limt!<sup>∞</sup> v tðÞ¼ 0.

TrM<sup>k</sup> <sup>¼</sup> <sup>∑</sup>

a. M<sup>k</sup> is a semi-stable matrix;

Assume that Tr Mk <sup>¼</sup> <sup>∑</sup><sup>n</sup>

Hence we have equalities ∣λ<sup>k</sup>

10.1 Cyclotomic Nakayama algebras

n vertices with radical rad<sup>A</sup> of nilpotency index r.

j ¼ 1, …, n, adding up, we get

(resp. semi-stable) matrix.

10. Nakayama algebras

26

<sup>j</sup>¼<sup>1</sup> <sup>μ</sup><sup>j</sup> k

n j¼1 ρk <sup>j</sup> cos kθ<sup>j</sup>

∣ cos kθ<sup>j</sup> ∣.

<sup>≥</sup>0. Since <sup>M</sup> is a real matrix then Tr Mk <sup>¼</sup> <sup>∑</sup><sup>n</sup>

Tr <sup>M</sup><sup>k</sup> <sup>¼</sup> <sup>∑</sup>

j¼1 ρk <sup>j</sup> cos kθ<sup>j</sup>

<sup>j</sup> k cos kθ<sup>j</sup>

<sup>j</sup>¼<sup>1</sup> λ<sup>j</sup> k

Tr Mk ≥ ∑<sup>n</sup>

Proof. If Mk is a semi-stable matrix, then <sup>μ</sup><sup>k</sup> <sup>¼</sup> <sup>ρ</sup><sup>k</sup>

We recall here the celebrated. Lyapunov criterion: The system y<sup>0</sup>

Polynomials - Theory and Application

9.2 Semi-stable powers

form. Observe that μ<sup>k</sup>

b. Tr <sup>M</sup><sup>k</sup> <sup>¼</sup> <sup>∑</sup><sup>n</sup>

cos kθ<sup>j</sup>

Starting with k ¼ 17 the sequence of traces repeats cyclically. Therefore, Tr χ<sup>k</sup> A � �≤6 for all 0 <sup>≤</sup>k. Then <sup>N</sup>ð Þ <sup>6</sup>; <sup>3</sup> is of cyclotomic type.

#### 10.2 An example

We recall in some length the argument given in [18] for the cyclotomicity of N nð Þ ; 3 , for all n ≥1.

Consider the algebra R2<sup>n</sup> with 2n vertices and whose quiver is given as


with all commutative relations. The corresponding Coxeter polynomial

$$
\chi\_{\mathbb{R}\_{2^n}} = \chi\_{\mathbb{V}\_n} \otimes \chi\_{\mathbb{V}\_2} = \upsilon\_{n+1} \otimes \upsilon\_3,
$$

is a product of cyclotomic polynomials, therefore χ<sup>R</sup>2<sup>n</sup> is a cyclotomic polynomial. In fact R2<sup>n</sup> ¼ A<sup>n</sup> ⊗ A2, where A<sup>s</sup> is the hereditary algebra associated to the linear quiver 1 ! 2 ! ⋯ ! s.

For 2m þ 1 odd, we consider.

$$R\_{2m+1} \Rightarrow \stackrel{\circ}{\uparrow} \rightsquigarrow \stackrel{\circ}{\uparrow} \rightsquigarrow \stackrel{\circ}{\uparrow} \rightsquigarrow \stackrel{\circ}{\uparrow} \rightsquigarrow \stackrel{\circ}{\uparrow}$$

The following holds for the sequence of algebras Rn and its Coxeter polynomials χRn :

a. Rn is derived equivalent to N nð Þ ; 3 .

b.χRn <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup>n�<sup>1</sup> � <sup>T</sup><sup>3</sup> <sup>χ</sup>Rn�<sup>6</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> 1, for all <sup>n</sup><sup>≥</sup> 6; c. M χRn � � <sup>¼</sup> 1.

Observe that the sequence of algebras ð Þ Rn forms an interlaced tower of algebras, that is, it is a sequence of triangular algebras R1, …, Rn, such that Rs is a basic algebra with s simple modules and, among others, the condition

$$\chi\_{\mathcal{R}\_{\iota+1}} = (T+\mathbf{1})\chi\_{\mathcal{R}\_{\iota}} - T\chi\_{\mathcal{R}\_{\iota-1}}$$

is satisfied for s ¼ 1, …, n � 1. Moreover, Asþ<sup>1</sup> is a one-point extension (or coextension) of an accessible algebra As by an exceptional As- module Ms such that the perpendicular category M<sup>⊥</sup> <sup>s</sup> formed in the derived category is triangular equivalent to modð Þ As�<sup>1</sup> , for s ¼ m þ 1, …, n � 1.

The following was shown in [18]: Consider an interlaced tower of algebras Am, …, An with m ≤ n � 2. If SpecϕAn is contained in the union of the unit circle and the semi-ray of positive real numbers then either all Ai are of cyclotomic type or M χAm � �<sup>&</sup>lt; <sup>M</sup> <sup>χ</sup>An � �. In the latter case, M <sup>χ</sup>An � �< Q<sup>n</sup>�<sup>1</sup> <sup>s</sup>¼<sup>m</sup> <sup>M</sup> <sup>χ</sup>As � �.

Since we know that M χ<sup>R</sup>2<sup>n</sup> � � <sup>¼</sup> 1, for all <sup>n</sup><sup>≥</sup> 0, we conclude that <sup>M</sup> <sup>χ</sup>Rn � � <sup>¼</sup> 1, for all n ≥0. That is the Nakayama algebras of the form N nð Þ ; 3 are of cyclotomic type.

#### 10.3 Non-cyclotomic Nakayama algebras

Calculation of Trϕ<sup>k</sup> <sup>A</sup> for A ¼ N nð Þ ;r and k in intervals, for data sets ð Þ n;r; k , yield interesting information. Namely,


N nð Þ ; 4 is of cyclotomic type for all 0 ≤n ≤100 except for n ¼ 10; 22; 30; 42; 50; 62; 70; 82 and 90

c. A canonical algebra C of weight p1; …; pt � � is strongly accessible if and only if t ¼ 3, in that case, C is derived-equivalent to a representation-finite algebra if and only if the weight type does not dominate 3ð Þ ; 3; 3 .

Author details

29

José-Antonio de la Peña1,2

2 El Colegio Nacional, México

\*Address all correspondence to: jantdelap@gmail.com

Cyclotomic and Littlewood Polynomials Associated to Algebras

DOI: http://dx.doi.org/10.5772/intechopen.82309

provided the original work is properly cited.

1 Instituto de Matemáticas, Universidad Nacional Autónoma de México, México

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

b.χRn <sup>¼</sup> <sup>T</sup><sup>n</sup> <sup>þ</sup> <sup>T</sup>n�<sup>1</sup> � <sup>T</sup><sup>3</sup> <sup>χ</sup>Rn�<sup>6</sup> <sup>þ</sup> <sup>T</sup> <sup>þ</sup> 1, for all <sup>n</sup><sup>≥</sup> 6;

with s simple modules and, among others, the condition

� �. In the latter case, M <sup>χ</sup>An

a. Many Nakayama algebras are of cyclotomic type;

n ¼ 10; 22; 30; 42; 50; 62; 70; 82 and 90

c. A canonical algebra C of weight p1; …; pt

N nð Þ ; 4 is of cyclotomic type for all 0 ≤n ≤100 except for

and only if the weight type does not dominate 3ð Þ ; 3; 3 .

Observe that the sequence of algebras ð Þ Rn forms an interlaced tower of algebras, that is, it is a sequence of triangular algebras R1, …, Rn, such that Rs is a basic algebra

χRsþ<sup>1</sup> ¼ ð Þ T þ 1 χRs � TχRs�<sup>1</sup>

<sup>s</sup> formed in the derived category is triangular equiv-

� � <sup>¼</sup> 1, for

<sup>s</sup>¼<sup>m</sup> <sup>M</sup> <sup>χ</sup>As � �.

<sup>A</sup> for A ¼ N nð Þ ;r and k in intervals, for data sets ð Þ n;r; k , yield

� � is strongly accessible if and only if

� � <sup>¼</sup> 1, for all <sup>n</sup><sup>≥</sup> 0, we conclude that <sup>M</sup> <sup>χ</sup>Rn

is satisfied for s ¼ 1, …, n � 1. Moreover, Asþ<sup>1</sup> is a one-point extension (or coextension) of an accessible algebra As by an exceptional As- module Ms such that

The following was shown in [18]: Consider an interlaced tower of algebras Am, …, An with m ≤ n � 2. If SpecϕAn is contained in the union of the unit circle and the

� �< Q<sup>n</sup>�<sup>1</sup>

all n ≥0. That is the Nakayama algebras of the form N nð Þ ; 3 are of cyclotomic type.

b.Not all Nakayama algebras are of cyclotomic type. The case r ¼ 4 illustrates

t ¼ 3, in that case, C is derived-equivalent to a representation-finite algebra if

semi-ray of positive real numbers then either all Ai are of cyclotomic type or

c. M χRn

M χAm

� �<sup>&</sup>lt; <sup>M</sup> <sup>χ</sup>An

� � <sup>¼</sup> 1.

Polynomials - Theory and Application

the perpendicular category M<sup>⊥</sup>

Since we know that M χ<sup>R</sup>2<sup>n</sup>

interesting information. Namely,

Calculation of Trϕ<sup>k</sup>

this claim:

28

alent to modð Þ As�<sup>1</sup> , for s ¼ m þ 1, …, n � 1.

10.3 Non-cyclotomic Nakayama algebras

### Author details

José-Antonio de la Peña1,2

1 Instituto de Matemáticas, Universidad Nacional Autónoma de México, México

2 El Colegio Nacional, México

\*Address all correspondence to: jantdelap@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Assem I, Simson D, Skowroński A. Elements of the representation theory of associative algebras 1: Techniques of representation theory. In: London Mathematical Society Student Texts 65. Cambridge University Press; 2006

[2] Boldt A. Methods to determine Coxeter polynomials. Linear Algebra and its Applications. 1995;230: 151-164

[3] Cross GW. Three types of matrix stability. Linear Algebra and its Applications. 1978;20:253-263

[4] Cvetković DM, Doob M, Sachs H. Spectra of Graphs, Theory and Application. Academic Press; 1979

[5] Dlab V, Ringel CM. Eigenvalues of Coxeter transformations and the Gelfand-Kirillov dimension of the preprojective algebras. Proceedings of the American Mathematical Society. 1981;83:228-232

[6] Ladkani S. On the periodicity of Coxeter transformations and the nonnegativity of their Euler forms. Linear Algebra and its Applications. 2008; 428(4):742-753

[7] Goodman FM, de la Harpe P, Jones VFR. Coxeter Graphs and Towers of Algebras. New York, Berlin, Heidelberg: Springer-Verlag; 1989

[8] Happel D. Hochschild Cohomology of Finite Dimensional Algebras. Lect. Notes Math. Vol. 1404. Springer Verlag; 1989. pp. 108-126

[9] Kronecker L. Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. Crelle's Journal. 1857; Ouvres I:105-108

[10] Lenzing H. A K-theoretic study of canonical algebras. Conference

Proceedings, Canadian Mathematical Society. 1996;18:433-454

generalized Cartan matrix.

with small growth numbers. Communications in Algebra. 1990;

18(10):3413-3420

331-339

31

Mathematische Annalen. 1994;300:

DOI: http://dx.doi.org/10.5772/intechopen.82309

Cyclotomic and Littlewood Polynomials Associated to Algebras

[21] Xi C. On wild hereditary algebras

[11] Lenzing H, de la Peña JA. Wild canonical algebras. Mathematische Zeitschrift. 1997;224:403-425

[12] Lenzing H, de la Peña JA. Concealed-canonical algebras and separating tubular families. Proceedings of the London Mathematical Society. 1999;78:513-540

[13] Lenzing H, de la Peña JA. Spectral analysis of finite dimensional algebras and singularities. In: Skowroński A, editor. Trends in Representation Theory of Algebras and Related Topics. Zürich: EMS Publishing House; 2008. pp. 541-588

[14] Lenzing H, de la Peña JA. Extended canonical algebras and Fuchsian singularities. Mathematische Zeitschrift. 2011;268(1–2):143-167

[15] Mahler K. Lectures on trascendental Number Theory. Lecture Notes in Mathematics. Vol. 546. Berlin: Springer Verlag; 1976

[16] Newman M. Integral Matrices. New York: Academic Press; 1972

[17] de la Peña JA, Takane M. Spectral properties of Coxeter transformations and applications. Archiv der Mathematik. 1990;55:120-134

[18] de la Peña JA. Algebras whose Coxeter polynomials are product of cyclotomic polynomials. Algebras and Representation Theory. 2014;17(3): 905-930

[19] Ringel CM. Tame Algebras and Integral Quadratic Forms. Lect. Notes Math. Vol. 1099. Berlin-Heidelberg-New York: Springer; 1984

[20] Ringel CM. The spectral radius of the Coxeter transformations for a

Cyclotomic and Littlewood Polynomials Associated to Algebras DOI: http://dx.doi.org/10.5772/intechopen.82309

generalized Cartan matrix. Mathematische Annalen. 1994;300: 331-339

References

2006

151-164

1981;83:228-232

428(4):742-753

Springer-Verlag; 1989

1989. pp. 108-126

Ouvres I:105-108

30

[1] Assem I, Simson D, Skowroński A. Elements of the representation theory of associative algebras 1: Techniques of representation theory. In: London Mathematical Society Student

Polynomials - Theory and Application

Proceedings, Canadian Mathematical

[11] Lenzing H, de la Peña JA. Wild canonical algebras. Mathematische Zeitschrift. 1997;224:403-425

[13] Lenzing H, de la Peña JA. Spectral analysis of finite dimensional algebras and singularities. In: Skowroński A, editor. Trends in Representation Theory of Algebras and Related Topics. Zürich: EMS Publishing House; 2008. pp. 541-588

[14] Lenzing H, de la Peña JA. Extended

singularities. Mathematische Zeitschrift.

[15] Mahler K. Lectures on trascendental Number Theory. Lecture Notes in Mathematics. Vol. 546. Berlin: Springer

[16] Newman M. Integral Matrices. New

[17] de la Peña JA, Takane M. Spectral properties of Coxeter transformations

York: Academic Press; 1972

and applications. Archiv der Mathematik. 1990;55:120-134

[18] de la Peña JA. Algebras whose Coxeter polynomials are product of cyclotomic polynomials. Algebras and Representation Theory. 2014;17(3):

[19] Ringel CM. Tame Algebras and Integral Quadratic Forms. Lect. Notes Math. Vol. 1099. Berlin-Heidelberg-

[20] Ringel CM. The spectral radius of the Coxeter transformations for a

New York: Springer; 1984

canonical algebras and Fuchsian

2011;268(1–2):143-167

Verlag; 1976

905-930

[12] Lenzing H, de la Peña JA. Concealed-canonical algebras and separating tubular families. Proceedings of the London Mathematical Society.

1999;78:513-540

Society. 1996;18:433-454

Texts 65. Cambridge University Press;

Algebra and its Applications. 1995;230:

[3] Cross GW. Three types of matrix stability. Linear Algebra and its Applications. 1978;20:253-263

[4] Cvetković DM, Doob M, Sachs H. Spectra of Graphs, Theory and Application. Academic Press; 1979

[5] Dlab V, Ringel CM. Eigenvalues of Coxeter transformations and the Gelfand-Kirillov dimension of the preprojective algebras. Proceedings of the American Mathematical Society.

[6] Ladkani S. On the periodicity of Coxeter transformations and the nonnegativity of their Euler forms. Linear Algebra and its Applications. 2008;

[7] Goodman FM, de la Harpe P, Jones VFR. Coxeter Graphs and Towers of Algebras. New York, Berlin, Heidelberg:

[8] Happel D. Hochschild Cohomology of Finite Dimensional Algebras. Lect. Notes Math. Vol. 1404. Springer Verlag;

[9] Kronecker L. Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. Crelle's Journal. 1857;

[10] Lenzing H. A K-theoretic study of canonical algebras. Conference

[2] Boldt A. Methods to determine Coxeter polynomials. Linear

[21] Xi C. On wild hereditary algebras with small growth numbers. Communications in Algebra. 1990; 18(10):3413-3420

Chapter 2

Abstract

of Signs

middle of the seventeenth century.

of signs, Rolle's theorem

1. Introduction

33

New Aspects of Descartes' Rule

Below, we summarize some new developments in the area of distribution of roots and signs of real univariate polynomials pioneered by R. Descartes in the

Keywords: real univariate polynomial, sign pattern, admissible pair, Descartes' rule

The classical Descartes' rule of signs claims that the number of positive roots of a

2010 Mathematics Subject Classication: Primary 26C10; Secondary 30C15

real univariate polynomial is bounded by the number of sign changes in the sequence of its coefficients and it coincides with the latter number modulo 2. It was published in French (instead of the usual at that time Latin) as a small portion of Sur la construction de problèmes solides ou plus que solide which is the third book of Descartes' fundamental treatise La Géométrie which, in its turn, is an appendix to his famous Discours de la méthode. It is in the latter chef d'oeuvre that Descartes developed his analytic approach to geometric problems leaving practically all proofs and details to an interested reader. This interested reader turned out to be Frans van Schooten, a professor of mathematics at Leiden who together with his students undertook a tedious work of making Descartes' writings understandable, translating and publishing them in the proper language, that is, Latin. (For the electronic version of this book, see [13].) Mathematical achievements of Descartes form a small fraction of his overall scientific and philosophical legacy, and Descartes' rule

of signs is a small but important fraction of his mathematical heritage.

contributions, see [1, 2, 6, 10, 12, 14], to mention a few.)

nonvanishing coefficients. For a polynomial P ≔ ∑<sup>d</sup>

To René Descartes, a polymath in philosophy and science.

Descartes' rule of signs has been studied and generalized by many authors over the years; one of the earliest can be found in [7], see also [4, 11]. (For some recent

In the present survey, we summarize a relatively new development in this area which, to the best of our knowledge, was initiated only in the 1990s (see [12]). For simplicity, we consider below only real univariate polynomials with all

coefficients, Descartes' rule of signs tells us what possible values the number of its real positive roots can have. For P as above, we define the sequence of � signs of

<sup>j</sup>¼<sup>0</sup>ajxj with fixed signs of its

Vladimir Petrov Kostov and Boris Shapiro

#### Chapter 2

## New Aspects of Descartes' Rule of Signs

Vladimir Petrov Kostov and Boris Shapiro

#### Abstract

Below, we summarize some new developments in the area of distribution of roots and signs of real univariate polynomials pioneered by R. Descartes in the middle of the seventeenth century.

Keywords: real univariate polynomial, sign pattern, admissible pair, Descartes' rule of signs, Rolle's theorem

2010 Mathematics Subject Classication: Primary 26C10; Secondary 30C15

#### 1. Introduction

The classical Descartes' rule of signs claims that the number of positive roots of a real univariate polynomial is bounded by the number of sign changes in the sequence of its coefficients and it coincides with the latter number modulo 2. It was published in French (instead of the usual at that time Latin) as a small portion of Sur la construction de problèmes solides ou plus que solide which is the third book of Descartes' fundamental treatise La Géométrie which, in its turn, is an appendix to his famous Discours de la méthode. It is in the latter chef d'oeuvre that Descartes developed his analytic approach to geometric problems leaving practically all proofs and details to an interested reader. This interested reader turned out to be Frans van Schooten, a professor of mathematics at Leiden who together with his students undertook a tedious work of making Descartes' writings understandable, translating and publishing them in the proper language, that is, Latin. (For the electronic version of this book, see [13].) Mathematical achievements of Descartes form a small fraction of his overall scientific and philosophical legacy, and Descartes' rule of signs is a small but important fraction of his mathematical heritage.

Descartes' rule of signs has been studied and generalized by many authors over the years; one of the earliest can be found in [7], see also [4, 11]. (For some recent contributions, see [1, 2, 6, 10, 12, 14], to mention a few.)

In the present survey, we summarize a relatively new development in this area which, to the best of our knowledge, was initiated only in the 1990s (see [12]).

For simplicity, we consider below only real univariate polynomials with all nonvanishing coefficients. For a polynomial P ≔ ∑<sup>d</sup> <sup>j</sup>¼<sup>0</sup>ajxj with fixed signs of its coefficients, Descartes' rule of signs tells us what possible values the number of its real positive roots can have. For P as above, we define the sequence of � signs of

To René Descartes, a polymath in philosophy and science.

length d þ 1 which we call the sign pattern (SP for short) of P, namely, we say that a polynomial P with all nonvanishing coefficients defines the sign pattern σ ≔ ðsd, sd�1, …, s0Þ if sj ¼ sgn aj. Since the roots of the polynomials P and �P are the same, we can, without loss of generality, assume that the first sign of a SP is always a þ.

One should also add that there is a number of completely different directions in which mathematicians are trying to extend Descartes' rule of signs. They include, for example, rule of signs for other univariate analytic functions including exponential functions, trigonometric functions and orthogonal polynomials, multivariate Descartes' rule of signs, tropical rule of signs, rule of signs in the complex domain, etc. (see, e.g., [6, 10, 14]) and references therein. But we think that Problem 1 is the closest one to the original investigations by Des-

The structure of this chapter is as follows. In Section 2, we provide the information about the solution of Problem 1 in degrees up to 11. In Section 3, we present several infinite series of non-realizable couples (SP, AP). Finally, in Section 4 we

To shorten the list of cases (SP, AP) under consideration, we can use the

P xð Þ <sup>↦</sup> ð Þ �<sup>1</sup> <sup>d</sup>

tive roots. Therefore, the SPs which they define have the same AP.

components of the SPs defined by the polynomials P xð Þ and ð Þ �<sup>1</sup> <sup>d</sup>

different. When an orbit of length 2 occurs and d is even, then both SPs are symmetric w.r.t. their middle points (hence their last component equal þ). Similarly, when d is odd, then one of the two SPs is symmetric w.r.t. its middle (with the last component equal to þ), and the other one is antisymmetric. Thus, its

Obviously, the first generator exchanges the components of the AP. Concerning the second generator, to obtain the SP defined by the polynomial P<sup>R</sup>, one has to read the SP defined by P xð Þ backward. The roots of <sup>P</sup><sup>R</sup> are the reciprocals of these of <sup>P</sup> which implies that both polynomials have the same numbers of positive and nega-

Remark 1. A priori the length of an orbit of any Z<sup>2</sup> � Z2-action could be 1, 2, or 4, but for the above action, orbits of length 1 do not exist since the second

It is obvious that all pairs or quadruples (SP, AP) constituting a given orbit are

As a warm-up exercise, let us consider degrees d ¼ 1, 2 and 3. In these cases, the answer to Problem 1 is positive. We give the list of SPs, with the respective values c and p of their APs and examples of polynomials realizing the couples (SP, AP). In order to shorten the list, we consider only SPs beginning with two þ signs; the cases when these signs are ð Þ þ; � are realized by the

Pð Þ �x , (1)

Pð Þ �x are always

P xð Þ <sup>↦</sup> <sup>P</sup><sup>R</sup>ð Þ <sup>x</sup> <sup>≔</sup> xdPð Þ <sup>1</sup>=<sup>x</sup> <sup>=</sup>Pð Þ <sup>0</sup> : (2)

Pð Þ �x . All quadratic factors in the table below have

discuss two generalizations of Problem 1 and their partial solutions.

2. Solution of the realization problem 1 in small degrees

2.1 Natural Z<sup>2</sup> � Z2-action and degrees d ¼ 1, 2, and 3

Let us start with the following useful observation.

following Z<sup>2</sup> � Z2-action whose first generator acts by.

and the second one acts by

last components equal �.

simultaneously (non-)realizable.

respective polynomials ð Þ �<sup>1</sup> <sup>d</sup>

no real roots.

35

cartes himself.

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

It is true that for a given SP with c sign changes (and hence with p ¼ d � c sign preservations), there always exist polynomials P defining this sign pattern and having exactly pos positive roots, where pos ¼ 0, 2, …, c if c is even and pos ¼ 1, 3, …, c if c is odd (see, e.g., [1, 3]). (Observe that we do not impose any restriction on the number of negative roots of these polynomials.)

One can apply Descartes' rule of signs to the polynomial ð Þ �<sup>1</sup> <sup>d</sup> Pð Þ �x which has p sign changes and c sign preservations in the sequence of its coefficients and whose leading coefficient is positive. The roots of ð Þ �<sup>1</sup> <sup>d</sup> Pð Þ �x are obtained from the roots of P xð Þ by changing their sign. Applying the above result of [1] to ð Þ �<sup>1</sup> <sup>d</sup> Pð Þ �x , one obtains the existence of polynomials P with exactly neg negative roots, where neg ¼ 0, 2, …, p if p is even and neg ¼ 1, 3, …, p if p is odd. (Here again we impose no requirement on the number of positive roots.)

A natural question apparently for the first time raised in [12] is whether one can freely combine these two results about the numbers of positive and negative roots. Namely, given a SP σ with c sign changes and p ¼ d � c sign preservations, we define its admissible pair (AP for short) as ð Þ pos; neg , where pos ≤ c, neg ≤ p, and the differences c � pos and p � neg are even. For the SP σ as above, we call ð Þ c; p the Descartes' pair of σ. The main question under consideration in this paper is as follows.

Problem 1. Given a couple (SP, AP), does there exist a polynomial of degree d with this SP and having exactly pos positive and exactly neg negative roots (and hence exactly ð Þ d � pos � neg =2 complex conjugate pairs)?

If such a polynomial exists, then we say that it realizes a given couple (SP, AP). The present paper discusses the current status of knowledge in this realization problem.

Example 1. For <sup>d</sup> <sup>¼</sup> 4 and for the sign pattern <sup>σ</sup><sup>0</sup> <sup>≔</sup> ð Þ <sup>þ</sup>; �; �; �; <sup>þ</sup> , the following pairs and only them are admissible: 2ð Þ ; 2 , 2ð Þ ; 0 , 0ð Þ ; 2 , and 0ð Þ ; 0 . The first of them is the Descartes' pair of σ0.

It is clear that if a couple (SP, AP) is realizable, then it can be realized by a polynomial with all simple roots, because the property of having nonvanishing coefficients is preserved under small perturbations of the roots.

In this short survey, we present what is currently known about Problem 1. After the pioneering observations of Grabiner [12] which started this line of research, important contributions to Problem 1 have been made by Albouy and Fu [1] who, in particular, described all non-realizable combinations of the numbers of positive and negative roots and respective sign patterns up to degree 6. Our results on this topic which we summarize below can be found in [5, 8, 9] and [15–19]. On the other hand, we find it surprising that such a natural classical question has not deserved any attention in the past, and we hope that this survey will help to change the situation. The current status of Problem 1 is not very satisfactory in spite of the complete results in degrees up to 8 as well as several series of non-realizable cases in all degrees. There is still no general conjecture describing all non-realizable cases. It might happen that the answer to Problem 1 in sufficiently high degrees is very complicated.

On the other hand, besides Problem 1 as it is stated, there is a significant number of related basic questions which can be posed in connection to the latter Problem and are still waiting for their researchers. (Very few of them are listed in Section 5.) New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

length d þ 1 which we call the sign pattern (SP for short) of P, namely, we say that a polynomial P with all nonvanishing coefficients defines the sign pattern σ ≔ ðsd, sd�1, …, s0Þ if sj ¼ sgn aj. Since the roots of the polynomials P and �P are the same, we can, without loss of generality, assume that the first sign of a SP is always a þ. It is true that for a given SP with c sign changes (and hence with p ¼ d � c sign preservations), there always exist polynomials P defining this sign pattern and

having exactly pos positive roots, where pos ¼ 0, 2, …, c if c is even and

restriction on the number of negative roots of these polynomials.) One can apply Descartes' rule of signs to the polynomial ð Þ �<sup>1</sup> <sup>d</sup>

whose leading coefficient is positive. The roots of ð Þ �<sup>1</sup> <sup>d</sup>

Polynomials - Theory and Application

we impose no requirement on the number of positive roots.)

ð Þ �<sup>1</sup> <sup>d</sup>

paper is as follows.

is the Descartes' pair of σ0.

problem.

34

ð Þ d � pos � neg =2 complex conjugate pairs)?

pos ¼ 1, 3, …, c if c is odd (see, e.g., [1, 3]). (Observe that we do not impose any

p sign changes and c sign preservations in the sequence of its coefficients and

the roots of P xð Þ by changing their sign. Applying the above result of [1] to

Pð Þ �x , one obtains the existence of polynomials P with exactly neg negative roots, where neg ¼ 0, 2, …, p if p is even and neg ¼ 1, 3, …, p if p is odd. (Here again

A natural question apparently for the first time raised in [12] is whether one can freely combine these two results about the numbers of positive and negative roots. Namely, given a SP σ with c sign changes and p ¼ d � c sign preservations, we define its admissible pair (AP for short) as ð Þ pos; neg , where pos ≤ c, neg ≤ p, and the differences c � pos and p � neg are even. For the SP σ as above, we call ð Þ c; p the Descartes' pair of σ. The main question under consideration in this

Problem 1. Given a couple (SP, AP), does there exist a polynomial of degree d with this SP and having exactly pos positive and exactly neg negative roots (and hence exactly

If such a polynomial exists, then we say that it realizes a given couple (SP, AP). The present paper discusses the current status of knowledge in this realization

Example 1. For <sup>d</sup> <sup>¼</sup> 4 and for the sign pattern <sup>σ</sup><sup>0</sup> <sup>≔</sup> ð Þ <sup>þ</sup>; �; �; �; <sup>þ</sup> , the following pairs and only them are admissible: 2ð Þ ; 2 , 2ð Þ ; 0 , 0ð Þ ; 2 , and 0ð Þ ; 0 . The first of them

In this short survey, we present what is currently known about Problem 1. After the pioneering observations of Grabiner [12] which started this line of research, important contributions to Problem 1 have been made by Albouy and Fu [1] who, in particular, described all non-realizable combinations of the numbers of positive and negative roots and respective sign patterns up to degree 6. Our results on this topic which we summarize below can be found in [5, 8, 9] and [15–19]. On the other hand, we find it surprising that such a natural classical question has not deserved any attention in the past, and we hope that this survey will help to change the situation. The current status of Problem 1 is not very satisfactory in spite of the complete results in degrees up to 8 as well as several series of non-realizable cases in all degrees. There is still no general conjecture describing all non-realizable cases. It might happen that the answer to Problem 1 in sufficiently high degrees is very complicated. On the other hand, besides Problem 1 as it is stated, there is a significant number of related basic questions which can be posed in connection to the latter Problem and are still waiting for their researchers. (Very few of them are listed in Section 5.)

It is clear that if a couple (SP, AP) is realizable, then it can be realized by a polynomial with all simple roots, because the property of having nonvanishing

coefficients is preserved under small perturbations of the roots.

Pð Þ �x which has

Pð Þ �x are obtained from

One should also add that there is a number of completely different directions in which mathematicians are trying to extend Descartes' rule of signs. They include, for example, rule of signs for other univariate analytic functions including exponential functions, trigonometric functions and orthogonal polynomials, multivariate Descartes' rule of signs, tropical rule of signs, rule of signs in the complex domain, etc. (see, e.g., [6, 10, 14]) and references therein. But we think that Problem 1 is the closest one to the original investigations by Descartes himself.

The structure of this chapter is as follows. In Section 2, we provide the information about the solution of Problem 1 in degrees up to 11. In Section 3, we present several infinite series of non-realizable couples (SP, AP). Finally, in Section 4 we discuss two generalizations of Problem 1 and their partial solutions.

#### 2. Solution of the realization problem 1 in small degrees

### 2.1 Natural Z<sup>2</sup> � Z2-action and degrees d ¼ 1, 2, and 3

Let us start with the following useful observation.

To shorten the list of cases (SP, AP) under consideration, we can use the following Z<sup>2</sup> � Z2-action whose first generator acts by.

$$P(\mathfrak{x}) \mapsto (-1)^d P(-\mathfrak{x}),\tag{1}$$

and the second one acts by

$$P(\mathfrak{x}) \mapsto P^{\mathbb{R}}(\mathfrak{x}) \, := \, \, \varkappa^d P(\mathfrak{1}/\mathfrak{x})/P(\mathfrak{0}). \tag{2}$$

Obviously, the first generator exchanges the components of the AP. Concerning the second generator, to obtain the SP defined by the polynomial P<sup>R</sup>, one has to read the SP defined by P xð Þ backward. The roots of <sup>P</sup><sup>R</sup> are the reciprocals of these of <sup>P</sup> which implies that both polynomials have the same numbers of positive and negative roots. Therefore, the SPs which they define have the same AP.

Remark 1. A priori the length of an orbit of any Z<sup>2</sup> � Z2-action could be 1, 2, or 4, but for the above action, orbits of length 1 do not exist since the second components of the SPs defined by the polynomials P xð Þ and ð Þ �<sup>1</sup> <sup>d</sup> Pð Þ �x are always different. When an orbit of length 2 occurs and d is even, then both SPs are symmetric w.r.t. their middle points (hence their last component equal þ). Similarly, when d is odd, then one of the two SPs is symmetric w.r.t. its middle (with the last component equal to þ), and the other one is antisymmetric. Thus, its last components equal �.

It is obvious that all pairs or quadruples (SP, AP) constituting a given orbit are simultaneously (non-)realizable.

As a warm-up exercise, let us consider degrees d ¼ 1, 2 and 3. In these cases, the answer to Problem 1 is positive. We give the list of SPs, with the respective values c and p of their APs and examples of polynomials realizing the couples (SP, AP). In order to shorten the list, we consider only SPs beginning with two þ signs; the cases when these signs are ð Þ þ; � are realized by the respective polynomials ð Þ �<sup>1</sup> <sup>d</sup> Pð Þ �x . All quadratic factors in the table below have no real roots.

d SP cp AP P 1 ð Þ þ; þ 01 0ð Þ ; 1 x þ 1 <sup>2</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup> 02 0ð Þ ; <sup>2</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> <sup>þ</sup> <sup>2</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>0</sup>; <sup>0</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; � 11 1ð Þ ; <sup>1</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> � <sup>2</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup> <sup>3</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup> 03 0ð Þ ; <sup>3</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>6</sup>x<sup>2</sup> <sup>þ</sup> <sup>11</sup><sup>x</sup> <sup>þ</sup> <sup>6</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>3</sup> ð Þ <sup>0</sup>; <sup>1</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>x</sup> <sup>þ</sup> <sup>2</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> <sup>x</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>x</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup>; � 12 1ð Þ ; <sup>2</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>x<sup>2</sup> <sup>þ</sup> <sup>x</sup> � <sup>6</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>3</sup> ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>5</sup>x<sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>x</sup> � <sup>10</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>x</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>x</sup> <sup>þ</sup> <sup>10</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; �; <sup>þ</sup> 21 2ð Þ ; <sup>1</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> � <sup>24</sup><sup>x</sup> <sup>þ</sup> <sup>36</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>6</sup> ð Þ <sup>x</sup> � <sup>2</sup> ð Þ <sup>x</sup> � <sup>3</sup> ð Þ <sup>0</sup>; <sup>1</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>x<sup>2</sup> � <sup>19</sup><sup>x</sup> <sup>þ</sup> <sup>30</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>6</sup> <sup>x</sup>ð Þ <sup>2</sup> � <sup>4</sup><sup>x</sup> <sup>þ</sup> <sup>5</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; �; � 12 1ð Þ ; <sup>2</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> � <sup>4</sup><sup>x</sup> � <sup>4</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>2</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>x<sup>2</sup> � <sup>3</sup><sup>x</sup> � <sup>10</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>2</sup> <sup>x</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>x</sup> <sup>þ</sup> <sup>5</sup>

realizes the second of these couples and has two positive roots α , β and no negative roots, then for any <sup>u</sup> <sup>∈</sup> ð Þ <sup>α</sup>; <sup>β</sup> , the values of the monomials <sup>x</sup>4, <sup>a</sup>2x2, and <sup>a</sup><sup>0</sup> are the same at <sup>u</sup> and �u, while the monomials <sup>a</sup>3x<sup>3</sup> and <sup>a</sup>1<sup>x</sup> are positive at <sup>u</sup> and negative at �u. Hence, Pð Þ �u , P uð Þ , 0. As Pð Þ 0 . 0 and limx!�<sup>∞</sup>P xð Þ¼þ∞, the

For d ¼ 4, realizability of all other couples (SP, AP) can be proven by producing

Remark 2. In [19] a geometric illustration of the non-realizability of the two cases mentioned in Theorem 1 is proposed. Namely, one considers the family of

<sup>Δ</sup> <sup>≔</sup> ð Þ <sup>a</sup>; <sup>b</sup>;<sup>c</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup> <sup>j</sup>Res <sup>Q</sup>; <sup>Q</sup><sup>0</sup> ð Þ¼ <sup>0</sup> ,

face <sup>Δ</sup> <sup>¼</sup> 0 partitions <sup>R</sup><sup>3</sup> into three open domains, in which the polynomial <sup>Q</sup> has 0, 1, or 2 complex conjugate pairs of roots, respectively. These domains intersect the 8 open orthants of <sup>R</sup><sup>3</sup> defined by the coordinate system ð Þ <sup>a</sup>; <sup>b</sup>;<sup>c</sup> , and in each of these intersections, the polynomial Q has one and the same number of positive, negative, and complex roots, as well as the same signs of its coefficients. The nonrealizability of the couple ð Þ ð Þ þ; þ; �; þ; þ ;ð Þ 2; 0 can be interpreted as the fact that the corresponding intersection is empty. Pictures of discriminant sets allow to construct easily the numerical examples mentioned in the proof of Theorem 1. It remains to be noticed that for α . 0 and β . 0, the polynomials P xð Þ and βPð Þ αx have one and the same numbers of positive, negative, and complex roots. Therefore, it suffices to consider the family of polynomials Q in order to cover all SPs beginning with ð Þ þ; þ . The ones beginning with ð Þ þ; � will be covered by the

. The hypersur-

polynomial P has two negative roots as well—a contradiction.

polynomials <sup>Q</sup> <sup>≔</sup> <sup>x</sup><sup>4</sup> <sup>þ</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> ax<sup>2</sup> <sup>þ</sup> bx <sup>þ</sup> <sup>c</sup> and the discriminant set

where Res Q; Q<sup>0</sup> ð Þ is the resultant of the polynomials Q and Q<sup>0</sup>

For degrees d ¼ 5 and 6, the following result can be found in [1].

each of the last two cases defines an orbit of length 4. For d ¼ 7, the following theorem is contained in [8].

Theorem 2. (1) The only two couples (SP, AP) which are non-realizable by uni-

ð Þ ð Þ þ; �; �; �; �; þ ;ð Þ 0; 3 and ð Þ ð Þ þ; þ; �; þ; �; � ;ð Þ 3; 0 :

ð Þ ð Þ þ; �; �; �; �; �; þ ;ð Þ 0; 2 ; ð Þ ð Þ þ; �; �; �; �; �; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; þ; �; �; �; þ ;ð Þ 0; 2 ; ð Þ ð Þ þ; þ; �; �; �; �; þ ;ð Þ 0; 4 :

The two cases of Part (1) of Theorem 2 also form an orbit of the Z<sup>2</sup> � Z2-action of length 2. Each of the first two cases of Part (2) defines an orbit of length 2, while

Theorem 3. For univariate polynomials of degree 7, among their 1472 possible couples (SP, AP) (up to the Z<sup>2</sup> � Z2-action), exactly the following 6 are non-realizable:

ð Þ ð Þ þ; þ; �; �; �; �; �; þ ;ð Þ 0; 5 ; ð Þ ð Þ þ; þ; �; �; �; �; þ; þ ;ð Þ 0; 5 ; ð Þ ð Þ þ; �; �; �; �; þ; �; þ ;ð Þ 0; 3 ; ð Þ ð Þ þ; þ; þ; �; �; �; �; þ ;ð Þ 0; 5 ; ð Þ ð Þ þ; �; �; �; �; �; �; þ ;ð Þ 0; 3 ; ð Þ ð Þ þ; �; �; �; �; �; �; þ ;ð Þ 0; 5 : The lengths of the respective orbits in these 6 cases are 4, 2, 4, 4, 2, and 2. The case d ¼ 8 has been partially solved in [8] and completely in [16]:

(2) For degree d ¼ 6, up to the above Z<sup>2</sup> � Z2-action, the only non-realizable couples

explicit examples.

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

family Qð Þ �x .

(SP, AP) are:

37

variate polynomials of degree 5 are:

Example 2. For d ¼ 4, an example of an orbit of length 2 is given by the couples

ð Þ ð Þ þ; �; �; �; þ ;ð Þ 2; 2 and ð Þ ð Þ þ; þ; �; þ; þ ;ð Þ 2; 2 :

Here, both SPs are symmetric w.r.t. its middle. For d ¼ 5, such an example is given by the couples

$$((+,-,-,-,-,+),(2,3)) \quad \text{and} \quad ((+,+,-,+,-,-),(3,2)).$$

The first of the SPs is symmetric, and the second one is antisymmetric w.r.t. their middles.

Finally, for d ¼ 3, the following four couples (SP, AP)

$$\begin{array}{ll} ((+,+,+,-),(\mathbf{1},\mathbf{2})); & ((+,-,+,+),(\mathbf{2},\mathbf{1}));\\ ((+,-,-,-),(\mathbf{1},\mathbf{2})); & ((+,+,-,+),(\mathbf{2},\mathbf{1})).\end{array}$$

constitute one orbit for d ¼ 3. In this example all admissible pairs are Descartes' pairs.

#### 2.2 Degrees d ≥ 4

It turns out that for d ≥ 4, it is no longer true that all couples (SP, AP) are realizable by polynomials of degree d. Namely, the following result can be found in [12]:

Theorem 1. The only couples (SP, AP) which are non-realizable by univariate polynomials of degree 4 are

ð Þ ð Þ þ; �; �; �; þ ;ð Þ 0; 2 and ð Þ ð Þ þ; þ; �; þ; þ ;ð Þ 2; 0 :

It is clear that these two cases constitute one orbit of the Z<sup>2</sup> � Z2-action of length 2 (the SPs are the same when read the usual way and backward).

Proof. The argument showing non-realizability in Theorem 1 is easy. Namely, if a polynomial

$$P \quad := \ \varkappa^4 + \mathfrak{a}\_3 \varkappa^3 + \mathfrak{a}\_2 \varkappa^2 + \mathfrak{a}\_1 \mathfrak{x} + \mathfrak{a}\_0 \mathfrak{x}$$

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

d SP cp AP P 1 ð Þ þ; þ 01 0ð Þ ; 1 x þ 1

Polynomials - Theory and Application

<sup>2</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup> 02 0ð Þ ; <sup>2</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> <sup>þ</sup> <sup>2</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup>

ð Þ <sup>0</sup>; <sup>0</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; � 11 1ð Þ ; <sup>1</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> � <sup>2</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup>

<sup>3</sup> ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup> 03 0ð Þ ; <sup>3</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>6</sup>x<sup>2</sup> <sup>þ</sup> <sup>11</sup><sup>x</sup> <sup>þ</sup> <sup>6</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>3</sup>

ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup>; � 12 1ð Þ ; <sup>2</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>x<sup>2</sup> <sup>þ</sup> <sup>x</sup> � <sup>6</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>3</sup>

ð Þ <sup>þ</sup>; <sup>þ</sup>; �; <sup>þ</sup> 21 2ð Þ ; <sup>1</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> � <sup>24</sup><sup>x</sup> <sup>þ</sup> <sup>36</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>6</sup> ð Þ <sup>x</sup> � <sup>2</sup> ð Þ <sup>x</sup> � <sup>3</sup>

ð Þ <sup>þ</sup>; <sup>þ</sup>; �; � 12 1ð Þ ; <sup>2</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> � <sup>4</sup><sup>x</sup> � <sup>4</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>2</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>2</sup>

Example 2. For d ¼ 4, an example of an orbit of length 2 is given by the couples

ð Þ ð Þ þ; �; �; �; þ ;ð Þ 2; 2 and ð Þ ð Þ þ; þ; �; þ; þ ;ð Þ 2; 2 :

ð Þ ð Þ þ; �; �; �; �; þ ;ð Þ 2; 3 and ð Þ ð Þ þ; þ; �; þ; �; � ;ð Þ 3; 2 : The first of the SPs is symmetric, and the second one is antisymmetric w.r.t.

> ð Þ ð Þ þ; þ; þ; � ;ð Þ 1; 2 ; ð Þ ð Þ þ; �; þ; þ ;ð Þ 2; 1 ; ð Þ ð Þ þ; �; �; � ;ð Þ 1; 2 ; ð Þ ð Þ þ; þ; �; þ ;ð Þ 2; 1 :

It turns out that for d ≥ 4, it is no longer true that all couples (SP, AP) are realizable by polynomials of degree d. Namely, the following result can be found

Theorem 1. The only couples (SP, AP) which are non-realizable by univariate

ð Þ ð Þ þ; �; �; �; þ ;ð Þ 0; 2 and ð Þ ð Þ þ; þ; �; þ; þ ;ð Þ 2; 0 :

Proof. The argument showing non-realizability in Theorem 1 is easy. Namely, if a

<sup>P</sup> <sup>≔</sup> <sup>x</sup><sup>4</sup> <sup>þ</sup> <sup>a</sup>3x<sup>3</sup> <sup>þ</sup> <sup>a</sup>2x<sup>2</sup> <sup>þ</sup> <sup>a</sup>1<sup>x</sup> <sup>þ</sup> <sup>a</sup><sup>0</sup>

It is clear that these two cases constitute one orbit of the Z<sup>2</sup> � Z2-action of

length 2 (the SPs are the same when read the usual way and backward).

constitute one orbit for d ¼ 3. In this example all admissible pairs are Descartes'

Here, both SPs are symmetric w.r.t. its middle. For d ¼ 5, such an example is given by the couples

Finally, for d ¼ 3, the following four couples (SP, AP)

their middles.

2.2 Degrees d ≥ 4

polynomials of degree 4 are

pairs.

in [12]:

polynomial

36

ð Þ <sup>0</sup>; <sup>1</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>x</sup> <sup>þ</sup> <sup>2</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> <sup>x</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>x</sup> <sup>þ</sup> <sup>2</sup>

ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>5</sup>x<sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>x</sup> � <sup>10</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>x</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>x</sup> <sup>þ</sup> <sup>10</sup>

ð Þ <sup>0</sup>; <sup>1</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>x<sup>2</sup> � <sup>19</sup><sup>x</sup> <sup>þ</sup> <sup>30</sup> <sup>¼</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>6</sup> <sup>x</sup>ð Þ <sup>2</sup> � <sup>4</sup><sup>x</sup> <sup>þ</sup> <sup>5</sup>

ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>x<sup>2</sup> � <sup>3</sup><sup>x</sup> � <sup>10</sup> <sup>¼</sup> ð Þ <sup>x</sup> � <sup>2</sup> <sup>x</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>x</sup> <sup>þ</sup> <sup>5</sup>

realizes the second of these couples and has two positive roots α , β and no negative roots, then for any <sup>u</sup> <sup>∈</sup> ð Þ <sup>α</sup>; <sup>β</sup> , the values of the monomials <sup>x</sup>4, <sup>a</sup>2x2, and <sup>a</sup><sup>0</sup> are the same at <sup>u</sup> and �u, while the monomials <sup>a</sup>3x<sup>3</sup> and <sup>a</sup>1<sup>x</sup> are positive at <sup>u</sup> and negative at �u. Hence, Pð Þ �u , P uð Þ , 0. As Pð Þ 0 . 0 and limx!�<sup>∞</sup>P xð Þ¼þ∞, the polynomial P has two negative roots as well—a contradiction.

For d ¼ 4, realizability of all other couples (SP, AP) can be proven by producing explicit examples.

Remark 2. In [19] a geometric illustration of the non-realizability of the two cases mentioned in Theorem 1 is proposed. Namely, one considers the family of polynomials <sup>Q</sup> <sup>≔</sup> <sup>x</sup><sup>4</sup> <sup>þ</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> ax<sup>2</sup> <sup>þ</sup> bx <sup>þ</sup> <sup>c</sup> and the discriminant set

$$\Delta \coloneqq \left\{ (a,b,c) \in \mathbb{R}^3 \, | \, \text{Res}(Q, Q') = \mathbf{0} \right\},$$

where Res Q; Q<sup>0</sup> ð Þ is the resultant of the polynomials Q and Q<sup>0</sup> . The hypersurface <sup>Δ</sup> <sup>¼</sup> 0 partitions <sup>R</sup><sup>3</sup> into three open domains, in which the polynomial <sup>Q</sup> has 0, 1, or 2 complex conjugate pairs of roots, respectively. These domains intersect the 8 open orthants of <sup>R</sup><sup>3</sup> defined by the coordinate system ð Þ <sup>a</sup>; <sup>b</sup>;<sup>c</sup> , and in each of these intersections, the polynomial Q has one and the same number of positive, negative, and complex roots, as well as the same signs of its coefficients. The nonrealizability of the couple ð Þ ð Þ þ; þ; �; þ; þ ;ð Þ 2; 0 can be interpreted as the fact that the corresponding intersection is empty. Pictures of discriminant sets allow to construct easily the numerical examples mentioned in the proof of Theorem 1.

It remains to be noticed that for α . 0 and β . 0, the polynomials P xð Þ and βPð Þ αx have one and the same numbers of positive, negative, and complex roots. Therefore, it suffices to consider the family of polynomials Q in order to cover all SPs beginning with ð Þ þ; þ . The ones beginning with ð Þ þ; � will be covered by the family Qð Þ �x .

For degrees d ¼ 5 and 6, the following result can be found in [1].

Theorem 2. (1) The only two couples (SP, AP) which are non-realizable by univariate polynomials of degree 5 are:

ð Þ ð Þ þ; �; �; �; �; þ ;ð Þ 0; 3 and ð Þ ð Þ þ; þ; �; þ; �; � ;ð Þ 3; 0 :

(2) For degree d ¼ 6, up to the above Z<sup>2</sup> � Z2-action, the only non-realizable couples (SP, AP) are:

$$\begin{array}{ll} ((+,-,-,-,-,-,+),(\mathbf{0},\mathbf{2})); & ((+,-,-,-,-,-,+),(\mathbf{0},\mathbf{4}));\\ ((+,-,+,-,-,-,+),(\mathbf{0},\mathbf{2})); & ((+,+,-,-,-,-,+),(\mathbf{0},\mathbf{4})).\end{array}$$

The two cases of Part (1) of Theorem 2 also form an orbit of the Z<sup>2</sup> � Z2-action of length 2. Each of the first two cases of Part (2) defines an orbit of length 2, while each of the last two cases defines an orbit of length 4.

For d ¼ 7, the following theorem is contained in [8].

Theorem 3. For univariate polynomials of degree 7, among their 1472 possible couples (SP, AP) (up to the Z<sup>2</sup> � Z2-action), exactly the following 6 are non-realizable:

$$\begin{array}{llll} \text{((+,+,-,-,-,-,-,+),(\mathbf{0},\mathbf{5}));} & ((+,+,-,-,-,-,-,+,+),(\mathbf{0},\mathbf{5}));\\ \text{((+,-,-,-,-,-,+,-,+),(\mathbf{0},\mathbf{3}));} & ((+,+,+,-,-,-,-,-,+),(\mathbf{0},\mathbf{5}));} \\ \text{((+,-,-,-,-,-,-,-,+),(\mathbf{0},\mathbf{3}));} & ((+,-,-,-,-,-,-,-,+),(\mathbf{0},\mathbf{5})).\end{array}$$

The lengths of the respective orbits in these 6 cases are 4, 2, 4, 4, 2, and 2. The case d ¼ 8 has been partially solved in [8] and completely in [16]:

Theorem 4. For degree d ¼ 8, among the 3648 possible couples (SP, AP) (up to the Z<sup>2</sup> � Z2-action), exactly the following 19 are non-realizable:

3.1 Some examples of realizability and a concatenation lemma

a þ (resp. is a �) is realizable with the AP ð Þ 0; 1 (resp. ð Þ 1; 0 ).

Proposition 2. Any SP is realizable with its Descartes' pair.

special cases cannot be concluded using this lemma.

When τ ¼ þ, then one has, respectively,

When τ ¼ �, then one has, respectively,

and the SP of <sup>ε</sup><sup>d</sup>2P1ð Þ <sup>x</sup> <sup>P</sup>2ð Þ <sup>x</sup>=<sup>ε</sup> equals

and the SP of <sup>ε</sup><sup>d</sup>2P1ð Þ <sup>x</sup> <sup>P</sup>2ð Þ <sup>x</sup>=<sup>ε</sup> equals

sible number of real roots:

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

in [18].

can be found in [8].

pos2; neg<sup>2</sup>

39

is deleted. Then:

Our first examples of realizability deal with polynomials with the minimal pos-

Proof. Indeed, for any given SP, it suffices to choose any polynomial defining this SP and to increase (resp. decrease) its constant term sufficiently much if the latter is positive (resp. negative). The resulting polynomial will have the required number of real roots. □ Our next example deals with hyperbolic polynomials, that is, real polynomials with all real roots. Several topics concerning hyperbolic polynomials are developed

Proposition 2 will follow from the following concatenation lemma whose proof

Lemma 1. Suppose that monic polynomials P<sup>1</sup> and P2, of degrees d<sup>1</sup> and d<sup>2</sup> resp., realize the SPs ð Þ þ; σ^<sup>1</sup> and ð Þ þ; σ^<sup>2</sup> , where σ^<sup>j</sup> are the SPs defined by Pj in which the first þ

1. If the last position of σ^<sup>1</sup> is a þ, then for any ε . 0 small enough, the polynomial <sup>ε</sup><sup>d</sup>2P1ð Þ <sup>x</sup> <sup>P</sup>2ð Þ <sup>x</sup>=<sup>ε</sup> realizes the SP ð Þ <sup>þ</sup>; <sup>σ</sup>^1; <sup>σ</sup>^<sup>2</sup> and the AP pos<sup>1</sup> <sup>þ</sup> pos2; neg<sup>1</sup> <sup>þ</sup> neg<sup>2</sup>

2.If the last position of σ^<sup>1</sup> is a �, then for any ε . 0 small enough, the polynomial <sup>ε</sup><sup>d</sup>2P1ð Þ <sup>x</sup> <sup>P</sup>2ð Þ <sup>x</sup>=<sup>ε</sup> realizes the SP ð Þ <sup>þ</sup>; <sup>σ</sup>^1; �σ^<sup>2</sup> and the AP pos<sup>1</sup> <sup>þ</sup> pos2; neg<sup>1</sup> <sup>þ</sup> neg<sup>2</sup>

(Here �σ^<sup>2</sup> is the SP obtained from σ^<sup>2</sup> by changing each þ by a � and vice versa.)

The concatenation lemma allows to deduce the realizability of couples (SP, AP) with higher values of d from that of couples with smaller d in which cases explicit constructions are usually easier to obtain. On the other hand, non-realizability of

<sup>P</sup>2ðÞ ¼ <sup>x</sup> <sup>x</sup> � <sup>1</sup>, x <sup>þ</sup> <sup>1</sup>, x<sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>x</sup> <sup>þ</sup> <sup>2</sup>, x<sup>2</sup> � <sup>2</sup><sup>x</sup> <sup>þ</sup> 2 with

<sup>¼</sup> ð Þ <sup>1</sup>; <sup>0</sup> , ð Þ <sup>0</sup>; <sup>1</sup> , ð Þ <sup>0</sup>; <sup>0</sup> , ð Þ <sup>0</sup>; <sup>0</sup> resp:

σ^<sup>2</sup> ¼ �ð Þ,ð Þ þ ,ð Þ þ; þ ,ð Þ �; þ ,

ð Þ þ; σ^1; � , ð Þ þ; σ^1; þ , ð Þ þ; σ^1; þ; þ , ð Þ þ; σ^1; �; þ :

σ^<sup>2</sup> ¼ þð Þ, ð Þ � , ð Þ �; � , ð Þ þ; � ,

ð Þ þ; σ^1; þ , ð Þ þ; σ^1; � , ð Þ þ; σ^1; �; � , ð Þ þ; σ^1; þ; � :

Example 3. Denote by τ the last entry of the SP σ^1. We consider the cases

.

:

Proposition 1. For d even, any SP whose last component is a þ (resp. is a �) is realizable with the AP ð Þ 0; 0 (resp. ð Þ 1; 1 ). For d odd, any SP whose last component is

ð Þ ð Þ þ; þ; �; �; �; �; �; þ; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; þ; �; �; �; �; �; �; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; þ; þ; �; �; �; �; �; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; þ; þ; þ; �; �; �; �; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; �; þ; �; �; �; þ; �; þ ;ð Þ 0; 2 ; ð Þ ð Þ þ; �; þ; �; þ; �; �; �; þ ;ð Þ 0; 2 ; ð Þ ð Þ þ; �; þ; �; �; �; �; �; þ ;ð Þ 0; 2 ; ð Þ ð Þ þ; �; þ; �; �; �; �; �; þ ;ð Þ 0; 4 ; ð þð Þ ; �; �; �; þ; �; �; �; þ ,ð Þ 0; 2 ; ð Þ ð Þ þ; �; �; �; þ; �; �; �; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; �; �; �; �; �; �; þ ;ð Þ 0; 2 ; ð þð Þ ; �; �; �; �; �; �; �; þ ,ð Þ 0; 4 ; ð Þ ð Þ þ; �; �; �; �; �; �; �; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; þ; þ; �; �; �; �; þ; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; �; �; �; �; þ; �; �; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; �; �; �; �; �; þ; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; þ; þ; �; �; �; �; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; þ; �; �; �; �; þ; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; �; �; �; þ; �; þ; þ ;ð Þ 0; 4 :

The lengths of the respective orbits are 2, 4, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, and 4.

Remark 3. As we see above, for d ¼ 4, 5, 6, 7, and 8, up to the Z<sup>2</sup> � Z2-action, the numbers of non-realizable cases are 1, 1, 4, 6, and 19, respectively. The fact that these numbers increase more when d ¼ 5 and d ¼ 7 than when d ¼ 4 and d ¼ 6 could be related to the fact that the maximal possible number of complex conjugate pairs of roots of a real univariate degree d polynomial is ½ � d=2 . This number increases w.r.t. ½ � ð Þ d � 1 =2 when d is even and does not increase when d is odd.

Observe that for d ≤ 8, all examples of couples (SP, AP) which are nonrealizable are with APs of the form ð Þ ν; 0 or 0ð Þ ; ν and ν∈ N. Initially, we thought that this is always the case. However, recently it was proven that, for higher degrees, this fact is no longer true (see [17]):

Theorem 5. For d ¼ 11, the following couple (SP, AP)

$$\{ (+, -, -, -, -, -, +, +, +, +, +, +, -), (1, 8) \}$$

is non-realizable. The Descartes' pair in this case equals ð Þ 3; 8 . There is a strong evidence that for d ¼ 9, the similar couple (SP, AP)

ð Þ ð Þ þ; �; �; �; �; þ; þ; þ; þ; � ;ð Þ 1; 6

is also non-realizable. (Its Descartes' pair equals 3ð Þ ; 6 .) If this were true, then 9 would be the smallest degree with an example of a non-realizable couple (SP, AP) for which both components of the AP are nonzero. When studying the cases d ¼ 8 and d ¼ 11 (see [16] and [17]), discriminant sets have been considered (see Remark 2).

Summarizing the above, we have to admit that the information in low degrees available at the moment does not allow us to formulate a consistent conjecture describing all non-realizable couples in an arbitrary degree which we could consider as sufficiently well motivated.

#### 3. Series of examples of (non-)realizable couples (SP, AP)

In this section we present a series of couples (non-)realizable for infinitely many degrees. We decided to include those proofs of the statements formulated below which are short and instructive.

Theorem 4. For degree d ¼ 8, among the 3648 possible couples (SP, AP) (up to the

ð Þ ð Þ þ; þ; �; �; �; �; �; þ; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; þ; �; �; �; �; �; �; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; þ; þ; �; �; �; �; �; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; þ; þ; þ; �; �; �; �; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; �; þ; �; �; �; þ; �; þ ;ð Þ 0; 2 ; ð Þ ð Þ þ; �; þ; �; þ; �; �; �; þ ;ð Þ 0; 2 ; ð Þ ð Þ þ; �; þ; �; �; �; �; �; þ ;ð Þ 0; 2 ; ð Þ ð Þ þ; �; þ; �; �; �; �; �; þ ;ð Þ 0; 4 ; ð þð Þ ; �; �; �; þ; �; �; �; þ ,ð Þ 0; 2 ; ð Þ ð Þ þ; �; �; �; þ; �; �; �; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; �; �; �; �; �; �; þ ;ð Þ 0; 2 ; ð þð Þ ; �; �; �; �; �; �; �; þ ,ð Þ 0; 4 ; ð Þ ð Þ þ; �; �; �; �; �; �; �; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; þ; þ; �; �; �; �; þ; þ ;ð Þ 0; 6 ; ð Þ ð Þ þ; �; �; �; �; þ; �; �; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; �; �; �; �; �; þ; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; þ; þ; �; �; �; �; þ ;ð Þ 0; 4 ; ð Þ ð Þ þ; �; þ; �; �; �; �; þ; þ ;ð Þ 0; 4 ;

The lengths of the respective orbits are 2, 4, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 2, 4, 4, 4, 4,

Remark 3. As we see above, for d ¼ 4, 5, 6, 7, and 8, up to the Z<sup>2</sup> � Z2-action, the numbers of non-realizable cases are 1, 1, 4, 6, and 19, respectively. The fact that these numbers increase more when d ¼ 5 and d ¼ 7 than when d ¼ 4 and d ¼ 6 could be related to the fact that the maximal possible number of complex conjugate

ð Þ ð Þ þ; �; �; �; �; �; þ; þ; þ; þ; þ; � ;ð Þ 1; 8

ð Þ ð Þ þ; �; �; �; �; þ; þ; þ; þ; � ;ð Þ 1; 6

is also non-realizable. (Its Descartes' pair equals 3ð Þ ; 6 .) If this were true, then 9 would be the smallest degree with an example of a non-realizable couple (SP, AP) for which both components of the AP are nonzero. When studying the cases d ¼ 8 and d ¼ 11 (see [16] and [17]), discriminant sets have been consid-

Summarizing the above, we have to admit that the information in low degrees available at the moment does not allow us to formulate a consistent conjecture describing all non-realizable couples in an arbitrary degree which we could consider

In this section we present a series of couples (non-)realizable for infinitely many degrees. We decided to include those proofs of the statements formulated below

There is a strong evidence that for d ¼ 9, the similar couple (SP, AP)

pairs of roots of a real univariate degree d polynomial is ½ � d=2 . This number increases w.r.t. ½ � ð Þ d � 1 =2 when d is even and does not increase when d is odd. Observe that for d ≤ 8, all examples of couples (SP, AP) which are nonrealizable are with APs of the form ð Þ ν; 0 or 0ð Þ ; ν and ν∈ N. Initially, we thought that this is always the case. However, recently it was proven that, for higher

Z<sup>2</sup> � Z2-action), exactly the following 19 are non-realizable:

Polynomials - Theory and Application

ð Þ ð Þ þ; �; �; �; �; þ; �; þ; þ ;ð Þ 0; 4 :

degrees, this fact is no longer true (see [17]):

ered (see Remark 2).

as sufficiently well motivated.

which are short and instructive.

38

Theorem 5. For d ¼ 11, the following couple (SP, AP)

is non-realizable. The Descartes' pair in this case equals ð Þ 3; 8 .

3. Series of examples of (non-)realizable couples (SP, AP)

4, and 4.

#### 3.1 Some examples of realizability and a concatenation lemma

Our first examples of realizability deal with polynomials with the minimal possible number of real roots:

Proposition 1. For d even, any SP whose last component is a þ (resp. is a �) is realizable with the AP ð Þ 0; 0 (resp. ð Þ 1; 1 ). For d odd, any SP whose last component is a þ (resp. is a �) is realizable with the AP ð Þ 0; 1 (resp. ð Þ 1; 0 ).

Proof. Indeed, for any given SP, it suffices to choose any polynomial defining this SP and to increase (resp. decrease) its constant term sufficiently much if the latter is positive (resp. negative). The resulting polynomial will have the required number of real roots. □

Our next example deals with hyperbolic polynomials, that is, real polynomials with all real roots. Several topics concerning hyperbolic polynomials are developed in [18].

Proposition 2. Any SP is realizable with its Descartes' pair.

Proposition 2 will follow from the following concatenation lemma whose proof can be found in [8].

Lemma 1. Suppose that monic polynomials P<sup>1</sup> and P2, of degrees d<sup>1</sup> and d<sup>2</sup> resp., realize the SPs ð Þ þ; σ^<sup>1</sup> and ð Þ þ; σ^<sup>2</sup> , where σ^<sup>j</sup> are the SPs defined by Pj in which the first þ is deleted. Then:


The concatenation lemma allows to deduce the realizability of couples (SP, AP) with higher values of d from that of couples with smaller d in which cases explicit constructions are usually easier to obtain. On the other hand, non-realizability of special cases cannot be concluded using this lemma.

Example 3. Denote by τ the last entry of the SP σ^1. We consider the cases

$$\begin{array}{ccccccccc} P\_2(\infty) & = & \infty - \mathbf{1}, & \mathbf{x} + \mathbf{1}, & \mathbf{x}^2 + 2\mathbf{x} + 2, & \mathbf{x}^2 - 2\mathbf{x} + 2 & \text{with} \\\ (pos\_2, neg\_2) & = & (\mathbf{1}, \mathbf{0}), & (\mathbf{0}, \mathbf{1}), & (\mathbf{0}, \mathbf{0}), & (\mathbf{0}, \mathbf{0}) & \text{resp.} \end{array}$$

When τ ¼ þ, then one has, respectively,

$$
\hat{\sigma}\_2 = (-), (+), (+, +), (-, +),
$$

and the SP of <sup>ε</sup><sup>d</sup>2P1ð Þ <sup>x</sup> <sup>P</sup>2ð Þ <sup>x</sup>=<sup>ε</sup> equals

ð Þ þ; σ^1; � , ð Þ þ; σ^1; þ , ð Þ þ; σ^1; þ; þ , ð Þ þ; σ^1; �; þ :

When τ ¼ �, then one has, respectively,

$$
\hat{\sigma}\_2 = \begin{array}{c} \text{(+)}, \quad \text{(-)}, \quad \text{(-)}, \quad \text{(+,-)}, \quad \text{(+,-)}, \quad \text{(-)}
\end{array}
$$

and the SP of <sup>ε</sup><sup>d</sup>2P1ð Þ <sup>x</sup> <sup>P</sup>2ð Þ <sup>x</sup>=<sup>ε</sup> equals

$$(+,\hat{\sigma}\_1,+), \quad (+,\hat{\sigma}\_1,-), \quad (+,\hat{\sigma}\_1,-,-), \quad (+,\hat{\sigma}\_1,+,-).$$

Proof of Proposition 2. We will use induction on the degree d of the polynomial. For d ¼ 1, the SP ð Þ þ; � (resp. ð Þ þ; þ ) is realizable with the AP 1ð Þ ; 0 (resp. 0ð Þ ; 1 ) by the polynomial x � 1 (resp. x þ 1).

Then, for any such SP, the APs ð Þ 2; 0 ,ð Þ 4; 0 , …,ð Þ 2ℓ; 0 , and only they, are non-

Suppose now that the degree d ≥ 5 is odd. For 1 ≤ k ≤ ð Þ d � 3 =2, denote by σ<sup>k</sup> the SP beginning with two pluses followed by k pairs ð Þ �; þ and then by d � 2k � 1 minuses. Its Descartes' pair of σ<sup>k</sup> equals 2ð Þ k þ 1; d � 2k � 1 . The following propo-

Theorem 6. (1) The SP σ<sup>k</sup> is not realizable with any of the pairs ð Þ 3; 0 ,ð Þ 5; 0 , …, ð Þ 2k þ 1; 0 ; (2) The SP σ<sup>k</sup> is realizable with the pair ð Þ 1; 0 ; (3) The SP σ<sup>k</sup> is realizable with any of the APs ð Þ 2ℓ þ 1; 2r , ℓ ¼ 0, 1, …, k, and r ¼ 1, 2, …, dð Þ � 2k � 1 =2.

One can observe that Cases (1), (2), and (3) exhaust all possible APs ð Þ pos; neg .

In this section, we consider realization problems similar or motivated by Problem 1. A priori it is hard to tell which of these or similar problems might have a

Consider a real polynomial P of degree d and its derivative. By Rolle's theorem, if

P has exactly r real roots (counted with multiplicity), then the derivative P<sup>0</sup> has r � 1 þ 2ℓ real roots (counted with multiplicity), where ℓ∈ N ∪ 0. It is possible that <sup>P</sup><sup>0</sup> has more real roots than <sup>P</sup>. For example, for <sup>d</sup> <sup>¼</sup> 2 and <sup>P</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> 1, one gets

¼ 2x which has a real root at 0, while P has no real roots at all. For d ¼ 3, the polynomial <sup>P</sup> <sup>¼</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup>x<sup>2</sup> � <sup>8</sup><sup>x</sup> <sup>þ</sup> <sup>10</sup> <sup>¼</sup> ð Þð <sup>x</sup> <sup>þ</sup> <sup>5</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>1</sup><sup>Þ</sup> has one negative root and one complex conjugate pair, while its derivative <sup>P</sup><sup>0</sup> <sup>¼</sup> <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>x</sup> � 8 has one

Now, for j ¼ 0, …, and d � 1, denote by rj and cj the numbers of real roots and complex conjugate pairs of roots of the polynomial Pð Þ<sup>j</sup> (both counted with multi-

Definition 1. A sequence ðð Þ r0; 2c<sup>0</sup> , rð Þ <sup>1</sup>; 2c<sup>1</sup> , …, rd�<sup>1</sup> ð ÞÞ ; 2cd�<sup>1</sup> satisfying conditions (4) will be called a D-sequence of length d. We say that a given D-sequence of length d is realizable if there exists a real polynomial P of degree d with this D-

Example 4. One has rd�<sup>1</sup> ¼ 1 and cd�<sup>1</sup> ¼ 0. Clearly, one has either rd�<sup>2</sup> ¼ 2, cd�<sup>2</sup> ¼ 0 or rd�<sup>2</sup> ¼ 0, cd�<sup>2</sup> ¼ 1. For small values of d, one has the following D-

> ð Þ ð Þ <sup>0</sup>; <sup>2</sup> ;ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>d</sup> <sup>¼</sup> 3 3 ð Þ ð Þ ; <sup>0</sup> ;ð Þ <sup>2</sup>; <sup>0</sup> ;ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>3</sup> � <sup>x</sup>

> > ð Þ ð Þ <sup>1</sup>; <sup>2</sup> ;ð Þ <sup>0</sup>; <sup>2</sup> ;ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>x</sup>

Problem 2. Is it true that for any d∈ N, any D-sequence is realizable?

The following question where a positive answer to which can be found in [15]

ð Þ ð Þ <sup>1</sup>; <sup>2</sup> ;ð Þ <sup>2</sup>; <sup>0</sup> ;ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>10</sup>x<sup>2</sup> <sup>þ</sup> <sup>26</sup>x:

sequence, where for <sup>j</sup> <sup>¼</sup> <sup>0</sup>, …, d � 1, all roots of <sup>P</sup>ð Þ<sup>j</sup> are distinct.

d ¼ 1 1ð Þ ; 0 x <sup>d</sup> <sup>¼</sup> 2 2 ð Þ ð Þ ; <sup>0</sup> ;ð Þ <sup>1</sup>; <sup>0</sup> <sup>x</sup><sup>2</sup> � <sup>1</sup>

sequences and respective polynomials realizing them:

rj ≤ rjþ<sup>1</sup> þ 1, rj þ 2cj ¼ d � j: (4)

realizable.

sition is proven in [19].

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

reasonable answer.

4.1 D-Sequences

P 0

4. Similar realization problems

positive and one negative root.

seems very natural.

41

plicity). These numbers satisfy the conditions

For d ¼ 2, we apply Lemma 1. Set P<sup>1</sup> ≔ x þ 1 and P<sup>2</sup> ≔ x � 1. Then, for ε . 0 small enough, the polynomials

$$\begin{array}{rclrclrclrcl\end{array}\begin{array}{rclrcl}\mbox{\(\varkappa\)}P\_{2}(\mathsf{x}/\mathsf{e})& =& & (\mathsf{x}+\mathsf{1})(\mathsf{x}-\mathsf{e})& = & \mathsf{x}^{2}+(\mathsf{1}-\mathsf{e})\mathsf{x}-\mathsf{e} & \text{ and }\\\mbox{\(\varkappa P\_{2}(\mathsf{x})P\_{1}(\mathsf{x}/\mathsf{e})\)& = & (\mathsf{x}-\mathsf{1})(\mathsf{x}+\mathsf{e})& = & \mathsf{x}^{2}+(-\mathsf{1}+\mathsf{e})\mathsf{x}-\mathsf{e} \end{array}$$

define the SPsð Þ þ; þ; � and ð Þ þ; �; � , respectively, and realize them with the AP ð Þ 1; 1 . In the same way, one can concatenate P<sup>1</sup> (resp. P2) with itself to realize the SP ð Þ þ; þ; þ with the AP 0ð Þ ; 2 (resp. the SP ð Þ þ; �; þ with the AP 2ð Þ ; 0 ). These are all possible cases of monic hyperbolic degree 2 polynomials with nonvanishing coefficients.

For d ≥ 2, in order to realize a SP σ with its Descartes' pair ð Þ c; p , we represent σ in the form <sup>σ</sup>† ð Þ ; <sup>u</sup>; <sup>v</sup> , where <sup>u</sup> and <sup>v</sup> are the last two components of <sup>σ</sup> and <sup>σ</sup>† is the SP obtained from σ by deleting u and v. Then, we choose P<sup>1</sup> to be a monic polynomial realizing the SP <sup>σ</sup>† ð Þ ; <sup>u</sup> :


Our next result discusses (non-)realizability for polynomials with only two sign changes (see [8, 9]).

Proposition 3. Consider a sign pattern σ with 2 sign changes, consisting of m consecutive pluses followed by n consecutive minuses and then by q consecutive pluses, where m þ n þ q ¼ d þ 1: Then:

i. For the pair ð Þ 0; d � 2 , this sign pattern is not realizable if

$$\kappa := \frac{d - m - 1}{m} \cdot \frac{d - q - 1}{q} \ge 4; \tag{3}$$

ii. The sign pattern σ is realizable with any pair of the form ð Þ 2; v , except in the case when d and m are even, n ¼ 1 (hence q is even), and v ¼ 0.

Certain results about realizability are formulated in terms of the ratios between the quantities pos, neg, and d. The following proposition is proven in [8].

Proposition 4. For a given couple (SP, AP), if minð Þ pos; neg . ½ � ð Þ d � 4 =3 , then this couple is realizable.

#### 3.2 The even and the odd series

Suppose that the degree d is even. Then, the following result holds (see Proposition 4 in [8]):

Proposition 5. Consider the SPs satisfying the following three conditions:


New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

Then, for any such SP, the APs ð Þ 2; 0 ,ð Þ 4; 0 , …,ð Þ 2ℓ; 0 , and only they, are nonrealizable.

Suppose now that the degree d ≥ 5 is odd. For 1 ≤ k ≤ ð Þ d � 3 =2, denote by σ<sup>k</sup> the SP beginning with two pluses followed by k pairs ð Þ �; þ and then by d � 2k � 1 minuses. Its Descartes' pair of σ<sup>k</sup> equals 2ð Þ k þ 1; d � 2k � 1 . The following proposition is proven in [19].

Theorem 6. (1) The SP σ<sup>k</sup> is not realizable with any of the pairs ð Þ 3; 0 ,ð Þ 5; 0 , …, ð Þ 2k þ 1; 0 ; (2) The SP σ<sup>k</sup> is realizable with the pair ð Þ 1; 0 ; (3) The SP σ<sup>k</sup> is realizable with any of the APs ð Þ 2ℓ þ 1; 2r , ℓ ¼ 0, 1, …, k, and r ¼ 1, 2, …, dð Þ � 2k � 1 =2.

One can observe that Cases (1), (2), and (3) exhaust all possible APs ð Þ pos; neg .

#### 4. Similar realization problems

In this section, we consider realization problems similar or motivated by Problem 1. A priori it is hard to tell which of these or similar problems might have a reasonable answer.

#### 4.1 D-Sequences

Proof of Proposition 2. We will use induction on the degree d of the polynomial. For d ¼ 1, the SP ð Þ þ; � (resp. ð Þ þ; þ ) is realizable with the AP 1ð Þ ; 0 (resp. 0ð Þ ; 1 )

For d ¼ 2, we apply Lemma 1. Set P<sup>1</sup> ≔ x þ 1 and P<sup>2</sup> ≔ x � 1. Then, for ε . 0

<sup>ε</sup>P1ð Þ <sup>x</sup> <sup>P</sup>2ð Þ¼ <sup>x</sup>=<sup>ε</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> ð Þ¼ <sup>x</sup> � <sup>ε</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>ε</sup> <sup>x</sup> � <sup>ε</sup> and <sup>ε</sup>P2ð Þ <sup>x</sup> <sup>P</sup>1ð Þ¼ <sup>x</sup>=<sup>ε</sup> ð Þ <sup>x</sup> � <sup>1</sup> ð Þ¼ <sup>x</sup> <sup>þ</sup> <sup>ε</sup> <sup>x</sup><sup>2</sup> þ �ð Þ <sup>1</sup> <sup>þ</sup> <sup>ε</sup> <sup>x</sup> � <sup>ε</sup>

define the SPsð Þ þ; þ; � and ð Þ þ; �; � , respectively, and realize them with the AP ð Þ 1; 1 . In the same way, one can concatenate P<sup>1</sup> (resp. P2) with itself to realize the SP ð Þ þ; þ; þ with the AP 0ð Þ ; 2 (resp. the SP ð Þ þ; �; þ with the AP 2ð Þ ; 0 ). These are all possible cases of monic hyperbolic degree 2 polynomials with nonvanishing coefficients. For d ≥ 2, in order to realize a SP σ with its Descartes' pair ð Þ c; p , we represent σ in the form <sup>σ</sup>† ð Þ ; <sup>u</sup>; <sup>v</sup> , where <sup>u</sup> and <sup>v</sup> are the last two components of <sup>σ</sup> and <sup>σ</sup>† is the SP obtained from σ by deleting u and v. Then, we choose P<sup>1</sup> to be a monic polynomial

ii. With the AP ð Þ <sup>c</sup>; <sup>p</sup> � <sup>1</sup> , and we set <sup>P</sup><sup>2</sup> <sup>≔</sup> <sup>x</sup> <sup>þ</sup> 1, if <sup>u</sup> <sup>¼</sup> <sup>v</sup>. □

Our next result discusses (non-)realizability for polynomials with only two sign

Proposition 3. Consider a sign pattern σ with 2 sign changes, consisting of m consecutive pluses followed by n consecutive minuses and then by q consecutive pluses, where

> d � q � 1 q

ii. The sign pattern σ is realizable with any pair of the form ð Þ 2; v , except in the case

Certain results about realizability are formulated in terms of the ratios between

Proposition 4. For a given couple (SP, AP), if minð Þ pos; neg . ½ � ð Þ d � 4 =3 , then

Suppose that the degree d is even. Then, the following result holds (see Proposi-

iii. Among the remaining signs of even monomials, there are exactly ℓ ≥ 1 signs �

Proposition 5. Consider the SPs satisfying the following three conditions:

i. Their last entry (i.e., the sign of the constant term) is a þ.

ii. The signs of all odd monomials are þ.

(at arbitrary positions).

≥ 4; (3)

i. With the AP ð Þ c � 1; p , and we set P<sup>2</sup> ≔ x � 1, if u ¼ �v.

i. For the pair ð Þ 0; d � 2 , this sign pattern is not realizable if

<sup>κ</sup> <sup>≔</sup> <sup>d</sup> � <sup>m</sup> � <sup>1</sup>

m �

when d and m are even, n ¼ 1 (hence q is even), and v ¼ 0.

the quantities pos, neg, and d. The following proposition is proven in [8].

by the polynomial x � 1 (resp. x þ 1).

small enough, the polynomials

Polynomials - Theory and Application

realizing the SP <sup>σ</sup>† ð Þ ; <sup>u</sup> :

changes (see [8, 9]).

m þ n þ q ¼ d þ 1: Then:

this couple is realizable.

tion 4 in [8]):

40

3.2 The even and the odd series

Consider a real polynomial P of degree d and its derivative. By Rolle's theorem, if P has exactly r real roots (counted with multiplicity), then the derivative P<sup>0</sup> has r � 1 þ 2ℓ real roots (counted with multiplicity), where ℓ∈ N ∪ 0. It is possible that <sup>P</sup><sup>0</sup> has more real roots than <sup>P</sup>. For example, for <sup>d</sup> <sup>¼</sup> 2 and <sup>P</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> 1, one gets P 0 ¼ 2x which has a real root at 0, while P has no real roots at all. For d ¼ 3, the polynomial <sup>P</sup> <sup>¼</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup>x<sup>2</sup> � <sup>8</sup><sup>x</sup> <sup>þ</sup> <sup>10</sup> <sup>¼</sup> ð Þð <sup>x</sup> <sup>þ</sup> <sup>5</sup> ð Þ <sup>x</sup> � <sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>1</sup><sup>Þ</sup> has one negative root and one complex conjugate pair, while its derivative <sup>P</sup><sup>0</sup> <sup>¼</sup> <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>x</sup> � 8 has one positive and one negative root.

Now, for j ¼ 0, …, and d � 1, denote by rj and cj the numbers of real roots and complex conjugate pairs of roots of the polynomial Pð Þ<sup>j</sup> (both counted with multiplicity). These numbers satisfy the conditions

$$r\_j \le r\_{j+1} + \mathbf{1}, \quad r\_j + \mathbf{2}c\_j = d - j. \tag{4}$$

Definition 1. A sequence ðð Þ r0; 2c<sup>0</sup> , rð Þ <sup>1</sup>; 2c<sup>1</sup> , …, rd�<sup>1</sup> ð ÞÞ ; 2cd�<sup>1</sup> satisfying conditions (4) will be called a D-sequence of length d. We say that a given D-sequence of length d is realizable if there exists a real polynomial P of degree d with this Dsequence, where for <sup>j</sup> <sup>¼</sup> <sup>0</sup>, …, d � 1, all roots of <sup>P</sup>ð Þ<sup>j</sup> are distinct.

Example 4. One has rd�<sup>1</sup> ¼ 1 and cd�<sup>1</sup> ¼ 0. Clearly, one has either rd�<sup>2</sup> ¼ 2, cd�<sup>2</sup> ¼ 0 or rd�<sup>2</sup> ¼ 0, cd�<sup>2</sup> ¼ 1. For small values of d, one has the following Dsequences and respective polynomials realizing them:

$$\begin{array}{cccc} d=\mathbf{1} & (\mathbf{1},\mathbf{0}) & \mathbf{x} \\ d=\mathbf{2} & ((\mathbf{2},\mathbf{0}),(\mathbf{1},\mathbf{0})) & \mathbf{x}^2-\mathbf{1} \\ & ((\mathbf{0},\mathbf{2}),(\mathbf{1},\mathbf{0})) & \mathbf{x}^2+\mathbf{1} \\ d=\mathbf{3} & ((\mathbf{3},\mathbf{0}),(\mathbf{2},\mathbf{0}),(\mathbf{1},\mathbf{0})) & \mathbf{x}^3-\mathbf{x} \\ & ((\mathbf{1},\mathbf{2}),(\mathbf{0},\mathbf{2}),(\mathbf{1},\mathbf{0})) & \mathbf{x}^3+\mathbf{x} \\ & ((\mathbf{1},\mathbf{2}),(\mathbf{2},\mathbf{0}),(\mathbf{1},\mathbf{0})) & \mathbf{x}^3+\mathbf{1} \mathbf{0} \mathbf{x}^2+2\mathbf{6} \mathbf{x}. \end{array}$$

The following question where a positive answer to which can be found in [15] seems very natural.

Problem 2. Is it true that for any d∈ N, any D-sequence is realizable?

#### 4.2 Sequences of admissible pairs

Now, we are going to formulate a problem which is a refinement of both Problems 1 and 2.

Recall that for a real polynomial P of degree d, the signs of its coefficients aj define the sign patterns σ0, σ1, …, σd�<sup>1</sup> corresponding to P and to all its derivatives of order ≤ d � 1 since the SP σ<sup>j</sup> is obtained from σj�<sup>1</sup> by deleting the last component. We denote by ck; pk and posk; negk the Descartes' and admissible pairs for the SPs σk, k ¼ 0, …, d � 1. The following restrictions follow from Rolle's theorem:

$$\begin{aligned} pos\_{k+1} &\geq pos\_k - 1 \quad , \ \operatorname{neg}\_{k+1} &\geq \operatorname{neg}\_k - 1 \\ \text{and} & \quad pos\_{k+1} + \operatorname{neg}\_{k+1} &\geq pos\_k + \operatorname{neg}\_k - 1. \end{aligned} \tag{5}$$

For d ¼ 4; 5; 6; 7; 8; 9; 10, the numbers A dð Þ of SAPs compatible with the SP of

7, 12, 30, 55, 143, 273, and 728,

Example 6. There are two couples (SP, SAP) corresponding to the couple (SP, AP) C ≔ ðð Þ þ; þ; �; þ; þ , 0ð ÞÞ ; 2 ; we also say that the couple C can be extended into

ð þð Þ ; þ; �; þ; þ , ð Þ 0; 2 , ð Þ 2; 1 , ð Þ 1; 1 , ð ÞÞ 0; 1 and ð þð Þ ; þ; �; þ; þ , ð Þ 0; 2 , ð Þ 0; 1 , ð Þ 1; 1 , ð ÞÞ 0; 1 :

has at least one negative root. By conditions (7), this derivative (whose degree equals 3) has an even number of positive roots. This yields just two possibilities for

Problem 3. For a given degree d, which couples (SP, SAP) are realizable? Remarks 1. (1) This problem is a refinement of Problem 1, because one considers the APs of the derivatives of all orders and not just the one of the polynomial itself (see Remark 4). Therefore, if a given couple (SP, AP) is non-realizable, then all couples (SP, SAP) corresponding to it in the sense of Example 6 are automati-

(2) Obviously, Problem 3 is a refinement of Problem 2—in the latter case, one does not take into account the signs of the real roots of the polynomial and its

(3) When we deal with couples (SP, SAP), we can use the Z2-action defined by (1). Therefore, it suffices to consider the cases of SPs beginning with ð Þ þ; þ . The generator (2.2) of the Z<sup>2</sup> � Z2-action cannot be used, because when the derivatives of a polynomial are involved, the polynomial loses its last coefficients. Due to this

Proposition 6. For any given SP of length d þ 1 and d ≥ 1, there exists a unique SAP such that pos<sup>0</sup> þ neg<sup>0</sup> ¼ d. This SAP is realizable. For the given SP, this pair

Example 7. For even d, consider the SP with all pluses. Any hyperbolic polyno-

ð Þ ð Þ 0; d ;ð Þ 0; d � 1 ; …;ð Þ 0; 1 :

ð Þ ð Þ 0; d � 2ℓ ;ð Þ 0; d � 1 ;ð Þ 0; d � 2 ; …;ð Þ 0; 1 , ℓ ¼ 0, 1, … d=2:

In the same way, for odd d, the SP ð Þ þ; þ; …; þ; � is realizable with the SAP

One can choose such a polynomial P with all d � 1 distinct critical values. Hence, in the family of polynomials P þ t and t . 0, one encounters polynomials realizing

circumstance, the two ends of the SP cannot be treated equally.

mial with all negative and distinct roots realizes this SP with SAP

The following proposition is proven in [5]:

 , namely, 2ð Þ ; <sup>1</sup> and 0ð Þ ; <sup>1</sup> . The second derivative is a quadratic polynomial with positive leading coefficient and negative constant term. Hence, it has a positive and a negative root. The realizability of the above two couples (SP, SAP) is

Indeed, by Rolle's theorem, the derivative of a polynomial realizing the couple C

respectively. One can show that A dð Þ ≥ 2A dð Þ � 1 , if d ≥ 2 is even, and

length d þ 1 having all pluses are

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

these couples (SP, SAP). These are

pos1; neg<sup>1</sup>

proven in [5].

cally non-realizable.

derivatives.

pos0; neg<sup>0</sup>

43

is its Descartes' pair.

this SP with any of the SAPs

A dð Þ ≥ 3A dð Þ � 1 =2, if d ≥ 3 is odd (see [5]).

Our final realization problem is as follows:

It is always true that

$$p s s\_{k+1} + n \text{eg}\_{k+1} + \mathfrak{Z} - p s s\_k - n \text{eg}\_k \in \mathfrak{N}.\tag{6}$$

Definition 2. Given a sign pattern σ<sup>0</sup> of length d þ 1, suppose that for k ¼ 0, …, d � 1, the pair posk; negk satisfies the conditions

$$pos\_k \le c\_k, \quad c\_k - pos\_k \in 2\mathbb{Z},$$

$$neg\_k \le p\_k, \quad p\_k - neg\_k \in 2\mathbb{Z},\tag{7}$$

$$\text{and}\\ \qquad \text{sgn } a\_k = (-1)^{pos\_k}.$$

as well as the inequalities (5)–(6). Then, we say that

$$\left( \left( pos\_0, neg\_0 \right), \dots, \left( pos\_{d-1}, neg\_{d-1} \right) \right) \tag{8}$$

is a sequence of admissible pairs (SAPs). In other words, it is a sequence of pairs admissible for the sign pattern σ<sup>0</sup> in the sense of these conditions. We say that a given couple (SP, SAP) is realizable if there exists a polynomial P whose coefficients have signs given by the SP <sup>σ</sup>0, and such that for <sup>k</sup> <sup>¼</sup> <sup>0</sup>, …, d � 1, the polynomial <sup>P</sup>ð Þ<sup>k</sup> has exactly posk positive and negk negative roots, all of them being simple. Complex roots are also supposed to be distinct.

Remark 4. If one only knows the SAP 8ð Þ, the SP σ<sup>0</sup> can be restituted by the formula

$$
\sigma\_0 = \left( +, \ (-\mathbf{1})^{p \alpha\_{d-1}}, \ (-\mathbf{1})^{p \alpha\_{d-2}}, \dots, (-\mathbf{1})^{p \alpha\_0} \ ) . \right)
$$

Nevertheless, in order to make comparisons with Problem 1 more easily, we consider couples (SP, SAP) instead of just SAPs. But for a given SP, there are, in general, several possible SAPs which is illustrated by the following example.

Example 5. Consider the SP of length d þ 1 with all pluses. For d ¼ 2 and 3, there are, respectively, two and three possible SAPs:

$$\begin{array}{llll} ((\mathbf{0}, \mathbf{2}), (\mathbf{0}, \mathbf{1})) & \text{, } & ((\mathbf{0}, \mathbf{0}), (\mathbf{0}, \mathbf{1})) & \text{, } & \text{ for } d = 2 \\\ \text{and} & & & \\\ ((\mathbf{0}, \mathbf{3}), (\mathbf{0}, \mathbf{2}), (\mathbf{0}, \mathbf{1})) & \text{, } & ((\mathbf{0}, \mathbf{1}), (\mathbf{0}, \mathbf{2}), (\mathbf{0}, \mathbf{1})) & \text{, } & ((\mathbf{0}, \mathbf{1}), (\mathbf{0}, \mathbf{0}), (\mathbf{0}, \mathbf{1})) & \text{ for } d = 3. \end{array}$$

4.2 Sequences of admissible pairs

Polynomials - Theory and Application

component. We denote by ck; pk

It is always true that

k ¼ 0, …, d � 1, the pair posk; negk

roots are also supposed to be distinct.

are, respectively, two and three possible SAPs:

formula

and

42

Problems 1 and 2.

Rolle's theorem:

Now, we are going to formulate a problem which is a refinement of both

and posk; negk

pairs for the SPs σk, k ¼ 0, …, d � 1. The following restrictions follow from

poskþ<sup>1</sup> <sup>≥</sup> posk � <sup>1</sup> , negkþ<sup>1</sup> <sup>≥</sup> negk � <sup>1</sup>

Definition 2. Given a sign pattern σ<sup>0</sup> of length d þ 1, suppose that for

as well as the inequalities (5)–(6). Then, we say that

pos0; neg<sup>0</sup>

satisfies the conditions

posk ≤ ck, ck � posk ∈2Z, negk ≤ pk, pk � negk ∈2Z, and sgn ak ¼ �ð Þ<sup>1</sup> posk :

; …; posd�<sup>1</sup>; negd�<sup>1</sup>

is a sequence of admissible pairs (SAPs). In other words, it is a sequence of pairs admissible for the sign pattern σ<sup>0</sup> in the sense of these conditions. We say that a given couple (SP, SAP) is realizable if there exists a polynomial P whose coefficients have signs given by the SP <sup>σ</sup>0, and such that for <sup>k</sup> <sup>¼</sup> <sup>0</sup>, …, d � 1, the polynomial <sup>P</sup>ð Þ<sup>k</sup> has exactly posk positive and negk negative roots, all of them being simple. Complex

Remark 4. If one only knows the SAP 8ð Þ, the SP σ<sup>0</sup> can be restituted by the

<sup>σ</sup><sup>0</sup> ¼ þ; ð Þ �<sup>1</sup> posd�<sup>1</sup> ; ð Þ �<sup>1</sup> posd�<sup>2</sup> ; …;ð Þ �<sup>1</sup> pos<sup>0</sup> ð Þ:

Nevertheless, in order to make comparisons with Problem 1 more easily, we consider couples (SP, SAP) instead of just SAPs. But for a given SP, there are, in general, several possible SAPs which is illustrated by the following example.

ð Þ ð Þ 0; 2 ;ð Þ 0; 1 , ð Þ ð Þ 0; 0 ;ð Þ 0; 1 , for d ¼ 2

ð Þ ð Þ 0; 3 ;ð Þ 0; 2 ;ð Þ 0; 1 , ð Þ ð Þ 0; 1 ;ð Þ 0; 2 ;ð Þ 0; 1 , ð Þ ð Þ 0; 1 ;ð Þ 0; 0 ;ð Þ 0; 1 for d ¼ 3:

Example 5. Consider the SP of length d þ 1 with all pluses. For d ¼ 2 and 3, there

Recall that for a real polynomial P of degree d, the signs of its coefficients aj define the sign patterns σ0, σ1, …, σd�<sup>1</sup> corresponding to P and to all its derivatives of order ≤ d � 1 since the SP σ<sup>j</sup> is obtained from σj�<sup>1</sup> by deleting the last

and poskþ<sup>1</sup> <sup>þ</sup> negkþ<sup>1</sup> <sup>≥</sup> posk <sup>þ</sup> negk � <sup>1</sup>: (5)

poskþ<sup>1</sup> <sup>þ</sup> negkþ<sup>1</sup> <sup>þ</sup> <sup>3</sup> � posk � negk <sup>∈</sup>2N: (6)

(8)

the Descartes' and admissible

(7)

For d ¼ 4; 5; 6; 7; 8; 9; 10, the numbers A dð Þ of SAPs compatible with the SP of length d þ 1 having all pluses are

7, 12, 30, 55, 143, 273, and 728,

respectively. One can show that A dð Þ ≥ 2A dð Þ � 1 , if d ≥ 2 is even, and A dð Þ ≥ 3A dð Þ � 1 =2, if d ≥ 3 is odd (see [5]).

Example 6. There are two couples (SP, SAP) corresponding to the couple (SP, AP) C ≔ ðð Þ þ; þ; �; þ; þ , 0ð ÞÞ ; 2 ; we also say that the couple C can be extended into these couples (SP, SAP). These are

$$\begin{array}{ccccccccc} (&(+,+,-,+,+) &, &(0,2) &, &(2,1) &, &(1,1) &, &(0,1) & ) & \text{and} \\ (&(+,+,-,+,+) &, &(0,2) &, &(0,1) &, &(1,1) & , &(0,1) & ) & . \end{array}$$

Indeed, by Rolle's theorem, the derivative of a polynomial realizing the couple C has at least one negative root. By conditions (7), this derivative (whose degree equals 3) has an even number of positive roots. This yields just two possibilities for pos1; neg<sup>1</sup> , namely, 2ð Þ ; <sup>1</sup> and 0ð Þ ; <sup>1</sup> . The second derivative is a quadratic polynomial with positive leading coefficient and negative constant term. Hence, it has a positive and a negative root. The realizability of the above two couples (SP, SAP) is proven in [5].

Our final realization problem is as follows:

Problem 3. For a given degree d, which couples (SP, SAP) are realizable?

Remarks 1. (1) This problem is a refinement of Problem 1, because one considers the APs of the derivatives of all orders and not just the one of the polynomial itself (see Remark 4). Therefore, if a given couple (SP, AP) is non-realizable, then all couples (SP, SAP) corresponding to it in the sense of Example 6 are automatically non-realizable.

(2) Obviously, Problem 3 is a refinement of Problem 2—in the latter case, one does not take into account the signs of the real roots of the polynomial and its derivatives.

(3) When we deal with couples (SP, SAP), we can use the Z2-action defined by (1). Therefore, it suffices to consider the cases of SPs beginning with ð Þ þ; þ . The generator (2.2) of the Z<sup>2</sup> � Z2-action cannot be used, because when the derivatives of a polynomial are involved, the polynomial loses its last coefficients. Due to this circumstance, the two ends of the SP cannot be treated equally.

The following proposition is proven in [5]:

Proposition 6. For any given SP of length d þ 1 and d ≥ 1, there exists a unique SAP such that pos<sup>0</sup> þ neg<sup>0</sup> ¼ d. This SAP is realizable. For the given SP, this pair pos0; neg<sup>0</sup> is its Descartes' pair.

Example 7. For even d, consider the SP with all pluses. Any hyperbolic polynomial with all negative and distinct roots realizes this SP with SAP

$$((\mathbf{0}, d), (\mathbf{0}, d - \mathbf{1}), \dots, (\mathbf{0}, \mathbf{1})).$$

One can choose such a polynomial P with all d � 1 distinct critical values. Hence, in the family of polynomials P þ t and t . 0, one encounters polynomials realizing this SP with any of the SAPs

ð Þ ð Þ 0; d � 2ℓ ;ð Þ 0; d � 1 ;ð Þ 0; d � 2 ; …;ð Þ 0; 1 , ℓ ¼ 0, 1, … d=2:

In the same way, for odd d, the SP ð Þ þ; þ; …; þ; � is realizable with the SAP

ð Þ ð Þ 1; d � 1 ; ð Þ 0; d � 1 ; ð Þ 0; d � 2 ; …; ð Þ 0; 1

The coefficients of T define the SP

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

to find the exact value.

5. Outlook

(2 and 4).

present.

45

can have some nontrivial topology?

ð Þ <sup>þ</sup>; <sup>þ</sup>; �; <sup>þ</sup>; <sup>þ</sup>; � for <sup>a</sup><sup>∈</sup> <sup>3</sup>=2; <sup>3</sup> <sup>þ</sup> ffiffiffi

ð Þ <sup>þ</sup>; <sup>þ</sup>; �; �; <sup>þ</sup>; � for <sup>a</sup><sup>∈</sup> <sup>3</sup> <sup>þ</sup> ffiffiffi

ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup>; �; <sup>þ</sup>; � for <sup>a</sup> . <sup>3</sup> <sup>þ</sup> ffiffiffi

<sup>6</sup> � � <sup>p</sup> <sup>=</sup><sup>3</sup> � � ,

<sup>p</sup> :

<sup>6</sup> � � <sup>p</sup> <sup>=</sup>3; <sup>3</sup> <sup>þ</sup> ffiffiffi <sup>6</sup> � � <sup>p</sup> and

6

1. Our first open question deals with the limit of the ratio between the quantities R dð Þ of all realizable and A dð Þ of all possible cases of couples (SP, AP) as d ! ∞. In principle, one does not have to take into account the Z<sup>2</sup> � Z2-action in order not to face the problem of the two different possible lengths of orbits

A priori, for d ≥ 4, one has R dð Þ=A dð Þ∈ð Þ 0; 1 . It would be interesting to find out whether this ratio has a limit as d ! ∞ and, if "yes," whether this limit is 0 and 1 or belongs to 0ð Þ ; 1 . In the latter case, it would be interesting

A less ambitious open problem is to find an interval ½ � α; β ⊂ð Þ 0; 1 to which this ratio belongs for any d∈ N, d ≥ 4, or at least for d sufficiently large.

2. A related problem would be to find sufficient conditions for realizability based on the ratios between the quantities pos, neg, and d. On the one hand, when the

ratios pos=d and neg=d are both large enough, one has realizability (see Proposition 4). On the other hand, in all examples of non-realizability known up to now, one of the quantities pos and neg is either 0 or is very small compared to the other one. Thus, it would be interesting to understand the role of these ratios for the (non)-realizability of the couples (SP, AP).

3. Our third open question is about the realizability of couples (SP, SAP). For d ≤ 5, the non-realizability of all non-realizable couples (SP, SAP) results from the non-realizability of the corresponding couples (SP, AP). In principle, one could imagine a situation in which there exists a couple (SP, AP) extending into several couples (SP, SAP) some of which are realizable and the remaining are not. Whether, for d ≥ 6, such couples (SP, AP) exist or not is unknown at

4.Our final natural and important question deals with the topology of

of the coefficients). It is well known that the set of monic univariate polynomials of a given degree and with a given number of real roots is contractible. When we cut this set with the union of coordinate hyperplanes (coordinates being the coefficients of polynomials), then it splits into a number of connected components. In each such connected component, the number of positive and negative roots is fixed. But, in principle, it can happen that different connected components correspond to the same pair (pos, neg). Could this really happen? Are all such connected components contractible, or they

intersections of the set of real univariant polynomials with a given number of real roots with orthants in the coefficient space (which means fixing the signs

by some hyperbolic polynomial R with all distinct roots and critical values. In the family of polynomials R � s and s . 0, one encounters polynomials realizing this SP with any of the SAPs

ð Þ ð Þ 1; d � 1 � 2ℓ ; ð Þ 0; d � 1 ; ð Þ 0; d � 2 ; …; ð Þ 0; 1 , ℓ ¼ 0, 1, …ð Þ d � 1 =2:

For d ≤ 5, the following exhaustive answer to Problem 3 is given in [5]:

A. For d ¼ 1, 2, and 3, all couples (SP, SAP) are realizable.

B. For d ¼ 4, the couple (SP, SAP)

$$( (+, +, -, +, +), \ (2, \mathbf{0}), \ (2, \mathbf{1}), \ (\mathbf{1}, \mathbf{1}), \ (\mathbf{0}, \mathbf{1}) ),$$

and only it (up to the Z2-action), is non-realizable. Its non-realizability follows from one of the couples (SP, AP) <sup>C</sup>† <sup>≔</sup> ð Þ ð Þ <sup>þ</sup>; <sup>þ</sup>; �; <sup>þ</sup>; <sup>þ</sup> ;ð Þ <sup>2</sup>; <sup>0</sup> (see Theorem 1).

One can observe that the couple C† can be uniquely extended into a couple (SP, SAP). Indeed, the first derivative has a positive constant term hence an even number of positive roots. This number is positive by Rolle's theorem. Hence, the AP of the first derivative is 2ð Þ ; 1 . In the same way, one obtains the APs 1ð Þ ; 1 and 0ð Þ ; 1 for the second and third derivatives, respectively.

C. For d ¼ 5, the following couples (SP, SAP), and only they, are non-realizable:


The non-realizability of the first four of them follows from that of the couple C† . The last one is implied by part (1) of Theorem 2; it is true that the couple (SP, AP) ð Þ ð Þ þ; þ; �; þ; �; � ;ð Þ 3; 0 extends in a unique way into a couple (SP, SAP), and this is the fifth of the five such couples cited above.

One of the methods used in the study of couples (SP, AP) or (SP, SAP) is the explicit construction of polynomials with multiple roots which define a given SP. Such constructions are not difficult to carry out because one has to use families of polynomials with fewer parameters. Once a polynomial with multiple roots is constructed, one has to justify the possibility to deform it continuously into a nearby polynomial with all distinct roots. Multiple roots can give rise to complex conjugate pairs of roots. An example of such a construction is the following lemma from [5].

Lemma 2. Consider the polynomials S <sup>≔</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> <sup>3</sup> ð Þ x � a <sup>2</sup> and T <sup>≔</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>a</sup> 2 ð Þ <sup>x</sup> � <sup>1</sup> <sup>3</sup> and a . 0. Their coefficients of x<sup>4</sup> are positive if and only if, respectively, a , 3=2 and a . 3=2. The coefficients of the polynomial S define the SP

$$\begin{array}{llll} (+,+,+,+,-,+) & \text{for} & a \in \left(0, \left(3-\sqrt{6}\right)/3\right) & \\ (+,+,+,-,-,+) & \text{for} & a \in \left(\left(3-\sqrt{6}\right)/3, 3-\sqrt{6}\right) & \\ (+,+,-,-,-,+) & \text{for} & a \in \left(3-\sqrt{6}, 2/3\right) & \\ (+,+,-,-,+,+) & \text{for} & a \in \left(2/3, 3/2\right) & \end{array}$$

The coefficients of T define the SP

$$\begin{array}{llll}(+,+,-,+,+,-) & \text{for} & a \in \left(\ 3/2,\left(3+\sqrt{6}\right)/3\right) \\ (+,+,-,-,+,-) & \text{for} & a \in \left(\ (3+\sqrt{6})/3,3+\sqrt{6}\right) \\ (+,+,+,-,+,-) & \text{for} & a \ge 3+\sqrt{6} \\ \end{array}$$

### 5. Outlook

ð Þ ð Þ 1; d � 1 ; ð Þ 0; d � 1 ; ð Þ 0; d � 2 ; …; ð Þ 0; 1

ð Þ ð Þ 1; d � 1 � 2ℓ ; ð Þ 0; d � 1 ; ð Þ 0; d � 2 ; …; ð Þ 0; 1 , ℓ ¼ 0, 1, …ð Þ d � 1 =2:

ð Þ ð Þ þ; þ; �; þ; þ ; ð Þ 2; 0 ; ð Þ 2; 1 ; ð Þ 1; 1 ; ð Þ 0; 1 , and only it (up to the Z2-action), is non-realizable. Its non-realizability follows from one of the couples (SP, AP) <sup>C</sup>† <sup>≔</sup> ð Þ ð Þ <sup>þ</sup>; <sup>þ</sup>; �; <sup>þ</sup>; <sup>þ</sup> ;ð Þ <sup>2</sup>; <sup>0</sup> (see Theorem 1). One can observe that the couple C† can be uniquely extended into a couple (SP,

C. For d ¼ 5, the following couples (SP, SAP), and only they, are non-realizable:

.

ð þð Þ ; þ; �; þ; þ; þ , ð Þ 2; 1 , ð Þ 2; 0 , ð Þ 2; 1 , ð Þ 1; 1 , ð ÞÞ 0; 1 , ð þð Þ ; þ; �; þ; þ; þ , ð Þ 0; 1 , ð Þ 2; 0 , ð Þ 2; 1 , ð Þ 1; 1 , ð ÞÞ 0; 1 , ð þð Þ ; þ; �; þ; þ; � , ð Þ 3; 0 , ð Þ 2; 0 , ð Þ 2; 1 , ð Þ 1; 1 , ð ÞÞ 0; 1 , ð þð Þ ; þ; �; þ; þ; � , ð Þ 1; 0 , ð Þ 2; 0 , ð Þ 2; 1 , ð Þ 1; 1 , ð ÞÞ 0; 1 , ð þð Þ ; þ; �; þ; �; � , ð Þ 3; 0 , ð Þ 3; 1 , ð Þ 2; 1 , ð Þ 1; 1 , ð ÞÞ 0; 1 : The non-realizability of the first four of them follows from that of the couple C†

The last one is implied by part (1) of Theorem 2; it is true that the couple (SP, AP) ð Þ ð Þ þ; þ; �; þ; �; � ;ð Þ 3; 0 extends in a unique way into a couple (SP, SAP), and this

One of the methods used in the study of couples (SP, AP) or (SP, SAP) is the explicit construction of polynomials with multiple roots which define a given SP. Such constructions are not difficult to carry out because one has to use families of polynomials with fewer parameters. Once a polynomial with multiple roots is constructed, one has to justify the possibility to deform it continuously into a nearby polynomial with all distinct roots. Multiple roots can give rise to complex conjugate pairs of roots. An example of such a construction is the following lemma from [5].

and a . 0. Their coefficients of x<sup>4</sup> are positive if and only if, respectively, a , 3=2 and

ð Þ þ; þ; �; �; þ; þ for a∈ð Þ 2=3; 3=2 :

ð Þ x � a

<sup>6</sup> � � <sup>p</sup> <sup>=</sup><sup>3</sup> � � ,

<sup>p</sup> ; <sup>2</sup>=<sup>3</sup> � � and

<sup>6</sup> � � <sup>p</sup> <sup>=</sup>3; <sup>3</sup> � ffiffiffi <sup>6</sup> � � <sup>p</sup> ,

6

<sup>2</sup> and T <sup>≔</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>a</sup>

2 ð Þ <sup>x</sup> � <sup>1</sup> <sup>3</sup>

SAP). Indeed, the first derivative has a positive constant term hence an even number of positive roots. This number is positive by Rolle's theorem. Hence, the AP of the first derivative is 2ð Þ ; 1 . In the same way, one obtains the APs 1ð Þ ; 1 and 0ð Þ ; 1

For d ≤ 5, the following exhaustive answer to Problem 3 is given in [5]:

A. For d ¼ 1, 2, and 3, all couples (SP, SAP) are realizable.

B. For d ¼ 4, the couple (SP, SAP)

for the second and third derivatives, respectively.

is the fifth of the five such couples cited above.

Lemma 2. Consider the polynomials S <sup>≔</sup> ð Þ <sup>x</sup> <sup>þ</sup> <sup>1</sup> <sup>3</sup>

a . 3=2. The coefficients of the polynomial S define the SP

44

ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup>; �; <sup>þ</sup> for <sup>a</sup><sup>∈</sup> <sup>0</sup>; <sup>3</sup> � ffiffiffi

ð Þ <sup>þ</sup>; <sup>þ</sup>; <sup>þ</sup>; �; �; <sup>þ</sup> for <sup>a</sup><sup>∈</sup> <sup>3</sup> � ffiffiffi

ð Þ <sup>þ</sup>; <sup>þ</sup>; �; �; �; <sup>þ</sup> for <sup>a</sup><sup>∈</sup> <sup>3</sup> � ffiffiffi

with any of the SAPs

Polynomials - Theory and Application

by some hyperbolic polynomial R with all distinct roots and critical values. In the family of polynomials R � s and s . 0, one encounters polynomials realizing this SP

> 1. Our first open question deals with the limit of the ratio between the quantities R dð Þ of all realizable and A dð Þ of all possible cases of couples (SP, AP) as d ! ∞. In principle, one does not have to take into account the Z<sup>2</sup> � Z2-action in order not to face the problem of the two different possible lengths of orbits (2 and 4).

A priori, for d ≥ 4, one has R dð Þ=A dð Þ∈ð Þ 0; 1 . It would be interesting to find out whether this ratio has a limit as d ! ∞ and, if "yes," whether this limit is 0 and 1 or belongs to 0ð Þ ; 1 . In the latter case, it would be interesting to find the exact value.

A less ambitious open problem is to find an interval ½ � α; β ⊂ð Þ 0; 1 to which this ratio belongs for any d∈ N, d ≥ 4, or at least for d sufficiently large.


### Acknowledgements

The authors want to thank the Department of Mathematics of the Université Côte d'Azur and Stockholms Universitet for the hospitality, financial support, and nice working conditions during our several visits to each other.

References

105:447-451

324(10):2884-2892

arXiv. 1805:04261

Gauthier-Villars, 1890

Mathematics. 2018

15(2):31-60

438-448

47

[1] Albouy A, Fu Y. Some remarks about Descartes' rule of signs. Elemente der

[11] Gauss CF. Beweis eines

Göttingen, 1866

106:854-856

Hermann, 1886)

2011;218(4):1203-1207

60:12, 1255-1258

appear

algebraischen Lehrsatzes. Journal fur die Reine und Angewandte Mathematik. 1828;(3):1-4; Werke 3, 67–70,

[12] Grabiner DJ. Descartes'€™ Rule of Signs: Another Construction. The American Mathematical Monthly. 1999;

[13] Gutenberg Project, version PDF de La Géométrie (édition modernisée de

[14] Haukkanen P, Tossavainen T. A generalization of Descartes' rule of signs and fundamental theorem of algebra. Applied Mathematics and Computation.

[15] Kostov VP. A realization theorem about D-sequences, Comptes rendus de l'Académie bulgare des. Sciences. 2007;

[16] Kostov VP. On realizability of sign

Czechoslovak Mathematical Journal to

[17] Kostov VP. Polynomials, sign patterns and Descartes' rule of signs. Mathematica Bohemica. to appear

[18] Kostov VP. Topics on hyperbolic polynomials in one variable. Panoramas et Synthèses. 2011;33 vi + 141 p. SMF

[19] Kostov VP, Shapiro BZ. Something You Always Wanted to Know About Real Polynomials (But Were Afraid to

Ask). arxiv:1703.04436

patterns by real polynomials.

[2] Anderson B, Jackson J, Sitharam M. Descartes'€™ rule of signs revisited. The American Mathematical Monthly. 1998;

[3] Avendaño M. Descartes' rule of signs is exact! Journal of Algebra. 2010;

[4] Cajori F. A history of the arithmetical methods of approximation to the roots of numerical equations of one unknown quantity. Colorado College Publication. Science Series. 1910;12–7:171-215

[5] Gati Y, Cheriha H, Kostov VP. Descartes' rule of signs. Rolle's theorem and sequences of admissible pairs,

[6] Dimitrov DK, Rafaeli FR. Descartes' rule of signs for orthogonal polynomials. East Journal on Approximations. 2009;

[7] Fourier J. Sur l'usage du théorème de Descartes dans la recherche des limites des racines. Bulletin des sciences par la Société philomatique de Paris. 1820: 156-165, 181–187; œuvres 2, 291–309,

[8] Forsgård J, Kostov VP, Shapiro B. Could René Descartes have known this? Experimental Mathematics. 2015;24(4):

[9] Forsgård J, Kostov VP, Shapiro B. Corrigendum: Could René Descartes have known this? Experimental

[10] Forsgård J, Novikov D, Shapiro B. A tropical analog of Descartes' rule of signs. International Mathematics Research Notices. 2017;12:3726-3750

Mathematik. 2014;69:186-194

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

### Author details

Vladimir Petrov Kostov<sup>1</sup> and Boris Shapiro<sup>2</sup> \*

1 LJAD, Université Côte d'Azur, France

2 Stockholm University, Stockholm, Sweden

\*Address all correspondence to: shapiro@math.su.se

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

New Aspects of Descartes' Rule of Signs DOI: http://dx.doi.org/10.5772/intechopen.82040

#### References

Acknowledgements

Polynomials - Theory and Application

Author details

46

Vladimir Petrov Kostov<sup>1</sup> and Boris Shapiro<sup>2</sup>

2 Stockholm University, Stockholm, Sweden

\*Address all correspondence to: shapiro@math.su.se

1 LJAD, Université Côte d'Azur, France

provided the original work is properly cited.

\*

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

The authors want to thank the Department of Mathematics of the Université Côte d'Azur and Stockholms Universitet for the hospitality, financial support, and

nice working conditions during our several visits to each other.

[1] Albouy A, Fu Y. Some remarks about Descartes' rule of signs. Elemente der Mathematik. 2014;69:186-194

[2] Anderson B, Jackson J, Sitharam M. Descartes'€™ rule of signs revisited. The American Mathematical Monthly. 1998; 105:447-451

[3] Avendaño M. Descartes' rule of signs is exact! Journal of Algebra. 2010; 324(10):2884-2892

[4] Cajori F. A history of the arithmetical methods of approximation to the roots of numerical equations of one unknown quantity. Colorado College Publication. Science Series. 1910;12–7:171-215

[5] Gati Y, Cheriha H, Kostov VP. Descartes' rule of signs. Rolle's theorem and sequences of admissible pairs, arXiv. 1805:04261

[6] Dimitrov DK, Rafaeli FR. Descartes' rule of signs for orthogonal polynomials. East Journal on Approximations. 2009; 15(2):31-60

[7] Fourier J. Sur l'usage du théorème de Descartes dans la recherche des limites des racines. Bulletin des sciences par la Société philomatique de Paris. 1820: 156-165, 181–187; œuvres 2, 291–309, Gauthier-Villars, 1890

[8] Forsgård J, Kostov VP, Shapiro B. Could René Descartes have known this? Experimental Mathematics. 2015;24(4): 438-448

[9] Forsgård J, Kostov VP, Shapiro B. Corrigendum: Could René Descartes have known this? Experimental Mathematics. 2018

[10] Forsgård J, Novikov D, Shapiro B. A tropical analog of Descartes' rule of signs. International Mathematics Research Notices. 2017;12:3726-3750

[11] Gauss CF. Beweis eines algebraischen Lehrsatzes. Journal fur die Reine und Angewandte Mathematik. 1828;(3):1-4; Werke 3, 67–70, Göttingen, 1866

[12] Grabiner DJ. Descartes'€™ Rule of Signs: Another Construction. The American Mathematical Monthly. 1999; 106:854-856

[13] Gutenberg Project, version PDF de La Géométrie (édition modernisée de Hermann, 1886)

[14] Haukkanen P, Tossavainen T. A generalization of Descartes' rule of signs and fundamental theorem of algebra. Applied Mathematics and Computation. 2011;218(4):1203-1207

[15] Kostov VP. A realization theorem about D-sequences, Comptes rendus de l'Académie bulgare des. Sciences. 2007; 60:12, 1255-1258

[16] Kostov VP. On realizability of sign patterns by real polynomials. Czechoslovak Mathematical Journal to appear

[17] Kostov VP. Polynomials, sign patterns and Descartes' rule of signs. Mathematica Bohemica. to appear

[18] Kostov VP. Topics on hyperbolic polynomials in one variable. Panoramas et Synthèses. 2011;33 vi + 141 p. SMF

[19] Kostov VP, Shapiro BZ. Something You Always Wanted to Know About Real Polynomials (But Were Afraid to Ask). arxiv:1703.04436

Chapter 3

Abstract

Meixner polynomials.

compositional inverse

1. Introduction

and Wilf [6]).

[23–25].

49

Obtaining Explicit Formulas and

by Generating Functions of the

Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya

In this chapter, we study properties of polynomials defined by generating func-

theorem and the theorem of logarithmic derivative for generating functions, we obtain new properties related to the compositional inverse generating functions of those polynomials. Also we study the composition of generating functions R tA t ð Þ ð Þ ,

results for obtaining explicit formulas and identities for such polynomials as the generalized Bernoulli, generalized Euler, Frobenius-Euler, generalized Sylvester, generalized Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and

Keywords: polynomial, identity, generating function, composita, composition,

Generating functions are a powerful tool for solving problems in number theory, combinatorics, algebra, probability theory, and other fields of mathematics. One of the advantages of generating functions is that an infinite number sequence can be represented in a form of a single expression. Many authors have studied generating functions and their properties and found applications for them (for instance, Comtet [1], Flajolet and Sedgewick [2], Graham et al. [3], Robert [4], Stanley [5],

Generating functions have an important role in the study of polynomials. Vast investigations related to the generating functions for many polynomials can be

A special place in this area is occupied by research in the field of obtaining new identities for polynomials and special numbers with using their generating functions. Interesting results in the field of obtaining new identities for polynomials can be found in some recent works by Simsek [18–20], Kim et al. [21, 22], and Ryoo

Another trend in study of polynomials is getting new representation and explicit formulas for those polynomials. For instance, Qi has recently established explicit

where A tð Þ is the generating function of the form F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup>

found in many books and articles (e.g., see [7–17]).

. Based on the Lagrange inversion

. We apply those

Form F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup>

tions of the form A tð Þ ; <sup>x</sup>; <sup>α</sup> <sup>=</sup> F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup>

Identities for Polynomials Defined

#### Chapter 3

## Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions of the Form F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup>

Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya

#### Abstract

In this chapter, we study properties of polynomials defined by generating functions of the form A tð Þ ; <sup>x</sup>; <sup>α</sup> <sup>=</sup> F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup> . Based on the Lagrange inversion theorem and the theorem of logarithmic derivative for generating functions, we obtain new properties related to the compositional inverse generating functions of those polynomials. Also we study the composition of generating functions R tA t ð Þ ð Þ , where A tð Þ is the generating function of the form F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup> . We apply those results for obtaining explicit formulas and identities for such polynomials as the generalized Bernoulli, generalized Euler, Frobenius-Euler, generalized Sylvester, generalized Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials.

Keywords: polynomial, identity, generating function, composita, composition, compositional inverse

#### 1. Introduction

Generating functions are a powerful tool for solving problems in number theory, combinatorics, algebra, probability theory, and other fields of mathematics. One of the advantages of generating functions is that an infinite number sequence can be represented in a form of a single expression. Many authors have studied generating functions and their properties and found applications for them (for instance, Comtet [1], Flajolet and Sedgewick [2], Graham et al. [3], Robert [4], Stanley [5], and Wilf [6]).

Generating functions have an important role in the study of polynomials. Vast investigations related to the generating functions for many polynomials can be found in many books and articles (e.g., see [7–17]).

A special place in this area is occupied by research in the field of obtaining new identities for polynomials and special numbers with using their generating functions. Interesting results in the field of obtaining new identities for polynomials can be found in some recent works by Simsek [18–20], Kim et al. [21, 22], and Ryoo [23–25].

Another trend in study of polynomials is getting new representation and explicit formulas for those polynomials. For instance, Qi has recently established explicit

formulas for the generalized Motzkin numbers in [26] and the central Delannoy numbers in [27]. One can find interesting results in papers of Srivastava [28, 29], Cenkci [30], and Boyadzhiev [31].

In this chapter, we obtain some interesting properties of polynomials defined by generating functions of the form F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup> . As an application, we give some new identities for the Bernoulli, Euler, Frobenius-Euler, Sylvester, Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials.

According to Stanley [32], ordinary generating functions are defined as follows:

Definition 1. An ordinary generating function of the sequence að Þ<sup>n</sup> <sup>n</sup>≥<sup>0</sup> is the formal power series

$$A(\mathbf{x}) = a\_0 + a\_1 \mathbf{x} + a\_2 \mathbf{x}^2 + \dots = \sum\_{n \ge 0} a\_n \mathbf{x}^n. \tag{1}$$

an =

DOI: http://dx.doi.org/10.5772/intechopen.82370

∑ n k= 1

8 < :

composita <sup>A</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> by using the following formula ([35]):

product of the powers of generating functions F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup>

Dn = Dnð Þ x; α = ∑

B tð Þ= ∑ n > 0 1 n

An�<sup>m</sup>ð Þ mx; <sup>m</sup><sup>α</sup> <sup>k</sup>

B tð Þ= tA tð Þ and C tð Þ= ∑<sup>n</sup>≥<sup>0</sup> Cnt

3.We have the following identities

∑ n m = k

> ∑ n m = k

where δn,k is the Kronecker delta.

m n

ð Þ B tð Þ <sup>k</sup> <sup>=</sup> ð Þ tA tð Þ <sup>k</sup> <sup>=</sup> <sup>t</sup>

= t <sup>k</sup> ∑ n≥0

Hence, the composita of B tð Þ= tA tð Þ is

Theorem 1. If A tð Þ is a generating function of the following form:

A tð Þ<sup>=</sup> F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup> <sup>=</sup> <sup>∑</sup>

1. For the composition of generating functions D tð Þ= CBt ð Þ ð Þ = ∑<sup>n</sup>≥<sup>0</sup> Bnt

n k= 1

n, we have

2. For the compositional inverse generating function B tð Þ of B tð Þ= tA tð Þ, we have

m

Proof. First we get the k-th power of the generating function B tð Þ= tA tð Þ

Anð Þ kx; kα t

<sup>k</sup>ð Þ F tð Þ xkð Þ G tð Þ <sup>α</sup><sup>k</sup> <sup>=</sup>

<sup>n</sup> = ∑ n≥k

2. Main results

then:

and

51

<sup>A</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>∑</sup>

n m = k

r0, forn = 0;

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions…

If we consider the composition A xð Þ<sup>=</sup> RFx ð Þ ð Þ <sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 anxn of generating functions R xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 rnx<sup>n</sup> and F xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 <sup>f</sup> <sup>n</sup>xn, then we can get the values of the

Let us consider a special case of generating functions that can be presented as the

n≥0

An�<sup>1</sup>ð Þ �nx; �nα t

Anð Þ x; α t

An�<sup>k</sup>ð Þ kx; kα Ck, D<sup>0</sup> = C0; (9)

Am�<sup>k</sup>ð Þ �mx; �mα = δn,k (11)

An�<sup>m</sup>ð Þ �nx; �nα Am�<sup>k</sup>ð Þ kx; kα = δn,k, (12)

An�<sup>k</sup>ð Þ kx; kα t

n:

functions, we obtain several properties, which are given in the following theorem:

<sup>F</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> rk, otherwise:

<sup>F</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>m</sup> <sup>R</sup><sup>Δ</sup>ð Þ <sup>m</sup>; <sup>k</sup> : (7)

. For such generating

n, (8)

n; (10)

n, where

(6)

Kruchinin et al. [33–35] introduced the mathematical notion of the composita of a given generating function, which can be used for calculating the coefficients of a composition of generating functions.

Definition 2. The composita of the generating function F xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 <sup>f</sup> <sup>n</sup>x<sup>n</sup> is the function with two variables

$$F^{\Delta}(n,k) = \sum\_{\pi\_k \in \mathcal{C}\_n} f\_{\lambda\_1} f\_{\lambda\_2} \cdots f\_{\lambda\_k},\tag{2}$$

where Cn is the set of all compositions of an integer n and π<sup>k</sup> is the composition n into k parts such that ∑<sup>k</sup> <sup>i</sup><sup>=</sup> <sup>1</sup>λ<sup>i</sup> = n.

Using the expression of the composita of a given generating function <sup>F</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> , we can get powers of the generating function F xð Þ:

$$\left(F(\boldsymbol{\kappa})\right)^{k} = \sum\_{n \geq k} F^{\Delta}(n, k) \boldsymbol{\kappa}^{n}. \tag{3}$$

Compositae also can be used for calculating the coefficients of generating functions obtained by addition, multiplication, composition, reciprocation, and compositional inversion of generating functions (for details see [33–35]).

By the reciprocal generating function we mean the following [6]:

Definition 1. A reciprocal generating function A xð Þ of a generating function B xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup>≥<sup>0</sup>bnxn is a power series such that satisfies the following condition:

$$A(\mathfrak{x})B(\mathfrak{x}) = \mathbf{1}.\tag{4}$$

By the compositional inverse generating function we mean the following:

Definition 2. A compositional inverse F xð Þ of generating function F xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 <sup>f</sup> <sup>n</sup>xn with <sup>f</sup>ð Þ<sup>1</sup> 6¼ 0 is a power series such that satisfies the following condition:

$$F\left(\overline{F(\varkappa)}\right) = \infty.\tag{5}$$

Also the compositional inverse can be written as <sup>F</sup>½ � �<sup>1</sup> ð Þ <sup>x</sup> or F xð Þ <sup>=</sup> RevF. For example, we will use the following formulas:

If we consider the composition A xð Þ<sup>=</sup> RFx ð Þ ð Þ <sup>=</sup> <sup>∑</sup><sup>n</sup>≥<sup>0</sup> anx<sup>n</sup> of generating functions R xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup>≥<sup>0</sup> rnxn and F xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 <sup>f</sup> <sup>n</sup>xn, then we can get the values of the coefficients an by using the following formula ([35], Eq. (17)):

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions… DOI: http://dx.doi.org/10.5772/intechopen.82370

$$a\_n = \begin{cases} r\_0, & \text{for } n = 0; \\ \sum\_{k=1}^n F^\Delta(n,k)r\_k, & \text{otherwise.} \end{cases} \tag{6}$$

If we consider the composition A xð Þ<sup>=</sup> RFx ð Þ ð Þ <sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 anxn of generating functions R xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 rnx<sup>n</sup> and F xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 <sup>f</sup> <sup>n</sup>xn, then we can get the values of the composita <sup>A</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> by using the following formula ([35]):

$$A^{\Delta}(n,k) = \sum\_{m=k}^{n} F^{\Delta}(n,m) R^{\Delta}(m,k). \tag{7}$$

#### 2. Main results

formulas for the generalized Motzkin numbers in [26] and the central Delannoy numbers in [27]. One can find interesting results in papers of Srivastava [28, 29],

identities for the Bernoulli, Euler, Frobenius-Euler, Sylvester, Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials.

A xð Þ<sup>=</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1<sup>x</sup> <sup>þ</sup> <sup>a</sup>2x<sup>2</sup> <sup>þ</sup> … <sup>=</sup> <sup>∑</sup>

<sup>F</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>∑</sup>

ð Þ F xð Þ <sup>k</sup> <sup>=</sup> <sup>∑</sup>

sitional inversion of generating functions (for details see [33–35]). By the reciprocal generating function we mean the following [6]:

<sup>i</sup><sup>=</sup> <sup>1</sup>λ<sup>i</sup> = n.

we can get powers of the generating function F xð Þ:

In this chapter, we obtain some interesting properties of polynomials defined by

According to Stanley [32], ordinary generating functions are defined as follows: Definition 1. An ordinary generating function of the sequence að Þ<sup>n</sup> <sup>n</sup>≥<sup>0</sup> is the formal

Kruchinin et al. [33–35] introduced the mathematical notion of the composita of a given generating function, which can be used for calculating the coefficients of a

Definition 2. The composita of the generating function F xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 <sup>f</sup> <sup>n</sup>x<sup>n</sup> is the

where Cn is the set of all compositions of an integer n and π<sup>k</sup> is the composition n

Using the expression of the composita of a given generating function <sup>F</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> ,

n≥k

Definition 1. A reciprocal generating function A xð Þ of a generating function B xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup>≥<sup>0</sup>bnxn is a power series such that satisfies the following condition:

By the compositional inverse generating function we mean the following:

F xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 <sup>f</sup> <sup>n</sup>xn with <sup>f</sup>ð Þ<sup>1</sup> 6¼ 0 is a power series such that satisfies the following

F F xð Þ 

Also the compositional inverse can be written as <sup>F</sup>½ � �<sup>1</sup> ð Þ <sup>x</sup> or F xð Þ <sup>=</sup> RevF.

If we consider the composition A xð Þ<sup>=</sup> RFx ð Þ ð Þ <sup>=</sup> <sup>∑</sup><sup>n</sup>≥<sup>0</sup> anx<sup>n</sup> of generating functions R xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup>≥<sup>0</sup> rnxn and F xð Þ<sup>=</sup> <sup>∑</sup><sup>n</sup> > 0 <sup>f</sup> <sup>n</sup>xn, then we can get the values of the

For example, we will use the following formulas:

coefficients an by using the following formula ([35], Eq. (17)):

Definition 2. A compositional inverse F xð Þ of generating function

Compositae also can be used for calculating the coefficients of generating functions obtained by addition, multiplication, composition, reciprocation, and compo-

f <sup>λ</sup><sup>1</sup> f <sup>λ</sup><sup>2</sup>

⋯ f <sup>λ</sup><sup>k</sup>

πk∈Cn

n≥0

. As an application, we give some new

anx<sup>n</sup>: (1)

, (2)

<sup>F</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> xn: (3)

A xð ÞB xð Þ= 1: (4)

= x: (5)

Cenkci [30], and Boyadzhiev [31].

Polynomials - Theory and Application

composition of generating functions.

function with two variables

into k parts such that ∑<sup>k</sup>

condition:

50

power series

generating functions of the form F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup>

Let us consider a special case of generating functions that can be presented as the product of the powers of generating functions F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup> . For such generating functions, we obtain several properties, which are given in the following theorem:

Theorem 1. If A tð Þ is a generating function of the following form:

$$A(t) = F(t)^{\mathbf{x}} \cdot G(t)^{a} = \sum\_{n \ge 0} A\_{n}(\mathbf{x}, a) t^{n},\tag{8}$$

then:

1. For the composition of generating functions D tð Þ= CBt ð Þ ð Þ = ∑<sup>n</sup>≥<sup>0</sup> Bnt n, where B tð Þ= tA tð Þ and C tð Þ= ∑<sup>n</sup>≥<sup>0</sup> Cnt n, we have

$$D\_n = D\_n(\infty, a) = \sum\_{k=1}^n A\_{n-k}(k\infty, ka) \mathbf{C}\_k, \quad D\_0 = \mathbf{C}\_0;\tag{9}$$

2. For the compositional inverse generating function B tð Þ of B tð Þ= tA tð Þ, we have

$$\overline{B}(t) = \sum\_{n>0} \frac{1}{n} A\_{n-1}(-n\infty, -na)t^n;\tag{10}$$

3.We have the following identities

$$\sum\_{m=k}^{n} A\_{n-m}(m\infty, m\alpha) \frac{k}{m} A\_{m-k}(-m\infty, -m\alpha) = \delta\_{n,k} \tag{11}$$

and

$$\sum\_{m=k}^{n} \frac{m}{n} A\_{n-m}(-n\infty, -na) A\_{m-k}(k\infty, ka) = \delta\_{n,k} \tag{12}$$

where δn,k is the Kronecker delta. Proof. First we get the k-th power of the generating function B tð Þ= tA tð Þ

$$\begin{aligned} \left(B(t)\right)^k &= \left(tA(t)\right)^k = t^k \left(F(t)\right)^{\times k} \left(G(t)\right)^{ak} =\\ &= t^k \sum\_{n\geq 0} A\_n(k\infty, k\alpha) t^n = \sum\_{n\geq k} A\_{n-k}(k\infty, k\alpha) t^n. \end{aligned}$$

Hence, the composita of B tð Þ= tA tð Þ is

$$B^{\Delta}(n,k) = A\_{n-k}(k\infty, k\alpha). \tag{13}$$

2.1 Generalized Bernoulli polynomials

DOI: http://dx.doi.org/10.5772/intechopen.82370

B tð Þ ; x; α = e

n! ð Þ n þ i !

The triangular form of this composita is

1 2x � α 2

<sup>12</sup>x<sup>2</sup> � <sup>12</sup>α<sup>x</sup> <sup>þ</sup> <sup>3</sup>α<sup>2</sup> � <sup>α</sup> 24

<sup>8</sup>x<sup>3</sup> � <sup>12</sup>αx<sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> ð Þ <sup>2</sup> � <sup>2</sup><sup>α</sup> <sup>x</sup> � <sup>α</sup><sup>3</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> 48

The triangular form of this composita is

1 �2x þ α 2

<sup>36</sup>x<sup>2</sup> � <sup>36</sup>α<sup>x</sup> <sup>þ</sup> <sup>9</sup>a<sup>2</sup> <sup>þ</sup> <sup>α</sup>

�<sup>32</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>48</sup>αx<sup>2</sup> � <sup>24</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>α</sup> <sup>x</sup> <sup>þ</sup> <sup>4</sup>α<sup>3</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> 12

> ∑ n m = k

m n

DΔ

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

D tð Þ of D tð Þ= tB tð Þ ; x; α is

polynomials:

and

53

xt t et � 1 <sup>α</sup>

> n þ α n � i

i þ α � 1

According to Eq. (13), the composita for the generating function

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>B</sup>ð Þ <sup>k</sup><sup>α</sup>

Using Eq. (17), the composita for the compositional inverse generating function

Bð Þ �n<sup>α</sup> <sup>n</sup>�<sup>k</sup> ð Þ �nx

<sup>24</sup> �2<sup>x</sup> <sup>þ</sup> <sup>α</sup> <sup>1</sup>

Bð Þ <sup>k</sup><sup>α</sup> <sup>m</sup>�<sup>k</sup>ð Þ kx

Also we can get the following new identities for the generalized Bernoulli

<sup>B</sup>ð Þ �n<sup>α</sup> <sup>n</sup>�<sup>m</sup> ð Þ �nx ð Þ n � m !

function [37, 38]:

where

Bð Þ <sup>α</sup>

<sup>n</sup> ð Þ x = ∑

D tð Þ= tB tð Þ ; x; α is

n i = 0

The generalized Bernoulli polynomials are defined by the following generating

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions…

= ∑ n≥0 Bð Þ <sup>α</sup> <sup>n</sup> ð Þ x t n n!

ð Þ �<sup>1</sup> <sup>j</sup> <sup>i</sup> j 

1

<sup>24</sup>x<sup>2</sup> � <sup>24</sup>α<sup>x</sup> <sup>þ</sup> <sup>6</sup>a<sup>2</sup> � <sup>α</sup> 12

2x � α 1

ð Þ <sup>n</sup> � <sup>k</sup> ! : (24)

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k (25)

1

<sup>48</sup>x<sup>2</sup> � <sup>48</sup>α<sup>x</sup> <sup>þ</sup> <sup>12</sup>α<sup>2</sup> <sup>þ</sup> <sup>α</sup> 12

ð Þ <sup>x</sup> <sup>þ</sup> <sup>j</sup> <sup>n</sup> <sup>þ</sup> <sup>i</sup>

: (23)

6x � 3α 2

1

�6x þ 3α 2

1

i j = 0

<sup>n</sup>�<sup>k</sup>ð Þ kx ð Þ n � k !

, (21)

: (22)

= e <sup>t</sup> ð Þ<sup>x</sup> <sup>t</sup> et � 1 <sup>α</sup>

i <sup>∑</sup>

Using Eqs. (6) and (13), we get Eq. (9).

According to [36], the composita of the compositional inverse generating function A tð Þ of A tð Þ= ∑<sup>n</sup> > 0 ant <sup>n</sup> is

$$
\overline{A}^{\Delta}(n,k) = \frac{k}{n} R^{\Delta}(2n-k,n),\tag{14}
$$

where <sup>R</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> is the composita of the generating function R tð Þ<sup>=</sup> <sup>t</sup> 2 A tð Þ.

For getting the composita of the compositional inverse generating function B tð Þ of B tð Þ= tA tð Þ, we need to know the composita of the generating function

$$R(t) = \frac{t^2}{B(t)} = \frac{t^2}{tA(t)} = \frac{t}{A(t)}.\tag{15}$$

Then we get the <sup>k</sup>-th power of the generating function R tð Þ<sup>=</sup> <sup>t</sup> A tð Þ

$$\begin{split} \left( R(t) \right)^{k} &= \left( \frac{t}{A(t)} \right)^{k} = t^{k} (F(t))^{-\infty k} (G(t))^{-ak} = \\ &= t^{k} \sum\_{n \geq 0} A\_{n} (-k\infty, -ka) t^{n} = \sum\_{n \geq k} A\_{n-k} (-k\infty, -ka) t^{n}. \end{split} \tag{16}$$

Hence, the composita of Eq. (15) is

$$R^{\Delta}(n,k) = A\_{n-k}(-k\infty, -ka). \tag{17}$$

Using Eqs. (14) and (17), we get

$$\overline{B}^{\Delta}(n,k) = \frac{k}{n} R^{\Delta}(2n-k,n) = \frac{k}{n} A\_{2n-k-n}(-n\infty, -na) = \frac{k}{n} A\_{n-k}(-n\infty, -na). \tag{18}$$

For k= 1, we get Eq. (10).

Applying Eq. (7) for the composition C tð Þ<sup>=</sup> <sup>B</sup> B tð Þ <sup>=</sup> <sup>t</sup>, we get

$$\begin{aligned} C^{\Delta}(n,k) &= \sum\_{m=k}^{n} \overline{B}^{\Delta}(n,m) B^{\Delta}(m,k) = \\\\ &= \sum\_{m=k}^{n} \frac{m}{n} A\_{n-m}(-n\infty, -n\alpha) A\_{m-k}(k\infty, k\alpha) = \delta\_{n,k} \end{aligned} \tag{19}$$

Applying Eq. (7) for the composition D tð Þ= BBt ð Þ ð Þ = x, we get

$$\begin{split} D^{\Delta}(n,k) &= \sum\_{m=k}^{n} B^{\Delta}(n,m) \overline{B}^{\Delta}(m,k) = \\ &= \sum\_{m=k}^{n} A\_{n-m}(m\infty,m\alpha) \frac{k}{m} A\_{m-k}(-m\infty,-ma) = \delta\_{n,k} .\end{split} \tag{20}$$

□

As an application of Theorem 1, we present several examples of its usage for such polynomials as the Bernoulli, Euler, Frobenius-Euler, Sylvester, Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner.

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions… DOI: http://dx.doi.org/10.5772/intechopen.82370

#### 2.1 Generalized Bernoulli polynomials

The generalized Bernoulli polynomials are defined by the following generating function [37, 38]:

$$B(t, \infty, a) = e^{\operatorname{xt}} \left( \frac{t}{e^t - 1} \right)^a = (e^t)^{\operatorname{\boldsymbol{x}}} \left( \frac{t}{e^t - 1} \right)^a = \sum\_{n \ge 0} B\_n^{(a)}(\infty) \frac{t^n}{n!},\tag{21}$$

where

<sup>B</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> An�kð Þ kx; <sup>k</sup><sup>α</sup> : (13)

<sup>R</sup><sup>Δ</sup>ð Þ <sup>2</sup><sup>n</sup> � <sup>k</sup>; <sup>n</sup> , (14)

An � <sup>k</sup>ð Þ �kx; �kα t

<sup>R</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> An � <sup>k</sup>ð Þ �kx; �k<sup>α</sup> : (17)

n

Am�<sup>k</sup>ð Þ �mx; �mα = δn,k:

An�<sup>m</sup>ð Þ �nx; �nα Am�<sup>k</sup>ð Þ kx; kα = δn,k:

2 A tð Þ.

A tð Þ: (15)

A tð Þ

n:

An � <sup>k</sup>ð Þ �nx; �nα : (18)

(16)

(19)

(20)

□

According to [36], the composita of the compositional inverse generating func-

For getting the composita of the compositional inverse generating function B tð Þ

2 tA tð Þ <sup>=</sup> <sup>t</sup>

<sup>k</sup>ð Þ F tð Þ �xkð Þ G tð Þ �α<sup>k</sup> <sup>=</sup>

<sup>n</sup> = ∑ n≥k

<sup>A</sup>2<sup>n</sup> � <sup>k</sup> � <sup>n</sup>ð Þ �nx; �n<sup>α</sup> <sup>=</sup> <sup>k</sup>

Using Eqs. (6) and (13), we get Eq. (9).

ð Þ R tð Þ <sup>k</sup> <sup>=</sup> <sup>t</sup>

= t <sup>k</sup> ∑ n≥0

Hence, the composita of Eq. (15) is

Using Eqs. (14) and (17), we get

For k= 1, we get Eq. (10).

<sup>R</sup><sup>Δ</sup>ð Þ <sup>2</sup><sup>n</sup> � <sup>k</sup>; <sup>n</sup> <sup>=</sup> <sup>k</sup>

<sup>C</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>∑</sup>

n m = k B Δ

= ∑ n m = k

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>∑</sup>

B Δ

52

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

A tð Þ <sup>k</sup>

<sup>n</sup> is

AΔ

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

where <sup>R</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> is the composita of the generating function R tð Þ<sup>=</sup> <sup>t</sup>

of B tð Þ= tA tð Þ, we need to know the composita of the generating function

2 B tð Þ <sup>=</sup> <sup>t</sup>

Then we get the <sup>k</sup>-th power of the generating function R tð Þ<sup>=</sup> <sup>t</sup>

Anð Þ �kx; �kα t

R tð Þ<sup>=</sup> <sup>t</sup>

= t

n

m n

n m = k

= ∑ n m = k

Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner.

Applying Eq. (7) for the composition C tð Þ<sup>=</sup> <sup>B</sup> B tð Þ <sup>=</sup> <sup>t</sup>, we get

Applying Eq. (7) for the composition D tð Þ= BBt ð Þ ð Þ = x, we get

<sup>B</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>m</sup> <sup>B</sup>

ð Þ <sup>n</sup>; <sup>m</sup> <sup>B</sup><sup>Δ</sup>ð Þ <sup>m</sup>; <sup>k</sup> <sup>=</sup>

Δ

An�<sup>m</sup>ð Þ mx; <sup>m</sup><sup>α</sup> <sup>k</sup>

As an application of Theorem 1, we present several examples of its usage for such polynomials as the Bernoulli, Euler, Frobenius-Euler, Sylvester, Laguerre,

ð Þ m; k =

m

tion A tð Þ of A tð Þ= ∑<sup>n</sup> > 0 ant

Polynomials - Theory and Application

$$B\_n^{(a)}(\mathbf{x}) = \sum\_{i=0}^n \frac{n!}{(n+i)!} \binom{n+a}{n-i} \binom{i+a-1}{i} \sum\_{j=0}^i (-1)^j \binom{i}{j} (\mathbf{x}+j)^{n+i}.\tag{22}$$

According to Eq. (13), the composita for the generating function D tð Þ= tB tð Þ ; x; α is

$$D^{\Delta}(n,k) = \frac{B\_{n-k}^{(k\alpha)}(k\omega)}{(n-k)!}.\tag{23}$$

The triangular form of this composita is

1 2x � α 2 1 <sup>12</sup>x<sup>2</sup> � <sup>12</sup>α<sup>x</sup> <sup>þ</sup> <sup>3</sup>α<sup>2</sup> � <sup>α</sup> 24 2x � α 1 <sup>8</sup>x<sup>3</sup> � <sup>12</sup>αx<sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> ð Þ <sup>2</sup> � <sup>2</sup><sup>α</sup> <sup>x</sup> � <sup>α</sup><sup>3</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> 48 <sup>24</sup>x<sup>2</sup> � <sup>24</sup>α<sup>x</sup> <sup>þ</sup> <sup>6</sup>a<sup>2</sup> � <sup>α</sup> 12 6x � 3α 2 1

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tB tð Þ ; x; α is

$$\overline{D}^{\Delta}(n,k) = \frac{k}{n} \frac{B\_{n-k}^{(-n\alpha)}(-n\alpha)}{(n-k)!}.\tag{24}$$

The triangular form of this composita is

$$\begin{array}{cccc} 1 & & & 1 \\ \hline & 2x+a & & 1 \\ \hline & 2 & & & \\ \hline & 24 & & & -2x+a & & \\ \hline & & 12 & & & \\ \end{array}$$
  $3\csc^2 - 36x^2 + 9a^2 + a$  
$$\begin{array}{cccc} -32x^3 + 48a\csc^2 - (24a^2 + 2a)x + 4a^3 + a^2 & & 48x^2 - 48ax + 12a^2 + a & & 48x^2 - 6x + 3a \\ \hline & & 12 & & & 2 \\ \end{array}$$

Also we can get the following new identities for the generalized Bernoulli polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} \frac{B\_{n-m}^{(-n\alpha)}(-n\infty)}{(n-m)!} \frac{B\_{m-k}^{(k\alpha)}(k\omega)}{(m-k)!} = \delta\_{n,k} \tag{25}$$

and

Polynomials - Theory and Application

$$\sum\_{m=k}^{n} \frac{B\_{n-m}^{(ma)}(m\infty)}{(n-m)!} \frac{k}{m} \frac{B\_{m-k}^{(-ma)}(-m\infty)}{(m-k)!} = \delta\_{n,k} \,. \tag{26}$$

and

function [39]:

where

Hð Þ <sup>α</sup>

D tð Þ= tH tð Þ ; x; α; λ is

D tð Þ of D tð Þ= tH tð Þ ; x; α; λ is

55

∑ n m = k

xt 1 � λ et � λ <sup>α</sup>

> 1 ð Þ <sup>1</sup> � <sup>λ</sup> <sup>i</sup>

n i= 0

The triangular form of this composita is

2.3 Frobenius-Euler polynomials

DOI: http://dx.doi.org/10.5772/intechopen.82370

H tð Þ ; x; α; λ = e

<sup>n</sup> ð Þ x; λ = ∑

<sup>E</sup>ð Þ <sup>m</sup><sup>α</sup> <sup>n</sup>�mð Þ mx ð Þ n � m !

k m

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions…

The Frobenius-Euler polynomials are defined by the following generating

<sup>t</sup> ð Þ<sup>x</sup> <sup>1</sup> � <sup>λ</sup> et � λ <sup>α</sup>

i þ α � 1 i <sup>∑</sup> = ∑ n≥0

ð Þ �<sup>1</sup> <sup>j</sup> <sup>i</sup> j 

i j = 0

<sup>n</sup>�<sup>k</sup>ð Þ kx; <sup>λ</sup>

Using Eq. (17), the composita for the compositional inverse generating function

Hð Þ �n<sup>α</sup>

<sup>n</sup>�<sup>k</sup> ð Þ �nx; <sup>λ</sup>

<sup>2</sup>λ<sup>2</sup> � <sup>4</sup><sup>λ</sup> <sup>þ</sup> <sup>2</sup> � ð Þ <sup>2</sup><sup>λ</sup> � <sup>2</sup> <sup>x</sup> <sup>þ</sup> <sup>2</sup><sup>α</sup>

Also we can get the following new identities for the Frobenius-Euler polynomials:

Hð Þ <sup>k</sup><sup>α</sup> <sup>m</sup>�<sup>k</sup>ð Þ kx; <sup>λ</sup> Hð Þ <sup>α</sup> <sup>n</sup> ð Þ <sup>x</sup>; <sup>λ</sup> <sup>t</sup>

n n!

ð Þ <sup>x</sup> <sup>þ</sup> <sup>j</sup> <sup>n</sup>

ð Þ <sup>n</sup> � <sup>k</sup> ! : (35)

1

ð Þ 2λ � 2 x þ 2α λ � 1

ð Þ <sup>n</sup> � <sup>k</sup> ! : (36)

1

λ � 1

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k (37)

1

1

, (33)

: (34)

= e

According to Eq. (13), the composita for the generating function

1 ð Þ λ � 1 x þ α λ � 1

<sup>λ</sup><sup>2</sup> � <sup>2</sup><sup>λ</sup> <sup>þ</sup> <sup>1</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup><sup>λ</sup> � <sup>2</sup> <sup>α</sup><sup>x</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> λα <sup>2</sup>λ<sup>2</sup> � <sup>4</sup><sup>λ</sup> <sup>þ</sup> <sup>2</sup>

DΔ

1

� ð Þ <sup>λ</sup> � <sup>1</sup> <sup>x</sup> <sup>þ</sup> <sup>α</sup> λ � 1

<sup>3</sup>λ<sup>2</sup> � <sup>6</sup><sup>λ</sup> <sup>þ</sup> <sup>3</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>6</sup><sup>λ</sup> � <sup>6</sup> <sup>α</sup><sup>x</sup> <sup>þ</sup> <sup>3</sup>α<sup>2</sup> � λα

<sup>H</sup>ð Þ �n<sup>α</sup> <sup>n</sup>�<sup>m</sup> ð Þ �nx; <sup>λ</sup> ð Þ n � m !

The triangular form of this composita is

∑ n m = k m n

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>H</sup>ð Þ <sup>k</sup><sup>α</sup>

Eð Þ �m<sup>α</sup> <sup>m</sup>�<sup>k</sup> ð Þ �mx

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k: (32)

#### 2.2 Generalized Euler polynomials

The generalized Euler polynomials are defined by the following generating function [37]:

$$E(t, \boldsymbol{x}, a) = e^{\boldsymbol{x}t} \left(\frac{2}{e^t + 1}\right)^a = (e^t)^{\boldsymbol{x}} \left(\frac{2}{e^t + 1}\right)^a = \sum\_{n \ge 0} E\_n^{(a)}(\boldsymbol{x}) \frac{t^n}{n!},\tag{27}$$

where

$$E\_n^{(a)}(\mathbf{x}) = \sum\_{i=0}^n \frac{1}{\mathfrak{Z}^i} \binom{i+a-1}{i} \sum\_{j=0}^i (-1)^j \binom{i}{j} (\mathbf{x}+j)^n. \tag{28}$$

According to Eq. (13), the composita for the generating function D tð Þ= tE tð Þ ; x; α is

$$D^{\Delta}(n,k) = \frac{E\_{n-k}^{(ka)}(k\infty)}{(n-k)!}.\tag{29}$$

The triangular form of this composita is

1 2x � α 2 1 <sup>4</sup>x<sup>2</sup> � <sup>4</sup>α<sup>x</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> � <sup>α</sup> 8 2x � α 1 <sup>8</sup>x<sup>3</sup> � <sup>12</sup>αx<sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> ð Þ <sup>2</sup> � <sup>6</sup><sup>α</sup> <sup>x</sup> � <sup>α</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup>α<sup>2</sup> 48 <sup>8</sup>x<sup>2</sup> � <sup>8</sup>α<sup>x</sup> <sup>þ</sup> <sup>2</sup>a<sup>2</sup> � <sup>α</sup> 4 6x � 3α 2 1

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tE tð Þ ; x; α is

$$\overline{D}^{\Delta}(n,k) = \frac{k}{n} \frac{E\_{n-k}^{(-n\alpha)}(-n\alpha)}{(n-k)!}.\tag{30}$$

The triangular form of this composita is

$$\begin{array}{cccc} 1 & & & 1 \\ \hline -2\mathbf{x} + a & & & 1 \\ \hline 2\mathbf{x}^2 - 12a\mathbf{x} + 3a^2 + a & & & -2\mathbf{x} + a & & \\ \hline 8 & & & -2\mathbf{x} + a & & & \\ -32\mathbf{x}^3 + 48a\mathbf{x}^2 - (24a^2 + 6a)\mathbf{x} + 4a^3 + 3a^2 & \frac{16\mathbf{x}^2 - 16a\mathbf{x} + 4a^2 + a}{4} & \frac{-6\mathbf{x} + 3a}{2} & 1 \\ \end{array}$$

Also we can get the following new identities for the generalized Euler polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} \frac{E\_{n-m}^{(-n\alpha)}(-n\infty)}{(n-m)!} \frac{E\_{m-k}^{(k\alpha)}(k\omega)}{(m-k)!} = \delta\_{n,k} \tag{31}$$

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions… DOI: http://dx.doi.org/10.5772/intechopen.82370

and

∑ n m = k

> xt 2 et þ 1 <sup>α</sup>

n i= 0

1 2i

2.2 Generalized Euler polynomials

Polynomials - Theory and Application

E tð Þ ; x; α = e

Eð Þ <sup>α</sup>

<sup>n</sup> ð Þ x = ∑

The triangular form of this composita is

1 2x � α 2

<sup>4</sup>x<sup>2</sup> � <sup>4</sup>α<sup>x</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> � <sup>α</sup> 8

<sup>8</sup>x<sup>3</sup> � <sup>12</sup>αx<sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> ð Þ <sup>2</sup> � <sup>6</sup><sup>α</sup> <sup>x</sup> � <sup>α</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup>α<sup>2</sup> 48

The triangular form of this composita is

1 �2x þ α 2

<sup>12</sup>x<sup>2</sup> � <sup>12</sup>α<sup>x</sup> <sup>þ</sup> <sup>3</sup>a<sup>2</sup> <sup>þ</sup> <sup>α</sup>

�32x<sup>3</sup> <sup>þ</sup> <sup>48</sup>αx<sup>2</sup> � <sup>24</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> <sup>x</sup> <sup>þ</sup> <sup>4</sup>α<sup>3</sup> <sup>þ</sup> <sup>3</sup>α<sup>2</sup> 12

> ∑ n m = k

m n

<sup>E</sup>ð Þ �n<sup>α</sup> <sup>n</sup>�<sup>m</sup> ð Þ �nx ð Þ n � m !

DΔ

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

function [37]:

where

D tð Þ= tE tð Þ ; x; α is

D tð Þ of D tð Þ= tE tð Þ ; x; α is

54

<sup>B</sup>ð Þ <sup>m</sup><sup>α</sup> <sup>n</sup>�mð Þ mx ð Þ n � m !

k m

The generalized Euler polynomials are defined by the following generating

<sup>t</sup> ð Þ<sup>x</sup> <sup>2</sup>

et þ 1 <sup>α</sup>

> ∑ i j = 0

<sup>n</sup>�<sup>k</sup>ð Þ kx ð Þ n � k !

= ∑ n≥0 Eð Þ <sup>α</sup> <sup>n</sup> ð Þ x t n n!

ð Þ �<sup>1</sup> <sup>j</sup> <sup>i</sup> j 

1

<sup>8</sup>x<sup>2</sup> � <sup>8</sup>α<sup>x</sup> <sup>þ</sup> <sup>2</sup>a<sup>2</sup> � <sup>α</sup> 4

2x � α 1

ð Þ <sup>n</sup> � <sup>k</sup> ! : (30)

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k (31)

1

<sup>16</sup>x<sup>2</sup> � <sup>16</sup>α<sup>x</sup> <sup>þ</sup> <sup>4</sup>α<sup>2</sup> <sup>þ</sup> <sup>α</sup> 4

= e

i þ α � 1 i 

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>E</sup>ð Þ <sup>k</sup><sup>α</sup>

Using Eq. (17), the composita for the compositional inverse generating function

Eð Þ �n<sup>α</sup> <sup>n</sup>�<sup>k</sup> ð Þ �nx

<sup>8</sup> �2<sup>x</sup> <sup>þ</sup> <sup>α</sup> <sup>1</sup>

Also we can get the following new identities for the generalized Euler polynomials:

Eð Þ <sup>k</sup><sup>α</sup> <sup>m</sup>�<sup>k</sup>ð Þ kx

According to Eq. (13), the composita for the generating function

Bð Þ �m<sup>α</sup> <sup>m</sup>�<sup>k</sup> ð Þ �mx

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k: (26)

ð Þ <sup>x</sup> <sup>þ</sup> <sup>j</sup> <sup>n</sup>

: (29)

6x � 3α 2

1

�6x þ 3α 2

1

, (27)

: (28)

$$\sum\_{m=k}^{n} \frac{E\_{n-m}^{(ma)}(m\infty)}{(n-m)!} \frac{k}{m} \frac{E\_{m-k}^{(-ma)}(-m\infty)}{(m-k)!} = \delta\_{n,k} \,. \tag{32}$$

#### 2.3 Frobenius-Euler polynomials

The Frobenius-Euler polynomials are defined by the following generating function [39]:

$$H(t, \mathbf{x}, a, \lambda) = e^{\mathbf{x}t} \left(\frac{\mathbf{1} - \lambda}{e^t - \lambda}\right)^a = (e^t)^{\mathbf{x}} \left(\frac{\mathbf{1} - \lambda}{e^t - \lambda}\right)^a = \sum\_{n \ge 0} H\_n^{(a)}(\mathbf{x}, \lambda) \frac{t^n}{n!},\tag{33}$$

where

$$H\_n^{(a)}(\mathbf{x}, \boldsymbol{\lambda}) = \sum\_{i=0}^n \frac{\mathbf{1}}{(\mathbf{1} - \boldsymbol{\lambda})^i} \binom{i + a - \mathbf{1}}{i} \sum\_{j=0}^i (-\mathbf{1})^j \binom{i}{j} (\mathbf{x} + j)^n. \tag{34}$$

According to Eq. (13), the composita for the generating function D tð Þ= tH tð Þ ; x; α; λ is

$$D^{\Delta}(n,k) = \frac{H\_{n-k}^{(ka)}(k\infty,\lambda)}{(n-k)!}.\tag{35}$$

The triangular form of this composita is

$$\frac{1}{\frac{(\lambda - 1)\mathbf{x} + a}{\lambda - 1}} \quad \text{or} \quad \mathbf{1}$$

$$\frac{(\lambda^2 - 2\lambda + 1)\mathbf{x}^2 + (2\lambda - 2)a\mathbf{x} + a^2 + \lambda a}{2\lambda^2 - 4\lambda + 2} \quad \frac{(2\lambda - 2)\mathbf{x} + 2a}{\lambda - 1} \quad \mathbf{1}$$

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tH tð Þ ; x; α; λ is

$$\overline{D}^{\Delta}(n,k) = \frac{k}{n} \frac{H\_{n-k}^{(-n\alpha)}(-n\infty,\lambda)}{(n-k)!}.\tag{36}$$

The triangular form of this composita is

$$\begin{aligned} 1 \\ -\frac{(\lambda - 1)\mathbf{x} + a}{\lambda - 1} \\\\ \frac{(3\lambda^2 - 6\lambda + 3)\mathbf{x}^2 + (6\lambda - 6)a\mathbf{x} + 3a^2 - \lambda a}{2\lambda^2 - 4\lambda + 2} - \frac{(2\lambda - 2)\mathbf{x} + 2a}{\lambda - 1} \mathbf{1} \end{aligned}$$

Also we can get the following new identities for the Frobenius-Euler polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} \frac{H\_{n-m}^{(-n\alpha)}(-n\varkappa,\lambda)}{(n-m)!} \frac{H\_{m-k}^{(ka)}(k\varkappa,\lambda)}{(m-k)!} = \delta\_{n,k} \tag{37}$$

and

$$\sum\_{m=k}^{n} \frac{H\_{n-m}^{(ma)}(m\infty,\lambda)}{(n-m)!} \frac{k}{m} \frac{H\_{m-k}^{(-ma)}(-m\infty,\lambda)}{(m-k)!} = \delta\_{n,k}.\tag{38}$$

and

function [8]:

where

D tð Þ= tL tð Þ ; x; α is

D tð Þ of D tð Þ= tL tð Þ ; x; α is

nomials:

and

57

∑ n m = k

2.5 Generalized Laguerre polynomials

DOI: http://dx.doi.org/10.5772/intechopen.82370

L tð Þ ; <sup>x</sup>; <sup>α</sup> <sup>=</sup> ð Þ <sup>1</sup> � <sup>t</sup> �α�<sup>1</sup>

Fn�mð Þ mx; <sup>α</sup> <sup>k</sup>

e xt <sup>t</sup>�<sup>1</sup> = e <sup>t</sup> t�1 <sup>x</sup> 1

Lð Þ <sup>α</sup>

The triangular form of this composita is

<sup>n</sup> ð Þ x = ∑

1

<sup>x</sup><sup>2</sup> � ð Þ <sup>2</sup><sup>α</sup> <sup>þ</sup> <sup>4</sup> <sup>x</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>α</sup> <sup>þ</sup> <sup>2</sup>

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

1

<sup>3</sup>x<sup>2</sup> � ð Þ <sup>6</sup><sup>α</sup> <sup>þ</sup> <sup>4</sup> <sup>x</sup> <sup>þ</sup> <sup>3</sup>α<sup>2</sup> <sup>þ</sup> <sup>5</sup><sup>α</sup> <sup>þ</sup> <sup>2</sup> 2

DΔ

The triangular form of this composita is

∑ n m = k

∑ n m = k m n

<sup>L</sup>ð Þ <sup>m</sup>αþm�<sup>1</sup> <sup>n</sup>�<sup>m</sup> ð Þ mx

According to Eq. (13), the composita for the generating function

n i= 0

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>L</sup>ð Þ <sup>k</sup>αþk�<sup>1</sup>

�x þ α þ 1 1

Using Eq. (17), the composita for the compositional inverse generating function

Lð Þ �nα�n�<sup>1</sup>

x � α � 1 1

Also we can get the following new identities for the generalized Laguerre poly-

<sup>L</sup>ð Þ �nα�n�<sup>1</sup> <sup>n</sup>�<sup>m</sup> ð Þ �nx <sup>L</sup>ð Þ <sup>k</sup>αþk�<sup>1</sup>

k m

Lð Þ �mα�m�<sup>1</sup>

<sup>2</sup> �2<sup>x</sup> <sup>þ</sup> <sup>2</sup><sup>α</sup> <sup>þ</sup> 2 1

ð Þ �x i

i!

m

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions…

The generalized Laguerre polynomials are defined by the following generating

1 � t <sup>α</sup>þ<sup>1</sup>

Fm�kð Þ �mx; α = δn,k: (44)

: (46)

<sup>n</sup>�<sup>k</sup> ð Þ kx : (47)

<sup>n</sup>�<sup>k</sup> ð Þ �nx : (48)

2x � 2α � 2 1

<sup>m</sup>�<sup>k</sup> ð Þ kx <sup>=</sup> <sup>δ</sup>n,k (49)

<sup>m</sup>�<sup>k</sup> ð Þ �mx <sup>=</sup> <sup>δ</sup>n,k: (50)

n, (45)

= ∑ n≥0 Lð Þ <sup>α</sup> <sup>n</sup> ð Þ x t

n þ α n � i

#### 2.4 Generalized Sylvester polynomials

The generalized Sylvester polynomials are defined by the following generating function [40]:

$$F(t, \boldsymbol{x}, a) = \left(\mathbf{1} - t\right)^{-\chi} e^{\mathrm{act}} = \left(\frac{e^{at}}{\mathbf{1} - t}\right)^{\chi} = \sum\_{n \ge 0} F\_n(\boldsymbol{x}, a) t^n,\tag{39}$$

where

$$F\_n(\infty, a) = \sum\_{i=0}^n \frac{(a\infty)^{n-i}}{(n-i)!} \binom{i+\varkappa-1}{i}.\tag{40}$$

According to Eq. (13), the composita for the generating function D tð Þ= tF tð Þ ; x; α is

$$D^{\Delta}(n,k) = F\_{n-k}(k\infty,\alpha). \tag{41}$$

The triangular form of this composita is

$$\begin{array}{cccc} 1 & & & & 1 \\ & (a+1)\mathbf{x} & & & \mathbf{1} \\ & \frac{(a^2+2a+1)\mathbf{x}^2+\mathbf{x}}{2} & & & (2a+2)\mathbf{x} & & \mathbf{1} \\ \frac{(a^3+3a^2+3a+1)\mathbf{x}^3+(3a+3)\mathbf{x}^2+2\mathbf{x}}{6} & (2a^2+4a+2)\mathbf{x}^2+\mathbf{x} & (3a+3)\mathbf{x} & \mathbf{1} \\ \end{array}$$

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tF tð Þ ; x; α is

$$
\overline{D}^{\Delta}(n,k) = \frac{k}{n} F\_{n-k}(-n\infty,\alpha). \tag{42}
$$

The triangular form of this composita is

$$\begin{array}{cccc} \mathbf{1} & & & \mathbf{1} \\ & -(a+\mathbf{1})\mathbf{x} & & \mathbf{1} \\ & \frac{(3a^2+6a+3)\mathbf{x}^2-\mathbf{x}}{2} & & -(2a+2)\mathbf{x} & & \mathbf{1} \\ -\frac{(8a^3+24a^2+24a+8)\mathbf{x}^3-(6a+6)\mathbf{x}^2+\mathbf{x}}{3} & (4a^2+8a+4)\mathbf{x}^2-\mathbf{x} & -(3a+3)\mathbf{x} & \mathbf{1} \end{array}$$

Also we can get the following new identities for the generalized Sylvester polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} F\_{n-m}(-n\varkappa, a) F\_{m-k}(k\varkappa, a) = \delta\_{n,k} \tag{43}$$

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions… DOI: http://dx.doi.org/10.5772/intechopen.82370

and

and

function [40]:

where

D tð Þ= tF tð Þ ; x; α is

D tð Þ of D tð Þ= tF tð Þ ; x; α is

nomials:

56

∑ n m = k

F tð Þ ; <sup>x</sup>; <sup>α</sup> <sup>=</sup> ð Þ <sup>1</sup> � <sup>t</sup> �<sup>x</sup>

The triangular form of this composita is

1

<sup>α</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup>

<sup>α</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> ð Þ <sup>3</sup><sup>α</sup> <sup>þ</sup> <sup>3</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>x</sup>

The triangular form of this composita is

1

<sup>3</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> <sup>þ</sup> <sup>3</sup> <sup>x</sup><sup>2</sup> � <sup>x</sup>

� <sup>8</sup>α<sup>3</sup> <sup>þ</sup> <sup>24</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>24</sup><sup>α</sup> <sup>þ</sup> <sup>8</sup> <sup>x</sup><sup>3</sup> � ð Þ <sup>6</sup><sup>α</sup> <sup>þ</sup> <sup>6</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> 3

> ∑ n m = k

m n

DΔ

Fnð Þ x; α = ∑

2.4 Generalized Sylvester polynomials

Polynomials - Theory and Application

<sup>H</sup>ð Þ <sup>m</sup><sup>α</sup> <sup>n</sup>�mð Þ mx; <sup>λ</sup> ð Þ n � m !

k m

e

n i= 0

According to Eq. (13), the composita for the generating function

ð Þ α þ 1 x 1

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

�ð Þ α þ 1 x 1

<sup>2</sup> ð Þ <sup>2</sup><sup>α</sup> <sup>þ</sup> <sup>2</sup> <sup>x</sup> <sup>1</sup>

<sup>2</sup> �ð Þ <sup>2</sup><sup>α</sup> <sup>þ</sup> <sup>2</sup> <sup>x</sup> <sup>1</sup>

Fn�<sup>m</sup>ð Þ �nx; α Fm�<sup>k</sup>ð Þ kx; α = δn,k (43)

Also we can get the following new identities for the generalized Sylvester poly-

Using Eq. (17), the composita for the compositional inverse generating function

<sup>6</sup> <sup>2</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>α</sup> <sup>þ</sup> <sup>2</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> ð Þ <sup>3</sup><sup>α</sup> <sup>þ</sup> <sup>3</sup> <sup>x</sup> <sup>1</sup>

Fn�<sup>k</sup>ð Þ �nx; α : (42)

<sup>4</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>8</sup><sup>α</sup> <sup>þ</sup> <sup>4</sup> <sup>x</sup><sup>2</sup> � <sup>x</sup> �ð Þ <sup>3</sup><sup>α</sup> <sup>þ</sup> <sup>3</sup> <sup>x</sup> <sup>1</sup>

Hð Þ �m<sup>α</sup>

The generalized Sylvester polynomials are defined by the following generating

<sup>α</sup>xt <sup>=</sup> <sup>e</sup><sup>α</sup><sup>t</sup>

ð Þ <sup>α</sup><sup>x</sup> <sup>n</sup>�<sup>i</sup> ð Þ n � i !

1 � t <sup>x</sup>

= ∑ n ≥0

i þ x � 1 i 

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> Fn�<sup>k</sup>ð Þ kx; <sup>α</sup> : (41)

<sup>m</sup>�<sup>k</sup> ð Þ �mx; <sup>λ</sup>

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k: (38)

Fnð Þ x; α t

n, (39)

: (40)

$$\sum\_{m=k}^{n} F\_{n-m}(m\infty, a) \frac{k}{m} F\_{m-k}(-m\infty, a) = \delta\_{n,k}.\tag{44}$$

#### 2.5 Generalized Laguerre polynomials

The generalized Laguerre polynomials are defined by the following generating function [8]:

$$L(t, \boldsymbol{x}, \boldsymbol{a}) = (\mathbf{1} - t)^{-a - 1} \boldsymbol{e}^{\frac{\boldsymbol{x}}{t - 1}} = \left(\boldsymbol{e}^{\frac{t}{t - 1}}\right)^{\boldsymbol{x}} \left(\frac{\mathbf{1}}{\mathbf{1} - t}\right)^{a + 1} = \sum\_{n \ge 0} L\_n^{(a)}(\boldsymbol{x}) t^n,\tag{45}$$

where

$$L\_n^{(a)}(\infty) = \sum\_{i=0}^n \frac{(-\infty)^i}{i!} \binom{n+a}{n-i}.\tag{46}$$

According to Eq. (13), the composita for the generating function D tð Þ= tL tð Þ ; x; α is

$$D^{\Delta}(n,k) = L\_{n-k}^{(ka+k-1)}(k\infty). \tag{47}$$

The triangular form of this composita is

$$\begin{array}{cccc} \mathbf{1} \\ -\mathbf{x} + \mathbf{a} + \mathbf{1} & \mathbf{1} \\ \frac{\mathbf{x}^2 - (2a + 4)\mathbf{x} + a^2 + 3a + 2}{2} & -2\mathbf{x} + 2a + 2 & \mathbf{1} \end{array}$$

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tL tð Þ ; x; α is

$$
\overline{D}^{\Delta}(n,k) = \frac{k}{n} L\_{n-k}^{(-na-n-1)}(-n\infty). \tag{48}
$$

The triangular form of this composita is

$$\begin{array}{cccc} 1 & & & & 1 \\ & \varkappa - a - 1 & & & 1 \\ \hline 3\varkappa^2 - (6\alpha + 4)\varkappa + 3\alpha^2 + 5\alpha + 2 & & 2\varkappa - 2\alpha - 2 & 1 \\ \hline \end{array} \quad 2\varkappa - 2\alpha - 2 \quad \text{or} \quad \frac{1}{2}$$

Also we can get the following new identities for the generalized Laguerre polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} L\_{n-m}^{(-na-n-1)}(-n\infty) L\_{m-k}^{(ka+k-1)}(k\infty) = \delta\_{n,k} \tag{49}$$

and

$$\sum\_{m=k}^{n} L\_{n-m}^{(ma+m-1)}(m\infty) \frac{k}{m} L\_{m-k}^{(-ma-m-1)}(-m\infty) = \delta\_{n,k} \,. \tag{50}$$

#### 2.6 Abel polynomials

The Abel polynomials are defined by the following generating function [8, 41]:

$$A(t, \boldsymbol{x}, a) = e^{\frac{W(at)\boldsymbol{x}}{a}} = \left(e^{\frac{W(at)}{a}}\right)^{\boldsymbol{\mathcal{X}}} = \sum\_{n \ge 0} A\_n(\boldsymbol{\mathcal{x}}, a) \frac{t^n}{n!},\tag{51}$$

where W tð Þ is the Lambert W function and

$$A\_n(\varkappa, a) = \varkappa(\varkappa - a n)^{n-1}.\tag{52}$$

∑ n m = k

DOI: http://dx.doi.org/10.5772/intechopen.82370

B tð Þ ; x = e

Bnð Þ x =

The triangular form of this composita is

1

<sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> 2

<sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup>

<sup>x</sup><sup>4</sup> <sup>þ</sup> <sup>6</sup>x<sup>3</sup> <sup>þ</sup> <sup>15</sup>x<sup>2</sup> <sup>þ</sup> <sup>15</sup><sup>x</sup> 24

The triangular form of this composita is

1

<sup>3</sup>x<sup>2</sup> � <sup>x</sup>

� <sup>16</sup>x<sup>3</sup> � <sup>12</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup>

<sup>125</sup>x<sup>4</sup> � <sup>150</sup>x<sup>3</sup> <sup>þ</sup> <sup>75</sup>x<sup>2</sup> � <sup>15</sup><sup>x</sup>

D tð Þ of D tð Þ= tB tð Þ ; x is

59

2.7 Bessel polynomials

where

An�mð Þ mx; α ð Þ n � m !

<sup>x</sup> <sup>1</sup>� ffiffiffiffiffiffiffi <sup>1</sup>�2<sup>t</sup> <sup>p</sup> ð Þ <sup>=</sup> <sup>e</sup>

> ∑ n k= 1

2 1

DΔ

�x 1

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

<sup>2</sup> �2<sup>x</sup> <sup>1</sup>

<sup>24</sup> � <sup>25</sup>x<sup>3</sup> � <sup>15</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup>

Also we can get the following new identities for the Bessel polynomials:

<sup>6</sup> <sup>4</sup>x<sup>2</sup> � <sup>x</sup> �3<sup>x</sup> <sup>1</sup>

3

8 ><

>:

k m

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions…

The Bessel polynomials are defined by the following generating function [8]:

ð Þ 2n � k � 1 ! ð Þ n � k !ð Þ k � 1 !

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> Bn�<sup>k</sup>ð Þ kx

According to Eq. (13), the composita for the generating function D tð Þ= tB tð Þ ; x is

<sup>1</sup>� ffiffiffiffiffiffiffi <sup>1</sup>�2<sup>t</sup> � � <sup>p</sup> <sup>x</sup>

1, n = 0;

ð Þ n � k !

<sup>6</sup> <sup>2</sup>x<sup>2</sup> <sup>þ</sup> <sup>x</sup> <sup>3</sup><sup>x</sup> <sup>1</sup>

<sup>4</sup>x<sup>3</sup> <sup>þ</sup> <sup>6</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> 3

Using Eq. (17), the composita for the compositional inverse generating function

Bn�<sup>k</sup>ð Þ �nx

= ∑ n≥0

xk

2x 1

<sup>9</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> 2

ð Þ <sup>n</sup> � <sup>k</sup> ! : (60)

<sup>15</sup>x<sup>2</sup> � <sup>3</sup><sup>x</sup>

<sup>2</sup> �4<sup>x</sup> <sup>1</sup>

<sup>2</sup><sup>n</sup>�<sup>k</sup> , n > 0:

Am�kð Þ �mx; α

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k: (56)

Bnð Þ x t n n!

: (59)

4x 1

, (57)

(58)

According to Eq. (13), the composita for the generating function D tð Þ= tA tð Þ ; x; α is

$$D^{\Delta}(n,k) = \frac{A\_{n-k}(k\infty, a)}{(n-k)!}.\tag{53}$$

The triangular form of this composita is

$$\begin{array}{ccccccccc} 1 & & & & & & & \\ & x & & & & & & \\ & \underline{x^2 - 2ax} & & & & & \\ & \underline{x^3 - 6ax^2 + 9a^2x} & & & & & \\ & \underline{x^3 - 6ax^2 + 9a^2x} & & & & & 2x^2 - 2ax & & \end{array}$$

$$\begin{array}{ccccccccc} \underline{x^4 - 12ax^3 + 48a^2x^2 - 64a^3x} & & \underline{4x^3 - 12ax^2 + 9a^2x} & & \underline{9x^2 - 6ax} & & 4x & 1 \\ \end{array}$$

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tA tð Þ ; x; α is

$$\overline{D}^{\Delta}(n,k) = \frac{k}{n} \frac{A\_{n-k}(-n\varkappa, a)}{(n-k)!}.\tag{54}$$

The triangular form of this composita is

1 �x 1 <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>2</sup>α<sup>x</sup> <sup>2</sup> �2<sup>x</sup> <sup>1</sup> � <sup>16</sup>x<sup>3</sup> <sup>þ</sup> <sup>24</sup>αx<sup>2</sup> <sup>þ</sup> <sup>9</sup>α<sup>2</sup><sup>x</sup> <sup>6</sup> <sup>4</sup>x<sup>2</sup> <sup>þ</sup> <sup>2</sup>α<sup>x</sup> �3<sup>x</sup> <sup>1</sup> <sup>125</sup>x<sup>4</sup> <sup>þ</sup> <sup>300</sup>αx<sup>3</sup> <sup>þ</sup> <sup>240</sup>α<sup>2</sup>x<sup>2</sup> <sup>þ</sup> <sup>64</sup>α<sup>3</sup><sup>x</sup> <sup>24</sup> � <sup>25</sup>x<sup>3</sup> <sup>þ</sup> <sup>30</sup>αx<sup>2</sup> <sup>þ</sup> <sup>9</sup>α<sup>2</sup><sup>x</sup> 3 <sup>15</sup>x<sup>2</sup> <sup>þ</sup> <sup>6</sup>α<sup>x</sup> <sup>2</sup> �4<sup>x</sup> <sup>1</sup>

Also we can get the following new identities for the Abel polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} \frac{A\_{n-m}(-n\infty, a)}{(n-m)!} \frac{A\_{m-k}(k\infty, a)}{(m-k)!} = \delta\_{n,k} \tag{55}$$

and

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions… DOI: http://dx.doi.org/10.5772/intechopen.82370

$$\sum\_{m=k}^{n} \frac{A\_{n-m}(m\infty, a)}{(n-m)!} \frac{k}{m} \frac{A\_{m-k}(-m\infty, a)}{(m-k)!} = \delta\_{n,k}.\tag{56}$$

#### 2.7 Bessel polynomials

The Bessel polynomials are defined by the following generating function [8]:

$$B(t,\infty) = e^{\chi\left(1-\sqrt{1-2t}\right)} = \left(e^{1-\sqrt{1-2t}}\right)^{\chi} = \sum\_{n\geq 0} B\_n(\infty) \frac{t^n}{n!},\tag{57}$$

where

2.6 Abel polynomials

Polynomials - Theory and Application

D tð Þ= tA tð Þ ; x; α is

A tð Þ ; x; α = e

where W tð Þ is the Lambert W function and

The triangular form of this composita is

1

<sup>x</sup><sup>2</sup> � <sup>2</sup>α<sup>x</sup> 2

<sup>x</sup><sup>3</sup> � <sup>6</sup>αx<sup>2</sup> <sup>þ</sup> <sup>9</sup>α<sup>2</sup><sup>x</sup>

<sup>x</sup><sup>4</sup> � <sup>12</sup>αx<sup>3</sup> <sup>þ</sup> <sup>48</sup>α<sup>2</sup>x<sup>2</sup> � <sup>64</sup>α<sup>3</sup><sup>x</sup> 24

The triangular form of this composita is

1

<sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>2</sup>α<sup>x</sup>

� <sup>16</sup>x<sup>3</sup> <sup>þ</sup> <sup>24</sup>αx<sup>2</sup> <sup>þ</sup> <sup>9</sup>α<sup>2</sup><sup>x</sup>

<sup>125</sup>x<sup>4</sup> <sup>þ</sup> <sup>300</sup>αx<sup>3</sup> <sup>þ</sup> <sup>240</sup>α<sup>2</sup>x<sup>2</sup> <sup>þ</sup> <sup>64</sup>α<sup>3</sup><sup>x</sup>

and

58

∑ n m = k m n

D tð Þ of D tð Þ= tA tð Þ ; x; α is

Wð Þ αt x <sup>α</sup> = e

According to Eq. (13), the composita for the generating function

x 1

DΔ

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

�x 1

The Abel polynomials are defined by the following generating function [8, 41]:

= ∑ n≥0 Anð Þ <sup>x</sup>; <sup>α</sup> <sup>t</sup>

n n!

ð Þ <sup>n</sup> � <sup>k</sup> ! : (53)

<sup>9</sup>x<sup>2</sup> � <sup>6</sup>α<sup>x</sup> 2

ð Þ <sup>n</sup> � <sup>k</sup> ! : (54)

<sup>15</sup>x<sup>2</sup> <sup>þ</sup> <sup>6</sup>α<sup>x</sup>

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k (55)

<sup>2</sup> �4<sup>x</sup> <sup>1</sup>

4x 1

2x 1

<sup>6</sup> <sup>2</sup>x<sup>2</sup> � <sup>2</sup>α<sup>x</sup> <sup>3</sup><sup>x</sup> <sup>1</sup>

<sup>4</sup>x<sup>3</sup> � <sup>12</sup>αx<sup>2</sup> <sup>þ</sup> <sup>9</sup>α<sup>2</sup><sup>x</sup> 3

An�<sup>k</sup>ð Þ �nx; α

Using Eq. (17), the composita for the compositional inverse generating function

<sup>2</sup> �2<sup>x</sup> <sup>1</sup>

<sup>24</sup> � <sup>25</sup>x<sup>3</sup> <sup>þ</sup> <sup>30</sup>αx<sup>2</sup> <sup>þ</sup> <sup>9</sup>α<sup>2</sup><sup>x</sup>

Also we can get the following new identities for the Abel polynomials:

An�<sup>m</sup>ð Þ �nx; α ð Þ n � m !

<sup>6</sup> <sup>4</sup>x<sup>2</sup> <sup>þ</sup> <sup>2</sup>α<sup>x</sup> �3<sup>x</sup> <sup>1</sup>

3

Am�<sup>k</sup>ð Þ kx; α

, (51)

: (52)

Wð Þ αt α <sup>x</sup>

Anð Þ <sup>x</sup>; <sup>α</sup> <sup>=</sup> x xð Þ � <sup>α</sup><sup>n</sup> <sup>n</sup>�<sup>1</sup>

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> An�kð Þ kx; <sup>α</sup>

$$B\_n(\infty) = \begin{cases} 1, & n = 0; \\ \sum\_{k=1}^n \frac{(2n - k - 1)!}{(n - k)!(k - 1)!} \frac{\varkappa^k}{2^{n - k}}, & n > 0. \end{cases} \tag{58}$$

According to Eq. (13), the composita for the generating function D tð Þ= tB tð Þ ; x is

$$D^{\Delta}(n,k) = \frac{B\_{n-k}(k\infty)}{(n-k)!}.\tag{59}$$

The triangular form of this composita is

1 2 1 <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> 2 2x 1 <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> <sup>6</sup> <sup>2</sup>x<sup>2</sup> <sup>þ</sup> <sup>x</sup> <sup>3</sup><sup>x</sup> <sup>1</sup> <sup>x</sup><sup>4</sup> <sup>þ</sup> <sup>6</sup>x<sup>3</sup> <sup>þ</sup> <sup>15</sup>x<sup>2</sup> <sup>þ</sup> <sup>15</sup><sup>x</sup> 24 <sup>4</sup>x<sup>3</sup> <sup>þ</sup> <sup>6</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> 3 <sup>9</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> 2 4x 1

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tB tð Þ ; x is

$$
\overline{D}^{\Delta}(n,k) = \frac{k}{n} \frac{B\_{n-k}(-n\infty)}{(n-k)!}.\tag{60}
$$

The triangular form of this composita is

1 �x 1 <sup>3</sup>x<sup>2</sup> � <sup>x</sup> <sup>2</sup> �2<sup>x</sup> <sup>1</sup> � <sup>16</sup>x<sup>3</sup> � <sup>12</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> <sup>6</sup> <sup>4</sup>x<sup>2</sup> � <sup>x</sup> �3<sup>x</sup> <sup>1</sup> <sup>125</sup>x<sup>4</sup> � <sup>150</sup>x<sup>3</sup> <sup>þ</sup> <sup>75</sup>x<sup>2</sup> � <sup>15</sup><sup>x</sup> <sup>24</sup> � <sup>25</sup>x<sup>3</sup> � <sup>15</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> 3 <sup>15</sup>x<sup>2</sup> � <sup>3</sup><sup>x</sup> <sup>2</sup> �4<sup>x</sup> <sup>1</sup>

Also we can get the following new identities for the Bessel polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} \frac{B\_{n-m}(-n\infty)}{(n-m)!} \frac{B\_{m-k}(k\infty)}{(m-k)!} = \delta\_{n,k} \tag{61}$$

∑ n m = k

DOI: http://dx.doi.org/10.5772/intechopen.82370

∑ n m = k

ln 1ð Þ þ t <sup>α</sup>

> x n � i <sup>∑</sup>

The triangular form of this composita is

1 2x þ α 2

<sup>12</sup>x<sup>2</sup> <sup>þ</sup> ð Þ <sup>12</sup><sup>α</sup> � <sup>12</sup> <sup>x</sup> <sup>þ</sup> <sup>3</sup>α<sup>2</sup> � <sup>5</sup><sup>α</sup> 24

<sup>8</sup>x<sup>3</sup> <sup>þ</sup> ð Þ <sup>12</sup><sup>α</sup> � <sup>24</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> ð Þ <sup>2</sup> � <sup>22</sup><sup>α</sup> <sup>þ</sup> <sup>16</sup> <sup>x</sup> <sup>þ</sup> <sup>α</sup><sup>3</sup> � <sup>5</sup>α<sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> 48

The triangular form of this composita is

1 � <sup>2</sup><sup>x</sup> <sup>þ</sup> <sup>α</sup> 2

<sup>36</sup>x<sup>2</sup> <sup>þ</sup> ð Þ <sup>36</sup><sup>α</sup> <sup>þ</sup> <sup>12</sup> <sup>x</sup> <sup>þ</sup> <sup>9</sup>α<sup>2</sup> � <sup>5</sup><sup>α</sup>

� <sup>64</sup>x<sup>3</sup>ð Þ <sup>96</sup><sup>α</sup> <sup>þ</sup> <sup>48</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>48</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>44</sup><sup>α</sup> <sup>þ</sup> <sup>8</sup> <sup>x</sup> <sup>þ</sup> <sup>8</sup>α<sup>3</sup> <sup>þ</sup> <sup>10</sup>α<sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>α</sup> 24

> ∑ n m = k

m n

DΔ

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

Also we can get the following new identities for the Narumi polynomials:

Sn�<sup>m</sup>ð Þ �nx; �nα ð Þ n � m !

i j = 0

n i= 0

2.9 Narumi polynomials

S tð Þ ; <sup>x</sup>; <sup>α</sup> <sup>=</sup> <sup>t</sup>

Snð Þ x; α = n! ∑

D tð Þ= tS tð Þ ; x; α is

D tð Þ of D tð Þ= tS tð Þ ; x; α is

61

where

and

m n

Sn�mð Þ mx <sup>þ</sup> <sup>m</sup> � <sup>1</sup> <sup>k</sup>

m

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions…

ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>x</sup> <sup>=</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>x</sup> <sup>t</sup>

j þ α � 1 j <sup>∑</sup>

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> Sn�<sup>k</sup>ð Þ kx; <sup>k</sup><sup>α</sup>

Using Eq. (17), the composita for the compositional inverse generating function

Sn�<sup>k</sup>ð Þ �nx; �nα

<sup>24</sup> �2<sup>x</sup> � <sup>α</sup> <sup>1</sup>

Sm�<sup>k</sup>ð Þ kx; kα

According to Eq. (13), the composita for the generating function

The Narumi polynomials are defined by the following generating function [8]:

ln 1ð Þ þ t <sup>α</sup>

> ð Þ �<sup>1</sup> <sup>l</sup> <sup>j</sup> l l!

j l= 0

Sn�mð Þ �nx � n � 1 Sm�kð Þ kx þ k � 1 = δn,k (67)

Sm�kð Þ �mx � m � 1 = δn,k: (68)

= ∑ n≥0

ð Þ l þ i !

ð Þ <sup>n</sup> � <sup>k</sup> ! : (71)

1

<sup>24</sup>x<sup>2</sup> <sup>þ</sup> ð Þ <sup>24</sup><sup>α</sup> � <sup>12</sup> <sup>x</sup> <sup>þ</sup> <sup>6</sup>α<sup>2</sup> � <sup>5</sup><sup>α</sup> 12

ð Þ <sup>n</sup> � <sup>k</sup> ! : (72)

1

<sup>48</sup>x<sup>2</sup> <sup>þ</sup> ð Þ <sup>48</sup><sup>α</sup> <sup>þ</sup> <sup>12</sup> <sup>x</sup> <sup>þ</sup> <sup>12</sup>α<sup>2</sup> <sup>þ</sup> <sup>5</sup><sup>α</sup>

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k (73)

<sup>12</sup> � <sup>6</sup><sup>x</sup> <sup>þ</sup> <sup>3</sup><sup>α</sup>

2

1

2x þ α 1

6x þ 3α 2

1

Snð Þ <sup>x</sup>; <sup>α</sup> <sup>t</sup>

l þ i l

: (70)

n n!

, (69)

and

$$\sum\_{m=k}^{n} \frac{B\_{n-m}(m\infty)}{(n-m)!} \frac{k}{m} \frac{B\_{m-k}(-m\infty)}{(m-k)!} = \delta\_{n,k} \,. \tag{62}$$

#### 2.8 Stirling polynomials

The Stirling polynomials are defined by the following generating function [8, 42]:

$$S(t,\infty) = \left(\frac{t}{1-e^{-t}}\right)^{\chi} = \sum\_{n\geq 0} S\_n(\infty) \frac{t^n}{n!},\tag{63}$$

where

$$S\_n(\mathbf{x}) = \sum\_{i=0}^n \binom{\mathbf{x} + i}{i} \sum\_{j=0}^i \frac{j!}{(n+j)!} (-1)^{n+j} \binom{i}{j} \binom{n+j}{j}.\tag{64}$$

According to Eq. (13), the composita for the generating function D tð Þ= tS tð Þ ; x is

$$D^{\Delta}(n,k) = \mathbb{S}\_{n-k}(k\mathbb{x} + k - \mathbb{1}).\tag{65}$$

The triangular form of this composita is

$$\begin{array}{cccc} 1\\ \underline{x+1} & 1\\ \underline{3x^2+5x+2} & \underline{x}+1 & 1\\ \underline{x^3+2x^2+x} & \underline{6x^2+11x+5} & \underline{3x+3} \\ 15x^4+30x^3+5x^2-18x-8 & 2x^3+5x^2+4x+1 & 9x^2+17x+8 & 2x+2 & 1 \\ \end{array}$$

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tS tð Þ ; x is

$$
\overline{D}^{\Delta}(n,k) = \frac{k}{n} \mathbb{S}\_{n-k}(-n\mathbb{x} - n - 1). \tag{66}
$$

The triangular form of this composita is

$$\begin{array}{cccc} 1 & & & 1 \\ -\frac{\mathbf{x} + 2}{2} & & & 1 \\ \frac{9\mathbf{x}^2 + 19\mathbf{x} + 10}{24} & & & -\mathbf{x} - 1 & & 1 \\ -\frac{4\mathbf{x}^3 + 13\mathbf{x}^2 + 14\mathbf{x} + 5}{12} & & & \frac{12\mathbf{x}^2 + 25\mathbf{x} + 13}{12} & & -\frac{3\mathbf{x} + 3}{2} & & 1 \\ 1875\mathbf{x}^4 + 8250\mathbf{x}^3 + 13525\mathbf{x}^2 + 9798\mathbf{x} + 2648 & & -25\mathbf{x}^3 + 80\mathbf{x}^2 + 85\mathbf{x} + 30 & & \frac{15\mathbf{x}^2 + 31\mathbf{x} + 16}{8} & -2\mathbf{x} - 2 & 1 \\ \end{array}$$

Also we can get the following new identities for the Stirling polynomials:

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions… DOI: http://dx.doi.org/10.5772/intechopen.82370

$$\sum\_{m=k}^{n} \frac{m}{n} \mathbf{S}\_{n-m}(-n\mathbf{x} - n - \mathbf{1}) \mathbf{S}\_{m-k}(k\mathbf{x} + k - \mathbf{1}) = \delta\_{n,k} \tag{67}$$

and

∑ n m = k

∑ n m = k

Snð Þ x = ∑

n i= 0

The triangular form of this composita is

DΔ

The triangular form of this composita is

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

1 x þ 1 2

<sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>5</sup><sup>x</sup> <sup>þ</sup> <sup>2</sup>

<sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>x<sup>2</sup> <sup>þ</sup> <sup>x</sup> 48

<sup>15</sup>x<sup>4</sup> <sup>þ</sup> <sup>30</sup>x<sup>3</sup> <sup>þ</sup> <sup>5</sup>x<sup>2</sup> � <sup>18</sup><sup>x</sup> � <sup>8</sup> 5760

> 1 � <sup>x</sup> <sup>þ</sup> <sup>2</sup> 2

<sup>9</sup>x<sup>2</sup> <sup>þ</sup> <sup>19</sup><sup>x</sup> <sup>þ</sup> <sup>10</sup>

� <sup>4</sup>x<sup>3</sup> <sup>þ</sup> <sup>13</sup>x<sup>2</sup> <sup>þ</sup> <sup>14</sup><sup>x</sup> <sup>þ</sup> <sup>5</sup> 12

<sup>1875</sup>x<sup>4</sup> <sup>þ</sup> <sup>8250</sup>x<sup>3</sup> <sup>þ</sup> <sup>13525</sup>x<sup>2</sup> <sup>þ</sup> <sup>9798</sup><sup>x</sup> <sup>þ</sup> <sup>2648</sup>

60

D tð Þ of D tð Þ= tS tð Þ ; x is

and

where

2.8 Stirling polynomials

Polynomials - Theory and Application

m n

Bn�mð Þ mx ð Þ n � m !

S tð Þ ; <sup>x</sup> <sup>=</sup> <sup>t</sup>

x þ i i 

Bn�mð Þ �nx ð Þ n � m !

> k m

1 � e�<sup>t</sup> <sup>x</sup>

> ∑ i j = 0

Bm�kð Þ kx

Bm�kð Þ �mx

The Stirling polynomials are defined by the following generating function [8, 42]:

j! ð Þ n þ j !

According to Eq. (13), the composita for the generating function D tð Þ= tS tð Þ ; x is

1

<sup>6</sup>x<sup>2</sup> <sup>þ</sup> <sup>11</sup><sup>x</sup> <sup>þ</sup> <sup>5</sup> 12

<sup>2</sup>x<sup>3</sup> <sup>þ</sup> <sup>5</sup>x<sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>x</sup> <sup>þ</sup> <sup>1</sup> 12

Using Eq. (17), the composita for the compositional inverse generating function

1

<sup>12</sup>x<sup>2</sup> <sup>þ</sup> <sup>25</sup><sup>x</sup> <sup>þ</sup> <sup>13</sup>

24

<sup>24</sup> �<sup>x</sup> � 1 1

<sup>5760</sup> � <sup>25</sup>x<sup>3</sup> <sup>þ</sup> <sup>80</sup>x<sup>2</sup> <sup>þ</sup> <sup>85</sup><sup>x</sup> <sup>þ</sup> <sup>30</sup>

Also we can get the following new identities for the Stirling polynomials:

<sup>24</sup> <sup>x</sup> <sup>þ</sup> 1 1

= ∑ n≥0 Snð Þ x t n n!

ð Þ �<sup>1</sup> <sup>n</sup>þ<sup>j</sup> <sup>i</sup>

j

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> Sn�<sup>k</sup>ð Þ kx <sup>þ</sup> <sup>k</sup> � <sup>1</sup> : (65)

n þ j

3x þ 3 2

<sup>9</sup>x<sup>2</sup> <sup>þ</sup> <sup>17</sup><sup>x</sup> <sup>þ</sup> <sup>8</sup> 8

Sn�<sup>k</sup>ð Þ �nx � n � 1 : (66)

<sup>12</sup> � <sup>3</sup><sup>x</sup> <sup>þ</sup> <sup>3</sup>

2

<sup>15</sup>x<sup>2</sup> <sup>þ</sup> <sup>31</sup><sup>x</sup> <sup>þ</sup> <sup>16</sup>

j 

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k (61)

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k: (62)

, (63)

: (64)

1

2x þ 2 1

1

<sup>8</sup> �2<sup>x</sup> � 2 1

$$\sum\_{m=k}^{n} \mathcal{S}\_{n-m}(m\boldsymbol{\omega} + m - 1) \frac{k}{m} \mathcal{S}\_{m-k}(-m\boldsymbol{\omega} - m - 1) = \delta\_{n,k}.\tag{68}$$

#### 2.9 Narumi polynomials

The Narumi polynomials are defined by the following generating function [8]:

$$\mathcal{S}(t,\boldsymbol{x},a) = \left(\frac{t}{\ln\left(\mathbf{1}+t\right)}\right)^a \left(\mathbf{1}+t\right)^\mathbf{x} = \left(\mathbf{1}+t\right)^\mathbf{x} \left(\frac{t}{\ln\left(\mathbf{1}+t\right)}\right)^a = \sum\_{n\geq 0} \mathcal{S}\_n(\mathbf{x},a) \frac{t^n}{n!},\tag{69}$$

where

$$S\_n(\mathbf{x}, a) = n! \sum\_{i=0}^n \binom{\mathbf{x}}{n-i} \sum\_{j=0}^i \binom{j+a-1}{j} \sum\_{l=0}^j (-1)^l \binom{j}{l} \frac{l!}{(l+i)!} \begin{bmatrix} l+i \\ l \end{bmatrix}.\tag{70}$$

According to Eq. (13), the composita for the generating function D tð Þ= tS tð Þ ; x; α is

$$D^{\Delta}(n,k) = \frac{\mathbb{S}\_{n-k}(k\infty, ka)}{(n-k)!}.\tag{71}$$

The triangular form of this composita is

$$\begin{array}{cccc} 1 & & & 1 \\ \hline 2x+a & & & 1 \\ \hline 2x^2+(12a-12)x+3a^2-5a & & & 2x+a & & 1 \\ \hline 24 & & & 2x+a & & & \\ \hline \end{array}$$
  $\frac{8x^3+(12a-24)x^2+(6a^2-22a+16)x+a^3-5a^2+6a}{48}$   $\frac{24x^2+(24a-12)x+6a^2-5a}{12}$   $\begin{array}{cccc} 6x+3a & & & 1 \\ \hline & & & 2 \\ \hline \end{array}$ 

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tS tð Þ ; x; α is

$$\overline{D}^{\Delta}(n,k) = \frac{k}{n} \frac{S\_{n-k}(-n\infty, -n\alpha)}{(n-k)!}.\tag{72}$$

The triangular form of this composita is

$$\begin{aligned} \frac{1}{2\kappa + a} & \quad \text{1} \\ \frac{36\kappa^2 + (36a + 12)\kappa + 9a^2 - 5a}{24} & \quad \text{-2x} - a & \quad \text{1} \\ -\frac{64a^3(96a + 48)\kappa^2 + (48a^2 + 44a + 8)\kappa + 8a^3 + 10a^2 + 3a}{24} & \quad \frac{48a^2 + (48a + 12)\kappa + 12a^2 + 5a}{12} & \quad -\frac{6\kappa + 3a}{2} & \quad \text{1} \end{aligned}$$

Also we can get the following new identities for the Narumi polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} \frac{\mathbf{S}\_{n-m}(-n\boldsymbol{\kappa}, -n\boldsymbol{\alpha})}{(n-m)!} \frac{\mathbf{S}\_{m-k}(k\boldsymbol{\kappa}, k\boldsymbol{\alpha})}{(m-k)!} = \delta\_{n,k} \tag{73}$$

and

$$\sum\_{m=k}^{n} \frac{\mathbb{S}\_{n-m}(m\infty, ma)}{(n-m)!} \frac{k}{m} \frac{\mathbb{S}\_{m-k}(-m\infty, -ma)}{(m-k)!} = \delta\_{n,k}.\tag{74}$$

2.11 Gegenbauer polynomials

where

is

C tð Þ ; x; α = 1 � 2xt þ t

DOI: http://dx.doi.org/10.5772/intechopen.82370

Cð Þ <sup>α</sup>

D tð Þ of D tð Þ= tC tð Þ ; x; α is

and

63

ing function [8, 44]:

<sup>n</sup> ð Þ x = ∑

The triangular form of this composita is

The triangular form of this composita is

1

<sup>64</sup>α<sup>3</sup> � <sup>48</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>8</sup><sup>α</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>24</sup><sup>α</sup> ð Þ <sup>2</sup> � <sup>6</sup><sup>α</sup> <sup>x</sup> 3

> ∑ n m = k

∑ n m = k

2.12 Meixner polynomials of the first kind

m n

<sup>C</sup>ð Þ <sup>m</sup><sup>α</sup> <sup>n</sup>�<sup>m</sup>ð Þ <sup>x</sup>

1

<sup>4</sup>α<sup>3</sup> <sup>þ</sup> <sup>12</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>8</sup><sup>α</sup> <sup>x</sup><sup>3</sup> � <sup>6</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>6</sup><sup>α</sup> <sup>x</sup>

DΔ

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n Cð Þ �n<sup>α</sup>

�2αx 1

<sup>6</sup><sup>α</sup> ð Þ <sup>2</sup> � <sup>2</sup><sup>α</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>α</sup> �4α<sup>x</sup> <sup>1</sup>

Also we can get the following new identities for the Gegenbauer polynomials:

<sup>C</sup>ð Þ �n<sup>α</sup> <sup>n</sup>�<sup>m</sup> ð Þ <sup>x</sup> <sup>C</sup>ð Þ <sup>k</sup><sup>α</sup>

k m

Cð Þ �m<sup>α</sup>

The Meixner polynomials of the first kind are defined by the following generat-

<sup>2</sup> �<sup>α</sup>

n i= 0

The Gegenbauer polynomials are defined by the following generating function [43]:

1 � 2xt þ t<sup>2</sup> <sup>α</sup>

i þ α � 1

i 

= ∑ n≥0 Cð Þ <sup>α</sup> <sup>n</sup> ð Þ x t

ð Þ 2x 2i�n

<sup>n</sup>�<sup>k</sup>ð Þ <sup>x</sup> : (83)

<sup>n</sup>�<sup>k</sup> ð Þ <sup>x</sup> : (84)

<sup>16</sup><sup>α</sup> ð Þ <sup>2</sup> � <sup>4</sup><sup>α</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>α</sup> �6α<sup>x</sup> <sup>1</sup>

<sup>m</sup>�<sup>k</sup>ð Þ <sup>x</sup> <sup>=</sup> <sup>δ</sup>n,k (85)

<sup>m</sup>�<sup>k</sup> ð Þ <sup>x</sup> <sup>=</sup> <sup>δ</sup>n,k: (86)

n, (81)

: (82)

<sup>=</sup> <sup>1</sup>

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions…

n � i

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>C</sup>ð Þ <sup>k</sup><sup>α</sup>

2αx 1

<sup>2</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>α</sup> <sup>x</sup><sup>2</sup> � <sup>α</sup> <sup>4</sup>α<sup>x</sup> <sup>1</sup>

Using Eq. (17), the composita for the compositional inverse generating function

<sup>3</sup> <sup>8</sup><sup>α</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>α</sup> <sup>x</sup><sup>2</sup> � <sup>2</sup><sup>α</sup> <sup>6</sup>α<sup>x</sup> <sup>1</sup>

According to Eq. (13), the composita for the generating function D tð Þ= tC tð Þ ; x; α

ð Þ �<sup>1</sup> <sup>n</sup>�<sup>i</sup> <sup>i</sup>

#### 2.10 Peters polynomials

The Peters polynomials are defined by the following generating function [8]:

$$S(t, \mathbf{x}, \boldsymbol{\mu}, \boldsymbol{\lambda}) = \left(\mathbf{1} + (\mathbf{1} + t)^{\boldsymbol{\lambda}}\right)^{-\mu} (\mathbf{1} + t)^{\mathbf{x}} = (\mathbf{1} + t)^{\mathbf{x}} \left(\frac{\mathbf{1}}{\mathbf{1} + (\mathbf{1} + t)^{\boldsymbol{\lambda}}}\right)^{\mu} = \sum\_{n \ge 0} S\_n(\mathbf{x}, \boldsymbol{\mu}, \boldsymbol{\lambda}) \frac{t^n}{n!},\tag{75}$$

where

$$\mathbf{S}\_{n}(\mathbf{x},\boldsymbol{\mu},\boldsymbol{\lambda}) = n! \sum\_{i=0}^{n} \binom{\boldsymbol{\chi}}{n-i}\_{j} \sum\_{j=0}^{i} \frac{\mathbf{1}}{2^{j+\mu}} \binom{j+\mu-\mathbf{1}}{j}\_{l} \sum\_{l=0}^{j} (-\mathbf{1})^{l} \binom{j}{l} \binom{l\boldsymbol{\lambda}}{i} . \tag{76}$$

According to Eq. (13), the composita for the generating function D tð Þ= tS tð Þ ; x; μ; λ is

$$D^{\Delta}(n,k) = \frac{\mathbb{S}\_{n-k}(k\infty, k\mu, \lambda)}{(n-k)!}. \tag{77}$$

The triangular form of this composita is

$$\mathcal{Z}^{-\mu}$$

$$\mathcal{Z}^{-\mu-1}(2\mathfrak{x}-\lambda\mu) \quad \quad \quad \quad \quad \quad \quad \mathcal{Z}^{-2\mu}$$

$$\mathcal{Z}^{-\mu-3}(4\mathfrak{x}^2 - (4\lambda\mu + 4)\mathfrak{x} + \lambda^2\mu^2 - \lambda^2\mu + 2\lambda\mu) \quad \mathcal{Z}^{-2\mu}(2\mathfrak{x}-\lambda\mu) \quad \mathcal{Z}^{-3\mu}$$

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tS tð Þ ; x; μ; λ is

$$\overline{D}^{\Delta}(n,k) = \frac{k}{n} \frac{\mathbb{S}\_{n-k}(-n\infty, -n\mu, \lambda)}{(n-k)!}.\tag{78}$$

The triangular form of this composita is

$$2^{\mu}$$

$$2^{2\mu-1}(-2\mathfrak{x}+\lambda\mu) \qquad\qquad 2^{2\mu}$$

$$2^{3\mu-3}\left(12\mathfrak{x}^2 + (4-12\lambda\mu)\mathfrak{x} + 3\lambda^2\mu^2 + \lambda^2\mu - 2\lambda\mu\right) \quad \Im^{3\mu}(\lambda\mu - 2\mathfrak{x}) \quad \Im^{3\mu}$$

Also we can get the following new identities for the Peters polynomials:

$$\sum\_{m=k}^{n} \frac{m \, \mathrm{S}\_{n-m}(-n\infty, -n\mu, \lambda)}{(n-m)!} \frac{\mathrm{S}\_{m-k}(k\varkappa, k\mu, \lambda)}{(m-k)!} = \delta\_{n,k} \tag{79}$$

and

$$\sum\_{m=k}^{n} \frac{\mathbb{S}\_{n-m}(m\boldsymbol{\omega}, m\boldsymbol{\mu}, \boldsymbol{\lambda})}{(n-m)!} \frac{k}{m} \frac{\mathbb{S}\_{m-k}(-m\boldsymbol{\omega}, -m\boldsymbol{\mu}, \boldsymbol{\lambda})}{(m-k)!} = \delta\_{n,k}.\tag{80}$$

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions… DOI: http://dx.doi.org/10.5772/intechopen.82370

#### 2.11 Gegenbauer polynomials

The Gegenbauer polynomials are defined by the following generating function [43]:

$$\mathbf{C}(t,\mathbf{x},a) = \left(\mathbf{1} - 2\mathbf{x}t + t^2\right)^{-a} = \left(\frac{\mathbf{1}}{\mathbf{1} - 2\mathbf{x}t + t^2}\right)^a = \sum\_{n \ge 0} \mathbf{C}\_n^{(a)}(\mathbf{x})t^n,\tag{81}$$

where

and

where

∑ n m = k

Polynomials - Theory and Application

2.10 Peters polynomials

Snð Þ x; μ; λ = n! ∑

D tð Þ of D tð Þ= tS tð Þ ; x; μ; λ is

D tð Þ= tS tð Þ ; x; μ; λ is

n i = 0

The triangular form of this composita is

<sup>2</sup>�μ�<sup>3</sup> <sup>4</sup>x<sup>2</sup> � ð Þ <sup>4</sup>λμ <sup>þ</sup> <sup>4</sup> <sup>x</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

The triangular form of this composita is

2�μ�<sup>1</sup>

DΔ

22<sup>μ</sup>�<sup>1</sup>

23<sup>μ</sup>�<sup>3</sup> <sup>12</sup>x<sup>2</sup> <sup>þ</sup> ð Þ <sup>4</sup> � <sup>12</sup>λμ <sup>x</sup> <sup>þ</sup> <sup>3</sup>λ<sup>2</sup>

m n

∑ n m = k

∑ n m = k

and

62

ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

2μ

x n � i � �

∑ i j= 0

2�<sup>μ</sup>

1 2<sup>j</sup>þ<sup>μ</sup>

According to Eq. (13), the composita for the generating function

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> Sn�<sup>k</sup>ð Þ kx; <sup>k</sup>μ; <sup>λ</sup>

<sup>μ</sup><sup>2</sup> � <sup>λ</sup><sup>2</sup> <sup>μ</sup> <sup>þ</sup> <sup>2</sup>λμ � � <sup>2</sup>�2<sup>μ</sup>ð Þ <sup>2</sup><sup>x</sup> � λμ <sup>2</sup>�3<sup>μ</sup>

Using Eq. (17), the composita for the compositional inverse generating function

S tð Þ ; <sup>x</sup>; <sup>μ</sup>; <sup>λ</sup> <sup>=</sup> <sup>1</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>λ</sup> � ��<sup>μ</sup>

Sn�mð Þ mx; mα ð Þ n � m !

k m

The Peters polynomials are defined by the following generating function [8]:

ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>x</sup> <sup>=</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>x</sup> <sup>1</sup>

j þ μ � 1 j � �

ð Þ <sup>2</sup><sup>x</sup> � λμ <sup>2</sup>�2<sup>μ</sup>

Sn�<sup>k</sup>ð Þ �nx; �nμ; λ

ð Þ �2<sup>x</sup> <sup>þ</sup> λμ 22<sup>μ</sup>

Sm�<sup>k</sup>ð Þ kx; kμ; λ

Sm�<sup>k</sup>ð Þ �mx; �mμ; λ

<sup>μ</sup><sup>2</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> <sup>μ</sup> � <sup>2</sup>λμ � � 23<sup>μ</sup>ð Þ λμ � <sup>2</sup><sup>x</sup> <sup>2</sup><sup>3</sup><sup>μ</sup>

Also we can get the following new identities for the Peters polynomials:

k m

Sn�<sup>m</sup>ð Þ �nx; �nμ; λ ð Þ n � m !

Sn�<sup>m</sup>ð Þ mx; mμ; λ ð Þ n � m !

Sm�kð Þ �mx; �mα

<sup>1</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>λ</sup>

∑ j l= 0 ð Þ �<sup>1</sup> <sup>l</sup> <sup>j</sup> l � � lλ

ð Þ <sup>n</sup> � <sup>k</sup> ! : (77)

ð Þ <sup>n</sup> � <sup>k</sup> ! : (78)

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k (79)

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k: (80)

!<sup>μ</sup>

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k: (74)

= ∑ n≥0

> i � �

Snð Þ <sup>x</sup>; <sup>μ</sup>; <sup>λ</sup> <sup>t</sup>

n n! ,

(75)

: (76)

$$C\_n^{(a)}(\mathbf{x}) = \sum\_{i=0}^n (-\mathbf{1})^{n-i} \binom{i}{n-i} \binom{i+a-1}{i} (2\mathbf{x})^{2i-n}.\tag{82}$$

According to Eq. (13), the composita for the generating function D tð Þ= tC tð Þ ; x; α is

$$D^{\Delta}(n,k) = \mathcal{C}^{(ka)}\_{n-k}(\infty). \tag{83}$$

The triangular form of this composita is

$$\begin{array}{cccc} 1 & & & & 1 & & & & & \\ & 2a\infty & & & & 1 & & & \\ & (2a^2 + 2a)\mathfrak{x}^2 - a & & & 4a\infty & & & 1 \\ \frac{(4a^3 + 12a^2 + 8a)\mathfrak{x}^3 - (6a^2 + 6a)\mathfrak{x}}{3} & (8a^2 + 4a)\mathfrak{x}^2 - 2a & & 6ax & 1 \\ \end{array}$$

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tC tð Þ ; x; α is

$$
\overline{D}^{\Delta}(n,k) = \frac{k}{n} C\_{n-k}^{(-na)}(\infty). \tag{84}
$$

The triangular form of this composita is

$$\begin{array}{cccc} 1 & & & & 1 & & & & & \\ & -2a\infty & & & & 1 & & & \\ & (6a^2 - 2a)\mathbf{x}^2 + a & & & & -4a\infty & & & \\ & (64a^3 - 48a^2 + 8a)\mathbf{x}^3 + (24a^2 - 6a)\mathbf{x} & & & (16a^2 - 4a)\mathbf{x}^2 + 2a & & -6a\infty & 1 \\ \end{array}$$

Also we can get the following new identities for the Gegenbauer polynomials:

$$\sum\_{m=k}^{n} \frac{m}{n} \mathbf{C}\_{n-m}^{(-na)}(\boldsymbol{\omega}) \mathbf{C}\_{m-k}^{(ka)}(\boldsymbol{\omega}) = \delta\_{n,k} \tag{85}$$

and

$$\sum\_{m=k}^{n} \mathcal{C}\_{n-m}^{(ma)}(\infty) \frac{k}{m} \mathcal{C}\_{m-k}^{(-ma)}(\infty) = \delta\_{n,k} \,. \tag{86}$$

#### 2.12 Meixner polynomials of the first kind

The Meixner polynomials of the first kind are defined by the following generating function [8, 44]:

$$M(t, \mathbf{x}, \boldsymbol{\beta}, c) = \left(\mathbf{1} - \frac{t}{c}\right)^{\mathbf{x}} (\mathbf{1} - t)^{-\mathbf{x} - \boldsymbol{\beta}} = \left(\frac{c - t}{c(\mathbf{1} - t)}\right)^{\mathbf{x}} \left(\frac{\mathbf{1}}{\mathbf{1} - t}\right)^{\boldsymbol{\beta}} = \sum\_{n \ge 0} M\_n(\mathbf{x}, \boldsymbol{\beta}, c) \frac{t^n}{n!},\tag{87}$$

where

$$M\_n(\infty, \beta, c) = (-1)^n n! \sum\_{i=0}^n \binom{\infty}{i} \binom{-\infty - \beta}{n-i} c^{-i}. \tag{88}$$

According to Eq. (13), the composita for the generating function D tð Þ= tM tð Þ ; x; β;c is

$$D^{\Delta}(n,k) = \frac{M\_{n-k}(k\infty, k\beta, c)}{(n-k)!}.\tag{89}$$

Sylvester, generalized Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials that are defined by generating functions of the form

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions…

A lot of studies have recently showed that polynomials are a solution for practical problems related to modeling, quantum mechanics, and other areas. So a study of obtaining explicit formulas and representations of polynomials will be important and influential. Also the further research can be conducted to find practical means

This work was partially funded by the Russian Foundation for Basic Research and the government of the Tomsk region of Russian Federation (Grants No. 18-41- 703006) and the Ministry of Education and Science of Russian Federation (Gov-

A tð Þ ; <sup>x</sup>; <sup>α</sup> <sup>=</sup> F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup>

of obtained properties.

Acknowledgements

Author details

65

.

DOI: http://dx.doi.org/10.5772/intechopen.82370

ernment Order No. 2.8172.2017/8.9, TUSUR).

Dmitry Kruchinin\*, Vladimir Kruchinin and Yuriy Shablya

\*Address all correspondence to: kruchinindm@gmail.com

provided the original work is properly cited.

Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

The triangular form of this composita is

$$\frac{1}{(c-1)x+\beta c} \\ \frac{(c-1)x+\beta c}{c} \\ \frac{1}{2c} \\ \frac{1}{(2c-1)x+(c^2-2\beta c-1)x+(\beta^2+\beta)c^2} \\ \frac{(2c-2)x+2\beta c}{c} \\ \frac{1}{2c}$$

Using Eq. (17), the composita for the compositional inverse generating function D tð Þ of D tð Þ= tM tð Þ ; x; β;c is

$$\overline{D}^{\Delta}(n,k) = \frac{k}{n} \frac{M\_{n-k}(-n\infty, -n\beta, c)}{(n-k)!}.\tag{90}$$

The triangular form of this composita is

$$\frac{1}{(1-c)\mathbf{x}-\beta\mathbf{c}} \quad \text{or} \quad \mathbf{1}$$

$$\frac{(3\mathbf{c}^2 - 6\mathbf{c} + 3)\mathbf{x}^2 + ((6\beta - 1)\mathbf{c}^2 - 6\beta\mathbf{c} + 1)\mathbf{x} + (3\rho^2 - \rho)\mathbf{c}^2}{2c^2} \quad \frac{(2-2c)\mathbf{x} - 2\beta\mathbf{c}}{c} \quad \mathbf{1}$$

Also we can get the following new identities for the Meixner polynomials of the first kind:

$$\sum\_{m=k}^{n} \frac{m}{n} \frac{M\_{n-m}(-n\infty, -n\beta, c)}{(n-m)!} \frac{M\_{m-k}(k\infty, k\beta, c)}{(m-k)!} = \delta\_{n,k} \tag{91}$$

and

$$\sum\_{m=k}^{n} \frac{M\_{n-m}(m\infty, m\beta, c)}{(n-m)!} \frac{k}{m} \frac{M\_{m-k}(-m\infty, -m\beta, c)}{(m-k)!} = \delta\_{n,k}.\tag{92}$$

#### 3. Conclusions and future developments

In this chapter, we find new explicit formulas and identities for such polynomials as the generalized Bernoulli, generalized Euler, Frobenius-Euler, generalized Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions… DOI: http://dx.doi.org/10.5772/intechopen.82370

Sylvester, generalized Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials that are defined by generating functions of the form A tð Þ ; <sup>x</sup>; <sup>α</sup> <sup>=</sup> F tð Þ<sup>x</sup> � G tð Þ<sup>α</sup> .

A lot of studies have recently showed that polynomials are a solution for practical problems related to modeling, quantum mechanics, and other areas. So a study of obtaining explicit formulas and representations of polynomials will be important and influential. Also the further research can be conducted to find practical means of obtained properties.

#### Acknowledgements

M tð Þ ; <sup>x</sup>; <sup>β</sup>;<sup>c</sup> <sup>=</sup> <sup>1</sup> � <sup>t</sup>

Polynomials - Theory and Application

where

D tð Þ= tM tð Þ ; x; β;c is

D tð Þ of D tð Þ= tM tð Þ ; x; β;c is

first kind:

and

64

c <sup>x</sup>

The triangular form of this composita is

Mnð Þ <sup>x</sup>; <sup>β</sup>;<sup>c</sup> <sup>=</sup> ð Þ �<sup>1</sup> <sup>n</sup>

1 ð Þ c � 1 x þ βc c

<sup>c</sup>ð Þ <sup>2</sup> � <sup>2</sup><sup>c</sup> <sup>þ</sup> <sup>1</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup><sup>β</sup> <sup>þ</sup> <sup>1</sup> <sup>c</sup> ð Þ <sup>2</sup> � <sup>2</sup>β<sup>c</sup> � <sup>1</sup> <sup>x</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup> c<sup>2</sup> 2c<sup>2</sup>

> ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> <sup>k</sup> n

1 ð Þ 1 � c x � βc c

<sup>3</sup><sup>c</sup> ð Þ <sup>2</sup> � <sup>6</sup><sup>c</sup> <sup>þ</sup> <sup>3</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>6</sup><sup>β</sup> � <sup>1</sup> <sup>c</sup> ð Þ <sup>2</sup> � <sup>6</sup>β<sup>c</sup> <sup>þ</sup> <sup>1</sup> <sup>x</sup> <sup>þ</sup> <sup>3</sup>β<sup>2</sup> � <sup>β</sup> c<sup>2</sup> 2c<sup>2</sup>

> Mn�<sup>m</sup>ð Þ �nx; �nβ;c ð Þ n � m !

> > k m

In this chapter, we find new explicit formulas and identities for such polynomials as the generalized Bernoulli, generalized Euler, Frobenius-Euler, generalized

Mn�<sup>m</sup>ð Þ mx; mβ;c ð Þ n � m !

DΔ

The triangular form of this composita is

∑ n m = k

∑ n m = k m n

3. Conclusions and future developments

ð Þ <sup>1</sup> � <sup>t</sup> �x�<sup>β</sup> <sup>=</sup> <sup>c</sup> � <sup>t</sup>

According to Eq. (13), the composita for the generating function

cð Þ 1 � t

n! ∑ n i= 0

<sup>D</sup><sup>Δ</sup>ð Þ <sup>n</sup>; <sup>k</sup> <sup>=</sup> Mn�kð Þ kx; <sup>k</sup>β;<sup>c</sup>

Using Eq. (17), the composita for the compositional inverse generating function

Also we can get the following new identities for the Meixner polynomials of the

Mm�<sup>k</sup>ð Þ kx; kβ;c

Mm�<sup>k</sup>ð Þ �mx; �mβ;c

Mn�<sup>k</sup>ð Þ �nx; �nβ;c

<sup>x</sup> 1

x i

�x � β

1 � t <sup>β</sup>

n � i 

= ∑ n≥0

> c �i

ð Þ <sup>n</sup> � <sup>k</sup> ! : (89)

ð Þ <sup>n</sup> � <sup>k</sup> ! : (90)

1

ð Þ 2c � 2 x þ 2βc c

1

ð Þ 2 � 2c x � 2βc c

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k (91)

ð Þ <sup>m</sup> � <sup>k</sup> ! <sup>=</sup> <sup>δ</sup>n,k: (92)

1

1

Mnð Þ x; β;c

: (88)

t n n! ,

(87)

This work was partially funded by the Russian Foundation for Basic Research and the government of the Tomsk region of Russian Federation (Grants No. 18-41- 703006) and the Ministry of Education and Science of Russian Federation (Government Order No. 2.8172.2017/8.9, TUSUR).

#### Author details

Dmitry Kruchinin\*, Vladimir Kruchinin and Yuriy Shablya Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia

\*Address all correspondence to: kruchinindm@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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438p

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[13] Ozden H, Simsek Y, Srivastava HM.

generating functions of the generalized

Mathematcs with Applications. 2010; 60(10):2779-2787. DOI: 10.1016/j.

[14] Simsek Y, Kim D. Identities and recurrence relations of special numbers and polynomials of higher order by analysis of their generating functions. Journal of Inequalities and Applications.

[15] Simsek Y. On generating functions for the special polynomials. Univerzitet u Nišu. 2017;31(1):9-16. DOI: 10.2298/

[16] Ozdemir G, Simsek Y, Milovanovic GV. Generating functions for special polynomials and numbers including Apostol-type and Humbert-type polynomials. Mediterranean Journal of Mathematics. 2017;14(3):1-17. DOI: 10.1007/s00009-017-0918-6

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polynomials. Journal of Computational Analysis and Applications. 2019;26(5):

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Appell polynomials to Sheffer

2018;2018:1-9. DOI: 10.1186/

A unified presentation of the

Bernoulli, Euler and Genocchi polynomials. Computers &

camwa.2010.09.031

s13660-018-1815-7

FIL1701009S

889-898

axioms7020022

769095

[2] Flajolet P, Sedgewick R. Analytic Combinatorics. United Kingdom: Cambridge University Press; 2009. 810p. DOI: 10.1017/CBO9780511801655

[3] Graham RL, Knuth DE, Patashnik O. Concrete Mathematics. USA: Addison-

[5] Stanley RP. Generating functions, in MAA Studies in Combinatorics. In: Rota G-C, editor. Studies in Combinatorics. Mathematical Association of America, Washington DC. 1978. pp. 100-141

[6] Wilf HS. Generating functionology. USA: Academic Press; 1994. 226p

[7] Boas RP Jr, Buck RC. Polynomial Expansions of Analytic Functions. Berlin, Heidelberg, Germany: Springer-Verlag; 1964. DOI: 10.1007/978-3-

[8] Roman S. The Umbral Calculus. New York, USA: Academic Press; 1984. 195p

[9] Srivastava HM, Manocha HL. A Treatise on Generating Functions. New York, USA: Halsted Press; 1984. 569p

[10] Srivastava HM, Choi J. Zeta and q-Zeta Functions and Associated Series and Integrals. Amsterdam, The Netherlands: Elsevier; 2012. 674p

[11] Srivastava HM. Some

2011;5(3):390-444

66

generalizations and basic (or q) extensions of the Bernoulli, Euler and Genocchi polynomials. Applied Mathematics & Information Sciences.

662-25170-6. 77p

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Dordrecht, Holland: D. Reidel Publishing Company; 1974. 343p

Polynomials - Theory and Application

[20] Simsek Y. Construction method for generating functions of special numbers and polynomials arising from analysis of new operators. Mathematical Methods in the Applied Sciences. 2018;41(16):6934-6954. DOI: 10.1002/mma.5207

[21] Kim DS, Kim T. On degenerate Bell numbers and polynomials. Iranian Journal of Science and Technology. 2017;41(3):749-753. DOI: 10.1007/ s13398-016-0304-4

[22] Kim T, Kim DS, Kwon H-I. Some identities of Carlitz degenerate Bernoulli numbers and polynomials. Rev.Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas RACSAM. 2017;111(2):435-446. DOI: 10.1007/s40995-017-0286-x

[23] Kim T, Ryoo CS. Some identities for Euler and Bernoulli polynomials and their zeros. Axioms. 2018;7(3):1-19. DOI: 10.3390/axioms7030056

[24] Ryoo CS. Some identities involving generalized degenerate tangent polynomials arising from differential equations. Journal of Computational Analysis and Applications. 2019;26(6): 975-984

[25] Ryoo CS. Some identities involving modified degenerate tangent numbers and polynomials. Global Journal of Pure and Applied Mathematics. 2016;12(3): 2621-2630

[26] Zhao J-L, Qi F. Two explicit formulas for the generalized Motzkin numbers. Journal of Inequalities and Applications. 2017;2017(1):1-8. DOI: 10.1186/s13660-017-1313-3

[27] Qi F, Cernanova V, Shi X-T, Guo B-N. Some properties of central Delannoy numbers. Journal of Computational and Applied Mathematics. 2018;328:101-115. DOI: 10.1016/j.cam.2017.07.013

[28] Liu G-D, Srivastava HM. Explicit formulas for the Norlund polynomials Bn (x) and bn (x). Computers & Mathematcs with Applications. 2006;51 (9–10):1377-1384. DOI: 10.1016/j. camwa.2006.02.003

[29] Srivastava HM, Todorov PG. An explicit formula for the generalized Bernoulli polynomials. Journal of Mathematical Analysis and Applications. 1988;130(2):509-513. DOI: 10.1016/0022-247X(88)90326-5

[30] Cenkci M. An explicit formula for generalized potential polynomials and its applications. Discrete Mathematics. 2009;309:1498-1510. DOI: 10.1016/j. disc.2008.02.021

[31] Boyadzhiev KN. Derivative polynomials for Tanh, Tan, Sech and Sec in explicit form. The Fibonacci Quarterly. 2007;45(4):291-303

[32] Stanley RP. Enumerative Combinatorics. Vol. 2. Cambridge University Press; 1999. 600p

[33] Kruchinin DV, Kruchinin VV. A method for obtaining generating functions for central coefficients of triangles. Journal of Integer Sequences. 2012;15:1-10

[34] Kruchinin DV, Kruchinin VV. Application of a composition of generating functions for obtaining explicit formulas of polynomials. Journal of Mathematical Analysis and Applications. 2013;404(1):161-171. DOI: 10.1016/j.jmaa.2013.03.009

[35] Kruchinin VV, Kruchinin DV. Composita and its properties. Journal of Analysis and Number Theory. 2014; 2:37-44

[36] Kruchinin DV, Shablya YV, Kruchinin VV, Shelupanov AA. A method for obtaining coefficients of compositional inverse generating functions. In: International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2015); 23–29 September 2015; Rodos: American Institute of Physics Inc.; 2016. pp. 1-4

[37] Boutiche MA, Rahmani M, Srivastava HM. Explicit formulas associated with some families of generalized Bernoulli and Euler polynomials. Mediterranean Journal of Mathematics. 2017;14(2):1-10. DOI: 10.1007/s00009-017-0891-0

[38] Elezovic N. Generalized Bernoulli polynomials and numbers, revisited. Mediterranean Journal of Mathematics. 2016;13(1):141-151. DOI: 10.1007/ s00009-014-0498-7

[39] Srivastava HM, Boutiche MA, Rahmani M. Some explicit formulas for the Frobenius-Euler polynomials of higher order. Applied Mathematics & Information Sciences. 2017;11(2): 621-626. DOI: 10.18576/amis/110234

[40] Agarwal AK. A note on generalized Sylvester polynomials. Indian Journal of Pure and Applied Mathematics. 1984;15: 431-434

[41] Kim DS, Kim T, Lee S-H, Rim S-H. Some identities of Bernoulli, Euler and Abel polynomials arising from umbral calculus. Adv. Difference Equ. 2013; 2013:1-8. DOI: 10.1186/1687-1847- 2013-15

[42] Kang JY, Ryoo CS. A research on the some properties and distribution of zeros for Stirling polynomials. Journal of Nonlinear Sciences and Applications. 2016;9(4):1735-1747

[43] Prajapati JC, Choi J, Kachhia KB, Agarwal P. Certain properties of Gegenbauer polynomials via Lie algebra. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas RACSAM. 2017;111(4): 1031-1037. DOI: 10.1007/s13398-016- 0343-x

Chapter 4

Zeros

Abstract

1. Introduction

the bibliography to this end.

69

We begin with some definitions:

Ricardo Vieira

Polynomials with Symmetric

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

Keywords: self-inversive polynomials, self-reciprocal polynomials, Pisot and Salem polynomials, Möbius transformations, knot theory, Bethe equations

properties of these polynomials can be found in the books of Marden [1], Milovanović et al. [2], and Sheil-Small [3]. Although these polynomials are very important in both mathematics and physics, it seems that there is no specific review about them; in this work, we present a bird's eye view to this theory, focusing on the zeros of such polynomials. Other properties of these polynomials (e.g., irreducibility, norms, analytical properties, etc.) are not covered here due to short space, nonetheless, the interested reader can check many of the references presented in

2. Self-conjugate, self-reciprocal, and self-inversive polynomials

Definition 1. Let p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>p</sup>1<sup>z</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pn�<sup>1</sup>z<sup>n</sup>�<sup>1</sup> <sup>þ</sup> pnzn be a polynomial of degree n with complex coefficients. We shall introduce three polynomials, namely the conjugate polynomial p zð Þ, the reciprocal polynomial p<sup>∗</sup> ð Þ<sup>z</sup> , and the inversive polynomial p†ð Þ<sup>z</sup> , which are, respectively, defined in terms of p zð Þ as follows:

> p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>p</sup><sup>1</sup> <sup>z</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pn�<sup>1</sup> <sup>z</sup><sup>n</sup>�<sup>1</sup> <sup>þ</sup> pn <sup>z</sup>n, <sup>p</sup><sup>∗</sup> ð Þ¼ <sup>z</sup> pn <sup>þ</sup> pn�<sup>1</sup><sup>z</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>p</sup>1z<sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>p</sup>0zn, <sup>p</sup>†ð Þ¼ <sup>z</sup> pn <sup>þ</sup> pn�<sup>1</sup> <sup>z</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>p</sup><sup>1</sup> <sup>z</sup><sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>p</sup>0zn,

(1)

In this work, we consider the theory of self-conjugate (SC), self-reciprocal (SR), and self-inversive (SI) polynomials. These are polynomials whose zeros are symmetric either to the real line R or to the unit circle S ¼ f g z∈ C : jzj ¼ 1 . The basic

[44] Kruchinin DV, Shablya YV. Explicit formulas for Meixner polynomials. International Journal of Mathematics and Mathematical Sciences. 2015;2015: 1-5. DOI: 10.1155/2015/620569

#### Chapter 4

Analysis and Number Theory. 2014;

Polynomials - Theory and Application

[43] Prajapati JC, Choi J, Kachhia KB, Agarwal P. Certain properties of

0343-x

Gegenbauer polynomials via Lie algebra. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas RACSAM. 2017;111(4): 1031-1037. DOI: 10.1007/s13398-016-

[44] Kruchinin DV, Shablya YV. Explicit formulas for Meixner polynomials. International Journal of Mathematics and Mathematical Sciences. 2015;2015:

1-5. DOI: 10.1155/2015/620569

[36] Kruchinin DV, Shablya YV, Kruchinin VV, Shelupanov AA. A method for obtaining coefficients of compositional inverse generating functions. In: International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2015); 23–29 September 2015; Rodos: American Institute of Physics Inc.; 2016. pp. 1-4

[37] Boutiche MA, Rahmani M, Srivastava HM. Explicit formulas associated with some families of generalized Bernoulli and Euler polynomials. Mediterranean Journal of Mathematics. 2017;14(2):1-10. DOI: 10.1007/s00009-017-0891-0

[38] Elezovic N. Generalized Bernoulli polynomials and numbers, revisited. Mediterranean Journal of Mathematics. 2016;13(1):141-151. DOI: 10.1007/

[39] Srivastava HM, Boutiche MA, Rahmani M. Some explicit formulas for the Frobenius-Euler polynomials of higher order. Applied Mathematics & Information Sciences. 2017;11(2): 621-626. DOI: 10.18576/amis/110234

[40] Agarwal AK. A note on generalized Sylvester polynomials. Indian Journal of Pure and Applied Mathematics. 1984;15:

[41] Kim DS, Kim T, Lee S-H, Rim S-H. Some identities of Bernoulli, Euler and Abel polynomials arising from umbral calculus. Adv. Difference Equ. 2013; 2013:1-8. DOI: 10.1186/1687-1847-

[42] Kang JY, Ryoo CS. A research on the some properties and distribution of zeros for Stirling polynomials. Journal of Nonlinear Sciences and Applications.

s00009-014-0498-7

431-434

2013-15

68

2016;9(4):1735-1747

2:37-44

## Polynomials with Symmetric Zeros

Ricardo Vieira

#### Abstract

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

Keywords: self-inversive polynomials, self-reciprocal polynomials, Pisot and Salem polynomials, Möbius transformations, knot theory, Bethe equations

#### 1. Introduction

In this work, we consider the theory of self-conjugate (SC), self-reciprocal (SR), and self-inversive (SI) polynomials. These are polynomials whose zeros are symmetric either to the real line R or to the unit circle S ¼ f g z∈ C : jzj ¼ 1 . The basic properties of these polynomials can be found in the books of Marden [1], Milovanović et al. [2], and Sheil-Small [3]. Although these polynomials are very important in both mathematics and physics, it seems that there is no specific review about them; in this work, we present a bird's eye view to this theory, focusing on the zeros of such polynomials. Other properties of these polynomials (e.g., irreducibility, norms, analytical properties, etc.) are not covered here due to short space, nonetheless, the interested reader can check many of the references presented in the bibliography to this end.

#### 2. Self-conjugate, self-reciprocal, and self-inversive polynomials

We begin with some definitions:

Definition 1. Let p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>p</sup>1<sup>z</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pn�<sup>1</sup>z<sup>n</sup>�<sup>1</sup> <sup>þ</sup> pnzn be a polynomial of degree n with complex coefficients. We shall introduce three polynomials, namely the conjugate polynomial p zð Þ, the reciprocal polynomial p<sup>∗</sup> ð Þ<sup>z</sup> , and the inversive polynomial p†ð Þ<sup>z</sup> , which are, respectively, defined in terms of p zð Þ as follows:

$$\begin{aligned} \overline{p}(\mathbf{z}) &= \overline{p\_0} + \overline{p\_1}\mathbf{z} + \dots + \overline{p\_{n-1}}\mathbf{z}^{n-1} + \overline{p\_n}\mathbf{z}^n, \\ p^\*(\mathbf{z}) &= p\_n + p\_{n-1}\mathbf{z} + \dots + p\_1\mathbf{z}^{n-1} + p\_0\mathbf{z}^n, \\ p^\dagger(\mathbf{z}) &= \overline{p\_n} + \overline{p\_{n-1}}\mathbf{z} + \dots + \overline{p\_1}\mathbf{z}^{n-1} + \overline{p\_0}\mathbf{z}^n, \end{aligned} \tag{1}$$

where the bar means complex conjugation. Notice that the conjugate, reciprocal, and inversive polynomials can also be defined without making reference to the coefficients of p zð Þ:

$$
\overline{p}(\mathbf{z}) = \overline{p(\overline{\mathbf{z}})}, \qquad p^\*(\mathbf{z}) = \mathbf{z}^\eta p(\mathbf{1}/\mathbf{z}), \qquad p^\dagger(\mathbf{z}) = \mathbf{z}^\eta \overline{p(\mathbf{1}/\overline{\mathbf{z}})}.\tag{2}
$$

p zð Þ¼ pn

Polynomials with Symmetric Zeros

Yn k¼1

following additional definitions:

p zð Þ¼ pn

nomials.

always zero.

such polynomials.

71

written in the following form:

1 zm � �

<sup>þ</sup> <sup>p</sup><sup>1</sup> <sup>z</sup><sup>m</sup>�<sup>1</sup> <sup>þ</sup>

p zð Þ¼ <sup>z</sup><sup>m</sup> <sup>p</sup><sup>0</sup> <sup>z</sup><sup>m</sup> <sup>þ</sup>

with <sup>ω</sup> ¼ �ð Þ<sup>1</sup> <sup>n</sup>

Yn k¼1

DOI: http://dx.doi.org/10.5772/intechopen.82728

with ω ¼ pn=pn so that ∣ω∣ ¼ pn=pn

<sup>z</sup> � <sup>1</sup> ζk � �

z � ζ<sup>k</sup> � � <sup>¼</sup> pn Yn k¼1

Then, for any zero ζ of p zð Þ, the reciprocal number 1=ζ is also a zero of it; thus,

Yn k¼1

sarily different from zero if p zð Þ is SR), there will be another zero whose value is 1=ζ so that ζ1…ζ<sup>n</sup> j j ¼ 1, which implies ∣ω∣ ¼ 1. The proof for SI polynomials is analogous and will be concealed; it follows that <sup>ω</sup> <sup>¼</sup> pn=p<sup>0</sup> in this case. □ Now from (1), (2) and (3), we can conclude that the coefficients of an SC, an SR,

pk <sup>¼</sup> <sup>ω</sup>pk , pk <sup>¼</sup> <sup>ω</sup>pn�<sup>k</sup>, pk <sup>¼</sup> <sup>ω</sup>pn�<sup>k</sup> , <sup>∣</sup>ω<sup>∣</sup> <sup>¼</sup> <sup>1</sup>, <sup>0</sup><sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>n</sup>: (6)

We highlight that any real polynomial is SC—in fact, many theorems which are valid for real polynomials are also valid for, or can be easily extended to, SC poly-

There also exist polynomials whose zeros are symmetric with respect to both the real line R and the unit circle S. A polynomial p zð Þ with this double symmetry is, at the same time, SC and SI (and, hence, SR as well). This is only possible if all the coefficients of p zð Þ are real, which implies that ω ¼ �1. This suggests the

Definition 3. A real self-reciprocal polynomial p zð Þ that satisfies the relation p zð Þ¼ <sup>ω</sup>znpð Þ <sup>1</sup>=<sup>z</sup> will be called a positive self-reciprocal (PSR) polynomial if <sup>ω</sup> <sup>¼</sup> <sup>1</sup>

Thus, the coefficients of any PSR polynomial p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pnz<sup>n</sup> of degree <sup>n</sup>

Some elementary properties of PSR and NSR polynomials are the following: first, notice that, if ζ is a zero of any PSR or NSR polynomial p zð Þ of degree n ⩾4, then the three complex numbers 1=ζ, ζ and 1=ζ are also zeros of p zð Þ. In particular, the number of zeros of such polynomials which are neither in S or in R is always a multiple of 4. Besides, any NSR polynomial has z ¼ 1 as a zero and p zð Þ=ð Þ z � 1 is PSR; further, if p zð Þ has even degree then <sup>z</sup> ¼ �1 is also a zero of it and p zð Þ<sup>=</sup> <sup>z</sup>ð Þ <sup>2</sup> � <sup>1</sup> is a PSR polynomial of even degree. In a similar way, any PSR polynomial p zð Þ of odd degree has z ¼ �1 as a zero and p zð Þ=ð Þ z þ 1 is also PSR. The product of two PSR, or two NSR, polynomials is PSR, while the product of a PSR polynomial with an NSR polynomial is NSR. These statements follow directly from the definitions of

We also mention that any PSR polynomial of even degree (say, n ¼ 2m) can be

1 zm�<sup>1</sup>

<sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pm�<sup>1</sup> <sup>z</sup> <sup>þ</sup>

1 z

þ pm, (7)

� �

� � � �

satisfy the relations pk <sup>¼</sup> pn�<sup>k</sup> for 0<sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>n</sup>, while the coefficients of any NSR polynomial p zð Þ of degree <sup>n</sup> satisfy the relations pk ¼ �pn�<sup>k</sup> for 0<sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>n</sup>; this last condition implies that the middle coefficient of an NSR polynomial of even degree is

and a negative self-reciprocal (NSR) polynomial if ω ¼ �1.

and an SI polynomial of degree n satisfy, respectively, the following relations:

1 z � ζ<sup>k</sup> � �

= ζ1…ζ<sup>n</sup> ð Þ¼ pn=p0; now, for any zero ζ of p zð Þ (which is neces-

znpn ζ1⋯ζ<sup>n</sup>

� � �

<sup>¼</sup> ð Þ �<sup>1</sup> <sup>n</sup>

z � ζ<sup>k</sup> ð Þ¼ pn=pn

� �pð Þ¼ <sup>z</sup> <sup>ω</sup>p zð Þ, (4)

� ¼ 1. Now, let us suppose that p zð Þ is SR.

zn ζ1⋯ζ<sup>n</sup>

p 1 z � �

<sup>¼</sup> <sup>ω</sup>p<sup>∗</sup> ð Þ<sup>z</sup> ,

(5)

<sup>¼</sup> ð Þ �<sup>1</sup> <sup>n</sup>

From these relations, we plainly see that if ζ1, …, ζ<sup>n</sup> are the zeros of a complex polynomial p zð Þ of degree <sup>n</sup>, then, the zeros of p zð Þ are <sup>ζ</sup><sup>1</sup> , …, <sup>ζ</sup><sup>n</sup> , the zeros of <sup>p</sup><sup>∗</sup> ð Þ<sup>z</sup> are 1=ζ1, …, <sup>1</sup>=ζn, and finally, the zeros of <sup>p</sup>†ð Þ<sup>z</sup> are 1=ζ<sup>1</sup> , …, <sup>1</sup>=ζ<sup>n</sup> . Thus, if p zð Þ has <sup>k</sup> zeros on R, l zeros on the upper half-plane C<sup>þ</sup> ¼ f g z∈ C : Imð Þz . 0 , and m zeros in the lower half-plane C� ¼ f g z∈ C : Imð Þz , 0 so that k þ l þ m ¼ n, then p zð Þ will have the same number k of zeros on R, l zeros in C� and m zeros in Cþ. Similarly, if p zð Þ has k zeros on S, l zeros inside S and m zeros outside S, so that k þ l þ m ¼ n, then both <sup>p</sup><sup>∗</sup> ð Þ<sup>z</sup> as <sup>p</sup>†ð Þ<sup>z</sup> will have the same number <sup>k</sup> of zeros on <sup>S</sup>, <sup>l</sup> zeros outside <sup>S</sup> and m zeros inside S.

These properties encourage us to introduce the following classes of polynomials:

Definition 2. A complex polynomial p zð Þ is called<sup>1</sup> self-conjugate (SC), selfreciprocal (SR), or self-inversive (SI) if, for any zero ζ of p zð Þ, the complex-conjugate ζ, the reciprocal 1=ζ, or the reciprocal of the complex-conjugate 1=ζ is also a zero of p zð Þ, respectively.

Thus, the zeros of any SC polynomial are all symmetric to the real line R, while the zeros of the any SI polynomial are symmetric to the unit circle S. The zeros of any SR polynomial are obtained by an inversion with respect to the unit circle followed by a reflection in the real line. From this, we can establish the following:

Theorem 1. If p zð Þ is an SC polynomial of odd degree, then it necessarily has at least one zero on R. Similarly, if p zð Þ is an SR or SI polynomial of odd degree, then it necessarily has at least one zero on S.

Proof. From Definition 2 it follows that the number of non-real zeros of an SC polynomial p zð Þ can only occur in (conjugate) pairs; thus, if p zð Þ has odd degree, then at least one zero of it must be real. Similarly, the zeros of <sup>p</sup>†ð Þ<sup>z</sup> or <sup>p</sup><sup>∗</sup> ð Þ<sup>z</sup> that have modulus different from 1 can only occur in (inversive or reciprocal) pairs as well; thus, if p zð Þ has odd degree then at least one zero of it must lie on <sup>S</sup>. □

Theorem 2. The necessary and sufficient condition for a complex polynomial p zð Þ to be SC, SR, or SI is that there exists a complex number ω of modulus 1 so that one of the following relations, respectively, holds:

$$p(\mathbf{z}) = a\overline{p}(\mathbf{z}), \qquad p(\mathbf{z}) = a p^\*(\mathbf{z}), \qquad p(\mathbf{z}) = a p^\dagger(\mathbf{z}).\tag{3}$$

Proof. It is clear in view of (1) and (2) that these conditions are sufficient. We need to show, therefore, that these conditions are also necessary. Let us suppose first that p zð Þ is SC. Then, for any zero ζ of p zð Þ the complex-conjugate number ζ is also a zero of it. Thus, we can write

<sup>1</sup> The reader should be aware that there is no standard in naming these polynomials. For instance, what we call here self-inversive polynomials are sometimes called self-reciprocal polynomials. What we mean positive self-reciprocal polynomials are usually just called self-reciprocal or yet palindrome polynomials (because their coefficients are the same whether they are read from forwards or backwards), as well as, negative self-reciprocal polynomials are usually called skew-reciprocal, anti-reciprocal, or yet antipalindrome polynomials.

Polynomials with Symmetric Zeros DOI: http://dx.doi.org/10.5772/intechopen.82728

where the bar means complex conjugation. Notice that the conjugate, reciprocal,

From these relations, we plainly see that if ζ1, …, ζ<sup>n</sup> are the zeros of a complex polynomial p zð Þ of degree <sup>n</sup>, then, the zeros of p zð Þ are <sup>ζ</sup><sup>1</sup> , …, <sup>ζ</sup><sup>n</sup> , the zeros of <sup>p</sup><sup>∗</sup> ð Þ<sup>z</sup> are 1=ζ1, …, <sup>1</sup>=ζn, and finally, the zeros of <sup>p</sup>†ð Þ<sup>z</sup> are 1=ζ<sup>1</sup> , …, <sup>1</sup>=ζ<sup>n</sup> . Thus, if p zð Þ has <sup>k</sup> zeros on R, l zeros on the upper half-plane C<sup>þ</sup> ¼ f g z∈ C : Imð Þz . 0 , and m zeros in the lower half-plane C� ¼ f g z∈ C : Imð Þz , 0 so that k þ l þ m ¼ n, then p zð Þ will have the same number k of zeros on R, l zeros in C� and m zeros in Cþ. Similarly, if p zð Þ has k zeros on S, l zeros inside S and m zeros outside S, so that k þ l þ m ¼ n, then both <sup>p</sup><sup>∗</sup> ð Þ<sup>z</sup> as <sup>p</sup>†ð Þ<sup>z</sup> will have the same number <sup>k</sup> of zeros on <sup>S</sup>, <sup>l</sup> zeros outside <sup>S</sup>

These properties encourage us to introduce the following classes of polynomials: Definition 2. A complex polynomial p zð Þ is called<sup>1</sup> self-conjugate (SC), selfreciprocal (SR), or self-inversive (SI) if, for any zero ζ of p zð Þ, the complex-conjugate ζ, the reciprocal 1=ζ, or the reciprocal of the complex-conjugate 1=ζ is also a zero of

Thus, the zeros of any SC polynomial are all symmetric to the real line R, while the zeros of the any SI polynomial are symmetric to the unit circle S. The zeros of any SR polynomial are obtained by an inversion with respect to the unit circle followed by a reflection in the real line. From this, we can establish the following: Theorem 1. If p zð Þ is an SC polynomial of odd degree, then it necessarily has at least

Proof. From Definition 2 it follows that the number of non-real zeros of an SC polynomial p zð Þ can only occur in (conjugate) pairs; thus, if p zð Þ has odd degree, then at least one zero of it must be real. Similarly, the zeros of <sup>p</sup>†ð Þ<sup>z</sup> or <sup>p</sup><sup>∗</sup> ð Þ<sup>z</sup> that have modulus different from 1 can only occur in (inversive or reciprocal) pairs as well; thus, if p zð Þ has odd degree then at least one zero of it must lie on <sup>S</sup>. □ Theorem 2. The necessary and sufficient condition for a complex polynomial p zð Þ to be SC, SR, or SI is that there exists a complex number ω of modulus 1 so that one of the

p zð Þ¼ <sup>ω</sup>p zð Þ, pzð Þ¼ <sup>ω</sup>p<sup>∗</sup> ð Þ<sup>z</sup> , pzð Þ¼ <sup>ω</sup>p†

Proof. It is clear in view of (1) and (2) that these conditions are sufficient. We need to show, therefore, that these conditions are also necessary. Let us suppose first that p zð Þ is SC. Then, for any zero ζ of p zð Þ the complex-conjugate number ζ is

<sup>1</sup> The reader should be aware that there is no standard in naming these polynomials. For instance, what we call here self-inversive polynomials are sometimes called self-reciprocal polynomials. What we mean positive self-reciprocal polynomials are usually just called self-reciprocal or yet palindrome polynomials (because their coefficients are the same whether they are read from forwards or backwards), as well as, negative self-reciprocal polynomials are usually called skew-reciprocal, anti-reciprocal, or yet anti-

one zero on R. Similarly, if p zð Þ is an SR or SI polynomial of odd degree, then it

ð Þ¼ <sup>z</sup> <sup>z</sup>npð Þ <sup>1</sup>=<sup>z</sup> : (2)

ð Þz : (3)

and inversive polynomials can also be defined without making reference to the

p zð Þ¼ <sup>p</sup>ð Þ<sup>z</sup> , p<sup>∗</sup> ð Þ¼ <sup>z</sup> <sup>z</sup>npð Þ <sup>1</sup>=<sup>z</sup> , p†

coefficients of p zð Þ:

Polynomials - Theory and Application

and m zeros inside S.

p zð Þ, respectively.

necessarily has at least one zero on S.

following relations, respectively, holds:

also a zero of it. Thus, we can write

palindrome polynomials.

70

$$p(\mathbf{z}) = p\_n \prod\_{k=1}^n \left(\mathbf{z} - \overline{\zeta\_k}\right) = p\_n \prod\_{k=1}^n \overline{(\overline{\mathbf{z}} - \zeta\_k)} = (p\_n / \overline{p\_n}) \overline{p(\overline{\mathbf{z}})} = \omega \overline{p}(\mathbf{z}),\tag{4}$$

with ω ¼ pn=pn so that ∣ω∣ ¼ pn=pn � � � � ¼ 1. Now, let us suppose that p zð Þ is SR. Then, for any zero ζ of p zð Þ, the reciprocal number 1=ζ is also a zero of it; thus,

$$p(\mathbf{z}) = p\_n \prod\_{k=1}^{n} \left(\mathbf{z} - \frac{\mathbf{1}}{\zeta\_k}\right) = \frac{(-1)^n z^n p\_n}{\zeta\_1 \cdots \zeta\_n} \prod\_{k=1}^{n} \left(\frac{\mathbf{1}}{\mathbf{z}} - \zeta\_k\right) = \frac{(-1)^n z^n}{\zeta\_1 \cdots \zeta\_n} p\left(\frac{\mathbf{1}}{\mathbf{z}}\right) = a p^\*(\mathbf{z}),\tag{5}$$

with <sup>ω</sup> ¼ �ð Þ<sup>1</sup> <sup>n</sup> = ζ1…ζ<sup>n</sup> ð Þ¼ pn=p0; now, for any zero ζ of p zð Þ (which is necessarily different from zero if p zð Þ is SR), there will be another zero whose value is 1=ζ so that ζ1…ζ<sup>n</sup> j j ¼ 1, which implies ∣ω∣ ¼ 1. The proof for SI polynomials is analogous and will be concealed; it follows that <sup>ω</sup> <sup>¼</sup> pn=p<sup>0</sup> in this case. □

Now from (1), (2) and (3), we can conclude that the coefficients of an SC, an SR, and an SI polynomial of degree n satisfy, respectively, the following relations:

$$p\_k = a\overline{p\_k}, \qquad p\_k = a p\_{n-k}, \qquad p\_k = a\overline{p\_{n-k}}, \qquad |a| = \mathbb{1}, \qquad 0 \lessapprox n. \tag{6}$$

We highlight that any real polynomial is SC—in fact, many theorems which are valid for real polynomials are also valid for, or can be easily extended to, SC polynomials.

There also exist polynomials whose zeros are symmetric with respect to both the real line R and the unit circle S. A polynomial p zð Þ with this double symmetry is, at the same time, SC and SI (and, hence, SR as well). This is only possible if all the coefficients of p zð Þ are real, which implies that ω ¼ �1. This suggests the following additional definitions:

Definition 3. A real self-reciprocal polynomial p zð Þ that satisfies the relation p zð Þ¼ <sup>ω</sup>znpð Þ <sup>1</sup>=<sup>z</sup> will be called a positive self-reciprocal (PSR) polynomial if <sup>ω</sup> <sup>¼</sup> <sup>1</sup> and a negative self-reciprocal (NSR) polynomial if ω ¼ �1.

Thus, the coefficients of any PSR polynomial p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pnz<sup>n</sup> of degree <sup>n</sup> satisfy the relations pk <sup>¼</sup> pn�<sup>k</sup> for 0<sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>n</sup>, while the coefficients of any NSR polynomial p zð Þ of degree <sup>n</sup> satisfy the relations pk ¼ �pn�<sup>k</sup> for 0<sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>n</sup>; this last condition implies that the middle coefficient of an NSR polynomial of even degree is always zero.

Some elementary properties of PSR and NSR polynomials are the following: first, notice that, if ζ is a zero of any PSR or NSR polynomial p zð Þ of degree n ⩾4, then the three complex numbers 1=ζ, ζ and 1=ζ are also zeros of p zð Þ. In particular, the number of zeros of such polynomials which are neither in S or in R is always a multiple of 4. Besides, any NSR polynomial has z ¼ 1 as a zero and p zð Þ=ð Þ z � 1 is PSR; further, if p zð Þ has even degree then <sup>z</sup> ¼ �1 is also a zero of it and p zð Þ<sup>=</sup> <sup>z</sup>ð Þ <sup>2</sup> � <sup>1</sup> is a PSR polynomial of even degree. In a similar way, any PSR polynomial p zð Þ of odd degree has z ¼ �1 as a zero and p zð Þ=ð Þ z þ 1 is also PSR. The product of two PSR, or two NSR, polynomials is PSR, while the product of a PSR polynomial with an NSR polynomial is NSR. These statements follow directly from the definitions of such polynomials.

We also mention that any PSR polynomial of even degree (say, n ¼ 2m) can be written in the following form:

$$p(\mathbf{z}) = \mathbf{z}^m \left[ p\_0 \left( \mathbf{z}^m + \frac{\mathbf{1}}{\mathbf{z}^m} \right) + p\_1 \left( \mathbf{z}^{m-1} + \frac{\mathbf{1}}{\mathbf{z}^{m-1}} \right) + \dots + p\_{m-1} \left( \mathbf{z} + \frac{\mathbf{1}}{\mathbf{z}} \right) \right] + p\_m,\tag{7}$$

an expression that is obtained by using the relations pk <sup>¼</sup> <sup>p</sup>2m�k, 0 <sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>2</sup>m, and gathering the terms of p zð Þ with the same coefficients. Furthermore, the expression Zsð Þ¼ <sup>z</sup> <sup>z</sup><sup>s</sup> <sup>þ</sup> <sup>z</sup>�<sup>s</sup> ð Þ for any integer <sup>s</sup> can be written as a polynomial of degree <sup>s</sup> in the new variable x ¼ z þ 1=z (the proof follows easily by induction over s); thus, we can write p zð Þ¼ <sup>z</sup>mq xð Þ, where q xð Þ¼ <sup>q</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> qmxm is such that the coefficients q0, …, qm are certain functions of p0, …, pm. From this we can state the following:

Given a polynomial p zð Þ of degree n, we define two Möbius-transformed

Q zð Þ. Provided pð Þ1 6¼ 0, we have that η<sup>1</sup> ¼ W ζ<sup>1</sup> ð Þ, …, η<sup>n</sup> ¼ W ζ<sup>n</sup> ð Þ. Similarly, if τ1, …τ<sup>n</sup> are the zeros of T zð Þ, then we have τ<sup>1</sup> ¼ M ζ<sup>1</sup> ð Þ, …, τ<sup>n</sup> ¼ M ζ<sup>n</sup> ð Þ, provided that

ð Þ <sup>ζ</sup><sup>k</sup> � <sup>1</sup> <sup>n</sup>

Theorem 5. Let p zð Þ be an SI polynomial. Then, the transformed polynomial

1 ζ 

Thus, any pair of zeros of p zð Þ that are symmetric to the unit circle are mapped in zeros of Q zð Þ that are symmetric to the real line; because p zð Þ is SI, it follows that Q zð Þ is SC. Conversely, let ζ and ζ be two zeros of an SC polynomial p zð Þ; then the

> <sup>¼</sup> <sup>ζ</sup> � <sup>i</sup> ζ þ i

mial through M zð Þ and any real polynomial is mapped to an SI polynomial with ω ¼ 1 through W zð Þ. Thus, the set of SI polynomials with ω ¼ 1 is isomorphic to the set of real polynomials. Besides, an SI polynomial with ω 6¼ 1 can be transformed into another one with ω ¼ 1 by performing a suitable uniform rotation of its zeros. It can also be shown that the action of the Möbius transformation over a PSR polynomial

leads to a real polynomial that has only even powers. See [4] for more.

Thus, any pair of zeros of p zð Þ that are symmetric to the real line are mapped in zeros of T zð Þ that are symmetric to the unit circle. Because p zð Þ is SC, it follows that T zð Þ is SI. □ We can also verify that any SI polynomial with ω ¼ 1 is mapped to a real polyno-

pWz ð Þ ð Þ will be an SI polynomial. Proof. Let ζ and 1=ζ be two inversive zeros an SI polynomial p zð Þ. Then,

according to Theorem 4, the corresponding zeros of Q zð Þ will be:

<sup>ζ</sup> � <sup>1</sup> <sup>¼</sup> <sup>η</sup> and <sup>W</sup>

¼ τ and M ζ

pMz ð Þ ð Þ is an SC polynomial. Similarly, if p zð Þ is an SC polynomial,

¼ �i

1=ζ þ 1 1=ζ � 1

¼ � <sup>1</sup>=<sup>ζ</sup> <sup>þ</sup> <sup>i</sup> 1=ζ � i

¼ i

ζ þ 1 ζ � 1

<sup>¼</sup> <sup>1</sup>Mð Þ¼ <sup>ζ</sup> <sup>1</sup>

τ

: (13)

¼ Wð Þ¼ ζ η:

(12)

pMz ð Þ ð Þ , and T zð Þ¼ ð Þ <sup>z</sup> � <sup>1</sup> <sup>n</sup>

The following theorem shows us how the zeros of Q zð Þ and T zð Þ are related with

Theorem 4. Let ζ1, …, ζ<sup>n</sup> denote the zeros of p zð Þ and η1, …, η<sup>n</sup> the respective zeros of

Proof. In fact, inverting the expression for Q zð Þ and evaluating it in any zero ζ<sup>k</sup>

that z ¼ 1 is not a zero of p zð Þ we get that η<sup>k</sup> ¼ W ζ<sup>k</sup> ð Þ is a zero of Q zð Þ. The proof for the zeros of T zð Þ is analogous. □ This result also shows that Q zð Þ and T zð Þ have the same degree as p zð Þ whenever pð Þ1 6¼ 0 or pð Þ �i 6¼ 0, respectively. In fact, if p zð Þ has a zero at z ¼ 1 of multiplicity m then Q zð Þ will be a polynomial of degree n � m, the same being true for T zð Þ if p zð Þ has a zero of multiplicity m at z ¼ �i. This can be explained by the fact that the points z ¼ 1 and z ¼ �i are mapped to infinity by W zð Þ and M zð Þ, respectively. The following theorem shows that the set of SI polynomials are isomorphic to

pWz ð Þ ð Þ : (11)

Q W ζ<sup>k</sup> ð Þ¼ ð Þ 0 for 0⩽ k ⩽ n. Provided

polynomials, namely

the zeros of p zð Þ:

pð Þ �i 6¼ 0.

Q zð Þ¼ ð Þ <sup>z</sup> <sup>þ</sup> <sup>i</sup> <sup>n</sup>

Polynomials with Symmetric Zeros

DOI: http://dx.doi.org/10.5772/intechopen.82728

of p zð Þ we get that <sup>p</sup> <sup>ζ</sup><sup>k</sup> ð Þ¼ �ð Þ <sup>i</sup>=<sup>2</sup> <sup>n</sup>

the set of SC polynomials:

Q zð Þ¼ ð Þ <sup>z</sup> <sup>þ</sup> <sup>i</sup> <sup>n</sup>

then T zð Þ¼ ð Þ <sup>z</sup> � <sup>1</sup> <sup>n</sup>

Wð Þ¼� ζ i

<sup>M</sup>ð Þ¼ <sup>ζ</sup> <sup>ζ</sup> � <sup>i</sup>

73

ζ þ i

ζ þ 1

corresponding zeros of T zð Þ will be:

Theorem 3. Let p zð Þ be a PSR polynomial of even degree n ¼ 2m. For each pair ζ and 1=ζ of self-reciprocal zeros of p zð Þ that lie on S, there is a corresponding zero ξ of the polynomial q xð Þ, as defined above, in the interval ½ � �2; 2 of the real line.

Proof. For each zero <sup>ζ</sup> of p xð Þ that lie on <sup>S</sup>, write <sup>ζ</sup> <sup>¼</sup> <sup>e</sup>i<sup>θ</sup> for some <sup>θ</sup> <sup>∈</sup> <sup>R</sup>. Thereby, as q xð Þ¼ q zð Þ¼ <sup>þ</sup> <sup>1</sup>=<sup>z</sup> p zð Þ=zm, it follows that <sup>ξ</sup> <sup>¼</sup> <sup>ζ</sup> <sup>þ</sup> <sup>1</sup>=<sup>ζ</sup> <sup>¼</sup> 2 cos <sup>θ</sup> will be a zero of q xð Þ. This shows us that ξ is limited to the interval ½ � �2; 2 of the real line. Finally, notice that the reciprocal zero 1=<sup>ζ</sup> of p zð Þ is mapped to the same zero <sup>ξ</sup> of q xð Þ. □

Finally, remembering that the Chebyshev polynomials of first kind, Tnð Þz , are defined by the formula Tn <sup>1</sup> <sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>z</sup>�<sup>1</sup> ð Þ <sup>¼</sup> <sup>1</sup> <sup>2</sup> zn <sup>þ</sup> <sup>z</sup>�<sup>n</sup> ð Þ for <sup>z</sup><sup>∈</sup> <sup>C</sup>, it follows as well that q xð Þ, and hence any PSR polynomial, can be written as a linear combination of Chebyshev polynomials:

$$q(\mathbf{x}) = 2\left[p\_0 T\_m(\mathbf{x}) + p\_1 T\_{m-1}(\mathbf{x}) + \dots + p\_{m-1} T\_1(\mathbf{x}) + \frac{1}{2} p\_m T\_0(\mathbf{x})\right].\tag{8}$$

#### 3. How these polynomials are related to each other?

In this section, we shall analyze how SC, SR, and SI polynomials are related to each other. Let us begin with the relationship between the SR and SI polynomials, which is actually very simple: indeed, from (1), (2), and (3) we can see that each one is nothing but the conjugate polynomial of the other, that is

$$p^\dagger(\mathbf{z}) = \overline{p^\*}\,(\mathbf{z}) = \overline{p^\*(\overline{\mathbf{z}})}, \qquad \text{and} \qquad p^\*(\mathbf{z}) = \overline{p^\*}(\mathbf{z}) = \overline{p^\*(\overline{\mathbf{z}})}.\tag{9}$$

Thus, if p zð Þ is an SR (SI) polynomial, then p zð Þ will be SI (SR) polynomial. Because of this simple relationship, several theorems which are valid for SI polynomials are also valid for SR polynomials and vice versa.

The relationship between SC and SI polynomials is not so easy to perceive. A way of revealing their connection is to make use of a suitable pair of Möbius transformations, that maps the unit circle onto the real line and vice versa, which is often called Cayley transformations, defined through the formulas:

$$M(\mathbf{z}) = (\mathbf{z} - i)/(\mathbf{z} + i), \qquad \text{and} \qquad W(\mathbf{z}) = -i(\mathbf{z} + \mathbf{1})/(\mathbf{z} - \mathbf{1}).\tag{10}$$

This approach was developed in [4], where some algorithms for counting the number of zeros that a complex polynomial has on the unit circle were also formulated.

It is an easy matter to verify that M zð Þ maps R onto S while W zð Þ maps S onto R. Besides, M zð Þ maps the upper (lower) half-plane to the interior (exterior) of S, while W zð Þ maps the interior (exterior) of S onto the upper (lower) half-plane. Notice that W zð Þ can be thought as the inverse of M zð Þ in the Riemann sphere C<sup>∞</sup> ¼ C∪ ∞f g, if we further assume that Mð Þ¼ �i ∞, Mð Þ¼ ∞ 1, Wð Þ¼ 1 ∞, and Wð Þ¼� ∞ i.

Polynomials with Symmetric Zeros DOI: http://dx.doi.org/10.5772/intechopen.82728

an expression that is obtained by using the relations pk <sup>¼</sup> <sup>p</sup>2m�k, 0 <sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>2</sup>m, and gathering the terms of p zð Þ with the same coefficients. Furthermore, the expression Zsð Þ¼ <sup>z</sup> <sup>z</sup><sup>s</sup> <sup>þ</sup> <sup>z</sup>�<sup>s</sup> ð Þ for any integer <sup>s</sup> can be written as a polynomial of degree <sup>s</sup> in the new variable x ¼ z þ 1=z (the proof follows easily by induction over s); thus, we can write p zð Þ¼ <sup>z</sup>mq xð Þ, where q xð Þ¼ <sup>q</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> qmxm is such that the coefficients q0, …, qm are certain functions of p0, …, pm. From this we can state the following: Theorem 3. Let p zð Þ be a PSR polynomial of even degree n ¼ 2m. For each pair ζ and 1=ζ of self-reciprocal zeros of p zð Þ that lie on S, there is a corresponding zero ξ of the

Proof. For each zero <sup>ζ</sup> of p xð Þ that lie on <sup>S</sup>, write <sup>ζ</sup> <sup>¼</sup> <sup>e</sup>i<sup>θ</sup> for some <sup>θ</sup> <sup>∈</sup> <sup>R</sup>. Thereby, as q xð Þ¼ q zð Þ¼ <sup>þ</sup> <sup>1</sup>=<sup>z</sup> p zð Þ=zm, it follows that <sup>ξ</sup> <sup>¼</sup> <sup>ζ</sup> <sup>þ</sup> <sup>1</sup>=<sup>ζ</sup> <sup>¼</sup> 2 cos <sup>θ</sup> will be a zero of q xð Þ. This shows us that ξ is limited to the interval ½ � �2; 2 of the real line. Finally, notice that the reciprocal zero 1=<sup>ζ</sup> of p zð Þ is mapped to the same zero <sup>ξ</sup> of q xð Þ. □ Finally, remembering that the Chebyshev polynomials of first kind, Tnð Þz , are

<sup>2</sup> zn <sup>þ</sup> <sup>z</sup>�<sup>n</sup> ð Þ for <sup>z</sup><sup>∈</sup> <sup>C</sup>, it follows as well that

pmT0ð Þ x

: (8)

1 2

polynomial q xð Þ, as defined above, in the interval ½ � �2; 2 of the real line.

<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>z</sup>�<sup>1</sup> ð Þ <sup>¼</sup> <sup>1</sup>

q xð Þ¼ <sup>2</sup> <sup>p</sup>0Tmð Þþ <sup>x</sup> <sup>p</sup>1Tm�<sup>1</sup>ð Þþ <sup>x</sup> <sup>⋯</sup> <sup>þ</sup> pm�<sup>1</sup>T1ð Þþ <sup>x</sup>

3. How these polynomials are related to each other?

one is nothing but the conjugate polynomial of the other, that is

nomials are also valid for SR polynomials and vice versa.

called Cayley transformations, defined through the formulas:

q xð Þ, and hence any PSR polynomial, can be written as a linear combination of

In this section, we shall analyze how SC, SR, and SI polynomials are related to each other. Let us begin with the relationship between the SR and SI polynomials, which is actually very simple: indeed, from (1), (2), and (3) we can see that each

Thus, if p zð Þ is an SR (SI) polynomial, then p zð Þ will be SI (SR) polynomial. Because of this simple relationship, several theorems which are valid for SI poly-

The relationship between SC and SI polynomials is not so easy to perceive. A way of revealing their connection is to make use of a suitable pair of Möbius transformations, that maps the unit circle onto the real line and vice versa, which is often

M zð Þ¼ ð Þ z � i =ð Þ z þ i , and W zð Þ¼�i zð Þ þ 1 =ð Þ z � 1 : (10)

This approach was developed in [4], where some algorithms for counting the number of zeros that a complex polynomial has on the unit circle were also formu-

It is an easy matter to verify that M zð Þ maps R onto S while W zð Þ maps S onto R.

Besides, M zð Þ maps the upper (lower) half-plane to the interior (exterior) of S, while W zð Þ maps the interior (exterior) of S onto the upper (lower) half-plane. Notice that W zð Þ can be thought as the inverse of M zð Þ in the Riemann sphere C<sup>∞</sup> ¼ C∪ ∞f g, if we further assume that Mð Þ¼ �i ∞, Mð Þ¼ ∞ 1, Wð Þ¼ 1 ∞, and

ð Þ¼ <sup>z</sup> <sup>p</sup><sup>∗</sup> ð Þ¼ <sup>z</sup> <sup>p</sup><sup>∗</sup> ð Þ<sup>z</sup> , and <sup>p</sup><sup>∗</sup> ð Þ¼ <sup>z</sup> <sup>p</sup>† ð Þ¼ <sup>z</sup> <sup>p</sup>†ð Þ<sup>z</sup> : (9)

defined by the formula Tn <sup>1</sup>

Polynomials - Theory and Application

Chebyshev polynomials:

p†

lated.

72

Wð Þ¼� ∞ i.

Given a polynomial p zð Þ of degree n, we define two Möbius-transformed polynomials, namely

$$Q(\mathbf{z}) = (\mathbf{z} + \mathbf{i})^\mathbf{n} p(\mathbf{M}(\mathbf{z})), \qquad \text{and} \qquad T(\mathbf{z}) = (\mathbf{z} - \mathbf{1})^\mathbf{n} p(\mathbf{W}(\mathbf{z})).\tag{11}$$

The following theorem shows us how the zeros of Q zð Þ and T zð Þ are related with the zeros of p zð Þ:

Theorem 4. Let ζ1, …, ζ<sup>n</sup> denote the zeros of p zð Þ and η1, …, η<sup>n</sup> the respective zeros of Q zð Þ. Provided pð Þ1 6¼ 0, we have that η<sup>1</sup> ¼ W ζ<sup>1</sup> ð Þ, …, η<sup>n</sup> ¼ W ζ<sup>n</sup> ð Þ. Similarly, if τ1, …τ<sup>n</sup> are the zeros of T zð Þ, then we have τ<sup>1</sup> ¼ M ζ<sup>1</sup> ð Þ, …, τ<sup>n</sup> ¼ M ζ<sup>n</sup> ð Þ, provided that pð Þ �i 6¼ 0.

Proof. In fact, inverting the expression for Q zð Þ and evaluating it in any zero ζ<sup>k</sup> of p zð Þ we get that <sup>p</sup> <sup>ζ</sup><sup>k</sup> ð Þ¼ �ð Þ <sup>i</sup>=<sup>2</sup> <sup>n</sup> ð Þ <sup>ζ</sup><sup>k</sup> � <sup>1</sup> <sup>n</sup> Q W ζ<sup>k</sup> ð Þ¼ ð Þ 0 for 0⩽ k ⩽ n. Provided that z ¼ 1 is not a zero of p zð Þ we get that η<sup>k</sup> ¼ W ζ<sup>k</sup> ð Þ is a zero of Q zð Þ. The proof for the zeros of T zð Þ is analogous. □

This result also shows that Q zð Þ and T zð Þ have the same degree as p zð Þ whenever pð Þ1 6¼ 0 or pð Þ �i 6¼ 0, respectively. In fact, if p zð Þ has a zero at z ¼ 1 of multiplicity m then Q zð Þ will be a polynomial of degree n � m, the same being true for T zð Þ if p zð Þ has a zero of multiplicity m at z ¼ �i. This can be explained by the fact that the points z ¼ 1 and z ¼ �i are mapped to infinity by W zð Þ and M zð Þ, respectively.

The following theorem shows that the set of SI polynomials are isomorphic to the set of SC polynomials:

Theorem 5. Let p zð Þ be an SI polynomial. Then, the transformed polynomial Q zð Þ¼ ð Þ <sup>z</sup> <sup>þ</sup> <sup>i</sup> <sup>n</sup> pMz ð Þ ð Þ is an SC polynomial. Similarly, if p zð Þ is an SC polynomial, then T zð Þ¼ ð Þ <sup>z</sup> � <sup>1</sup> <sup>n</sup> pWz ð Þ ð Þ will be an SI polynomial.

Proof. Let ζ and 1=ζ be two inversive zeros an SI polynomial p zð Þ. Then, according to Theorem 4, the corresponding zeros of Q zð Þ will be:

$$W(\zeta) = -i\frac{\zeta + 1}{\overline{\zeta} - 1} = \eta \qquad \text{and} \qquad W\left(\frac{1}{\overline{\zeta}}\right) = -i\frac{1/\overline{\zeta} + 1}{1/\overline{\zeta} - 1} = i\frac{\overline{\zeta} + 1}{\overline{\zeta} - 1} = \overline{W(\zeta)} = \overline{\eta}. \tag{12}$$

Thus, any pair of zeros of p zð Þ that are symmetric to the unit circle are mapped in zeros of Q zð Þ that are symmetric to the real line; because p zð Þ is SI, it follows that Q zð Þ is SC. Conversely, let ζ and ζ be two zeros of an SC polynomial p zð Þ; then the corresponding zeros of T zð Þ will be:

$$M(\zeta) = \frac{\zeta - i}{\zeta + i} = \tau \qquad \text{and} \qquad M(\overline{\zeta}) = \frac{\overline{\zeta} - i}{\overline{\zeta} + i} = -\frac{\mathbf{1}/\overline{\zeta} + i}{\mathbf{1}/\overline{\zeta} - i} = \mathbf{1} \\ \overline{M(\zeta)} = \frac{\mathbf{1}}{\overline{\tau}}.\tag{13}$$

Thus, any pair of zeros of p zð Þ that are symmetric to the real line are mapped in zeros of T zð Þ that are symmetric to the unit circle. Because p zð Þ is SC, it follows that T zð Þ is SI. □

We can also verify that any SI polynomial with ω ¼ 1 is mapped to a real polynomial through M zð Þ and any real polynomial is mapped to an SI polynomial with ω ¼ 1 through W zð Þ. Thus, the set of SI polynomials with ω ¼ 1 is isomorphic to the set of real polynomials. Besides, an SI polynomial with ω 6¼ 1 can be transformed into another one with ω ¼ 1 by performing a suitable uniform rotation of its zeros. It can also be shown that the action of the Möbius transformation over a PSR polynomial leads to a real polynomial that has only even powers. See [4] for more.

#### 4. Zeros location theorems

In this section, we shall discuss some theorems regarding the distribution of the zeros of SC, SR, and SI polynomials on the complex plane. Some general theorems relying on the number of zeros that an arbitrary complex polynomial has inside, on, or outside S are also discussed. To save space, we shall not present the proofs of these theorems, which can be found in the original works. Other related theorems can be found in Marden's book [1].

A simple necessary and sufficient condition for all the zeros of a complex poly-

Theorem 11. (Chen) A necessary and sufficient condition for all the zeros of a complex polynomial p zð Þ of degree n to lie on S is that there exists a polynomial q zð Þ of degree n � m whose zeros are all in or on <sup>S</sup> and such that p zð Þ¼ zmq zð Þþ <sup>ω</sup>q†ð Þ<sup>z</sup> for

We close this section by mentioning that there exist many other well-known theorems regarding the distribution of the zeros of complex polynomials. We can cite, for example, the famous rule of Descartes (the number of positive zeros of a real polynomial is limited from above by the number of sign variations in the ordered sequence of its coefficients), the Sturm Theorem (the exact number of zeros that a real polynomial has in a given interval ð � a; b of the real line is determined by the formula N ¼ var½ �� S bð Þ var½ � S að Þ , where var½ � Sð Þξ means the number of sign variations of the Sturm sequence S xð Þ evaluated at x ¼ ξ) and Kronecker Theorem (if all the zeros of a monic polynomial with integer coefficients lie on the unit circle, then all these zeros are indeed roots of unity), see [1] for more. There are still other important theorems relying on matrix methods and quadratic forms that were developed by several authors as Cohn, Schur, Hermite, Sylvester, Hurwitz, Krein,

Let us now consider real SR polynomials. The theorems below are usually applied to PSR polynomials, but some of them can be extended to NSR polynomials

An analog of Eneström-Kakeya theorem for PSR polynomials was found by

 

Lakatos discussed the separation of the zeros on the unit circle of PSR polynomials in [16]; she also found several sufficient conditions for their zeros to be all

For PSR polynomials of odd degree, Lakatos and Losonczi [17] found a stronger

Theorem 15. (Lakatos and Losonczi) Let p zð Þ be a PSR polynomial of odd degree,

Theorem 16. (Lakatos and Losonczi) All zeros of a PSR polynomial p zð Þ of degree

then all the zeros of p zð Þ lie on S. The zeros are simple except when the equality is strict. Theorem 14 was generalized further by Lakatos and Losonczi in [18]:

<sup>k</sup>¼<sup>1</sup> <sup>p</sup>2mþ<sup>1</sup> � pk 

<sup>⩾</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup>

, then all the zeros of p zð Þ lie on S. Moreover, the zeros of p zð Þ are

Theorem 14. (Lakatos) Let p zð Þ be a PSR polynomial of degree n . 2. If

for <sup>0</sup><sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>n</sup> � <sup>1</sup>. Then, all the zeros of p zð Þ

, where ϕ<sup>m</sup> ¼ π=½ � 4ð Þ m þ 1 ,

, pnr⩾0, and

<sup>k</sup>¼<sup>1</sup> pk � pn <sup>þ</sup> <sup>r</sup> 

Theorem 12. (Chen and Chinen) Let p zð Þ be a PSR polynomial of degree n that is written in the form p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>p</sup>1<sup>z</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pkz<sup>k</sup> <sup>þ</sup> pkzn�<sup>k</sup> <sup>þ</sup> pk�<sup>1</sup>z<sup>n</sup>�kþ<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>p</sup>0z<sup>n</sup> and such that <sup>0</sup> , pk , pk�<sup>1</sup> , <sup>⋯</sup> , <sup>p</sup><sup>1</sup> , <sup>p</sup>0. Then all the zeros of p zð Þ are on <sup>S</sup>. Going in the same direction, Choo found in [15] the following condition: Theorem 13. (Choo) Let p zð Þ be a PSR polynomial of degree n and such that its coefficients satisfy the following conditions: npn <sup>⩾</sup> ð Þ <sup>n</sup> � <sup>1</sup> pn�<sup>1</sup> <sup>⩾</sup> <sup>⋯</sup> <sup>⩾</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> pkþ<sup>1</sup> . <sup>0</sup>

Chen in [12] and then, in a slightly stronger version, by Chinen in [14]:

<sup>j</sup>¼<sup>0</sup> ð Þ <sup>j</sup> <sup>þ</sup> <sup>1</sup> pjþ<sup>1</sup> � jpj

<sup>⩾</sup> cos <sup>2</sup>ð Þ <sup>ϕ</sup><sup>m</sup> <sup>∑</sup><sup>2</sup><sup>m</sup>

 

on S. One of the main theorems is the following:

all simple, except when the equality takes place.

<sup>n</sup> . <sup>2</sup> lie on <sup>S</sup> if the following conditions hold: pn <sup>þ</sup> <sup>r</sup>

 

nomial to lie on S was presented by Chen in [12]:

some complex number ω of modulus 1.

DOI: http://dx.doi.org/10.5772/intechopen.82728

Polynomials with Symmetric Zeros

among others, see [13].

and kð Þ <sup>þ</sup> <sup>1</sup> pkþ<sup>1</sup> <sup>⩾</sup> <sup>∑</sup><sup>k</sup>

<sup>k</sup>¼<sup>1</sup> pn � pk 

version of this result:

say n <sup>¼</sup> <sup>2</sup><sup>m</sup> <sup>þ</sup> <sup>1</sup>. If p2mþ<sup>1</sup>

⩾ j jr , for r∈ R.

as well.

are on S.

pn 

pn 

75

<sup>⩾</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup>

4.2 Real self-reciprocal polynomials

#### 4.1 Polynomials that do not necessarily have symmetric zeros

The following theorems are classics (see [1] for the proofs):

Theorem 6. (Rouché). Let q zð Þ and r zð Þ be polynomials such that ∣q zð Þ∣ , ∣r zð Þ∣ along all points of S. Then, the polynomial p zð Þ¼ q zð Þþ r zð Þ has the same number of zeros inside S as the polynomial r zð Þ, counted with multiplicity.

Thus, if a complex polynomial p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pkz<sup>k</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pnz<sup>n</sup> of degree <sup>n</sup> is such that pk . <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pk�<sup>1</sup> <sup>þ</sup> pkþ<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pn , then p zð Þ will have exactly k zeros inside S, counted with multiplicity.

Theorem 7. (Gauss and Lucas) The zeros of the derivative p<sup>0</sup> ð Þz of a polynomial p zð Þ lie all within the convex hull of the zeros of the p zð Þ.

Thereby, if a polynomial p zð Þ has all its zeros on S, then all the zeros of p<sup>0</sup> ð Þz will lie in or on S. In particular, the zeros of p<sup>0</sup> ð Þz will lie on S if, and only if, they are multiple zeros of p zð Þ.

Theorem 8. (Cohn) A necessary and sufficient condition for all the zeros of a complex polynomial p zð Þ to lie on S is that p zð Þ is SI and that its derivative p<sup>0</sup> ð Þz does not have any zero outside S.

Cohn introduced his theorem in [5]. Bonsall and Marden presented a simpler proof of Conh's theorem in [6] (see also [7]) and applied it to SI polynomials—in fact, this was probably the first paper to use the expression "self-inversive." Other important result of Cohn is the following: all the zeros of a complex polynomial p zð Þ¼ pnz<sup>n</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>p</sup><sup>0</sup> will lie on <sup>S</sup> if, and only if, <sup>∣</sup>pn<sup>∣</sup> <sup>¼</sup> <sup>∣</sup>p0<sup>∣</sup> and all the zeros of p zð Þ do not lie outside S.

Restricting ourselves to polynomials with real coefficients, Eneström and Kakeya [8–10] independently presented the following theorem:

Theorem 9. (Eneström and Kakeya) Let p zð Þ be a polynomial of degree n with real coefficients. If its coefficients are such that <sup>0</sup> , <sup>p</sup><sup>0</sup> <sup>⩽</sup> <sup>p</sup><sup>1</sup> <sup>⩽</sup> <sup>⋯</sup> <sup>⩽</sup> pn�<sup>1</sup> <sup>⩽</sup> pn, then all the zeros of p zð Þ lie in or on S. Likewise, if the coefficients of p zð Þ are such that <sup>0</sup> , pn <sup>⩽</sup> pn�<sup>1</sup> <sup>⩽</sup> <sup>⋯</sup> <sup>⩽</sup> <sup>p</sup><sup>1</sup> <sup>⩽</sup> <sup>p</sup>0, then all the zeros of p zð Þ lie on or outside <sup>S</sup>.

The following theorems are relatively more recent. The distribution of the zeros of a complex polynomial regarding the unit circle S was presented by Marden in [1] and slightly enhanced by Jury in [11]:

Theorem 10. (Marden and Jury) Let p zð Þ be a complex polynomial of degree n and p<sup>∗</sup> ð Þ<sup>z</sup> its reciprocal. Construct the sequence of polynomials Pjð Þ¼ <sup>z</sup> <sup>∑</sup><sup>n</sup>�<sup>j</sup> <sup>k</sup>¼<sup>0</sup>Pj,kz<sup>k</sup> such that P0ð Þ¼ <sup>z</sup> p zð Þ and Pjþ<sup>1</sup>ð Þ¼ <sup>z</sup> pj,0Pjð Þ� <sup>z</sup> pj,n�<sup>j</sup> P∗ <sup>j</sup> ð Þz for 0⩽ j⩽ n � 1 so that we have the relations pjþ1,k <sup>¼</sup> pj,0pj,k � pj,n�<sup>j</sup> pj,n�j�<sup>k</sup> . Let <sup>δ</sup><sup>j</sup> denote the constant terms of the polynomials Pjð Þz , i.e., δ<sup>j</sup> ¼ pj,<sup>0</sup> and Δ<sup>k</sup> ¼ δ1⋯δk. Thus, if N of the products Δ<sup>k</sup> are negative and n � N of the products Δ<sup>k</sup> are positive so that none of them are zero, then p zð Þ has N zeros inside S, n � N zeros outside S and no zero on S. On the other hand, if Δ<sup>k</sup> 6¼ 0 for some k , n but Pkþ<sup>1</sup>ð Þ¼ z 0, then p zð Þ has either n � k zeros on S or n � k zeros symmetric to S. It has additionally N zeros inside S and k � N zeros outside S.

#### Polynomials with Symmetric Zeros DOI: http://dx.doi.org/10.5772/intechopen.82728

4. Zeros location theorems

Polynomials - Theory and Application

can be found in Marden's book [1].

zeros inside S, counted with multiplicity.

lie in or on S. In particular, the zeros of p<sup>0</sup>

and slightly enhanced by Jury in [11]:

have the relations pjþ1,k <sup>¼</sup> pj,0pj,k � pj,n�<sup>j</sup>

that P0ð Þ¼ <sup>z</sup> p zð Þ and Pjþ<sup>1</sup>ð Þ¼ <sup>z</sup> pj,0Pjð Þ� <sup>z</sup> pj,n�<sup>j</sup>

such that pk 

multiple zeros of p zð Þ.

have any zero outside S.

do not lie outside S.

74

In this section, we shall discuss some theorems regarding the distribution of the zeros of SC, SR, and SI polynomials on the complex plane. Some general theorems relying on the number of zeros that an arbitrary complex polynomial has inside, on, or outside S are also discussed. To save space, we shall not present the proofs of these theorems, which can be found in the original works. Other related theorems

Theorem 6. (Rouché). Let q zð Þ and r zð Þ be polynomials such that ∣q zð Þ∣ , ∣r zð Þ∣ along all points of S. Then, the polynomial p zð Þ¼ q zð Þþ r zð Þ has the same number of

Thus, if a complex polynomial p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pkz<sup>k</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pnz<sup>n</sup> of degree <sup>n</sup> is

Thereby, if a polynomial p zð Þ has all its zeros on S, then all the zeros of p<sup>0</sup>

Theorem 8. (Cohn) A necessary and sufficient condition for all the zeros of a

Cohn introduced his theorem in [5]. Bonsall and Marden presented a simpler proof of Conh's theorem in [6] (see also [7]) and applied it to SI polynomials—in fact, this was probably the first paper to use the expression "self-inversive." Other important result of Cohn is the following: all the zeros of a complex polynomial p zð Þ¼ pnz<sup>n</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>p</sup><sup>0</sup> will lie on <sup>S</sup> if, and only if, <sup>∣</sup>pn<sup>∣</sup> <sup>¼</sup> <sup>∣</sup>p0<sup>∣</sup> and all the zeros of p zð Þ

Restricting ourselves to polynomials with real coefficients, Eneström and

Theorem 9. (Eneström and Kakeya) Let p zð Þ be a polynomial of degree n with real coefficients. If its coefficients are such that <sup>0</sup> , <sup>p</sup><sup>0</sup> <sup>⩽</sup> <sup>p</sup><sup>1</sup> <sup>⩽</sup> <sup>⋯</sup> <sup>⩽</sup> pn�<sup>1</sup> <sup>⩽</sup> pn, then all the

The following theorems are relatively more recent. The distribution of the zeros of a complex polynomial regarding the unit circle S was presented by Marden in [1]

Theorem 10. (Marden and Jury) Let p zð Þ be a complex polynomial of degree n

polynomials Pjð Þz , i.e., δ<sup>j</sup> ¼ pj,<sup>0</sup> and Δ<sup>k</sup> ¼ δ1⋯δk. Thus, if N of the products Δ<sup>k</sup> are negative and n � N of the products Δ<sup>k</sup> are positive so that none of them are zero, then p zð Þ has N zeros inside S, n � N zeros outside S and no zero on S. On the other hand, if Δ<sup>k</sup> 6¼ 0 for some k , n but Pkþ<sup>1</sup>ð Þ¼ z 0, then p zð Þ has either n � k zeros on S or n � k zeros symmetric to S. It has additionally N zeros inside S and k � N zeros outside S.

P∗

Kakeya [8–10] independently presented the following theorem:

zeros of p zð Þ lie in or on S. Likewise, if the coefficients of p zð Þ are such that <sup>0</sup> , pn <sup>⩽</sup> pn�<sup>1</sup> <sup>⩽</sup> <sup>⋯</sup> <sup>⩽</sup> <sup>p</sup><sup>1</sup> <sup>⩽</sup> <sup>p</sup>0, then all the zeros of p zð Þ lie on or outside <sup>S</sup>.

and p<sup>∗</sup> ð Þ<sup>z</sup> its reciprocal. Construct the sequence of polynomials Pjð Þ¼ <sup>z</sup> <sup>∑</sup><sup>n</sup>�<sup>j</sup>

complex polynomial p zð Þ to lie on S is that p zð Þ is SI and that its derivative p<sup>0</sup>

, then p zð Þ will have exactly k

ð Þz will lie on S if, and only if, they are

ð Þz of a polynomial

ð Þz will

ð Þz does not

<sup>k</sup>¼<sup>0</sup>Pj,kz<sup>k</sup> such

<sup>j</sup> ð Þz for 0⩽ j⩽ n � 1 so that we

pj,n�j�<sup>k</sup> . Let <sup>δ</sup><sup>j</sup> denote the constant terms of the

4.1 Polynomials that do not necessarily have symmetric zeros

The following theorems are classics (see [1] for the proofs):

zeros inside S as the polynomial r zð Þ, counted with multiplicity.

. <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pk�<sup>1</sup> <sup>þ</sup> pkþ<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pn

p zð Þ lie all within the convex hull of the zeros of the p zð Þ.

Theorem 7. (Gauss and Lucas) The zeros of the derivative p<sup>0</sup>

A simple necessary and sufficient condition for all the zeros of a complex polynomial to lie on S was presented by Chen in [12]:

Theorem 11. (Chen) A necessary and sufficient condition for all the zeros of a complex polynomial p zð Þ of degree n to lie on S is that there exists a polynomial q zð Þ of degree n � m whose zeros are all in or on <sup>S</sup> and such that p zð Þ¼ zmq zð Þþ <sup>ω</sup>q†ð Þ<sup>z</sup> for some complex number ω of modulus 1.

We close this section by mentioning that there exist many other well-known theorems regarding the distribution of the zeros of complex polynomials. We can cite, for example, the famous rule of Descartes (the number of positive zeros of a real polynomial is limited from above by the number of sign variations in the ordered sequence of its coefficients), the Sturm Theorem (the exact number of zeros that a real polynomial has in a given interval ð � a; b of the real line is determined by the formula N ¼ var½ �� S bð Þ var½ � S að Þ , where var½ � Sð Þξ means the number of sign variations of the Sturm sequence S xð Þ evaluated at x ¼ ξ) and Kronecker Theorem (if all the zeros of a monic polynomial with integer coefficients lie on the unit circle, then all these zeros are indeed roots of unity), see [1] for more. There are still other important theorems relying on matrix methods and quadratic forms that were developed by several authors as Cohn, Schur, Hermite, Sylvester, Hurwitz, Krein, among others, see [13].

#### 4.2 Real self-reciprocal polynomials

Let us now consider real SR polynomials. The theorems below are usually applied to PSR polynomials, but some of them can be extended to NSR polynomials as well.

An analog of Eneström-Kakeya theorem for PSR polynomials was found by Chen in [12] and then, in a slightly stronger version, by Chinen in [14]:

Theorem 12. (Chen and Chinen) Let p zð Þ be a PSR polynomial of degree n that is written in the form p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>p</sup>1<sup>z</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pkz<sup>k</sup> <sup>þ</sup> pkzn�<sup>k</sup> <sup>þ</sup> pk�<sup>1</sup>z<sup>n</sup>�kþ<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>p</sup>0z<sup>n</sup> and such that <sup>0</sup> , pk , pk�<sup>1</sup> , <sup>⋯</sup> , <sup>p</sup><sup>1</sup> , <sup>p</sup>0. Then all the zeros of p zð Þ are on <sup>S</sup>.

Going in the same direction, Choo found in [15] the following condition:

Theorem 13. (Choo) Let p zð Þ be a PSR polynomial of degree n and such that its coefficients satisfy the following conditions: npn <sup>⩾</sup> ð Þ <sup>n</sup> � <sup>1</sup> pn�<sup>1</sup> <sup>⩾</sup> <sup>⋯</sup> <sup>⩾</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> pkþ<sup>1</sup> . <sup>0</sup> and kð Þ <sup>þ</sup> <sup>1</sup> pkþ<sup>1</sup> <sup>⩾</sup> <sup>∑</sup><sup>k</sup> <sup>j</sup>¼<sup>0</sup> ð Þ <sup>j</sup> <sup>þ</sup> <sup>1</sup> pjþ<sup>1</sup> � jpj for <sup>0</sup><sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>n</sup> � <sup>1</sup>. Then, all the zeros of p zð Þ are on S.

Lakatos discussed the separation of the zeros on the unit circle of PSR polynomials in [16]; she also found several sufficient conditions for their zeros to be all on S. One of the main theorems is the following:

Theorem 14. (Lakatos) Let p zð Þ be a PSR polynomial of degree n . 2. If pn <sup>⩾</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup> <sup>k</sup>¼<sup>1</sup> pn � pk , then all the zeros of p zð Þ lie on S. Moreover, the zeros of p zð Þ are all simple, except when the equality takes place.

For PSR polynomials of odd degree, Lakatos and Losonczi [17] found a stronger version of this result:

Theorem 15. (Lakatos and Losonczi) Let p zð Þ be a PSR polynomial of odd degree, say n <sup>¼</sup> <sup>2</sup><sup>m</sup> <sup>þ</sup> <sup>1</sup>. If p2mþ<sup>1</sup> <sup>⩾</sup> cos <sup>2</sup>ð Þ <sup>ϕ</sup><sup>m</sup> <sup>∑</sup><sup>2</sup><sup>m</sup> <sup>k</sup>¼<sup>1</sup> <sup>p</sup>2mþ<sup>1</sup> � pk , where ϕ<sup>m</sup> ¼ π=½ � 4ð Þ m þ 1 , then all the zeros of p zð Þ lie on S. The zeros are simple except when the equality is strict.

Theorem 14 was generalized further by Lakatos and Losonczi in [18]:

Theorem 16. (Lakatos and Losonczi) All zeros of a PSR polynomial p zð Þ of degree <sup>n</sup> . <sup>2</sup> lie on <sup>S</sup> if the following conditions hold: pn <sup>þ</sup> <sup>r</sup> <sup>⩾</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup> <sup>k</sup>¼<sup>1</sup> pk � pn <sup>þ</sup> <sup>r</sup> , pnr⩾0, and pn ⩾ j jr , for r∈ R.

Other conditions for all the zeros of a PSR polynomial to lie on S were presented by Kwon in [19]. In its simplest form, Kown's theorem can be enunciated as follows: Theorem 20. (Schinzel) Let p zð Þ be an SI polynomial of degree n. If the inequality

Theorem 21. (Losonczi and Schinzel) Let p zð Þ be an SI polynomial of odd degree,

Another sufficient condition for all the zeros of an SI polynomial to lie on S was

Theorem 22. (Lakatos and Losonczi) Let p zð Þ be an SI polynomial of degree n

In [45], Lakatos and Losonczi also formulated a theorem that contains as special

Theorem 23. (Lakatos and Losonczi) Let p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pnz<sup>n</sup> be an SI polynomial of degree n⩾2 and a, b, and c be complex numbers such that a 6¼ 0, ∣b∣ ¼ 1, and

Theorem 24. (Losonczi) Let p zð Þ be a monic complex SR polynomial of even degree, say n ¼ 2m. Then, all the zeros of p zð Þ will lie on S if, and only if, there exist real numbers α1, …, α2m, all with moduli less than or equal to 2, that satisfy the inequalities:

pn

, then, all the zeros of p zð Þ lie on S. Moreover, these zeros are simple if the

In [46], Losonczi presented the following necessary and sufficient conditions for

Losonczi, in [46], also showed that if all the zeros of a complex monic reciprocal

The theorems above give conditions for all the zeros of SI or SR polynomials to

Theorem 25. (Vieira) Let p zð Þ be an SI polynomial of degree n. If the inequality

on S; besides, all these zeros are simple when the inequality is strict. Moreover, p zð Þ will

The case m ¼ 0 corresponds to Lakatos and Losonczi Theorem 14 for all the zeros of p zð Þ to lie on S. The necessary counterpart of this theorem was considered by Stankov in [48], with an application to the theory of Salem numbers—see

Other results on the distribution of zeros of SI polynomials include the following:

Sinclair and Vaaler [49] showed that a monic SI polynomial p zð Þ of degree n

Lp z ½ � ð Þ , m , <sup>n</sup>=2, holds true, then p zð Þ will have exactly n � <sup>2</sup>m zeros

 . <sup>1</sup>

<sup>þ</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup>

ð Þ <sup>α</sup>1; …; <sup>α</sup>2<sup>m</sup> , <sup>0</sup><sup>⩽</sup> <sup>k</sup><sup>⩽</sup> m, where <sup>σ</sup><sup>2</sup><sup>m</sup>

 

<sup>2</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup> <sup>k</sup>¼<sup>1</sup> pk 

<sup>⩾</sup> ap<sup>0</sup> � <sup>b</sup><sup>n</sup>

all the zeros of a (complex) SR polynomial of even degree to lie on S:

k�2l

denotes the kth elementary symmetric function in the 2m variables α1, …, α2m.

polynomial are on S, then its coefficients are all real and satisfy the inequality

lie on S. In many cases, however, we need to verify if a polynomial has a given number of zeros (or none) on the unit circle. Considering this problem, Vieira in [47] found sufficient conditions for an SI polynomial of degree n to have a determined

number of zeros on the unit circle. In terms of the length, Lp z ½ �¼ ð Þ p<sup>0</sup>

of a polynomial p zð Þ of degree n, this theorem can be stated as follows:

have no zero on S if, for n even and m ¼ n=2, the inequality pm

In a similar way, Losonczi and Schinzel [43] generalized theorem 15 for the

<sup>⩾</sup> cos <sup>2</sup>ð Þ <sup>ϕ</sup><sup>m</sup> infa,b<sup>∈</sup> <sup>C</sup>:∣b∣¼<sup>1</sup>∑<sup>2</sup>mþ<sup>1</sup>

ϕ<sup>m</sup> ¼ π=½ � 4ð Þ m þ 1 , then all the zeros of p zð Þ lie on S. The zeros are simple except when

, then all the zeros of p zð Þ lie on <sup>S</sup>. These zero are

<sup>k</sup>¼<sup>1</sup> apk � <sup>b</sup>2mþ1�<sup>k</sup>

holds. Then, all the zeros of p zð Þ lie on

<sup>k</sup>¼<sup>1</sup> apk � bn�<sup>k</sup> <sup>c</sup> � pn 

 

<sup>p</sup>2mþ<sup>1</sup>

 þ

<sup>k</sup> ð Þ α1; …; α2<sup>m</sup>

 

 þ ⋯ þ pn 

<sup>2</sup> Lp z ½ � ð Þ is satisfied.

 , where

pn

 

pn 

SI case:

apn � pn

pk ¼ �ð Þ<sup>1</sup> <sup>k</sup>

n k

pn ⩽

pn�<sup>m</sup> ⩾ <sup>1</sup> 4 n n�m

Section 5.1.

77

inequality is strict.

<sup>⩾</sup> infa,b <sup>∈</sup> <sup>C</sup>:∣b∣¼<sup>1</sup>∑<sup>n</sup>

Polynomials with Symmetric Zeros

i.e., n <sup>¼</sup> <sup>2</sup><sup>m</sup> <sup>þ</sup> <sup>1</sup>. If p2mþ<sup>1</sup>

the equality is strict.

simple whenever the equality is strict.

DOI: http://dx.doi.org/10.5772/intechopen.82728

 

and suppose that the inequality pn

cases many of the previous results:

<sup>c</sup>=pn <sup>∈</sup> <sup>R</sup>, <sup>0</sup><sup>⩽</sup> <sup>c</sup>=pn <sup>⩽</sup> <sup>1</sup>. If pn <sup>þ</sup> <sup>c</sup>

∑½ � <sup>k</sup>=<sup>2</sup> l¼0

for 0<sup>⩽</sup> <sup>k</sup><sup>⩽</sup> <sup>n</sup>.

presented by Lakatos and Losonczi in [44]:

<sup>k</sup>¼<sup>0</sup> apk � bn�<sup>k</sup>

 ⩾ <sup>1</sup>

m � k þ 2l l σ<sup>2</sup><sup>m</sup>

S. Moreover, the zeros are all simple except when an equality takes place.

 

Theorem 17. (Kwon) Let p zð Þ be a PSR polynomial of even degree n⩾2 whose leading coefficient pn is positive and p<sup>0</sup> ⩽ p<sup>1</sup> ⩽ ⋯ ⩽ pn. In this case, all the zeros of p zð Þ will lie on <sup>S</sup> if, either pn<sup>=</sup><sup>2</sup> <sup>⩾</sup> <sup>∑</sup><sup>n</sup> <sup>k</sup>¼<sup>0</sup> pk � pn<sup>=</sup><sup>2</sup> � � � � � �, or pð Þ<sup>1</sup> <sup>⩾</sup><sup>0</sup> and pn <sup>⩾</sup> <sup>1</sup> <sup>2</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup> <sup>k</sup>¼<sup>1</sup> pk � pn<sup>=</sup><sup>2</sup> � � � � � �.

Modified forms of this theorem hold for PSR polynomials of odd degree and for the case where the coefficients of p zð Þ do not have the ordination above—see [19] for these cases. Kwon also found conditions for all but two zeros of p zð Þ to lie on S in [20], which is relevant to the theory of Salem polynomials—see Section 5.

Other interesting results are the following: Konvalina and Matache [21] found conditions under which a PSR polynomial has at least one non-real zero on S. Kim and Park [22] and then Kim and Lee [23] presented conditions for which all the zeros of certain PSR polynomials lie on S (some open cases were also addressed by Botta et al. in [24]). Suzuki [25] presented necessary and sufficient conditions, relying on matrix algebra and differential equations, for all the zeros of PSR polynomials to lie on S. In [26] Botta et al. studied the distribution of the zeros of PSR polynomials with a small perturbation in their coefficients. Real SR polynomials of height 1—namely, special cases of Littlewood, Newman, and Borwein polynomials were studied by several authors, see [27–35] and references therein.<sup>2</sup> Zeros of the so-called Ramanujan Polynomials and generalizations were analyzed in [37–39]. Finally, the Galois theory of PSR polynomials was studied in [40] by Lindstrøm, who showed that any PSR polynomial of degree less than 10 can be solved by radicals.

#### 4.3 Complex self-reciprocal and self-inversive polynomials

Let us consider now the case of complex SR polynomials and SI polynomials. Here, we remark that many of the theorems that hold for SI polynomials either also hold for SR polynomials or can be easily adapted to this case (the opposite is also true).

Theorem 18. (Cohn) An SI polynomial p zð Þ has as many zeros outside S as does its derivative p0 ð Þz .

This follows directly from Cohn's Theorem 8 for the case where p zð Þ is SI. Besides, we can also conclude from this that the derivative of p zð Þ has no zeros on S except at the multiple zeros of p zð Þ. Furthermore, if an SI polynomial p zð Þ of degree n has exactly k zeros on S, while its derivative has exactly l zeros in or on S, both counted with multiplicity, then n ¼ 2ð Þ� l þ 1 k.

O'Hara and Rodriguez [41] showed that the following conditions are always satisfied by SI polynomials whose zeros are all on S:

Theorem 19. (O'Hara and Rodriguez) Let p zð Þ be an SI polynomial of degree n whose zeros are all on S. Then, the following inequality holds: ∑<sup>n</sup> <sup>j</sup>¼<sup>0</sup> pj � � � � � � 2 <sup>⩽</sup> k k p zð Þ <sup>2</sup> , where k k p zð Þ denotes the maximum modulus of p zð Þ on the unit circle; besides, if this inequality is strict then the zeros of p zð Þ are rotations of nth roots of unity. Moreover, the following inequalities are also satisfied: aj j <sup>k</sup> <sup>⩽</sup> <sup>1</sup> <sup>2</sup> k k p zð Þ if <sup>k</sup> 6¼ <sup>n</sup>=2 and j j ak <sup>⩽</sup> ffiffi 2 p <sup>2</sup> k k p zð Þ for k ¼ n=2.

Schinzel in [42], generalized Lakatos Theorem 14 for SI polynomials:

<sup>2</sup> The zeros of such polynomials present a fractal behavior, as was first discovered by Odlyzko and Poonen in [36].

Other conditions for all the zeros of a PSR polynomial to lie on S were presented by Kwon in [19]. In its simplest form, Kown's theorem can be enunciated as follows: Theorem 17. (Kwon) Let p zð Þ be a PSR polynomial of even degree n⩾2 whose leading coefficient pn is positive and p<sup>0</sup> ⩽ p<sup>1</sup> ⩽ ⋯ ⩽ pn. In this case, all the zeros of p zð Þ

� �

Modified forms of this theorem hold for PSR polynomials of odd degree and for the case where the coefficients of p zð Þ do not have the ordination above—see [19] for these cases. Kwon also found conditions for all but two zeros of p zð Þ to lie on S in

Other interesting results are the following: Konvalina and Matache [21] found conditions under which a PSR polynomial has at least one non-real zero on S. Kim and Park [22] and then Kim and Lee [23] presented conditions for which all the zeros of certain PSR polynomials lie on S (some open cases were also addressed by Botta et al. in [24]). Suzuki [25] presented necessary and sufficient conditions, relying on matrix algebra and differential equations, for all the zeros of PSR polynomials to lie on S. In [26] Botta et al. studied the distribution of the zeros of PSR polynomials with a small perturbation in their coefficients. Real SR polynomials of height 1—namely, special cases of Littlewood, Newman, and Borwein polynomials were studied by several authors, see [27–35] and references therein.<sup>2</sup> Zeros of the so-called Ramanujan Polynomials and generalizations were analyzed in [37–39]. Finally, the Galois theory of PSR polynomials was studied in [40] by Lindstrøm, who showed that any PSR polynomial of degree less than 10 can be solved by

Let us consider now the case of complex SR polynomials and SI polynomials. Here, we remark that many of the theorems that hold for SI polynomials either also hold for SR polynomials or can be easily adapted to this case (the opposite is also

Theorem 18. (Cohn) An SI polynomial p zð Þ has as many zeros outside S as does its

This follows directly from Cohn's Theorem 8 for the case where p zð Þ is SI. Besides, we can also conclude from this that the derivative of p zð Þ has no zeros on S except at the multiple zeros of p zð Þ. Furthermore, if an SI polynomial p zð Þ of degree n has exactly k zeros on S, while its derivative has exactly l zeros in or on S, both

O'Hara and Rodriguez [41] showed that the following conditions are always

Theorem 19. (O'Hara and Rodriguez) Let p zð Þ be an SI polynomial of degree n

k k p zð Þ denotes the maximum modulus of p zð Þ on the unit circle; besides, if this inequality is strict then the zeros of p zð Þ are rotations of nth roots of unity. Moreover, the following

Schinzel in [42], generalized Lakatos Theorem 14 for SI polynomials:

<sup>2</sup> The zeros of such polynomials present a fractal behavior, as was first discovered by Odlyzko and

<sup>2</sup> k k p zð Þ if <sup>k</sup> 6¼ <sup>n</sup>=2 and j j ak <sup>⩽</sup> ffiffi

<sup>j</sup>¼<sup>0</sup> pj � � � � � � 2

> 2 p

<sup>⩽</sup> k k p zð Þ <sup>2</sup>

<sup>2</sup> k k p zð Þ for

, where

�, or pð Þ<sup>1</sup> <sup>⩾</sup><sup>0</sup> and pn <sup>⩾</sup> <sup>1</sup>

<sup>2</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup>

<sup>k</sup>¼<sup>1</sup> pk � pn<sup>=</sup><sup>2</sup> � � �

� � �.

<sup>k</sup>¼<sup>0</sup> pk � pn<sup>=</sup><sup>2</sup> � � �

[20], which is relevant to the theory of Salem polynomials—see Section 5.

4.3 Complex self-reciprocal and self-inversive polynomials

will lie on <sup>S</sup> if, either pn<sup>=</sup><sup>2</sup> <sup>⩾</sup> <sup>∑</sup><sup>n</sup>

Polynomials - Theory and Application

radicals.

true).

derivative p0

k ¼ n=2.

Poonen in [36].

76

ð Þz .

counted with multiplicity, then n ¼ 2ð Þ� l þ 1 k.

inequalities are also satisfied: aj j <sup>k</sup> <sup>⩽</sup> <sup>1</sup>

satisfied by SI polynomials whose zeros are all on S:

whose zeros are all on S. Then, the following inequality holds: ∑<sup>n</sup>

Theorem 20. (Schinzel) Let p zð Þ be an SI polynomial of degree n. If the inequality pn <sup>⩾</sup> infa,b <sup>∈</sup> <sup>C</sup>:∣b∣¼<sup>1</sup>∑<sup>n</sup> <sup>k</sup>¼<sup>0</sup> apk � bn�<sup>k</sup> pn , then all the zeros of p zð Þ lie on <sup>S</sup>. These zero are simple whenever the equality is strict.

In a similar way, Losonczi and Schinzel [43] generalized theorem 15 for the SI case:

Theorem 21. (Losonczi and Schinzel) Let p zð Þ be an SI polynomial of odd degree, i.e., n <sup>¼</sup> <sup>2</sup><sup>m</sup> <sup>þ</sup> <sup>1</sup>. If p2mþ<sup>1</sup> <sup>⩾</sup> cos <sup>2</sup>ð Þ <sup>ϕ</sup><sup>m</sup> infa,b<sup>∈</sup> <sup>C</sup>:∣b∣¼<sup>1</sup>∑<sup>2</sup>mþ<sup>1</sup> <sup>k</sup>¼<sup>1</sup> apk � <sup>b</sup>2mþ1�<sup>k</sup> <sup>p</sup>2mþ<sup>1</sup> , where ϕ<sup>m</sup> ¼ π=½ � 4ð Þ m þ 1 , then all the zeros of p zð Þ lie on S. The zeros are simple except when the equality is strict.

Another sufficient condition for all the zeros of an SI polynomial to lie on S was presented by Lakatos and Losonczi in [44]:

Theorem 22. (Lakatos and Losonczi) Let p zð Þ be an SI polynomial of degree n and suppose that the inequality pn ⩾ <sup>1</sup> <sup>2</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup> <sup>k</sup>¼<sup>1</sup> pk holds. Then, all the zeros of p zð Þ lie on S. Moreover, the zeros are all simple except when an equality takes place.

In [45], Lakatos and Losonczi also formulated a theorem that contains as special cases many of the previous results:

Theorem 23. (Lakatos and Losonczi) Let p zð Þ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> pnz<sup>n</sup> be an SI polynomial of degree n⩾2 and a, b, and c be complex numbers such that a 6¼ 0, ∣b∣ ¼ 1, and <sup>c</sup>=pn <sup>∈</sup> <sup>R</sup>, <sup>0</sup><sup>⩽</sup> <sup>c</sup>=pn <sup>⩽</sup> <sup>1</sup>. If pn <sup>þ</sup> <sup>c</sup> <sup>⩾</sup> ap<sup>0</sup> � <sup>b</sup><sup>n</sup> pn <sup>þ</sup> <sup>∑</sup><sup>n</sup>�<sup>1</sup> <sup>k</sup>¼<sup>1</sup> apk � bn�<sup>k</sup> <sup>c</sup> � pn þ apn � pn , then, all the zeros of p zð Þ lie on S. Moreover, these zeros are simple if the inequality is strict.

In [46], Losonczi presented the following necessary and sufficient conditions for all the zeros of a (complex) SR polynomial of even degree to lie on S:

Theorem 24. (Losonczi) Let p zð Þ be a monic complex SR polynomial of even degree, say n ¼ 2m. Then, all the zeros of p zð Þ will lie on S if, and only if, there exist real numbers α1, …, α2m, all with moduli less than or equal to 2, that satisfy the inequalities: pk ¼ �ð Þ<sup>1</sup> <sup>k</sup> ∑½ � <sup>k</sup>=<sup>2</sup> l¼0 m � k þ 2l l σ<sup>2</sup><sup>m</sup> k�2l ð Þ <sup>α</sup>1; …; <sup>α</sup>2<sup>m</sup> , <sup>0</sup><sup>⩽</sup> <sup>k</sup><sup>⩽</sup> m, where <sup>σ</sup><sup>2</sup><sup>m</sup> <sup>k</sup> ð Þ α1; …; α2<sup>m</sup> denotes the kth elementary symmetric function in the 2m variables α1, …, α2m.

Losonczi, in [46], also showed that if all the zeros of a complex monic reciprocal polynomial are on S, then its coefficients are all real and satisfy the inequality

$$|p\_n| \ll \binom{n}{k} \text{ for } 0 \ll k \ll n.$$

The theorems above give conditions for all the zeros of SI or SR polynomials to lie on S. In many cases, however, we need to verify if a polynomial has a given number of zeros (or none) on the unit circle. Considering this problem, Vieira in [47] found sufficient conditions for an SI polynomial of degree n to have a determined number of zeros on the unit circle. In terms of the length, Lp z ½ �¼ ð Þ p<sup>0</sup> þ ⋯ þ pn of a polynomial p zð Þ of degree n, this theorem can be stated as follows:

Theorem 25. (Vieira) Let p zð Þ be an SI polynomial of degree n. If the inequality pn�<sup>m</sup> ⩾ <sup>1</sup> 4 n n�m Lp z ½ � ð Þ , m , <sup>n</sup>=2, holds true, then p zð Þ will have exactly n � <sup>2</sup>m zeros on S; besides, all these zeros are simple when the inequality is strict. Moreover, p zð Þ will have no zero on S if, for n even and m ¼ n=2, the inequality pm . <sup>1</sup> <sup>2</sup> Lp z ½ � ð Þ is satisfied.

The case m ¼ 0 corresponds to Lakatos and Losonczi Theorem 14 for all the zeros of p zð Þ to lie on S. The necessary counterpart of this theorem was considered by Stankov in [48], with an application to the theory of Salem numbers—see Section 5.1.

Other results on the distribution of zeros of SI polynomials include the following: Sinclair and Vaaler [49] showed that a monic SI polynomial p zð Þ of degree n

satisfying the inequalities Lr ½ � p zð Þ <sup>⩽</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>r</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>1</sup>�<sup>r</sup> or Lr ½ � p zð Þ <sup>⩽</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>r</sup> ð Þ <sup>l</sup> � <sup>2</sup> <sup>1</sup>�<sup>r</sup> , where r⩾1, Lr ½ �¼ p zð Þ p<sup>0</sup> � � � � r þ ⋯ þ pn � � � � r , and l is the number of non-null terms of p zð Þ, has all their zeros on S; the authors also studied the geometry of SI polynomials whose zeros are all on S. Choo and Kim applied Theorem 11 to SI polynomials in [50]. Hypergeometric polynomials with all their zeros on S were considered in [51, 52]. Kim [53] also obtained SI polynomials which are related to Jacobi polynomials. Ito and Wimmer [54] studied SI polynomial operators in Hilbert space whose spectrum is on S.

σ ¼

DOI: http://dx.doi.org/10.5772/intechopen.82728

Polynomials with Symmetric Zeros

constant.

mial Lð Þz , then,

5.2 Knot theory

79

problem raised by Lehmer:

satisfies the inequality M p z ½ � ð Þ . 1 þ ϵ.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>27</sup> <sup>r</sup> <sup>3</sup>

ffiffiffiffiffi 23

þ

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The Mahler problem is, however, still open for SR polynomials. A monic integer SR polynomial with exactly two (real and positive) zeros (say, ζ and 1=ζ) not lying on S is called a Salem polynomial [62, 64]. It can be shown that a Pisot polynomial with at least one zero on S is also a Salem polynomial. The unique positive zero greater than one of a Salem polynomial is called its Salem number, which also equals the value of its Mahler measure. A Salem number s is said to be small if s , σ; up to date, only 47 small Salem numbers are known [65, 66] and the smallest known one was found about 80 years ago by Lehmer [67]. This gave place to the following: Conjecture 1. (Lehmer) The monic integer polynomial with the smallest Mahler measure is the Lehmer polynomial <sup>L</sup>ð Þ¼ <sup>z</sup> <sup>z</sup><sup>10</sup> <sup>þ</sup> <sup>z</sup><sup>9</sup> � <sup>z</sup><sup>7</sup> � <sup>z</sup><sup>6</sup> � <sup>z</sup><sup>5</sup> � <sup>z</sup><sup>4</sup> � <sup>z</sup><sup>3</sup> <sup>þ</sup> <sup>z</sup> <sup>þ</sup> 1, a Salem polynomial whose Mahler measure is Λ ≈ 1:17628081826, known as Lehmer's

The proof of this conjecture is also an open problem. To be fair, we do not even know if there exists a smallest Salem number at all. This is the content of another

Problem 2. (Lehmer) Answer whether there exists or not a positive number ϵ such that the Mahler measure of any monic, integer, and non-cyclotomic polynomial p zð Þ

Lehmer's polynomial also appears in connection with several fields of mathematics. Many examples are discussed in Hironaka's paper [68]; here we shall only present an amazing identity found by Bailey and Broadhurst in [69] in their works on polylogarithm ladders: if λ is any zero of the aforementioned Lehmer's polyno-

A knot is a closed, non-intersecting, one-dimensional curve embedded on R<sup>3</sup> [70]. Knot theory studies topological properties of knots as, for example, criteria under which a knot can be unknot, conditions for the equivalency between knots, the classification of prime knots, etc.; see [70] for the corresponding definitions. In

One of the most important questions in knot theory is to determine whether or not two knots are equivalent. This, however, is not an easy task. A way of attacking this question is to look for abstract objects—mainly the so-called knot invariants rather than to the knots themselves. A knot invariant is a (topologic, combinatorial, algebraic, etc.) quantity that can be computed for any knot and that is always the same for equivalents knots.<sup>4</sup> An important class of knot invariants is constituted by the so-called Knot Polynomials. Knot polynomials were introduced in 1928 by Alexander [71]. They consist in polynomials with integer coefficients that can be written down for every knot. For about 60 years since its creation, Alexander polynomials were the only known kind of knot polynomial. It was only in 1985 that Jones [72]

<sup>4</sup> We remark, however, that different knots can have the same knot invariant. Up to date, we do not know whether there exists a knot invariant that distinguishes all non-equivalent knots from each other (although there do exist some invariants that distinguish every knot from the trivial knot). Thus, until

now the concept of knot invariants only partially solves the problem of knot classification.

<sup>λ</sup><sup>315</sup> � <sup>1</sup> � � <sup>λ</sup><sup>210</sup> � <sup>1</sup> � � <sup>λ</sup><sup>126</sup> � <sup>1</sup> � �<sup>2</sup> <sup>λ</sup><sup>90</sup> � <sup>1</sup> � � <sup>λ</sup><sup>3</sup> � <sup>1</sup> � �<sup>3</sup> <sup>λ</sup><sup>2</sup> � <sup>1</sup> � �<sup>5</sup>

Figure 1, we plotted all prime knots up to six crossings.

<sup>λ</sup><sup>630</sup> � <sup>1</sup> � � <sup>λ</sup><sup>35</sup> � <sup>1</sup> � � <sup>λ</sup><sup>15</sup> � <sup>1</sup> � �<sup>2</sup> <sup>λ</sup><sup>14</sup> � <sup>1</sup> � �<sup>2</sup> <sup>λ</sup><sup>5</sup> � <sup>1</sup> � �<sup>6</sup>

<sup>27</sup> <sup>r</sup> <sup>3</sup>

ffiffiffiffiffi 23

≈ 1:32471795724: (15)

ð Þ <sup>λ</sup> � <sup>1</sup> <sup>3</sup>

<sup>λ</sup><sup>68</sup> <sup>¼</sup> <sup>1</sup>: (16)

s

#### 5. Where these polynomials are found?

In this section, we shall briefly discuss some important or recent applications of the theory of polynomials with symmetric zeros. We remark, however, that our selection is by no means exhaustive: for example, SR and SI polynomials also find applications in many fields of mathematics (e.g., information and coding theory [55], algebraic curves over a finite field and cryptography [56], elliptic functions [57], number theory [58], etc.) and physics (e.g., Lee-Yang theorem in statistical physics [59], Poincaré Polynomials defined on Calabi-Yau manifolds of superstring theory [60], etc.).

#### 5.1 Polynomials with small Mahler measure

Given a monic polynomial p zð Þ of degree n, with integer coefficients, the Mahler measure of p zð Þ, denoted by Mp z ½ � ð Þ , is defined as the product of the modulus of all those zeros of p zð Þ that lie in the exterior of S [61]. That is

$$M[p(\mathbf{z})] = \prod\_{i=1}^{n} \max\{\mathbf{1}, |\zeta\_i|\},\tag{14}$$

where <sup>ζ</sup>1, …, <sup>ζ</sup><sup>n</sup> are the zeros<sup>3</sup> of p zð Þ. Thus, if a monic integer polynomial p zð Þ has all its zeros in or on the unit circle, we have Mp z ½ �¼ ð Þ 1; in particular, all cyclotomic polynomials (which are PSR polynomials whose zeros are the primitive roots of unity, see [1]) have Mahler measure equal to 1. In a sense, the Mahler measure of a polynomial p zð Þ measures how close it is to the cyclotomic polynomials. Therefore, it is natural to raise the following:

Problem 1. (Mahler) Find the monic, integer, non-cyclotomic polynomial with the smallest Mahler measure.

This is an 80-year-old open problem of mathematics. Of course, we can expect that the polynomials with the smallest Mahler measure be among those with only a few number of zeros outside S, in particular among those with only one zero outside S. A monic integer polynomial that has exactly one zero outside S is called a Pisot polynomial and its unique zero of modulus greater than 1 is called its Pisot number [62]. A breakthrough towards the solution of Mahler's problem was given by Smyth in [63]:

Theorem 26. (Smyth) The Pisot polynomial S zð Þ¼ <sup>z</sup><sup>3</sup> � <sup>z</sup> � <sup>1</sup> is the polynomial with smallest Mahler measure among the set of all monic, integer, and non-SR polynomials. Its Mahler measure is given by the value of its Pisot number, which is,

<sup>3</sup> The Mahler measure of a monic integer polynomial p zð Þ can also be defined without making reference to its zeros through the formula Mpz ½ �¼ ð Þ exp <sup>Ð</sup> <sup>1</sup> <sup>0</sup> log p e<sup>2</sup>πit � � � � dt n o—see [61].

Polynomials with Symmetric Zeros DOI: http://dx.doi.org/10.5772/intechopen.82728

satisfying the inequalities Lr

Polynomials - Theory and Application

whose spectrum is on S.

theory [60], etc.).

smallest Mahler measure.

by Smyth in [63]:

78

½ �¼ p zð Þ p<sup>0</sup> � � � � r

5. Where these polynomials are found?

5.1 Polynomials with small Mahler measure

those zeros of p zð Þ that lie in the exterior of S [61]. That is

mials. Therefore, it is natural to raise the following:

to its zeros through the formula Mpz ½ �¼ ð Þ exp <sup>Ð</sup> <sup>1</sup>

Mp z ½ �¼ ð Þ <sup>Y</sup><sup>n</sup>

where r⩾1, Lr

½ � p zð Þ <sup>⩽</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>r</sup>

p zð Þ, has all their zeros on S; the authors also studied the geometry of SI polynomials whose zeros are all on S. Choo and Kim applied Theorem 11 to SI polynomials in [50]. Hypergeometric polynomials with all their zeros on S were considered in [51, 52]. Kim [53] also obtained SI polynomials which are related to Jacobi polynomials. Ito and Wimmer [54] studied SI polynomial operators in Hilbert space

In this section, we shall briefly discuss some important or recent applications of the theory of polynomials with symmetric zeros. We remark, however, that our selection is by no means exhaustive: for example, SR and SI polynomials also find applications in many fields of mathematics (e.g., information and coding theory [55], algebraic curves over a finite field and cryptography [56], elliptic functions [57], number theory [58], etc.) and physics (e.g., Lee-Yang theorem in statistical physics [59], Poincaré Polynomials defined on Calabi-Yau manifolds of superstring

Given a monic polynomial p zð Þ of degree n, with integer coefficients, the Mahler measure of p zð Þ, denoted by Mp z ½ � ð Þ , is defined as the product of the modulus of all

i¼1

has all its zeros in or on the unit circle, we have Mp z ½ �¼ ð Þ 1; in particular, all cyclotomic polynomials (which are PSR polynomials whose zeros are the primitive roots of unity, see [1]) have Mahler measure equal to 1. In a sense, the Mahler measure of a polynomial p zð Þ measures how close it is to the cyclotomic polyno-

where <sup>ζ</sup>1, …, <sup>ζ</sup><sup>n</sup> are the zeros<sup>3</sup> of p zð Þ. Thus, if a monic integer polynomial p zð Þ

Problem 1. (Mahler) Find the monic, integer, non-cyclotomic polynomial with the

This is an 80-year-old open problem of mathematics. Of course, we can expect that the polynomials with the smallest Mahler measure be among those with only a few number of zeros outside S, in particular among those with only one zero outside S. A monic integer polynomial that has exactly one zero outside S is called a Pisot polynomial and its unique zero of modulus greater than 1 is called its Pisot number [62]. A breakthrough towards the solution of Mahler's problem was given

Theorem 26. (Smyth) The Pisot polynomial S zð Þ¼ <sup>z</sup><sup>3</sup> � <sup>z</sup> � <sup>1</sup> is the polynomial with smallest Mahler measure among the set of all monic, integer, and non-SR polynomials. Its Mahler measure is given by the value of its Pisot number, which is,

<sup>3</sup> The Mahler measure of a monic integer polynomial p zð Þ can also be defined without making reference

<sup>0</sup> log p e<sup>2</sup>πit � � � � dt n o

—see [61].

þ ⋯ þ pn � � � � r ð Þ <sup>n</sup> � <sup>1</sup> <sup>1</sup>�<sup>r</sup> or Lr

½ � p zð Þ <sup>⩽</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>r</sup>

, and l is the number of non-null terms of

max 1; ζ<sup>i</sup> f g j j , (14)

ð Þ <sup>l</sup> � <sup>2</sup> <sup>1</sup>�<sup>r</sup>

,

$$
\sigma = \sqrt[3]{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{23}{27}}} + \sqrt[3]{\frac{1}{2} - \frac{1}{2}\sqrt{\frac{23}{27}}} \approx 1.32471795724.\tag{15}
$$

The Mahler problem is, however, still open for SR polynomials. A monic integer SR polynomial with exactly two (real and positive) zeros (say, ζ and 1=ζ) not lying on S is called a Salem polynomial [62, 64]. It can be shown that a Pisot polynomial with at least one zero on S is also a Salem polynomial. The unique positive zero greater than one of a Salem polynomial is called its Salem number, which also equals the value of its Mahler measure. A Salem number s is said to be small if s , σ; up to date, only 47 small Salem numbers are known [65, 66] and the smallest known one was found about 80 years ago by Lehmer [67]. This gave place to the following:

Conjecture 1. (Lehmer) The monic integer polynomial with the smallest Mahler measure is the Lehmer polynomial <sup>L</sup>ð Þ¼ <sup>z</sup> <sup>z</sup><sup>10</sup> <sup>þ</sup> <sup>z</sup><sup>9</sup> � <sup>z</sup><sup>7</sup> � <sup>z</sup><sup>6</sup> � <sup>z</sup><sup>5</sup> � <sup>z</sup><sup>4</sup> � <sup>z</sup><sup>3</sup> <sup>þ</sup> <sup>z</sup> <sup>þ</sup> 1, a Salem polynomial whose Mahler measure is Λ ≈ 1:17628081826, known as Lehmer's constant.

The proof of this conjecture is also an open problem. To be fair, we do not even know if there exists a smallest Salem number at all. This is the content of another problem raised by Lehmer:

Problem 2. (Lehmer) Answer whether there exists or not a positive number ϵ such that the Mahler measure of any monic, integer, and non-cyclotomic polynomial p zð Þ satisfies the inequality M p z ½ � ð Þ . 1 þ ϵ.

Lehmer's polynomial also appears in connection with several fields of mathematics. Many examples are discussed in Hironaka's paper [68]; here we shall only present an amazing identity found by Bailey and Broadhurst in [69] in their works on polylogarithm ladders: if λ is any zero of the aforementioned Lehmer's polynomial Lð Þz , then,

$$\frac{\left(\lambda^{315}-\mathbf{1}\right)\left(\lambda^{210}-\mathbf{1}\right)\left(\lambda^{126}-\mathbf{1}\right)^{2}\left(\lambda^{90}-\mathbf{1}\right)\left(\lambda^{3}-\mathbf{1}\right)^{3}\left(\lambda^{2}-\mathbf{1}\right)^{5}\left(\lambda-\mathbf{1}\right)^{3}}{\left(\lambda^{630}-\mathbf{1}\right)\left(\lambda^{35}-\mathbf{1}\right)\left(\lambda^{15}-\mathbf{1}\right)^{2}\left(\lambda^{14}-\mathbf{1}\right)^{2}\left(\lambda^{5}-\mathbf{1}\right)^{6}\lambda^{68}}=\mathbf{1}.\tag{16}$$

#### 5.2 Knot theory

A knot is a closed, non-intersecting, one-dimensional curve embedded on R<sup>3</sup> [70]. Knot theory studies topological properties of knots as, for example, criteria under which a knot can be unknot, conditions for the equivalency between knots, the classification of prime knots, etc.; see [70] for the corresponding definitions. In Figure 1, we plotted all prime knots up to six crossings.

One of the most important questions in knot theory is to determine whether or not two knots are equivalent. This, however, is not an easy task. A way of attacking this question is to look for abstract objects—mainly the so-called knot invariants rather than to the knots themselves. A knot invariant is a (topologic, combinatorial, algebraic, etc.) quantity that can be computed for any knot and that is always the same for equivalents knots.<sup>4</sup> An important class of knot invariants is constituted by the so-called Knot Polynomials. Knot polynomials were introduced in 1928 by Alexander [71]. They consist in polynomials with integer coefficients that can be written down for every knot. For about 60 years since its creation, Alexander polynomials were the only known kind of knot polynomial. It was only in 1985 that Jones [72]

<sup>4</sup> We remark, however, that different knots can have the same knot invariant. Up to date, we do not know whether there exists a knot invariant that distinguishes all non-equivalent knots from each other (although there do exist some invariants that distinguish every knot from the trivial knot). Thus, until now the concept of knot invariants only partially solves the problem of knot classification.

Bethe Ansatz. In fact, for the XXZ Heisenberg spin chain, the Bethe Equations consist in a coupled system of trigonometric equations; however, after a change of

> xixk � 2Δxi þ 1 xixk � 2Δxk þ 1

where L ∈ N is the length of the chain, N ∈ N is the excitation number and Δ ∈ R is the so-called spectral parameter. A solution of (18) consists in a (non-ordered) set X ¼ f g x1; …; xN of the unknowns x1, …, xN so that (18) is satisfied. Notice that the

In [76], Vieira and Lima-Santos showed that the solutions of (18), for N ¼ 2 and arbitrary L, are given in terms of the zeros of certain SI polynomials. In fact, (18)

<sup>1</sup> x<sup>L</sup> 2⋯x<sup>L</sup>

<sup>2</sup> ¼ � <sup>x</sup>1x<sup>2</sup> � <sup>2</sup>Δx<sup>2</sup> <sup>þ</sup> <sup>1</sup> x1x<sup>2</sup> � 2Δx<sup>1</sup> þ 1

<sup>2</sup> ¼ 1 we can eliminate one of the unknowns in (19)

L L � 2 � �

<sup>a</sup> , ∣Δ∣ , Δð Þ<sup>2</sup>

, 1⩽ i ⩽ N, (18)

<sup>N</sup> ¼ 1, which suggests an

: (19)

j j ω<sup>a</sup> þ 1 , (21)

m, n∈ N, (22)

<sup>a</sup> , then all the

<sup>a</sup> and more details).

variables is performed, we can write them in the following rational form:

becomes a system of two coupled algebraic equations for N ¼ 2, namely,

<sup>1</sup> xL

, and x<sup>L</sup>

—for instance, by setting x<sup>2</sup> ¼ ωa=x1, where ω<sup>a</sup> ¼ exp 2ð Þ πia=L , 1⩽ a⩽ L, are the roots of unity of degree L. Replacing these values for x<sup>2</sup> into (19), we obtain the

pað Þ¼ <sup>z</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>a</sup> <sup>z</sup><sup>L</sup> � <sup>2</sup>Δωaz<sup>L</sup>�<sup>1</sup> � <sup>2</sup>Δ<sup>z</sup> <sup>þ</sup> ð Þ¼ <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>a</sup> <sup>0</sup>, <sup>1</sup><sup>⩽</sup> <sup>a</sup><sup>⩽</sup> <sup>L</sup>: (20)

We can easily verify that the polynomial pað Þz is SI for each value of a. They also satisfy the relations pað Þ¼ <sup>z</sup> <sup>z</sup>Lpð Þ <sup>ω</sup>a=<sup>z</sup> , 1<sup>⩽</sup> <sup>a</sup><sup>⩽</sup> <sup>L</sup>, which means that the solutions of (19) have the general form X ¼ f g ζ;ωa=ζ for ζ any zero of pað Þz . In [76], the distribution of the zeros of the polynomials pað Þz was analyzed through an application of Vieira's Theorem 25. It was shown that the exact behavior of the zeros of the

> <sup>a</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>a</sup> , then all the zeros of pað Þ<sup>z</sup> are on <sup>S</sup>; if <sup>∣</sup>Δ∣⩾Δð Þ<sup>2</sup>

Finally, we highlight that the polynomial pað Þz becomes a Salem polynomial for a ¼ L and integer values of Δ. This was one of the first appearances of Salem poly-

An infinite sequence P ¼ f g Pnð Þz <sup>n</sup><sup>∈</sup> <sup>N</sup> of polynomials Pnð Þz of degree n is said to be an orthogonal polynomial sequence on the interval ð Þ l;r of the real line if there

where K0, K1, etc. are positive numbers. Orthogonal polynomial sequences on

<sup>0</sup>, m 6¼ n, �

polynomials pað Þz , for each a, depends on two critical values of Δ, namely

j j <sup>ω</sup><sup>a</sup> <sup>þ</sup> <sup>1</sup> , and <sup>Δ</sup>ð Þ<sup>2</sup>

Pmð Þ<sup>z</sup> Pnð Þ<sup>z</sup> w zð Þdz <sup>¼</sup> Kn, m <sup>¼</sup> n,

the real line have many interesting and important properties—see [77].

zeros of pað Þ<sup>z</sup> but two are on <sup>S</sup>; (see [76] for the case <sup>Δ</sup>ð Þ<sup>1</sup>

exists a function w xð Þ, positive in ð Þ l;r ∈ R, such that

N

<sup>k</sup>¼1, <sup>k</sup>6¼<sup>i</sup>

xL

Polynomials with Symmetric Zeros

DOI: http://dx.doi.org/10.5772/intechopen.82728

inversive symmetry of their zeros.

<sup>1</sup> ¼ � <sup>x</sup>1x<sup>2</sup> � <sup>2</sup>Δx<sup>1</sup> <sup>þ</sup> <sup>1</sup> x1x<sup>2</sup> � 2Δx<sup>2</sup> þ 1

following polynomial equations fixing x1:

Now, from the relation xL

Δð Þ<sup>1</sup> <sup>a</sup> <sup>¼</sup> <sup>1</sup> 2

as follows: if ∣Δ∣⩽ Δð Þ<sup>1</sup>

5.4 Orthogonal polynomials

ðr l

nomials in physics.

81

xL

<sup>i</sup> ¼ �ð Þ<sup>1</sup> <sup>N</sup>�<sup>1</sup> <sup>Y</sup>

Bethe equations satisfy the important relation xL

#### Figure 1.

A table of prime knots up to six crossings. In the Alexander-Briggs notation these knots are, in order, 01, 31, 41, 51, 52, 61, 62, and 63.

came up with a new kind of knot polynomials—today known as Jones polynomials and since then other kinds were discovered as well, see [70].

What is interesting for us here is that the Alexander polynomials are PSR polynomials of even degree (say, <sup>n</sup> <sup>¼</sup> <sup>2</sup>m) and with integer coefficients.<sup>5</sup> Thus, they have the following general form:

$$\Delta(t) = \delta\_0 + \delta\_1 \mathbf{z} + \dots + \delta\_{m-1} t^{m-1} + \delta\_m t^m + \delta\_{m-1} t^{m+1} + \dots + \delta\_1 t^{2m-1} + \delta\_0 t^{2m}, \tag{17}$$

where δ<sup>i</sup> ∈ N, 0⩽ i⩽ m. In Table 1, we present the δ<sup>m</sup>�1Alexander polynomials for the prime knots up to six crossings.

Knots theory finds applications in many fields of mathematics in physics—see [70]. In mathematics, we can cite a very interesting connection between Alexander polynomials and the theory of Salem numbers: more precisely, the Alexander polynomial associated with the so-called Pretzel Knot Pð Þ �2; 3; 7 is nothing but the Lehmer polynomial Lð Þz introduced in Section 5.1; it is indeed the Alexander polynomial with the smallest Mahler measure [73]. In physics, knot theory is connected with quantum groups and it also can be used to one construct solutions of the Yang-Baxter equation [74] through a method called baxterization of braid groups.

#### 5.3 Bethe equations

Bethe equations were introduced in 1931 by Hans Bethe [75], together with his powerful method—the so-called Bethe Ansatz Method—for solving spectral problems associated with exactly integrable models of statistical mechanics. They consist in a system of coupled and non-linear equations that ensure the consistency of the


#### Table 1.

Alexander polynomials for prime knots up to six crossings.

<sup>5</sup> Alexander polynomials can also be defined as Laurent polynomials, see [70].

Polynomials with Symmetric Zeros DOI: http://dx.doi.org/10.5772/intechopen.82728

Bethe Ansatz. In fact, for the XXZ Heisenberg spin chain, the Bethe Equations consist in a coupled system of trigonometric equations; however, after a change of variables is performed, we can write them in the following rational form:

$$\mathbf{x}\_{i}^{L} = \left(-\mathbf{1}\right)^{N-1} \prod\_{\substack{k=1,\,k\neq i}}^{N} \frac{\mathbf{x}\_{i}\mathbf{x}\_{k} - 2\Delta\mathbf{x}\_{i} + \mathbf{1}}{\mathbf{x}\_{i}\mathbf{x}\_{k} - 2\Delta\mathbf{x}\_{k} + \mathbf{1}}, \qquad \mathbf{1} \lessdot i \leqslant N,\tag{18}$$

where L ∈ N is the length of the chain, N ∈ N is the excitation number and Δ ∈ R is the so-called spectral parameter. A solution of (18) consists in a (non-ordered) set X ¼ f g x1; …; xN of the unknowns x1, …, xN so that (18) is satisfied. Notice that the Bethe equations satisfy the important relation xL <sup>1</sup> x<sup>L</sup> 2⋯x<sup>L</sup> <sup>N</sup> ¼ 1, which suggests an inversive symmetry of their zeros.

In [76], Vieira and Lima-Santos showed that the solutions of (18), for N ¼ 2 and arbitrary L, are given in terms of the zeros of certain SI polynomials. In fact, (18) becomes a system of two coupled algebraic equations for N ¼ 2, namely,

$$\boldsymbol{\alpha}\_{1}^{L} = -\frac{\boldsymbol{\alpha}\_{1}\boldsymbol{\alpha}\_{2} - 2\Delta\boldsymbol{\alpha}\_{1} + 1}{\boldsymbol{\alpha}\_{1}\boldsymbol{\alpha}\_{2} - 2\Delta\boldsymbol{\alpha}\_{2} + 1}, \qquad \text{and} \qquad \boldsymbol{\alpha}\_{2}^{L} = -\frac{\boldsymbol{\alpha}\_{1}\boldsymbol{\alpha}\_{2} - 2\Delta\boldsymbol{\alpha}\_{2} + 1}{\boldsymbol{\alpha}\_{1}\boldsymbol{\alpha}\_{2} - 2\Delta\boldsymbol{\alpha}\_{1} + 1}. \tag{19}$$

Now, from the relation xL <sup>1</sup> xL <sup>2</sup> ¼ 1 we can eliminate one of the unknowns in (19) —for instance, by setting x<sup>2</sup> ¼ ωa=x1, where ω<sup>a</sup> ¼ exp 2ð Þ πia=L , 1⩽ a⩽ L, are the roots of unity of degree L. Replacing these values for x<sup>2</sup> into (19), we obtain the following polynomial equations fixing x1:

$$p\_a(\mathbf{z}) = (\mathbf{1} + \alpha\_a)\mathbf{z}^L - 2\Delta\alpha\_d \mathbf{z}^{L-1} - 2\Delta\mathbf{z} + (\mathbf{1} + \alpha\_d) = \mathbf{0}, \qquad \mathbf{1} \lessapprox a \lessapprox L. \tag{20}$$

We can easily verify that the polynomial pað Þz is SI for each value of a. They also satisfy the relations pað Þ¼ <sup>z</sup> <sup>z</sup>Lpð Þ <sup>ω</sup>a=<sup>z</sup> , 1<sup>⩽</sup> <sup>a</sup><sup>⩽</sup> <sup>L</sup>, which means that the solutions of (19) have the general form X ¼ f g ζ;ωa=ζ for ζ any zero of pað Þz . In [76], the distribution of the zeros of the polynomials pað Þz was analyzed through an application of Vieira's Theorem 25. It was shown that the exact behavior of the zeros of the polynomials pað Þz , for each a, depends on two critical values of Δ, namely

$$
\Delta\_a^{(1)} = \frac{1}{2} |\alpha\_a + \mathbf{1}|, \qquad \text{and} \qquad \Delta\_a^{(2)} = \frac{1}{2} \left( \frac{L}{L-2} \right) |\alpha\_a + \mathbf{1}|, \tag{21}
$$

as follows: if ∣Δ∣⩽ Δð Þ<sup>1</sup> <sup>a</sup> , then all the zeros of pað Þ<sup>z</sup> are on <sup>S</sup>; if <sup>∣</sup>Δ∣⩾Δð Þ<sup>2</sup> <sup>a</sup> , then all the zeros of pað Þ<sup>z</sup> but two are on <sup>S</sup>; (see [76] for the case <sup>Δ</sup>ð Þ<sup>1</sup> <sup>a</sup> , ∣Δ∣ , Δð Þ<sup>2</sup> <sup>a</sup> and more details).

Finally, we highlight that the polynomial pað Þz becomes a Salem polynomial for a ¼ L and integer values of Δ. This was one of the first appearances of Salem polynomials in physics.

#### 5.4 Orthogonal polynomials

An infinite sequence P ¼ f g Pnð Þz <sup>n</sup><sup>∈</sup> <sup>N</sup> of polynomials Pnð Þz of degree n is said to be an orthogonal polynomial sequence on the interval ð Þ l;r of the real line if there exists a function w xð Þ, positive in ð Þ l;r ∈ R, such that

$$\int\_{l}^{r} P\_{m}(z)P\_{n}(z)w(z)dz = \begin{cases} K\_{m} & m=n, \\ 0, & m \neq n, \end{cases} \qquad m, n \in \mathbb{N}, \tag{22}$$

where K0, K1, etc. are positive numbers. Orthogonal polynomial sequences on the real line have many interesting and important properties—see [77].

came up with a new kind of knot polynomials—today known as Jones polynomials—

A table of prime knots up to six crossings. In the Alexander-Briggs notation these knots are, in order, 01, 31, 41,

What is interesting for us here is that the Alexander polynomials are PSR polynomials of even degree (say, <sup>n</sup> <sup>¼</sup> <sup>2</sup>m) and with integer coefficients.<sup>5</sup> Thus, they

<sup>m</sup> <sup>þ</sup> <sup>δ</sup><sup>m</sup>�<sup>1</sup><sup>t</sup>

where δ<sup>i</sup> ∈ N, 0⩽ i⩽ m. In Table 1, we present the δ<sup>m</sup>�1Alexander polynomials

Knots theory finds applications in many fields of mathematics in physics—see [70]. In mathematics, we can cite a very interesting connection between Alexander polynomials and the theory of Salem numbers: more precisely, the Alexander polynomial associated with the so-called Pretzel Knot Pð Þ �2; 3; 7 is nothing but the Lehmer polynomial Lð Þz introduced in Section 5.1; it is indeed the Alexander polynomial with the smallest Mahler measure [73]. In physics, knot theory is connected with quantum groups and it also can be used to one construct solutions of the Yang-Baxter equation [74] through a method called baxterization of braid groups.

Bethe equations were introduced in 1931 by Hans Bethe [75], together with his powerful method—the so-called Bethe Ansatz Method—for solving spectral problems associated with exactly integrable models of statistical mechanics. They consist in a system of coupled and non-linear equations that ensure the consistency of the

Knot Alexander polynomial Δð Þt Knot Alexander polynomial Δð Þt

<sup>2</sup> 61 <sup>2</sup> � <sup>5</sup><sup>t</sup> <sup>þ</sup> <sup>2</sup><sup>t</sup>

<sup>4</sup> 63 <sup>1</sup> � <sup>3</sup><sup>t</sup> <sup>þ</sup> <sup>5</sup><sup>t</sup>

<sup>2</sup> 62 <sup>1</sup> � <sup>3</sup><sup>t</sup> <sup>þ</sup> <sup>3</sup><sup>t</sup>

01 1 52 2 � 3t þ 2t

<sup>m</sup>þ<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>δ</sup>1<sup>t</sup>

<sup>2</sup>m�<sup>1</sup> <sup>þ</sup> <sup>δ</sup>0<sup>t</sup>

<sup>2</sup>m, (17)

2

2

<sup>2</sup> � <sup>3</sup><sup>t</sup> <sup>3</sup> <sup>þ</sup> <sup>t</sup> 4

<sup>2</sup> � <sup>3</sup><sup>t</sup> <sup>3</sup> <sup>þ</sup> <sup>t</sup> 4

and since then other kinds were discovered as well, see [70].

<sup>m</sup>�<sup>1</sup> <sup>þ</sup> <sup>δ</sup>mt

have the following general form:

Polynomials - Theory and Application

Figure 1.

51, 52, 61, 62, and 63.

ΔðÞ¼ t δ<sup>0</sup> þ δ1z þ ⋯ þ δ<sup>m</sup>�<sup>1</sup>t

5.3 Bethe equations

31 1 � t þ t

41 1 � 3t þ t

<sup>2</sup> � <sup>t</sup> <sup>3</sup> <sup>þ</sup> <sup>t</sup>

<sup>5</sup> Alexander polynomials can also be defined as Laurent polynomials, see [70].

Alexander polynomials for prime knots up to six crossings.

51 1 � t þ t

Table 1.

80

for the prime knots up to six crossings.


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self-reciprocal polynomials.

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[9] Eneström G. Remarque sur un théorème relatif aux racines de <sup>l</sup>'équation anxn <sup>þ</sup> an�<sup>1</sup>xn�<sup>1</sup><sup>þ</sup> ⋯ þ a1x þ a<sup>0</sup> ¼ 0 où tous les coefficientes a sont réels et positifs. Tohoku Mathematical Journal, First

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1902.04231

2031905

Table 2.

Hermite and Möbius-transformed Hermite polynomials, up to 4th degree.

Very recently, Vieira and Botta [78, 79] studied the action of Möbius transformations over orthogonal polynomial sequences on the real line. In particular, they showed that the infinite sequence T ¼ f g Tnð Þz <sup>n</sup><sup>∈</sup> <sup>N</sup> of the Möbius-transformed polynomials Tnð Þ¼ <sup>z</sup> ð Þ <sup>z</sup> � <sup>1</sup> <sup>n</sup> Pnð Þ W zð Þ , where W zð Þ¼�i zð Þ þ 1 =ð Þ z � 1 , is an SI polynomial sequence with all their zeros on the unit circle S—see Table 2 for an example. We highlight that the polynomials Tnð Þz ∈ T also have properties similar to the original polynomials Pnð Þz ∈P as, for instance, they satisfy an orthogonality condition on the unit circle and a three-term recurrence relation, their zeros lie all on S and are simple, for n ⩾1 the zeros of Tnð Þz interlaces with those of Tnþ<sup>1</sup>ð Þz and so on—see [78, 79] for more details.

#### 6. Conclusions

In this work, we reviewed the theory of self-conjugate, self-reciprocal, and selfinversive polynomials. We discussed their main properties, how they are related to each other, the main theorems regarding the distribution of their zeros and some applications of these polynomials both in physics and mathematics. We hope that this short review suits for a compact introduction of the subject, paving the way for further developments in this interesting field of research.

#### Acknowledgements

We thank the editorial staff for all the support during the publishing process and also the Coordination for the Improvement of Higher Education (CAPES).

#### Author details

Ricardo Vieira Faculty of Science and Technology, Department of Mathematics and Computer Science, São Paulo State University (UNESP), Presidente Prudente, SP, Brazil

\*Address all correspondence to: rs.vieira@unesp.br

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Polynomials with Symmetric Zeros DOI: http://dx.doi.org/10.5772/intechopen.82728

#### References

Very recently, Vieira and Botta [78, 79] studied the action of Möbius transformations over orthogonal polynomial sequences on the real line. In particular, they showed that the infinite sequence T ¼ f g Tnð Þz <sup>n</sup><sup>∈</sup> <sup>N</sup> of the Möbius-transformed

Hermite polynomials Möbius-transformed Hermite polynomials

H0ð Þ¼ z 1 H0ð Þ¼ z 1 H1ð Þ¼ z 2z H1ð Þ¼� z 2i � 2iz <sup>H</sup>2ð Þ¼� <sup>z</sup> <sup>2</sup> <sup>þ</sup> <sup>4</sup>z<sup>2</sup> <sup>H</sup>2ð Þ¼� <sup>z</sup> <sup>6</sup> � <sup>4</sup><sup>z</sup> � <sup>6</sup>z<sup>2</sup> <sup>H</sup>3ð Þ¼� <sup>z</sup> <sup>12</sup><sup>z</sup> <sup>þ</sup> <sup>8</sup>z<sup>3</sup> <sup>H</sup>3ð Þ¼� <sup>z</sup> <sup>20</sup><sup>i</sup> <sup>þ</sup> <sup>12</sup>iz <sup>þ</sup> <sup>12</sup>iz<sup>2</sup> <sup>þ</sup> <sup>20</sup>iz<sup>3</sup> <sup>H</sup>4ð Þ¼ <sup>z</sup> <sup>12</sup> � <sup>48</sup>z<sup>2</sup> <sup>þ</sup> <sup>16</sup>z<sup>4</sup> <sup>H</sup>4ð Þ¼ <sup>z</sup> <sup>76</sup> <sup>þ</sup> <sup>16</sup><sup>z</sup> <sup>þ</sup> <sup>72</sup>z<sup>2</sup> <sup>þ</sup> <sup>16</sup>z<sup>3</sup> <sup>þ</sup> <sup>76</sup>z<sup>4</sup>

Hermite and Möbius-transformed Hermite polynomials, up to 4th degree.

polynomial sequence with all their zeros on the unit circle S—see Table 2 for an example. We highlight that the polynomials Tnð Þz ∈ T also have properties similar to the original polynomials Pnð Þz ∈P as, for instance, they satisfy an orthogonality condition on the unit circle and a three-term recurrence relation, their zeros lie all on S and are simple, for n ⩾1 the zeros of Tnð Þz interlaces with those of Tnþ<sup>1</sup>ð Þz and

In this work, we reviewed the theory of self-conjugate, self-reciprocal, and selfinversive polynomials. We discussed their main properties, how they are related to each other, the main theorems regarding the distribution of their zeros and some applications of these polynomials both in physics and mathematics. We hope that this short review suits for a compact introduction of the subject, paving the way for

We thank the editorial staff for all the support during the publishing process and

Faculty of Science and Technology, Department of Mathematics and Computer Science, São Paulo State University (UNESP), Presidente Prudente, SP, Brazil

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

also the Coordination for the Improvement of Higher Education (CAPES).

Pnð Þ W zð Þ , where W zð Þ¼�i zð Þ þ 1 =ð Þ z � 1 , is an SI

polynomials Tnð Þ¼ <sup>z</sup> ð Þ <sup>z</sup> � <sup>1</sup> <sup>n</sup>

Polynomials - Theory and Application

6. Conclusions

Table 2.

Acknowledgements

Author details

Ricardo Vieira

82

so on—see [78, 79] for more details.

further developments in this interesting field of research.

\*Address all correspondence to: rs.vieira@unesp.br

provided the original work is properly cited.

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[57] Joyner D, Shaska T. Self-inversive polynomials, curves, and codes. In: Higher Genus Curves in Mathematical Physics and Arithmetic Geometry. Vol. 703. American Mathematical Society; 2018. pp. 189-208. DOI: 10.1090/conm/

[58] McKee J, McKee JF, Smyth C. Number Theory and Polynomials. Vol. 352. Cambridge: Cambridge University

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[60] He Y-H. Polynomial roots and Calabi-Yau geometries. Advances in High Energy Physics. 2011;2011:1-15.

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[77] Chihara TS. An Introduction to Orthogonal Polynomials. Dover Publications; 2011

[78] Vieira RS, Botta V. Möbius transformations and orthogonal polynomials (In preparation). New York: Gordon and Breach Science Publishers; 1978

[79] Vieira RS, Botta V. Möbius transformations, orthogonal polynomials and self-inversive polynomials (In preparation). New York: Gordon and Breach Science Publishers; 1978

Chapter 5

Abstract

system, fixed point

1. Introduction

and theorems (see [1–26]).

and Dq fð Þ¼ 0 f

Dqxn <sup>¼</sup> ½ � <sup>n</sup> <sup>q</sup>x<sup>n</sup>�1.

89

0

A Numerical Investigation on

Q-Tangent Polynomials

Jung Yoog Kang and Cheon Seoung Ryoo

polynomials and classical tangent polynomials.

For any n∈ C, the q-number is defined by

the Structure of the Zeros of the

We introduce q-tangent polynomials and their basic properties including q-derivative and q-integral. By using Mathematica, we find approximate roots of q-tangent polynomials. We also investigate relations of zeros between q-tangent

Keywords: q-tangent polynomials, q-derivative, q-integral, Newton dynamical

For a long time, studies on q-difference equations appeared in intensive works especially by F. H. Jackson [1, 2], R. D. Carmichael [3], T. E. Mason [4], and other authors [5–26]. An intensive and somewhat surprising interest in q-numbers appeared in many areas of mathematics and applications including q-difference equations, special functions, q-combinatorics, q-integrable systems, variational q-calculus, q-series, and so on. In this paper, we introduce some basic definitions

Definition 1.1. [1, 2, 9, 13] The q-derivative operator of any function f is defined by

ð Þ 1 � q x

f xð Þdqx ¼ ð Þ 1 � q b ∑

<sup>1</sup> � <sup>q</sup> , <sup>∣</sup>q<sup>∣</sup> < 1: (1)

ð Þ 0 . We can prove that f is differentiable at 0, and it is clear that

∞ j¼0 q j f q <sup>j</sup>

, x 6¼ 0, (2)

b � �: (3)

2000 Mathematics Subject Classification: 11B68, 11B75, 12D10

½ � <sup>n</sup> <sup>q</sup> <sup>¼</sup> <sup>1</sup> � <sup>q</sup><sup>n</sup>

Dq f xð Þ¼ f xð Þ� f qx ð Þ

Definition 1.2. [1, 2, 9, 13, 17] We define the q-integral as

ðb 0

#### Chapter 5

## A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

Jung Yoog Kang and Cheon Seoung Ryoo

#### Abstract

We introduce q-tangent polynomials and their basic properties including q-derivative and q-integral. By using Mathematica, we find approximate roots of q-tangent polynomials. We also investigate relations of zeros between q-tangent polynomials and classical tangent polynomials.

Keywords: q-tangent polynomials, q-derivative, q-integral, Newton dynamical system, fixed point

2000 Mathematics Subject Classification: 11B68, 11B75, 12D10

#### 1. Introduction

For a long time, studies on q-difference equations appeared in intensive works especially by F. H. Jackson [1, 2], R. D. Carmichael [3], T. E. Mason [4], and other authors [5–26]. An intensive and somewhat surprising interest in q-numbers appeared in many areas of mathematics and applications including q-difference equations, special functions, q-combinatorics, q-integrable systems, variational q-calculus, q-series, and so on. In this paper, we introduce some basic definitions and theorems (see [1–26]).

For any n∈ C, the q-number is defined by

$$[n]\_q = \frac{\mathbf{1} - q^n}{\mathbf{1} - q}, \qquad |q| < \mathbf{1}. \tag{1}$$

Definition 1.1. [1, 2, 9, 13] The q-derivative operator of any function f is defined by

$$D\_q f(\mathbf{x}) = \frac{f(\mathbf{x}) - f(q\mathbf{x})}{(\mathbf{1} - q)\mathbf{x}}, \qquad \mathbf{x} \neq \mathbf{0}, \tag{2}$$

and Dq fð Þ¼ 0 f 0 ð Þ 0 . We can prove that f is differentiable at 0, and it is clear that Dqxn <sup>¼</sup> ½ � <sup>n</sup> <sup>q</sup>x<sup>n</sup>�1.

Definition 1.2. [1, 2, 9, 13, 17] We define the q-integral as

$$\int\_{0}^{b} f(\mathbf{x})d\_{q}\mathbf{x} = (\mathbf{1} - q)b \sum\_{j=0}^{\infty} q^{j}f\left(q^{j}b\right). \tag{3}$$

If this function, f(x), is differentiable on the point x, the q-derivative in Definition 1.1 goes to the ordinary derivative in the classical analysis when q ! 1.

Definition 1.3. [5, 17, 18, 21] The Gaussian binomial coefficients are defined by

The main aim of this paper is to extend tangent numbers and polynomials, and study some of their properties. Our paper is organized as follows: In Section 2, we define q-tangent polynomials and find some properties of these polynomials. We consider q-tangent polynomials in two parameters and establish some relations between q-tangent polynomials and q-Euler or Bernoulli polynomials. In Section 3, we observe approximate roots distributions of q-tangent polynomials and demon-

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

In this section we define the q-tangent numbers and polynomials and establish

some of their basic properties. we shall also study the q-tangent polynomials involving two parameters. We shall find some important relations between these

> <sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

¼ ∑ ∞ n¼0 T n, <sup>q</sup> t n ½ � n <sup>q</sup>!

where T n, <sup>q</sup> is q-tangent number. If q ! 1, then it reduces to the classical tangent

eqð Þ tx , <sup>∣</sup>t<sup>∣</sup> <sup>&</sup>lt; <sup>π</sup>

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

<sup>T</sup> k, <sup>q</sup> <sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> if <sup>n</sup> <sup>¼</sup> <sup>0</sup>

(

<sup>2</sup><sup>n</sup>�<sup>k</sup><sup>T</sup> k, <sup>q</sup>ð Þ¼ <sup>x</sup> ½ � <sup>2</sup> <sup>q</sup>x<sup>n</sup>:

0 if n 6¼ 0

1 A t n n! :

<sup>n</sup> we find (i). For (ii) we use the relation

T k,qð Þ x

1 A t n n! ,

. ☐

,

2

: (11)

, (12)

(13)

(14)

(15)

Definition 2.1. For x, q∈ C, we define q-tangent polynomials as

t n ½ � n <sup>q</sup>!

n ½ � n <sup>q</sup>!

strate interesting phenomenon.

DOI: http://dx.doi.org/10.5772/intechopen.83497

polynomials and q-other polynomials.

∑ ∞ n¼0

From Definition 2.1, it follows that

∑ ∞ n¼0

polynomial(see [22–25]).

T n, <sup>q</sup>ð Þ x

<sup>T</sup> n, <sup>q</sup>ð Þ <sup>0</sup> <sup>t</sup>

Theorem 2.2. Let x, q∈ C. Then, the following hold.

n k¼0

n k � �

n k¼0

½ � <sup>2</sup> <sup>q</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> eqð Þ <sup>2</sup><sup>t</sup> � � <sup>∑</sup>

0 @

¼ ∑ ∞ n¼0

½ � <sup>2</sup> <sup>q</sup>eqð Þ¼ tx <sup>1</sup> <sup>þ</sup> eqð Þ <sup>2</sup><sup>t</sup> � � <sup>∑</sup>

0 @

¼ ∑ ∞ n¼0 q 2<sup>n</sup>�<sup>k</sup>

n k � �

q

∞ n¼0 T n, <sup>q</sup> t n n!

n

" #

k

T n,qð Þ x

n k¼0

n

q 2<sup>n</sup>�<sup>k</sup> T k, <sup>q</sup>

n

" #

k

q 2<sup>n</sup>�<sup>k</sup>

t n ½ � n <sup>q</sup>!

T n,q þ ∑ n k¼0

> ∞ n¼0

T n, <sup>q</sup>ð Þþ x ∑

i: T n, <sup>q</sup> þ ∑

ii: T n, <sup>q</sup>ð Þþ x ∑

Proof. From the Definition 2.1, we have

Now comparing the coefficients of t

and again compare the coefficients of t

91

2. Some properties of the q-tangent polynomials

$$
\begin{pmatrix} m \\ r \end{pmatrix}\_q = \begin{bmatrix} m \\ r \end{bmatrix}\_q = \begin{cases} 0 & \text{if } r > m \\ \frac{(\mathbf{1} - q^m)(\mathbf{1} - q^{m-1}) \cdots (\mathbf{1} - q^{m-r+1})}{(\mathbf{1} - q)(\mathbf{1} - q^2) \cdots (\mathbf{1} - q^r)} & \text{if } r \le m \end{cases} \tag{4}
$$

where m and r are non-negative integers. For r ¼ 0 the value is 1 since the numerator and the denominator are both empty products. Like the classical binomial coefficients, the Gaussian binomial coefficients are center-symmetric. There are analogues of the binomial formula, and this definition has a number of properties.

Theorem 1.4. Let n, k be non-negative integers. Then we get.

$$\text{i.i. } \prod\_{k=0}^{n-1} \left( \mathbf{1} + q^k t \right) = \sum\_{k=0}^{n} q^{\binom{k}{2}} \begin{bmatrix} n \\ k \end{bmatrix}\_q t^k,\tag{5}$$

$$\text{iii.} \prod\_{k=0}^{n-1} \frac{1}{(1 - q^k t)} = \sum\_{k=0}^{\infty} \begin{bmatrix} n+k-1\\k \end{bmatrix}\_q t^k.$$

Definition 1.5. [5, 26] Let z be any complex number with ∣z∣ < 1. Two forms of q-exponential functions are defined by

$$e\_q(z) = \sum\_{n=0}^{\infty} \frac{z^n}{[n]\_q!}, \quad e\_{q^{-1}}(z) = \sum\_{n=0}^{\infty} \frac{z^n}{[n]\_{q^{-1}}!} = \sum\_{n=0}^{\infty} q \binom{n}{2} \frac{z^n}{[n]\_q!}.\tag{6}$$

Bernoulli, Euler, and Genocchi polynomials have been studied extensively by many mathematicians(see [22–25]). In 2013, C. S. Ryoo introduced tangent polynomials and he developed several properties of these polynomials (see [22, 23]). The tangent numbers are closely related to Euler numbers.

Definition 1.6. [22–25] Tangent numbers Tn and tangent polynomials Tnð Þ x are defined by means of the generating functions

$$\begin{aligned} \sum\_{n=0}^{\infty} T\_n \frac{t^n}{n!} &= \frac{2}{e^{2t} + 1} = 2 \sum\_{m=0}^{\infty} (-1)^m e^{2mt}, \\ \sum\_{n=0}^{\infty} T\_n \frac{t^n}{n!} &= \frac{2}{e^{2t} + 1} e^{t\mathbf{x}} = 2 \sum\_{m=0}^{\infty} (-1)^m e^{(2m+\mathbf{x})\mathbf{t}}. \end{aligned} \tag{7}$$

Theorem 1.7. For any positive integer n, we have

$$T\_n(\mathfrak{x}) = (-\mathbf{1})^n T\_n(2-\mathfrak{x}).\tag{8}$$

Theorem 1.8. For any positive integer mð Þ ¼ odd , we have

$$T\_n(\varkappa) = m^n \sum\_{i=0}^{m-1} (-1)^i T\_n \left( \frac{2i + \varkappa}{m} \right), \quad n \in \mathbb{Z}\_+. \tag{9}$$

Theorem 1.9. For n∈Zþ, we have

$$T\_n(\varkappa + \jmath) = \sum\_{k=0}^n \binom{n}{k} T\_k(\varkappa) \jmath^{n-k}.\tag{10}$$

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials DOI: http://dx.doi.org/10.5772/intechopen.83497

The main aim of this paper is to extend tangent numbers and polynomials, and study some of their properties. Our paper is organized as follows: In Section 2, we define q-tangent polynomials and find some properties of these polynomials. We consider q-tangent polynomials in two parameters and establish some relations between q-tangent polynomials and q-Euler or Bernoulli polynomials. In Section 3, we observe approximate roots distributions of q-tangent polynomials and demonstrate interesting phenomenon.

#### 2. Some properties of the q-tangent polynomials

In this section we define the q-tangent numbers and polynomials and establish some of their basic properties. we shall also study the q-tangent polynomials involving two parameters. We shall find some important relations between these polynomials and q-other polynomials.

Definition 2.1. For x, q∈ C, we define q-tangent polynomials as

$$\sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\infty) \frac{t^n}{[n]\_q!} = \frac{[\mathfrak{I}]\_q}{e\_q(2t) + \mathbf{1}} e\_q(t\infty), \qquad |t| < \frac{\pi}{2}. \tag{11}$$

From Definition 2.1, it follows that

$$\sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{0}) \frac{t^n}{[n]\_q!} = \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q} \frac{t^n}{[n]\_q!} = \frac{[2]\_q}{e\_q(2t) + 1},\tag{12}$$

where T n, <sup>q</sup> is q-tangent number. If q ! 1, then it reduces to the classical tangent polynomial(see [22–25]).

Theorem 2.2. Let x, q∈ C. Then, the following hold.

$$\begin{aligned} \text{i. } \begin{aligned} \text{i. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{end{0.5} \end{cases} + \sum \end{aligned} \end{^{n} \begin{bmatrix} n \\ k \end{cases} \end{aligned} \end{aligned} \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } n = \mathbf{0} \\ \text{0. } \end{aligned} \end{aligned} \end{aligned} \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } n = \mathbf{0} \\ \text{0. } \end{aligned} \end{aligned} \end{aligned} \end{aligned} \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \begin{aligned} \text{7. } \end{aligned} \end{aligned} \end{aligned} \end{aligned} \end{aligned} \end{aligned} \end{aligned} \right] \end{aligned}$$

Proof. From the Definition 2.1, we have

$$\begin{split} \left[ \mathbf{2} \right]\_q &= \left( \mathbf{1} + \mathbf{e}\_q(2t) \right) \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q} \frac{t^n}{n!} \\ &= \sum\_{n=0}^{\infty} \left( \mathcal{T}\_{n,q} + \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_q \mathbf{2}^{n-k} \mathcal{T}\_{k,q} \right) \frac{t^n}{n!} . \end{split} \tag{14}$$

Now comparing the coefficients of t <sup>n</sup> we find (i). For (ii) we use the relation

$$\begin{split} [\mathbf{2}]\_q e\_q(\mathbf{x}) &= \left(\mathbf{1} + e\_q(\mathbf{2}t)\right) \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^n}{[n]\_q!} \\ &= \sum\_{n=0}^{\infty} \left(\mathcal{T}\_{n,q}(\mathbf{x}) + \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_q \mathcal{T}^{n-k} \mathcal{T}\_{k,q}(\mathbf{x})\right) \frac{t^n}{n!}, \end{split} \tag{15}$$

and again compare the coefficients of t n

If this function, f(x), is differentiable on the point x, the q-derivative in Defini-

Definition 1.3. [5, 17, 18, 21] The Gaussian binomial coefficients are defined by

<sup>1</sup> � qm ð Þ <sup>1</sup> � <sup>q</sup>m�<sup>1</sup> ð Þ<sup>⋯</sup> <sup>1</sup> � qm�rþ<sup>1</sup> ð Þ

where m and r are non-negative integers. For r ¼ 0 the value is 1 since the numerator and the denominator are both empty products. Like the classical binomial coefficients, the Gaussian binomial coefficients are center-symmetric. There are analogues of the binomial formula, and this definition has a number of properties.

Theorem 1.4. Let n, k be non-negative integers. Then we get.

q t k

q t k:

, eq�<sup>1</sup> ð Þ¼ z ∑

many mathematicians(see [22–25]). In 2013, C. S. Ryoo introduced tangent polynomials and he developed several properties of these polynomials (see [22, 23]). The tangent numbers are closely related to Euler numbers.

<sup>e</sup><sup>2</sup><sup>t</sup> <sup>þ</sup> <sup>1</sup> <sup>¼</sup> <sup>2</sup> <sup>∑</sup>

Tnð Þ¼ � <sup>x</sup> ð Þ<sup>1</sup> <sup>n</sup>

ð Þ �<sup>1</sup> <sup>i</sup> Tn

> n k¼0

n k � �

<sup>¼</sup> <sup>2</sup>

<sup>¼</sup> <sup>2</sup> e<sup>2</sup><sup>t</sup> þ 1 e

Theorem 1.8. For any positive integer mð Þ ¼ odd , we have

m�1 i¼0

Tnð Þ¼ x þ y ∑

Theorem 1.7. For any positive integer n, we have

Tnð Þ¼ <sup>x</sup> <sup>m</sup><sup>n</sup> <sup>∑</sup>

Definition 1.5. [5, 26] Let z be any complex number with ∣z∣ < 1. Two forms of

∞ n¼0

Bernoulli, Euler, and Genocchi polynomials have been studied extensively by

Definition 1.6. [22–25] Tangent numbers Tn and tangent polynomials Tnð Þ x are

∞ m¼0

tx <sup>¼</sup> <sup>2</sup> <sup>∑</sup> ∞ m¼0

ð Þ �<sup>1</sup> me

2i þ x m � �

Tkð Þ <sup>x</sup> <sup>y</sup><sup>n</sup>�<sup>k</sup>

<sup>2</sup>mt,

ð Þ 2mþx t :

Tnð Þ 2 � x : (8)

, n∈Zþ: (9)

: (10)

ð Þ �<sup>1</sup> me

zn ½ � n <sup>q</sup>�<sup>1</sup> ! ¼ ∑ ∞ n¼0 q

ð Þ <sup>1</sup> � <sup>q</sup> <sup>1</sup> � <sup>q</sup><sup>2</sup> ð Þ<sup>⋯</sup> <sup>1</sup> � qr ð Þ if <sup>r</sup><sup>≤</sup> <sup>m</sup> ,

0 if r > m

, (5)

n 2 � �

zn ½ � n <sup>q</sup>!

: (6)

(7)

(4)

tion 1.1 goes to the ordinary derivative in the classical analysis when q ! 1.

m r � �

i. <sup>Y</sup><sup>n</sup>�<sup>1</sup>

ii. <sup>n</sup> Q�1 k¼0

k¼0

1 <sup>1</sup>�qk ð Þ<sup>t</sup> <sup>¼</sup> <sup>∑</sup>

q

<sup>¼</sup> <sup>m</sup> r � �

Polynomials - Theory and Application

<sup>1</sup> <sup>þ</sup> <sup>q</sup>kt � � <sup>¼</sup> <sup>∑</sup>

∞ k¼0

q-exponential functions are defined by

eqð Þ¼ z ∑

∞ n¼0

defined by means of the generating functions

∑ ∞ n¼0 Tn t n n!

∑ ∞ n¼0 Tn t n n!

Theorem 1.9. For n∈Zþ, we have

90

zn ½ � n <sup>q</sup>!

q ¼ 8 ><

>:

n k¼0 q k 2 � � n k � �

n þ k � 1 k � � Theorem 2.3. Let n be a non-negative integer. Then, the following holds

$$\mathcal{T}\_{n,q}(\mathbf{x}) = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q \mathcal{T}\_{n-k,q} \mathbf{x}^k. \tag{16}$$

Theorem 2.5. For k∈ N, the following holds

DOI: http://dx.doi.org/10.5772/intechopen.83497

Dð Þ<sup>k</sup>

Proof. Considering q-derivative of eqð Þ tx , we find

t n ½ � n <sup>q</sup>!

T n,qð Þ x

ðb a

Proof. From Theorem 2.3, we find

ðb a

∑ ∞ n¼0

T n, <sup>q</sup>ð Þ x; y

From the Definition 2.7, it is clear that

parameters as

93

Dð Þ <sup>i</sup>þ<sup>1</sup> <sup>q</sup> ∑ ∞ n¼0 <sup>q</sup> T n, <sup>q</sup>ð Þ¼ x

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

¼ ∑ ∞ n¼0

¼ t

¼ ∑ ∞ n¼0

¼ ∑ ∞ n¼0

Theorem 2.6. Let a, b be any real numbers. Then, we have

nþ1 k¼0

> ðb a ∑ n k¼0

¼ ∑ n k¼0

> ¼ ∑ nþ1 k¼0

t n ½ � n <sup>q</sup>!

1 ½ � n þ 1 <sup>q</sup>

n

3 5 q

T k,qxn�<sup>k</sup>

T <sup>n</sup>þ1, <sup>q</sup>ð Þ� b T <sup>n</sup>þ1, <sup>q</sup>ð Þ a ½ � n þ 1 <sup>q</sup>

1 ½ � n � k þ 1 <sup>q</sup>

dqx

2 4

n

3 5 q T k, <sup>q</sup>

2 4

k

Definition 2.7. For x, y ∈ C, we define q-tangent polynomial with two

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

k

T n, <sup>q</sup>ð Þ x dqx ¼ ∑

T n, <sup>q</sup>ð Þ x dqx ¼

� T n, <sup>q</sup>ð Þ x

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

Dð Þ <sup>i</sup>þ<sup>1</sup>

<sup>i</sup>þ<sup>1</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

½ � n <sup>q</sup>! ½ � n � k <sup>q</sup>!

<sup>q</sup> T n, <sup>q</sup>ð Þ x

Dð Þ <sup>i</sup>þ<sup>1</sup> <sup>q</sup> eqð Þ tx

eqð Þ tx

½ � n þ ð Þ i þ 1 <sup>q</sup>⋯½ � n þ 2 <sup>q</sup>½ � n þ 1 <sup>q</sup>

T <sup>n</sup>�ð Þ <sup>i</sup>þ<sup>1</sup> , <sup>q</sup>ð Þ x

t n ½ � n <sup>q</sup>! ,

<sup>T</sup> <sup>n</sup>þ1, <sup>q</sup>ð Þ� <sup>b</sup> <sup>T</sup> <sup>n</sup>þ1, <sup>q</sup>ð Þ <sup>a</sup> � �: (24)

xn�kþ<sup>1</sup>

:

eqð Þ tx eqð Þ ty , <sup>∣</sup>t<sup>∣</sup> <sup>&</sup>lt; <sup>π</sup>

� � � � � � �

b

a

2

: (26)

(23)

(25)

☐

t <sup>n</sup>þiþ<sup>1</sup> ½ � n þ ð Þ i þ 1 <sup>q</sup>!

½ � n <sup>q</sup> ½ � n þ ð Þ i þ 1 <sup>q</sup>!

which immediately gives the required result. ☐

t n ½ � n <sup>q</sup>!

T <sup>n</sup>�k,qð Þ x : (22)

Proof. From the definition of the q-exponential function, we have

$$\begin{split} \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^n}{[n]\_q!} &= \frac{[\mathbf{2}]\_q}{e\_q(2t) + \mathbf{1}} e\_q(t\mathbf{x}) = \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q} \frac{t^n}{[n]\_q!} \sum\_{n=0}^{\infty} \boldsymbol{\mathcal{X}}^n \frac{t^n}{[n]\_q!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \binom{n}{k}\_q \mathcal{T}\_{n-k,q}(\mathbf{x}) \boldsymbol{\mathcal{X}}^k \right) \frac{t^n}{[n]\_q!} . \end{split} \tag{17}$$

The required relation now follows on comparing the coefficients of t <sup>n</sup> on both sides. ☐

Theorem 2.4. Let n be a non-negative integer. Then, the following holds

$$\mathcal{T}\_{n,q} = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q (-\mathbf{1})^{n-k} q^{\binom{n-k}{2}} \mathcal{T}\_{k,q}(\mathbf{x}) \mathbf{x}^{n-k}.\tag{18}$$

Proof. From the property of q-exponential function, it follows that

$$\begin{split} \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q} \frac{t^n}{[n]\_q!} &= \frac{[2]\_q}{e\_q(2t) + 1} e\_q(t\mathbf{x}) e\_{q^1}(-t\mathbf{x}) \\ &= \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^n}{[n]\_q!} \sum\_{n=0}^{\infty} q^{\binom{n}{2}} (-\mathbf{1})^n \mathbf{x}^n \frac{t^n}{[n]\_q!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \binom{n}{k}\_q (-\mathbf{1})^{n-k} q^{\binom{n-k}{2}} \mathcal{T}\_{k,q}(\mathbf{x}) \mathbf{x}^{n-k} \right) \frac{t^n}{[n]\_q!} .\end{split} \tag{19}$$

The required relation now follows immediately. ☐

In what follows, we consider q-derivative of eqð Þ tx . Using the Mathematical Induction, we find.

$$\text{i. } k = 1: \quad D\_q^{(1)} e\_q(t\infty) = \sum\_{n=1}^{\infty} \varkappa^{n-1} \frac{t^n}{[n-1]\_q!}. \tag{20}$$

$$\text{ii. } k = i: \quad D\_q^{(i)} e\_q(t\infty) = \sum\_{n=i}^{\infty} \varkappa^{n-i} \frac{t^n}{[n-i]\_q!}.$$

If (ii) is true, then it follows that.

$$\begin{split} \text{iii. } k = i + \mathbf{1}: \quad & D\_q^{(i+1)} e\_q(t\mathbf{x}) = D\_{q; \mathbf{x}}^{(1)} \left( \sum\_{n=i}^{\infty} \mathbf{x}^{n-i} \frac{t^n}{[n-i]\_q!} \right) \\ &= \sum\_{n=i+1}^{\infty} \mathbf{x}^{n-(i+1)} \frac{t^n}{[n-(i+1)]\_q!} \\ &= t^{i+1} e\_q(t\mathbf{x}). \end{split} \tag{21}$$

We are now in the position to prove the following theorem.

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials DOI: http://dx.doi.org/10.5772/intechopen.83497

Theorem 2.5. For k∈ N, the following holds

Theorem 2.3. Let n be a non-negative integer. Then, the following holds

n k¼0

n k � �

eqð Þ¼ tx ∑

q

∞ n¼0 T n, <sup>q</sup> t n ½ � n <sup>q</sup>! ∑ ∞ n¼0

<sup>T</sup> <sup>n</sup>�k, <sup>q</sup>ð Þ <sup>x</sup> xk

ð Þ �<sup>1</sup> <sup>n</sup>

xn <sup>t</sup> n ½ � n <sup>q</sup>!

<sup>x</sup><sup>n</sup>�<sup>1</sup> <sup>t</sup>

xn�<sup>i</sup> <sup>t</sup>

<sup>T</sup> k, <sup>q</sup>ð Þ <sup>x</sup> xn�<sup>k</sup>

n ½ � n � 1 <sup>q</sup>!

n ½ � n � i <sup>q</sup>!

<sup>x</sup><sup>n</sup>�<sup>i</sup> <sup>t</sup>

xn�ð Þ <sup>i</sup>þ<sup>1</sup> <sup>t</sup>

!

:

n ½ � n � i <sup>q</sup>!

n ½ � n � ð Þ i þ 1 <sup>q</sup>!

1 A t n ½ � n <sup>q</sup>! :

<sup>T</sup> <sup>n</sup>�k,qxk

1 A t n ½ � n <sup>q</sup>! :

<sup>T</sup> k, <sup>q</sup>ð Þ <sup>x</sup> xn�<sup>k</sup>

: (16)

(17)

(19)

(21)

<sup>n</sup> on both

: (18)

: (20)

xn <sup>t</sup> n ½ � n <sup>q</sup>!

T n,qð Þ¼ x ∑

∑ n k¼0

n k � �

q

Proof. From the property of q-exponential function, it follows that

t n ½ � n <sup>q</sup>! ∑ ∞ n¼0 q n 2 !

n

" #

k

q

eqð Þ tx eq<sup>1</sup> ð Þ �tx

ð Þ �<sup>1</sup> <sup>n</sup>�<sup>k</sup> q n � k 2 !

The required relation now follows immediately. ☐ In what follows, we consider q-derivative of eqð Þ tx . Using the Mathematical

> ∞ n¼1

> > ∞ n¼i

> > > q;x ∑ ∞ n¼i

¼ ∑ ∞ n¼iþ1

¼ t iþ1 eqð Þ tx :

<sup>q</sup> eqð Þ¼ tx ∑

<sup>q</sup> eqð Þ¼ tx ∑

<sup>q</sup> eqð Þ¼ tx <sup>D</sup>ð Þ<sup>1</sup>

0 @

n

" #

k

The required relation now follows on comparing the coefficients of t

q

sides. ☐ Theorem 2.4. Let n be a non-negative integer. Then, the following holds

> ð Þ �<sup>1</sup> <sup>n</sup>�<sup>k</sup> q n � k 2 � �

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

¼ ∑ ∞ n¼0

T n, <sup>q</sup> ¼ ∑ n k¼0

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2 t 1

¼ ∑ ∞ n¼0

¼ ∑ ∞ n¼0

If (ii) is true, then it follows that.

iii: <sup>k</sup> <sup>¼</sup> <sup>i</sup> <sup>þ</sup> <sup>1</sup> : <sup>D</sup>ð Þ <sup>i</sup>þ<sup>1</sup>

T n, <sup>q</sup>ð Þ x

∑ n k¼0

<sup>i</sup>: <sup>k</sup> <sup>¼</sup> <sup>1</sup> : <sup>D</sup>ð Þ<sup>1</sup>

ii: <sup>k</sup> <sup>¼</sup> <sup>i</sup> : <sup>D</sup>ð Þ<sup>i</sup>

We are now in the position to prove the following theorem.

0 @

∑ ∞ n¼0

∑ ∞ n¼0 T n, <sup>q</sup> t n ½ � n <sup>q</sup>!

Induction, we find.

92

T n, <sup>q</sup>ð Þ x

Polynomials - Theory and Application

t n ½ � n <sup>q</sup>!

Proof. From the definition of the q-exponential function, we have

$$D\_q^{(k)} \mathcal{T}\_{n,q}(\infty) = \frac{[n]\_q!}{[n-k]\_q!} \mathcal{T}\_{n-k,q}(\infty). \tag{22}$$

Proof. Considering q-derivative of eqð Þ tx , we find

$$\begin{split} D\_q^{(i+1)} \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^n}{[n]\_q!} &= \sum\_{n=0}^{\infty} D\_q^{(i+1)} \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^n}{[n]\_q!} \\ &= \frac{[2]\_q}{\epsilon\_q(2\mathbf{t}) + \mathbf{1}} D\_q^{(i+1)} e\_q(\mathbf{t}\mathbf{x}) \\ &= t^{i+1} \frac{[2]\_q}{\epsilon\_q(2\mathbf{t}) + \mathbf{1}} e\_q(\mathbf{t}\mathbf{x}) \\ &= \sum\_{n=0}^{\infty} [n + (i+1)]\_q \cdots [n+2]\_q [n+1]\_q \\ &\quad \times \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^{n+i+1}}{[n+(i+1)]\_q!} \\ &= \sum\_{n=0}^{\infty} \frac{[n]\_q}{[n+(i+1)]\_q!} \mathcal{T}\_{n-(i+1),q}(\mathbf{x}) \frac{t^n}{[n]\_q!}, \end{split} \tag{23}$$

which immediately gives the required result. ☐

Theorem 2.6. Let a, b be any real numbers. Then, we have

$$\int\_{a}^{b} \mathcal{T}\_{n,q}(\mathbf{x}) d\_{q}\mathbf{x} = \sum\_{k=0}^{n+1} \frac{\mathbf{1}}{[n+1]\_{q}} \left( \mathcal{T}\_{n+1,q}(b) - \mathcal{T}\_{n+1,q}(a) \right). \tag{24}$$

Proof. From Theorem 2.3, we find

$$\begin{aligned} \int\_{a}^{b} \mathcal{T}\_{n,q}(\mathbf{x}) d\_{q}\mathbf{x} &= \int\_{a}^{b} \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \\ k \end{bmatrix}\_{q} \mathcal{T}\_{k,q} \mathbf{x}^{n-k} d\_{q} \mathbf{x} \\ &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \\ k \end{bmatrix}\_{q} \mathcal{T}\_{k,q} \frac{1}{[n-k+1]\_{q}} \mathbf{x}^{n-k+1} \bigg|\_{a}^{b} \\ &= \sum\_{k=0}^{n+1} \frac{\mathcal{T}\_{n+1,q}(b) - \mathcal{T}\_{n+1,q}(a)}{[n+1]\_{q}}. \end{aligned} \tag{25}$$

Definition 2.7. For x, y ∈ C, we define q-tangent polynomial with two parameters as

$$\sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\boldsymbol{x}, \boldsymbol{y}) \frac{t^n}{[n]\_q!} = \frac{[\boldsymbol{2}]\_q}{e\_q(2t) + \mathbf{1}} e\_q(t\boldsymbol{\infty}) e\_q(t\boldsymbol{y}), \qquad |t| < \frac{\pi}{2}. \tag{26}$$

From the Definition 2.7, it is clear that

☐

$$\begin{aligned} \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}, \mathbf{0}) \frac{t^n}{[n]\_q!} &= \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^n}{[n]\_q!} = \frac{[2]\_q}{e\_q(2t) + \mathbf{1}} e\_q(t\mathbf{x}), \\ \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{0}, \mathbf{0}) \frac{t^n}{[n]\_q!} &= \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q} \frac{t^n}{[n]\_q!} = \frac{[2]\_q}{e\_q(2t) + \mathbf{1}}, \end{aligned} \tag{27}$$

where T n, <sup>q</sup> is q-tangent number. We also note that the original tangent number, T <sup>n</sup>,

$$\lim\_{q \to 1} \sum\_{n=0}^{\infty} T\_{n,q} \frac{t^n}{[n]\_q!} = \sum\_{n=0}^{\infty} T\_n \frac{t^n}{n!} = \frac{2}{e^{2t} + 1},\tag{28}$$

where q ! 1.

Theorem 2.8. Let x, y be any complex numbers. Then, the following hold.

$$\text{i. } \mathcal{T}\_{n,q}(\mathbf{x}, \boldsymbol{\mathcal{y}}) = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q \mathcal{T}\_{n-k,q}(\mathbf{x}) \boldsymbol{\mathcal{y}}^k,\tag{29}$$
 
$$\text{ii. } \mathcal{T}\_{n,q}(\mathbf{x}, \boldsymbol{\mathcal{y}}) = \sum\_{l=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q \mathcal{T}\_{n-l,q} \sum\_{k=0}^{l} \begin{bmatrix} l \\ k \end{bmatrix}\_q \boldsymbol{\mathcal{x}}^{l-k} \boldsymbol{\mathcal{y}}^k.$$

Proof. From the Definition 2.7, we have

$$\begin{split} \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\boldsymbol{x}, \boldsymbol{y}) \frac{t^{n}}{[n]\_{q}!} &= \frac{[\mathbf{2}]\_{q}}{e\_{q}(\mathbf{2}t) + \mathbf{1}} e\_{q}(t\boldsymbol{\infty}) e\_{q}(t\boldsymbol{y}) \\ &= \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\boldsymbol{\infty}) \frac{t^{n}}{[n]\_{q}!} \sum\_{n=0}^{\infty} \mathcal{Y}^{n} \frac{t^{n}}{[n]\_{q}!} . \end{split} \tag{30}$$

Using Cauchy's product and the method of coefficient comparison in the above relation, we find (i). Next, we transform q-tangent polynomials in two parameters as

$$\begin{split} \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\boldsymbol{x}, \boldsymbol{y}) \frac{t^{n}}{[n]\_{q}!} &= \frac{[2]\_{q}}{e\_{q}(2t) + 1} e\_{q}(t\boldsymbol{x}) e\_{q}(t\boldsymbol{y}) \\ &= \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q} \frac{t^{n}}{[n]\_{q}!} \sum\_{n=0}^{\infty} \boldsymbol{x}^{n} \frac{t^{n}}{[n]\_{q}!} \sum\_{n=0}^{\infty} \boldsymbol{y}^{n} \frac{t^{n}}{[n]\_{q}!} . \end{split} \tag{31}$$

Now following same procedure as in (i), we obtain (ii). ☐

Theorem 2.9. Setting y ¼ 2 in q-tangent polynomials with two parameters, the following relation holds

$$[\mathfrak{Z}]\_q \mathfrak{x}^n = \mathcal{T}\_{n,q}(\mathfrak{x}, \mathfrak{Z}) + \mathcal{T}\_{n,q}(\mathfrak{x}).\tag{32}$$

Proof. Using q-tangent polynomials and its polynomials with two parameters, we have

$$\sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}, \mathsf{2}) \frac{t^n}{[n]\_q!} + \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^n}{[n]\_q!} = \frac{[\mathbf{2}]\_q e\_q(\mathbf{2}t)}{e\_q(\mathbf{2}t) + \mathbf{1}} e\_q(\mathbf{t} \mathbf{x}) + \frac{[\mathbf{2}]\_q}{e\_q(\mathbf{2}t) + \mathbf{1}} e\_q(\mathbf{t} \mathbf{x}) \tag{33}$$
 
$$= [\mathbf{2}]\_q e\_q(\mathbf{t} \mathbf{x})$$

Now from the definition of q-exponential function, the required relation follows. ☐

Theorem 2.9 is interesting as it leads to the relation

DOI: http://dx.doi.org/10.5772/intechopen.83497

Theorem 2.10. Let j j q < 1. Then, the following holds

T n, <sup>q</sup>ð Þ¼ x ∑

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> 1 þ Eqð Þ �2t

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> e1 q

¼ ∑ ∞ n¼0 T n, <sup>1</sup> q

¼ ∑ ∞ n¼0

ð Þþ �2t 1

∑ n k¼0

8 ><

>:

t n ½ � n <sup>q</sup>!

t n ½ � n <sup>q</sup>!

Theorem 2.12. For x, y∈ C, the following relation holds

T <sup>n</sup>�l, <sup>q</sup>ð Þ x

Proof. To prove the relation, we note that

t n ½ � n <sup>q</sup>!

polynomials as

q-numbers.

polynomials, Bn, <sup>q</sup>ð Þ x , as

T n, <sup>q</sup>ð Þ¼ x; y

95

∑ ∞ n¼0

∑ ∞ n¼0

1 ½ � 2 <sup>q</sup> ∑ n l¼0 En,qð Þ x

Bn,qð Þ x

n k � �

q

∑ ∞ n¼0

T n,qð Þ x

n k¼0

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

e1 q

n k � �

q ð Þ �<sup>1</sup> <sup>k</sup>

where EqðÞ¼ t eq�<sup>1</sup> ð Þt . Using the above equation we can represent the q-tangent

Eqð Þ �2t eqð Þ tx

ð Þ �2t eqð Þ tx

xn <sup>t</sup> n ½ � n <sup>q</sup>!

Tk, <sup>1</sup> q ð Þ<sup>2</sup> <sup>x</sup><sup>n</sup>�<sup>k</sup>

eqð Þ tx , ∣t∣ < π,

eqð Þ tx , ∣t∣ < 2π:

q

T k,qð Þ x mn�<sup>k</sup>

9 >=

t n ½ � n <sup>q</sup>! ,

>;

eqð Þ tx

e1 q

ð Þ<sup>2</sup> ð Þ �<sup>t</sup> <sup>n</sup> ½ � <sup>n</sup> <sup>q</sup>! <sup>∑</sup> ∞ n¼0

n

3 5 q

ð Þ �<sup>1</sup> <sup>k</sup>

which leads to the required relation immediately. ☐

2 4

k

Now we shall find relations between q-tangent polynomials and others polynomials. For this, first we introduce well known polynomials by using

Definition 2.11. We define q-Euler polynomials, En, <sup>q</sup>ð Þ x , and q-Bernoulli

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqðÞþt 1

<sup>¼</sup> <sup>t</sup> eqðÞ�t 1

ml <sup>þ</sup> <sup>∑</sup>

n�l k¼0

Proof. Transforming q-tangent polynomials containing two parameters, we find

!

n � l k � �

T k, <sup>1</sup> q ð Þ<sup>2</sup> xn�<sup>k</sup>

ð Þ¼ �2t Eqð Þ �2t , (36)

xn <sup>¼</sup> <sup>T</sup> n, <sup>q</sup>ð Þþ <sup>x</sup>; <sup>2</sup> <sup>T</sup> n, <sup>q</sup>ð Þ <sup>x</sup> ½ � 2 <sup>q</sup>

: (34)

: (35)

(37)

(38)

El, <sup>q</sup>ð Þ my : (39)

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials DOI: http://dx.doi.org/10.5772/intechopen.83497

Theorem 2.9 is interesting as it leads to the relation

$$\boldsymbol{\mathfrak{x}}^{n} = \frac{\boldsymbol{\mathcal{T}}\_{n,q}(\boldsymbol{\mathfrak{x}}, \boldsymbol{\mathfrak{Z}}) + \boldsymbol{\mathcal{T}}\_{n,q}(\boldsymbol{\mathfrak{x}})}{[\boldsymbol{\mathfrak{Z}}]\_{q}}.\tag{34}$$

Theorem 2.10. Let j j q < 1. Then, the following holds

$$\mathcal{T}\_{n,q}(\mathbf{x}) = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q (-\mathbf{1})^k \mathcal{T}\_{k,\frac{1}{q}}(\mathbf{2}) \mathbf{x}^{n-k}.\tag{35}$$

Proof. To prove the relation, we note that

∑ ∞ n¼0

Polynomials - Theory and Application

∑ ∞ n¼0

where q ! 1.

T <sup>n</sup>,

<sup>T</sup> n, <sup>q</sup>ð Þ <sup>x</sup>; <sup>0</sup> <sup>t</sup>

<sup>T</sup> n, <sup>q</sup>ð Þ <sup>0</sup>; <sup>0</sup> <sup>t</sup>

lim <sup>q</sup>!<sup>1</sup> <sup>∑</sup> ∞ n¼0

Proof. From the Definition 2.7, we have

T n, <sup>q</sup>ð Þ x; y

∑ ∞ n¼0

∑ ∞ n¼0

following relation holds

<sup>T</sup> n,qð Þ <sup>x</sup>; <sup>2</sup> <sup>t</sup>

n ½ � n <sup>q</sup>! þ ∑ ∞ n¼0

T n,qð Þ x

t n ½ � n <sup>q</sup>!

have

94

∑ ∞ n¼0 T n, <sup>q</sup>ð Þ x; y

t n ½ � n <sup>q</sup>!

n ½ � n <sup>q</sup>!

n ½ � n <sup>q</sup>!

> T n, <sup>q</sup> t n ½ � n <sup>q</sup>!

i: T n, <sup>q</sup>ð Þ¼ x; y ∑

ii: T n, <sup>q</sup>ð Þ¼ x; y ∑

t n ½ � n <sup>q</sup>!

¼ ∑ ∞ n¼0

¼ ∑ ∞ n¼0 T n, <sup>q</sup> t n ½ � n <sup>q</sup>!

T n, <sup>q</sup>ð Þ x

where T n, <sup>q</sup> is q-tangent number. We also note that the original tangent number,

¼ ∑ ∞ n¼0 Tn t n n!

Theorem 2.8. Let x, y be any complex numbers. Then, the following hold.

n k¼0

> n l¼0

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

Using Cauchy's product and the method of coefficient comparison in the above relation, we find (i). Next, we transform q-tangent polynomials in two parameters as

Now following same procedure as in (i), we obtain (ii). ☐ Theorem 2.9. Setting y ¼ 2 in q-tangent polynomials with two parameters, the

Proof. Using q-tangent polynomials and its polynomials with two parameters, we

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup>eqð Þ <sup>2</sup><sup>t</sup> eqð Þþ 2t 1

¼ ½ � 2 <sup>q</sup>eqð Þ tx

Now from the definition of q-exponential function, the required relation follows. ☐

¼ ∑ ∞ n¼0

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

¼ ∑ ∞ n¼0 T n, <sup>q</sup> t n ½ � n <sup>q</sup>! ∑ ∞ n¼0

n k 

n k 

q

q

T n, <sup>q</sup>ð Þ x

t n ½ � n <sup>q</sup>! <sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2 t 1

,

<sup>¼</sup> ½ � <sup>2</sup> <sup>q</sup> eqð Þþ 2t 1

> <sup>¼</sup> <sup>2</sup> e2<sup>t</sup> þ 1

<sup>T</sup> <sup>n</sup>�k, <sup>q</sup>ð Þ <sup>x</sup> <sup>y</sup><sup>k</sup>

l k 

yn <sup>t</sup> n ½ � n <sup>q</sup>! :

> <sup>y</sup><sup>n</sup> <sup>t</sup> n ½ � n <sup>q</sup>! :

½ � 2 <sup>q</sup> eqð Þþ 2t 1

eqð Þ tx

q x<sup>l</sup>�<sup>k</sup> yk :

T <sup>n</sup>�l, <sup>q</sup> ∑ l k¼0

eqð Þ tx eqð Þ ty

t n ½ � n <sup>q</sup>! ∑ ∞ n¼0

eqð Þ tx eqð Þ ty

xn <sup>t</sup> n ½ � n <sup>q</sup>! ∑ ∞ n¼0

½ � <sup>2</sup> <sup>q</sup>x<sup>n</sup> <sup>¼</sup> <sup>T</sup> n, <sup>q</sup>ð Þþ <sup>x</sup>; <sup>2</sup> <sup>T</sup> n, <sup>q</sup>ð Þ <sup>x</sup> : (32)

eqð Þþ tx

eqð Þ tx ,

, (28)

, (29)

(27)

(30)

(31)

(33)

$$
\sigma\_{\frac{\mathbf{e}}{q}}(-\mathbf{2t}) = \mathcal{E}\_{q}(-\mathbf{2t}),\tag{36}
$$

where EqðÞ¼ t eq�<sup>1</sup> ð Þt . Using the above equation we can represent the q-tangent polynomials as

$$\begin{split} \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\mathbf{x}) \frac{t^n}{[n]\_q!} &= \frac{[2]\_q}{e\_q(2t) + 1} e\_q(t\mathbf{x}) \\ &= \frac{[2]\_q}{1 + \mathcal{E}\_q(-2t)} \mathcal{E}\_q(-2t) e\_q(t\mathbf{x}) \\ &= \frac{[2]\_q}{e\_q(-2t) + 1} e\_q(-2t) e\_q(t\mathbf{x}) \\ &= \sum\_{n=0}^{\infty} \mathcal{T}\_{n, \frac{1}{q}}(2) \frac{(-t)^n}{[n]\_q!} \sum\_{n=0}^{\infty} \mathbf{x}^n \frac{t^n}{[n]\_q!} \\ &= \sum\_{n=0}^{\infty} \left\{ \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_q (-1)^k T\_{k, \frac{1}{q}}(2) \mathbf{x}^{n-k} \right\} \frac{t^n}{[n]\_q!}, \end{split} \tag{37}$$

which leads to the required relation immediately. ☐

Now we shall find relations between q-tangent polynomials and others polynomials. For this, first we introduce well known polynomials by using q-numbers.

Definition 2.11. We define q-Euler polynomials, En, <sup>q</sup>ð Þ x , and q-Bernoulli polynomials, Bn, <sup>q</sup>ð Þ x , as

$$\begin{aligned} \sum\_{n=0}^{\infty} E\_{n,q}(\infty) \frac{t^n}{[n]\_q!} &= \frac{[2]\_q}{e\_q(t) + 1} e\_q(t\infty), \qquad |t| < \pi, \\\sum\_{n=0}^{\infty} B\_{n,q}(\infty) \frac{t^n}{[n]\_q!} &= \frac{t}{e\_q(t) - 1} e\_q(t\infty), \qquad |t| < 2\pi. \end{aligned} \tag{38}$$

Theorem 2.12. For x, y∈ C, the following relation holds

$$\mathcal{T}\_{n,q}(\mathbf{x},\mathbf{y}) = \frac{\mathbf{1}}{[\mathbf{2}]\_q} \sum\_{l=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_q \left( \frac{\mathcal{T}\_{n-l,q}(\mathbf{x})}{m^l} + \sum\_{k=0}^{n-l} \begin{bmatrix} n-l \\ k \end{bmatrix}\_q \frac{\mathcal{T}\_{k,q}(\mathbf{x})}{m^{n-k}} \right) \mathcal{E}\_{l,q}(m\mathbf{y}). \tag{39}$$

Proof. Transforming q-tangent polynomials containing two parameters, we find

Polynomials - Theory and Application

$$\frac{[2]\_q}{e\_q(2t) + \mathbf{1}} e\_q(t\mathbf{x}) e\_q(t\mathbf{y}) = \left(\frac{[2]\_q}{e\_q(\frac{t}{m}) + \mathbf{1}} e\_q(t\mathbf{y})\right) \left(\frac{e\_q(\frac{t}{m}) + \mathbf{1}}{[2]\_q}\right) \left(\frac{[2]\_q}{e\_q(2t) + \mathbf{1}} e\_q(t\mathbf{x})\right). \tag{40}$$

The required relation now follows on comparing the coefficients. ☐

q

T k, <sup>q</sup>ð Þ x

mn�<sup>k</sup> � <sup>T</sup> <sup>n</sup>�lð Þ <sup>x</sup> ml

<sup>x</sup> � <sup>1</sup> <sup>þ</sup> <sup>q</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> � �x<sup>2</sup> <sup>þ</sup> <sup>x</sup><sup>3</sup> � �,

� �Blð Þ my :

!

mn�<sup>k</sup> � <sup>T</sup> <sup>n</sup>�l, <sup>q</sup>ð Þ <sup>x</sup>

ml

Bl, <sup>q</sup>ð Þ my : (46)

(47)

Corollary 2.15. From the Theorem 2.14, the following relations hold.

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

n � l k � � <sup>T</sup> <sup>k</sup>ð Þ <sup>x</sup>

3. The observation of scattering zeros of the q-tangent polynomials

In this section, we will find the approximate structure and shape of the roots according to the changes in n and q. We will extend this to identify the fixed points and try to understand the structure of the composite function using the Newton

n � l k � �

i: T <sup>n</sup>�1, <sup>q</sup>ð Þ¼ x; y

ii: T <sup>n</sup>�1ð Þ¼ x; y

method.

T <sup>0</sup>,qð Þ¼ x

T <sup>1</sup>, <sup>q</sup>ð Þ¼ x

T <sup>2</sup>,qð Þ¼ x

T <sup>3</sup>, <sup>q</sup>ð Þ¼ x

T <sup>4</sup>, <sup>q</sup>ð Þ¼ x

Figure 1.

97

1 ½ � n <sup>q</sup> ∑ n l¼0

DOI: http://dx.doi.org/10.5772/intechopen.83497

1 <sup>n</sup> <sup>∑</sup> n l¼0

1 þ q <sup>2</sup> ,

1 2

1 2

1 2

> 1 2

�ð Þ 1 þ q

above conditions and fixed at q = 0.5.

Zeros of T n,0:1ð Þ x for n = 30, 40, 50.

2

and it has 2.0 as an approximate root, which is unusual.

n l � �

n l � � <sup>∑</sup> n�l k¼0

The first five q-tangent polynomials are:

ð Þ� 1 þ q ð Þ 1 þ x ,

ð Þ <sup>1</sup> <sup>þ</sup> <sup>q</sup> <sup>1</sup> <sup>þ</sup> <sup>q</sup>ð Þþ �<sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>x</sup> � <sup>x</sup><sup>2</sup> � �,

ð Þ� <sup>1</sup> <sup>þ</sup> <sup>q</sup> <sup>1</sup> <sup>þ</sup> <sup>q</sup>ð<sup>2</sup> � �ð Þ <sup>2</sup> <sup>þ</sup> <sup>q</sup> <sup>q</sup>Þ � <sup>x</sup> <sup>þ</sup> <sup>q</sup><sup>3</sup>

ð Þ <sup>1</sup> þ �ð Þ <sup>3</sup> <sup>þ</sup> <sup>q</sup> <sup>q</sup> <sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> ð Þ<sup>x</sup>

ð Þð � <sup>1</sup> <sup>þ</sup> <sup>q</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>q</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>q</sup> ð Þ <sup>1</sup> þ �ð Þ <sup>4</sup> <sup>þ</sup> <sup>q</sup> <sup>q</sup> <sup>1</sup> <sup>þ</sup> <sup>q</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> � �

þ �ð Þ <sup>1</sup> <sup>þ</sup> <sup>q</sup> <sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>q</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> ð Þx<sup>2</sup> � ð Þ <sup>1</sup> <sup>þ</sup> <sup>q</sup> <sup>1</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> ð Þx<sup>3</sup> <sup>þ</sup> <sup>x</sup><sup>4</sup>Þ:

Using Mathematica, we will examine the approximate movement of the roots. In

Figure 2 shows the shape of the approximate roots when n is changed to the

Figure 1, the x-axis means the numbers of real zeros and the y-axis means the numbers of complex zeros in the q-tangent polynomials. When it moves from left to right, it changes to n = 30, 40, 50, and when it is fixed at q = 0.1, the approximate shape of the root appears to be almost circular. The center is identified as the origin,

q ∑ n�l k¼0

Thus, for the relation between q-tangent polynomials of two parameters and q-Euler polynomials, we have

$$\begin{split} &\sum\_{n=0}^{\infty} \mathsf{T}\_{n,q}(\mathbf{x},\boldsymbol{\uprho}) \frac{t^{n}}{[n]\_{q}!} \\ &= \sum\_{n=0}^{\infty} E\_{n,q}(\boldsymbol{m}\mathbf{y}) \frac{t^{n}}{m^{n}[n]\_{q}!} \sum\_{n=0}^{\infty} \mathsf{T}\_{n,q}(\mathbf{x}) \frac{t^{n}}{[n]\_{q}!} \left( \sum\_{n=0}^{\infty} \frac{1}{[2]\_{q}} \frac{t^{n}}{m^{n}[n]\_{q}!} + \frac{1}{[2]\_{q}} \right) \\ &= \frac{1}{[2]\_{q}} \sum\_{n=0}^{\infty} \sum\_{l=0}^{n} \begin{bmatrix} n \\ l \end{bmatrix}\_{q} E\_{l,q}(\boldsymbol{m}\mathbf{y}) \sum\_{k=0}^{n-l} \begin{bmatrix} n-l \\ k \end{bmatrix}\_{q} \frac{\mathcal{T}\_{k,q}(\mathbf{x})}{m^{n-k}} \frac{t^{n}}{[n]\_{q}!} \\ &+ \frac{1}{[2]\_{q}} \sum\_{n=0}^{\infty} \sum\_{l=0}^{n} \begin{bmatrix} n \\ l \end{bmatrix}\_{q} E\_{l,q}(\boldsymbol{m}\mathbf{y}) \frac{\mathcal{T}\_{n-l,q}(\mathbf{x})}{m^{l}} \frac{t^{n}}{[n]\_{q}!}, \end{split} \tag{41}$$

which on comparing the coefficients immediately gives the required relation. ☐ Corollary 2.13. From Theorem 2.12, the following hold.

$$\text{i.i. } \mathcal{T}\_{n,q}(\mathbf{x}, \mathbf{y}) = \frac{1}{[2]\_q} \sum\_{l=0}^n \begin{bmatrix} n \\ l \end{bmatrix}\_q \left( \frac{\mathcal{T}\_{n-l,q}(\mathbf{x})}{m^l} + \sum\_{k=0}^{n-l} \begin{bmatrix} n-l \\ k \end{bmatrix}\_q \frac{\mathcal{T}\_{k,q}(\mathbf{x})}{m^{n-k}} \right) \mathbf{E}\_{l,q}(m\mathbf{y}). \tag{42}$$
 
$$\text{ii. } \mathcal{T}\_n(\mathbf{x}, \mathbf{y}) = \frac{1}{2} \sum\_{l=0}^n \binom{n}{l} \left( \frac{\mathcal{T}\_{n-l}(\mathbf{x})}{m^l} + \sum\_{k=0}^{n-l} \binom{n-l}{k} \frac{\mathcal{T}\_k(\mathbf{x})}{m^{n-k}} \right) \mathbf{E}\_l(m\mathbf{y}).$$

Theorem 2.14. For x, y∈ C, the following relation holds

$$\mathcal{T}\_{n-1,q}(\mathbf{x},\mathbf{y}) = \frac{\mathbf{1}}{[n]\_q} \sum\_{l=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_q \left( \sum\_{k=0}^{n-l} \begin{bmatrix} n-l \\ k \end{bmatrix}\_q \frac{\mathcal{T}\_{k,q}(\mathbf{x})}{m^{n-k}} - \frac{\mathcal{T}\_{n-l,q}(\mathbf{x})}{m^l} \right) B\_{l,q}(m\mathbf{y}). \tag{43}$$

Proof. We note that

$$\frac{[2]\_q}{e\_q(2t) + 1} e\_q(t\mathbf{x}) e\_q(t\mathbf{y}) = \left(\frac{t}{e\_q\left(\frac{t}{m}\right) - 1} e\_q(t\mathbf{y})\right) \left(\frac{e\_q\left(\frac{t}{m}\right) - 1}{t}\right) \left(\frac{[2]\_q}{e\_q(2t) + 1} e\_q(t\mathbf{x})\right). \tag{44}$$

Thus as in Theorem 2.12, we have

$$\begin{split} &\sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\boldsymbol{x},\boldsymbol{y}) \frac{t^{n}}{[n]\_{q}!} \\ &= \left(\sum\_{n=0}^{\infty} \frac{t^{n-1}}{m^{n}[n]\_{q}!} - \frac{1}{t}\right) \sum\_{n=0}^{\infty} B\_{n,q}(m\boldsymbol{y}) \frac{t^{n}}{m^{n}[n]\_{q}!} \sum\_{n=0}^{\infty} \mathcal{T}\_{n,q}(\boldsymbol{x}) \frac{t^{n}}{[n]\_{q}!} \\ &= \sum\_{n=0}^{\infty} \left(\sum\_{l=0}^{n} \begin{bmatrix} n \\ l \end{bmatrix}\_{q}\right) \sum\_{k=0}^{n-l} \begin{bmatrix} n-l \\ k \end{bmatrix}\_{q} \frac{\mathcal{T}\_{k,q}(\boldsymbol{x})}{m^{n-k}} B\_{l,q}(m\boldsymbol{y}) \right) \frac{t^{n-1}}{[n]\_{q}!} \\ &- \sum\_{n=0}^{\infty} \left(\sum\_{l=0}^{n} \begin{bmatrix} n \\ l \end{bmatrix}\_{q}\right) \frac{\mathcal{T}\_{n-l,q}(\boldsymbol{x})}{m^{l}} B\_{l,q}(m\boldsymbol{y}) \right) \frac{t^{n-1}}{[n]\_{q}!} . \end{split} \tag{45}$$

The required relation now follows on comparing the coefficients. ☐ Corollary 2.15. From the Theorem 2.14, the following relations hold.

$$\mathbb{T}$$

$$\text{i. } \ T\_{n-1,q}(\mathbf{x}, \boldsymbol{\eta}) = \frac{1}{[n]\_q} \sum\_{l=0}^n \begin{bmatrix} n \\ l \end{bmatrix}\_q \left( \sum\_{k=0}^{n-l} \begin{bmatrix} n-l \\ k \end{bmatrix}\_q \frac{\mathcal{T}\_{k,q}(\mathbf{x})}{m^{n-k}} - \frac{\mathcal{T}\_{n-l,q}(\mathbf{x})}{m^l} \right) \mathbf{B}\_{l,q}(m\mathbf{y}). \tag{46}$$
 
$$\text{ii. } \ T\_{n-1}(\mathbf{x}, \boldsymbol{\eta}) = \frac{1}{n} \sum\_{l=0}^n \binom{n}{l} \left( \sum\_{k=0}^{n-l} \binom{n-l}{k} \frac{\mathcal{T}\_k(\mathbf{x})}{m^{n-k}} - \frac{\mathcal{T}\_{n-l}(\mathbf{x})}{m^l} \right) \mathbf{B}\_l(m\mathbf{y}).$$

#### 3. The observation of scattering zeros of the q-tangent polynomials

In this section, we will find the approximate structure and shape of the roots according to the changes in n and q. We will extend this to identify the fixed points and try to understand the structure of the composite function using the Newton method.

The first five q-tangent polynomials are:

$$\begin{aligned} \mathcal{T}\_{0,q}(\mathbf{x}) &= \frac{1+q}{2}, \\ \mathcal{T}\_{1,q}(\mathbf{x}) &= \frac{1}{2}(1+q)(-\mathbf{1}+\mathbf{x}), \\ \mathcal{T}\_{2,q}(\mathbf{x}) &= \frac{1}{2}(1+q)\left(\mathbf{1}+q(-\mathbf{1}+\mathbf{x})+\mathbf{x}-\mathbf{x}^2\right), \\ \mathcal{T}\_{3,q}(\mathbf{x}) &= \frac{1}{2}(\mathbf{1}+q)\left(-\mathbf{1}+q(2-(-\mathbf{2}+q)q)-\mathbf{x}+q^3\mathbf{x}-(\mathbf{1}+q+q^2)\mathbf{x}^2+\mathbf{x}^3\right), \\ \mathcal{T}\_{4,q}(\mathbf{x}) &= \frac{1}{2}(\mathbf{1}+q)((-\mathbf{1}+q)(\mathbf{1}+q)(\mathbf{1}+(-\mathbf{4}+q)q)(\mathbf{1}+q+q^2) \\ &\quad - (\mathbf{1}+q)^2(\mathbf{1}+(-\mathbf{3}+q)q)(\mathbf{1}+q^2)\mathbf{x} \\ &\quad + (-\mathbf{1}+q)(\mathbf{1}+q^2)(\mathbf{1}+q+q^2)\mathbf{x}^2-(\mathbf{1}+q)(\mathbf{1}+q^2)\mathbf{x}^3+\mathbf{x}^4). \end{aligned} \tag{47}$$

Using Mathematica, we will examine the approximate movement of the roots. In Figure 1, the x-axis means the numbers of real zeros and the y-axis means the numbers of complex zeros in the q-tangent polynomials. When it moves from left to right, it changes to n = 30, 40, 50, and when it is fixed at q = 0.1, the approximate shape of the root appears to be almost circular. The center is identified as the origin, and it has 2.0 as an approximate root, which is unusual.

Figure 2 shows the shape of the approximate roots when n is changed to the above conditions and fixed at q = 0.5.

Figure 1. Zeros of T n,0:1ð Þ x for n = 30, 40, 50.

½ � 2 <sup>q</sup> eqð Þþ 2 t 1

eqð Þ tx eqð Þ¼ ty

Polynomials - Theory and Application

q-Euler polynomials, we have

T n, <sup>q</sup>ð Þ x; y

En, <sup>q</sup>ð Þ my

t n ½ � n <sup>q</sup>!

> n l

n l

1 ½ � 2 <sup>q</sup> ∑ n l¼0

1 2 ∑ n l¼0

1 ½ � n <sup>q</sup> ∑ n l¼0

eqð Þ tx eqð Þ¼ ty

∑ ∞ n¼0

¼ ∑ ∞ n¼0

¼ ∑ ∞ n¼0

� ∑ ∞ n¼0

Thus as in Theorem 2.12, we have

T n, <sup>q</sup>ð Þ x; y

t n�1 mn½ � n <sup>q</sup>!

∑ n l¼0

∑ n l¼0

0 @

0 @

!

n

" #

l

n

" #

l

q ∑ n�l k¼0

q

" #

" #

t n mn½ � n <sup>q</sup>!

q

q

n l � �

n k � �

q ∑ n�l k¼0

t n ½ � n <sup>q</sup>!

> � 1 t

∑ ∞ n¼0

n � l

" #

k

T <sup>n</sup>�l,qð Þ x

Bn, <sup>q</sup>ð Þ my

q

ml Bl, <sup>q</sup>ð Þ my

n l

q

� � <sup>T</sup> <sup>n</sup>�<sup>l</sup>ð Þ <sup>x</sup>

Theorem 2.14. For x, y∈ C, the following relation holds

t eq <sup>t</sup> m � � � <sup>1</sup>

∑ ∞ n¼0

¼ ∑ ∞ n¼0

<sup>¼</sup> <sup>1</sup> ½ � 2 <sup>q</sup> ∑ ∞ n¼0 ∑ n l¼0

þ 1 ½ � 2 <sup>q</sup> ∑ ∞ n¼0 ∑ n l¼0

i: T n,qð Þ¼ x; y

ii: T <sup>n</sup>ð Þ¼ x; y

T <sup>n</sup>�1, <sup>q</sup>ð Þ¼ x; y

½ � 2 <sup>q</sup> eqð Þþ 2t 1

96

Proof. We note that

½ � 2 <sup>q</sup> eq <sup>t</sup> m � � <sup>þ</sup> <sup>1</sup>

> ∑ ∞ n¼0

El, <sup>q</sup>ð Þ my ∑

El, <sup>q</sup>ð Þ my

Corollary 2.13. From Theorem 2.12, the following hold.

T n, <sup>q</sup>ð Þ x

n�l k¼0

T <sup>n</sup>�l, <sup>q</sup>ð Þ x

ml <sup>þ</sup> <sup>∑</sup>

n � l k � �

eqð Þ ty ! eq <sup>t</sup>

ml <sup>þ</sup> <sup>∑</sup>

n�l k¼0

� �

q

eqð Þ ty ! eq <sup>t</sup>

Thus, for the relation between q-tangent polynomials of two parameters and

t n ½ � <sup>n</sup> <sup>q</sup>! <sup>∑</sup> ∞ n¼0

n � l k

T <sup>n</sup>�l, <sup>q</sup>ð Þ x ml

q

t n ½ � n <sup>q</sup>! ,

which on comparing the coefficients immediately gives the required relation. ☐

n�l k¼0

!

n � l k � � <sup>T</sup> <sup>k</sup>ð Þ <sup>x</sup>

T k, <sup>q</sup>ð Þ x

m � � � <sup>1</sup> t

t n mn½ � n <sup>q</sup>!

mn�<sup>k</sup> Bl, <sup>q</sup>ð Þ my

1 A t n�1 ½ � n <sup>q</sup>! :

T k, <sup>q</sup>ð Þ x

∑ ∞ n¼0

T n,qð Þ x

1 A t n�1 ½ � n <sup>q</sup>!

t n ½ � n <sup>q</sup>!

!

" #

m � � <sup>þ</sup> <sup>1</sup> ½ � 2 <sup>q</sup>

! ½ � <sup>2</sup> <sup>q</sup>

1 ½ � 2 <sup>q</sup>

T k, <sup>q</sup>ð Þ x mn�<sup>k</sup>

n � l k � �

q

mn�<sup>k</sup>

mn�<sup>k</sup> � <sup>T</sup> <sup>n</sup>�l, <sup>q</sup>ð Þ <sup>x</sup>

� � ½ � 2 <sup>q</sup>

ml

eqð Þþ 2t 1

!

T k, <sup>q</sup>ð Þ x mn�<sup>k</sup>

Elð Þ my :

t n mn½ � n <sup>q</sup>!

t n ½ � n <sup>q</sup>!

!

eqð Þþ 2t 1

þ 1 ½ � 2 <sup>q</sup>

!

eqð Þ tx

: (40)

(41)

El,qð Þ my : (42)

Bl, <sup>q</sup>ð Þ my : (43)

: (44)

(45)

eqð Þ tx

The iterates of f are the functions f,f ∘f,f ∘f ∘f, …, which are denoted

, … If z∈ C, and then the orbit of z<sup>0</sup> under f is the sequence

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

We consider the Newton's dynamical system as follows [12, 15, 20]:

<sup>C</sup><sup>∞</sup> : R xð Þ¼ <sup>x</sup> � <sup>T</sup> ð Þ <sup>x</sup>

R is called the Newton iteration function of T . It can be considered that the fixed points of R are the zeros of T and all the fixed points of R are attracting. R may also

For x∈ C, we consider T <sup>4</sup>, <sup>q</sup>ð Þ x , and then this polynomial has four distinct complex numbers, aið Þ i ¼ 1; 2; 3; 4 such that T <sup>4</sup>, <sup>q</sup>ð Þ¼ ai 0. Using a computer, we obtain

In Newton's method, the generalized expectation is that a typical orbit {R(x)} will converge to one of the roots of T <sup>4</sup>, <sup>q</sup>ð Þ x for x<sup>0</sup> ∈ C. If we choose x0, which is

When it is given a point x<sup>0</sup> in the complex plane, we want to determine whether

The output in Figure 5 is the last calculated orbit value. We construct a function, which assigns one of four colors for each point according to the outcome of R in the plane. If an orbit of x<sup>0</sup> for q = 0.1 converges to �0.672809, �0.0821877 � 0.710388i, �0.0821877 + 0.710388i and 1.94818, then we denote the red, blue, yellow, and sky-blue, respectively(the left figure). For example, the yellow region for the left figure represents the part of the basin of attraction of a<sup>3</sup> = �0.0821877 + 0.710388i.

i q = 0.1 q = 0.5 q = 0.9 �0.672809 �0.581881 � 0.412941i �1.10249 �0.0821877 � 0.710388i �0.581881 + 0.412941i �0.158841 �0.0821877 + 0.710388i 0.907024 1.84004 1.94818 2.13174 2.86029

the orbit of x<sup>0</sup> under the action of R(x) converges to one of the roots of the equation. The orbit of x<sup>0</sup> under the action of R also appears by calculating until 30 iterations or the absolute difference value of the last two iterations is within 10�<sup>6</sup>

T 0 ð Þ x

: (49)

lim<sup>r</sup>!<sup>∞</sup> R xð Þ¼ <sup>0</sup> ai,for<sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup>: (50)

.

f 1 , f 2 , f <sup>3</sup>

Table 1.

Figure 5.

99

Approximate zeros of T <sup>4</sup>, <sup>q</sup>ð Þ x .

Orbit of x<sup>0</sup> under the action of R for T <sup>4</sup>, <sup>q</sup>ð Þ x for q = 0.1, 0.5, 0.9.

< z0,f zð Þ<sup>0</sup> ,f f z ð Þ ð Þ<sup>0</sup> , ⋯ > .

have one or more attracting cycles.

DOI: http://dx.doi.org/10.5772/intechopen.83497

the approximate zeros (Table 1) as follows:

sufficiently close to ai, then this proves that

Figure 2. Zeros of T <sup>n</sup>;0:5ð Þ x for n = 30, 40, 50.

In Figure 2, the shape of the root changes to an ellipse, unlike the q = 0.1 condition, and the widening phenomenon appears when the real number is 0.5. In addition, like the previous Figure 1, we can see that it has a common approximate root at 2.0. In the following Figure 3, n of the far-left figure is 30, and it increases by 10 while moving to the right, and the far-right figure shows the shape of the root when n = 50 and is fixed at q = 0.9.

In Figure 3, the roots have a general tangent polynomial shape with similar properties (see [22–25]). If each approximate root obtained in the previous step is piled up according to the value of n, it will appear as shown in Figure 4. The left Figure 4 is q = 0.1 with n from 1 to 50. The middle Figure 4 is q = 0.5 with n from 1 to 50. The right Figure 4 is q = 0.9 with n from 1 to 50.

Let f : D ! D be a complex function, with D as a subset of C. We define the iterated maps of the complex function as the following:

$$f\_r: z\_0 \mapsto \underbrace{f(f(\cdots(f\ (z\_0)\cdots)))}\_{r} \tag{48}$$

Figure 3. Zeros of T <sup>n</sup>;0:9ð Þ x for n = 30, 40, 50.

Figure 4. Zeros of T n, <sup>q</sup>ð Þ x for q = 0.1, 0.5, 0.9, 1 ≤ n ≤ 50.

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials DOI: http://dx.doi.org/10.5772/intechopen.83497

The iterates of f are the functions f,f ∘f,f ∘f ∘f, …, which are denoted f 1 , f 2 , f <sup>3</sup> , … If z∈ C, and then the orbit of z<sup>0</sup> under f is the sequence < z0,f zð Þ<sup>0</sup> ,f f z ð Þ ð Þ<sup>0</sup> , ⋯ > .

We consider the Newton's dynamical system as follows [12, 15, 20]:

$$\left\{ \mathbb{C}\_{\infty} : R(\boldsymbol{\omega}) = \boldsymbol{\infty} - \frac{\boldsymbol{\mathcal{T}}(\boldsymbol{\omega})}{\boldsymbol{\mathcal{T}}'(\boldsymbol{\omega})} \right\}. \tag{49}$$

R is called the Newton iteration function of T . It can be considered that the fixed points of R are the zeros of T and all the fixed points of R are attracting. R may also have one or more attracting cycles.

For x∈ C, we consider T <sup>4</sup>, <sup>q</sup>ð Þ x , and then this polynomial has four distinct complex numbers, aið Þ i ¼ 1; 2; 3; 4 such that T <sup>4</sup>, <sup>q</sup>ð Þ¼ ai 0. Using a computer, we obtain the approximate zeros (Table 1) as follows:

In Newton's method, the generalized expectation is that a typical orbit {R(x)} will converge to one of the roots of T <sup>4</sup>, <sup>q</sup>ð Þ x for x<sup>0</sup> ∈ C. If we choose x0, which is sufficiently close to ai, then this proves that

$$\lim\_{r \to \infty} R(\mathbf{x}\_0) = a\_{i\flat} \text{ for } i = \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}. \tag{50}$$

When it is given a point x<sup>0</sup> in the complex plane, we want to determine whether the orbit of x<sup>0</sup> under the action of R(x) converges to one of the roots of the equation. The orbit of x<sup>0</sup> under the action of R also appears by calculating until 30 iterations or the absolute difference value of the last two iterations is within 10�<sup>6</sup> .

The output in Figure 5 is the last calculated orbit value. We construct a function, which assigns one of four colors for each point according to the outcome of R in the plane. If an orbit of x<sup>0</sup> for q = 0.1 converges to �0.672809, �0.0821877 � 0.710388i, �0.0821877 + 0.710388i and 1.94818, then we denote the red, blue, yellow, and sky-blue, respectively(the left figure). For example, the yellow region for the left figure represents the part of the basin of attraction of a<sup>3</sup> = �0.0821877 + 0.710388i.


Table 1. Approximate zeros of T <sup>4</sup>, <sup>q</sup>ð Þ x .

Figure 5. Orbit of x<sup>0</sup> under the action of R for T <sup>4</sup>, <sup>q</sup>ð Þ x for q = 0.1, 0.5, 0.9.

In Figure 2, the shape of the root changes to an ellipse, unlike the q = 0.1 condition, and the widening phenomenon appears when the real number is 0.5. In addition, like the previous Figure 1, we can see that it has a common approximate root at 2.0. In the following Figure 3, n of the far-left figure is 30, and it increases by 10 while moving to the right, and the far-right figure shows the shape of the root

In Figure 3, the roots have a general tangent polynomial shape with similar properties (see [22–25]). If each approximate root obtained in the previous step is piled up according to the value of n, it will appear as shown in Figure 4. The left Figure 4 is q = 0.1 with n from 1 to 50. The middle Figure 4 is q = 0.5 with n from 1

Let f : D ! D be a complex function, with D as a subset of C. We define the


ð Þ z<sup>0</sup> ⋯ÞÞÞ (48)

fr : z<sup>0</sup> ↦ fðfð⋯ðf

when n = 50 and is fixed at q = 0.9.

Zeros of T <sup>n</sup>;0:5ð Þ x for n = 30, 40, 50.

Polynomials - Theory and Application

Figure 2.

Figure 3.

Figure 4.

98

Zeros of T <sup>n</sup>;0:9ð Þ x for n = 30, 40, 50.

Zeros of T n, <sup>q</sup>ð Þ x for q = 0.1, 0.5, 0.9, 1 ≤ n ≤ 50.

to 50. The right Figure 4 is q = 0.9 with n from 1 to 50.

iterated maps of the complex function as the following:

If we use T <sup>3</sup>;0:1ð Þ x to draw a figure using the Newton method, we can obtain Figure 6. The picture on the left shows three roots, and the colors are blue, red, and ivory in the counterclockwise direction. When we examine the area closely, we can see that it converges to an approximate value in each color area. The convergence value in the blue area is �0:379202 þ 0:523651i, that in the red area is �0:379202 � 0:523651i, and that in the ivory area is 1.8684. We can also see that it shows self-similarity at the boundary point as divided into three areas. The figure on the right is obtained by 2-times iterated q-tangent polynomials, T <sup>2</sup> <sup>3</sup>;0:1ð Þ x , and the area is divided into nine colors "gray (x ¼ 2:31831), scarlet (x ¼ 1:76736 þ 0:216319i), light brown (x ¼ 0:137247 þ 0:59473i), sky blue (x ¼ �0:604153 þ 1:19884i), blue (x ¼ �0:794606 þ 0:378411i), red (x ¼ �0:794606 � 0:378411i), ivory (x ¼ �0:604153 � 1:19884i), green (x ¼ 0:137247 � 0:59473i), and navy blue (x ¼ 1:76736 � 0:216319i) in the counterclockwise direction. This also shows self-similarity at the boundary.

In Figure 7, we express the coloring for T <sup>2</sup> <sup>3</sup>;0:1ð Þ x .

Conjecture 3.1. The q-tangent polynomials always have self-similarity at the boundary.

We know that the fixed point is divided as follows. Suppose that the complex function f is analytic in a region D of C, and f has a fixed point at z<sup>0</sup> ∈ D. Then z<sup>0</sup> is said to be (see [6, 16, 20]):

an attracting fixed point if ∣ f 0 ð Þ z<sup>0</sup> ∣ < 1; a repelling fixed point if ∣ f 0 ð Þ z<sup>0</sup> ∣ > 1; a neutral fixed point if ∣ f 0 ð Þ z<sup>0</sup> ∣ ¼ 1.

For example, T <sup>3</sup>;0:1ð Þ x has three points satisfying T <sup>3</sup>;0:1ð Þ¼ x x. That is, x<sup>0</sup> ¼ �0:967484, � 0:33466; 2:41214. Since

$$\left| \frac{d}{dt} \mathcal{T}\_{3,0.1}(-0.967484) \right| = 0 \prec 1, \quad \left| \frac{d}{dt} \mathcal{T}\_{3,0.1}(-0.93466) \right| = 0 \prec 1 \tag{51}$$

Theorem 3.2. T <sup>3</sup>;0:1ð Þ x for q = 0.1 has two attracting fixed points.

<sup>3</sup>;0:<sup>1</sup> ð Þ <sup>x</sup> for 1≤r ≤6.

132 232 332 4 23 2 522 611

1≤n ≤4. We can also reach Conjecture 3.3.

<sup>3</sup>;0:<sup>1</sup> ð Þ <sup>x</sup> and RF<sup>T</sup> <sup>r</sup>

Table 2.

Table 3.

Figure 8.

101

Stacks of fixed point of T <sup>r</sup>

<sup>3</sup>,0:1ð Þ x for 1 ≤ r ≤ 6.

The numbers of R<sup>T</sup> <sup>r</sup>

Numbers of fixed points of T <sup>n</sup>;0:1ð Þ x .

r R<sup>T</sup> <sup>r</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83497

Using Mathematica, we can separate the numerical results for fixed points of T <sup>n</sup>;0:1ð Þ x . From Table 2, we know that T <sup>n</sup>;0:1ð Þ x have no neutral fixed point for

Degree n Attractor Repellor Neutral 1 0 10 2 1 10 3 2 10 4 1 30 5 1 40

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

<sup>3</sup>;0:<sup>1</sup> ð Þ <sup>x</sup> RF<sup>T</sup> <sup>r</sup>

<sup>3</sup>;0:<sup>1</sup> ð Þ x

Figure 6. Orbit of <sup>x</sup><sup>0</sup> under the action of <sup>R</sup> for <sup>T</sup> <sup>3</sup>,0:1ð Þ <sup>x</sup> , <sup>T</sup> <sup>2</sup> <sup>3</sup>;0:1ð Þ x .

Figure 7. Palette for escaping points.

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials DOI: http://dx.doi.org/10.5772/intechopen.83497


Table 2.

If we use T <sup>3</sup>;0:1ð Þ x to draw a figure using the Newton method, we can obtain Figure 6. The picture on the left shows three roots, and the colors are blue, red, and ivory in the counterclockwise direction. When we examine the area closely, we can see that it converges to an approximate value in each color area. The convergence

�0:379202 � 0:523651i, and that in the ivory area is 1.8684. We can also see that it shows self-similarity at the boundary point as divided into three areas. The figure

Conjecture 3.1. The q-tangent polynomials always have self-similarity at the

We know that the fixed point is divided as follows. Suppose that the complex function f is analytic in a region D of C, and f has a fixed point at z<sup>0</sup> ∈ D. Then z<sup>0</sup> is

<sup>3</sup>;0:1ð Þ x .

dt <sup>T</sup> <sup>3</sup>;0:1ð Þ �0:<sup>33466</sup>

 

<sup>¼</sup> 0 < 1 (51)

<sup>3</sup>;0:1ð Þ x , and the

value in the blue area is �0:379202 þ 0:523651i, that in the red area is

on the right is obtained by 2-times iterated q-tangent polynomials, T <sup>2</sup>

0 ð Þ z<sup>0</sup> ∣ < 1;

ð Þ z<sup>0</sup> ∣ ¼ 1.

For example, T <sup>3</sup>;0:1ð Þ x has three points satisfying T <sup>3</sup>;0:1ð Þ¼ x x.

<sup>¼</sup> 0<1, <sup>d</sup>

 

<sup>3</sup>;0:1ð Þ x .

0 ð Þ z<sup>0</sup> ∣ > 1;

0

That is, x<sup>0</sup> ¼ �0:967484, � 0:33466; 2:41214. Since

 

(x ¼ 1:76736 þ 0:216319i), light brown (x ¼ 0:137247 þ 0:59473i), sky blue (x ¼ �0:604153 þ 1:19884i), blue (x ¼ �0:794606 þ 0:378411i), red (x ¼ �0:794606 � 0:378411i), ivory (x ¼ �0:604153 � 1:19884i), green (x ¼ 0:137247 � 0:59473i), and navy blue (x ¼ 1:76736 � 0:216319i) in the counterclockwise direction. This also shows self-similarity at the boundary.

area is divided into nine colors "gray (x ¼ 2:31831), scarlet

In Figure 7, we express the coloring for T <sup>2</sup>

boundary.

said to be (see [6, 16, 20]):

d

 

Figure 6.

Figure 7.

100

Palette for escaping points.

an attracting fixed point if ∣ f

dt <sup>T</sup> <sup>3</sup>;0:1ð Þ �0:<sup>967484</sup>

Orbit of <sup>x</sup><sup>0</sup> under the action of <sup>R</sup> for <sup>T</sup> <sup>3</sup>,0:1ð Þ <sup>x</sup> , <sup>T</sup> <sup>2</sup>

a repelling fixed point if ∣ f

a neutral fixed point if ∣ f

Polynomials - Theory and Application

Numbers of fixed points of T <sup>n</sup>;0:1ð Þ x .


#### Table 3.

The numbers of R<sup>T</sup> <sup>r</sup> <sup>3</sup>;0:<sup>1</sup> ð Þ <sup>x</sup> and RF<sup>T</sup> <sup>r</sup> <sup>3</sup>;0:<sup>1</sup> ð Þ <sup>x</sup> for 1≤r ≤6.

Theorem 3.2. T <sup>3</sup>;0:1ð Þ x for q = 0.1 has two attracting fixed points.

Using Mathematica, we can separate the numerical results for fixed points of T <sup>n</sup>;0:1ð Þ x . From Table 2, we know that T <sup>n</sup>;0:1ð Þ x have no neutral fixed point for 1≤n ≤4. We can also reach Conjecture 3.3.

Figure 8. Stacks of fixed point of T <sup>r</sup> <sup>3</sup>,0:1ð Þ x for 1 ≤ r ≤ 6.

Conjecture 3.3. The q-tangent polynomials for n≥2 have at least one attracting fixed point except for infinity.

In Table 3, we denote R<sup>T</sup> <sup>r</sup> n,qð Þ <sup>x</sup> as the numbers of real zeros for rth iteration and RF<sup>T</sup> <sup>r</sup> n,qð Þ <sup>x</sup> as the numbers of attracting fixed point on real number. From this table, we can know that number of real fixed points of T <sup>r</sup> <sup>3</sup>, <sup>q</sup>ð Þ x are less than two. Here, we can suggest Conjecture 3.4.

Conjecture 3.4. The q-tangent polynomials that are iterated, T <sup>r</sup> <sup>3</sup>;0:1ð Þ x , have real fixed point, α ¼ �0:33466.

In the top-left of Figure 8, we can see the forms of 3D structure related to stacks of fixed points of T <sup>r</sup> <sup>3</sup>;0:1ð Þ x for 1≤r≤6. When we look at the top-left of Figure 8 in the below position, we can draw the top-right figure. The bottom-left of Figure 8 shows that image and n-axes exist but not real axis in three dimensions. In three dimensions, the bottom-right of Figure 8 is the right orthographic viewpoint for the top-left figure,-that is, there exist real and n-axes but there is no image axis (Figure 8).

#### 4. Conclusion

We can see that when q comes closer to 0, the approximate shape of the roots become increasingly more circular. Also in this situation, we can observe scattering of zeros in q-tangent polynomials around 2 in three-dimension. When q comes closer to 1, it has properties that are more symmetrical. We can also assume that the property that appears when iterating T n, <sup>q</sup>ð Þ x has self-similarity. By iterating, we can conjecture some properties about fixed points. This property warrants further study so that we can create a new property.

#### Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (No. 2017R1E1A1A03070483).

#### Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Author details

Jung Yoog Kang<sup>1</sup>

103

\* and Cheon Seoung Ryoo<sup>2</sup>

\*Address all correspondence to: jykang@silla.ac.kr

provided the original work is properly cited.

1 Department of Mathematics Education, Silla University, Busan, Korea

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

DOI: http://dx.doi.org/10.5772/intechopen.83497

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 Department of Mathematics, Hanman University, Daejeon, Korea

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials DOI: http://dx.doi.org/10.5772/intechopen.83497

#### Author details

Conjecture 3.3. The q-tangent polynomials for n≥2 have at least one attracting

n,qð Þ <sup>x</sup> as the numbers of attracting fixed point on real number. From this table,

In the top-left of Figure 8, we can see the forms of 3D structure related to stacks

the below position, we can draw the top-right figure. The bottom-left of Figure 8 shows that image and n-axes exist but not real axis in three dimensions. In three dimensions, the bottom-right of Figure 8 is the right orthographic viewpoint for the top-left figure,-that is, there exist real and n-axes but there is no image axis

We can see that when q comes closer to 0, the approximate shape of the roots become increasingly more circular. Also in this situation, we can observe scattering of zeros in q-tangent polynomials around 2 in three-dimension. When q comes closer to 1, it has properties that are more symmetrical. We can also assume that the property that appears when iterating T n, <sup>q</sup>ð Þ x has self-similarity. By iterating, we can conjecture some properties about fixed points. This property warrants further

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science,

The authors declare that there is no conflict of interests regarding the publica-

<sup>3</sup>;0:1ð Þ x for 1≤r≤6. When we look at the top-left of Figure 8 in

Conjecture 3.4. The q-tangent polynomials that are iterated, T <sup>r</sup>

n,qð Þ <sup>x</sup> as the numbers of real zeros for rth iteration and

<sup>3</sup>, <sup>q</sup>ð Þ x are less than two. Here, we

<sup>3</sup>;0:1ð Þ x , have real

fixed point except for infinity. In Table 3, we denote R<sup>T</sup> <sup>r</sup>

Polynomials - Theory and Application

can suggest Conjecture 3.4.

fixed point, α ¼ �0:33466.

of fixed points of T <sup>r</sup>

(Figure 8).

4. Conclusion

Acknowledgements

Conflict of interests

tion of this paper.

102

we can know that number of real fixed points of T <sup>r</sup>

study so that we can create a new property.

ICT and Future Planning (No. 2017R1E1A1A03070483).

RF<sup>T</sup> <sup>r</sup>

Jung Yoog Kang<sup>1</sup> \* and Cheon Seoung Ryoo<sup>2</sup>


\*Address all correspondence to: jykang@silla.ac.kr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Jackson HF. q-Difference equations. American Journal of Mathematics. 1910;32:305-314

[2] Jackson HF. On q-functions and a certain difference operator. Transactions of the Royal Society of Edinburgh.46:253-281

[3] Carmichael RD. The general theory of linear q-difference equations. American Journal of Mathematics. 1912;34:147-168

[4] Mason TE. On properties of the solution of linear q-difference equations with entire function coefficients. American Journal of Mathematics. 1915;37:439-444

[5] Andrews GE, Askey R, Roy R. Special Functions. Cambridge, UK: Cambridge Press; 1999

[6] Agarwal RP, Meehan M, O'Regan D. Fixed Point Theory and Applications. Cambridge University Press; 2001 ISBN 0-521-80250-4

[7] Ayoub R. Euler and zeta function. American Mathematical Monthly. 1974;81:1067-1086

[8] Arik M, Demircan E, Turgut T, Ekinci L, Mungan M. Fibonacci oscillators. Zeitschrift für Physik C: Particles and Fields. 1992;55:89-95

[9] Bangerezako G. An Introduction to q-Difference Equations. Preprint. Bujumbura; 2008

[10] Comtet L. Advanced Combinatorics. Dordrecht, The Netherlands: Reidel; 1974

[11] Carlitz L. A note on Bernoulli and Euler polynomials of the second kind. Scripta Mathematica. 1961;25:323-330 [12] Endre S, David M. An Introduction to Numerical Analysis. Cambridge University Press; 2003 ISBN 0-521- 00794-1

Euler Polynomials, arXive:1201.6633v1

DOI: http://dx.doi.org/10.5772/intechopen.83497

A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials

[22] Ryoo CS. A numerical investigation on the zeros of the tangent polynomials. Journal of Applied Mathematics & Informatics. 2014;32:315-322

[23] Ryoo CS. Differential equations associated with tangent numbers. Journal of Applied Mathematics & Informatics. 2016;34:487-494

[24] Ryoo CS. Calculating zeros of the second kind Euler polynomials. Journal

[25] Ryoo CS, Kim T, Agarwal RP. A numerical investigation of the roots of q-polynomials, inter. Journal of Computational Mathematics. 2006;83:

[26] Trjitzinsky WJ. Analytic theory of linear q-difference equations. Acta

of Computational Analysis and Applications. 2010;12:828-833

223-234

105

Mathematica. 1933

[math. CA]Jan 2012. p. 31

[13] Exton H. q-Hypergeometric Functions and Applications. New York/ Chichester: Halstead Press/Ellis Horwood; 1983 ISBN 978-0470274538

[14] Jagannathan R, Rao KS. Twoparameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. arXiv:math/ 0602613[math.NT]

[15] Kelley CT. Solving Nonlinear Equations with Newton's Method, No. 1 in Fundamentals of Algorithms. SIAM; 2003 ISBN 0-89871-546-6

[16] Kirk WA, Goebel K. Topics in Metric Fixed Point Theory. Cambridge University Press; 1990 ISBN 0-521- 38289-0

[17] Kac V, Cheung P. Quantum Calculus, Universitext. Springer-Verlag; 2002 ISBN 0-387-95341-8

[18] Konvalina J. A unified interpretation of the binomial coefficients, the Stirling numbers, and the Gaussian coefficients. American Mathematical Monthly. 2000; 107(10):901910

[19] Kang JY, Ryoo CS. The structure of the zeros and fixed point for Genocchi polynomials. Journal of Computational Analysis and Applications. 2017;22(6): 1023-1034

[20] Kang JY, Ryoo CS. A research on the some properties and distribution of zeros for Stirling polynomials. Journal of Nonlinear Sciences and Applications. 2016;9:1735-1747

[21] Mahmudov NI. A New Class of Generalized Bernoulli Polynomials and A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials DOI: http://dx.doi.org/10.5772/intechopen.83497

Euler Polynomials, arXive:1201.6633v1 [math. CA]Jan 2012. p. 31

References

1910;32:305-314

1912;34:147-168

1915;37:439-444

Press; 1999

0-521-80250-4

1974;81:1067-1086

Bujumbura; 2008

104

[10] Comtet L. Advanced Combinatorics. Dordrecht, The Netherlands: Reidel; 1974

[1] Jackson HF. q-Difference equations. American Journal of Mathematics.

Polynomials - Theory and Application

[12] Endre S, David M. An Introduction to Numerical Analysis. Cambridge University Press; 2003 ISBN 0-521-

Functions and Applications. New York/

[13] Exton H. q-Hypergeometric

Chichester: Halstead Press/Ellis Horwood; 1983 ISBN 978-0470274538

[14] Jagannathan R, Rao KS. Twoparameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. arXiv:math/

[15] Kelley CT. Solving Nonlinear Equations with Newton's Method, No. 1 in Fundamentals of Algorithms. SIAM;

[16] Kirk WA, Goebel K. Topics in Metric Fixed Point Theory. Cambridge University Press; 1990 ISBN 0-521-

[17] Kac V, Cheung P. Quantum

2002 ISBN 0-387-95341-8

107(10):901910

1023-1034

2016;9:1735-1747

Calculus, Universitext. Springer-Verlag;

[18] Konvalina J. A unified interpretation of the binomial coefficients, the Stirling numbers, and the Gaussian coefficients. American Mathematical Monthly. 2000;

[19] Kang JY, Ryoo CS. The structure of the zeros and fixed point for Genocchi polynomials. Journal of Computational Analysis and Applications. 2017;22(6):

[20] Kang JY, Ryoo CS. A research on the some properties and distribution of zeros for Stirling polynomials. Journal of Nonlinear Sciences and Applications.

[21] Mahmudov NI. A New Class of Generalized Bernoulli Polynomials and

2003 ISBN 0-89871-546-6

0602613[math.NT]

38289-0

00794-1

[2] Jackson HF. On q-functions and a

Transactions of the Royal Society of

[3] Carmichael RD. The general theory of linear q-difference equations. American Journal of Mathematics.

[4] Mason TE. On properties of the solution of linear q-difference equations with entire function coefficients. American Journal of Mathematics.

[5] Andrews GE, Askey R, Roy R. Special Functions. Cambridge, UK: Cambridge

[6] Agarwal RP, Meehan M, O'Regan D. Fixed Point Theory and Applications. Cambridge University Press; 2001 ISBN

[7] Ayoub R. Euler and zeta function. American Mathematical Monthly.

[8] Arik M, Demircan E, Turgut T, Ekinci L, Mungan M. Fibonacci oscillators. Zeitschrift für Physik C: Particles and Fields. 1992;55:89-95

[9] Bangerezako G. An Introduction to q-Difference Equations. Preprint.

[11] Carlitz L. A note on Bernoulli and Euler polynomials of the second kind. Scripta Mathematica. 1961;25:323-330

certain difference operator.

Edinburgh.46:253-281

[22] Ryoo CS. A numerical investigation on the zeros of the tangent polynomials. Journal of Applied Mathematics & Informatics. 2014;32:315-322

[23] Ryoo CS. Differential equations associated with tangent numbers. Journal of Applied Mathematics & Informatics. 2016;34:487-494

[24] Ryoo CS. Calculating zeros of the second kind Euler polynomials. Journal of Computational Analysis and Applications. 2010;12:828-833

[25] Ryoo CS, Kim T, Agarwal RP. A numerical investigation of the roots of q-polynomials, inter. Journal of Computational Mathematics. 2006;83: 223-234

[26] Trjitzinsky WJ. Analytic theory of linear q-difference equations. Acta Mathematica. 1933

Section 2

Applications of Polynomials

107

Section 2

## Applications of Polynomials

Chapter 6

Nesenchuk Alla

polynomial family.

1. Introduction

109

Abstract

Investigation and Synthesis

of Robust Polynomials in

Uncertainty on the Basis

of the Root Locus Theory

The root locus method is proposed in the chapter for searching intervals of uncertainty for coefficients of the given (source) polynomial with constant or interval coefficients under perturbations, which ensures its robust stability regardless of whether the given polynomial is Hurwitz or not. The method is based on introduction and application of the "extended root locus" notion. Polynomial adjustment is performed by setting up each one of its coefficients separately and sequentially and determining permissible values of coefficient variation intervals (intervals of uncertainty). The effect of each coefficient variation upon the polynomial root dynamics (behavior) is considered and analyzed separately, and this influence could be observed in the root locus portraits. Root locus method is thus generalized to the cases when the number of polynomial variable coefficients is arbitrary. The root locus parameter distribution diagram along the asymptotic stability bound is introduced and applied for observing the roots behavior regularities. On this basis, the stability conditions are derived, and analytical and graphic-analytical methods are worked out for calculating intervals of variation for the 4th order polynomial family parameters ensuring its robust stability. It also allows to extract Hurwitz subfamilies from the non-Hurwitz families of interval polynomials and to determine whether there exists at least one stable polynomial in the unstable

Keywords: polynomial, dynamic system, uncertainty, stability, robustness, root locus portrait, extended root locus, root locus parameter function

As it is emphasized in [1, 2], the tasks of analysis and synthesis of control processes occurring in dynamic systems of different physical nature, operating in conditions of substantial plant parametric uncertainty, including the engineering ones, are currently the most urgent and challenging within the framework of the control theory. Among these tasks, one could mention the problem of flux control for the electric motor vector control systems operating in uncertainty because the flux control quality strongly affects the electromagnetic torque and speed control

### Chapter 6

## Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus Theory

Nesenchuk Alla

### Abstract

The root locus method is proposed in the chapter for searching intervals of uncertainty for coefficients of the given (source) polynomial with constant or interval coefficients under perturbations, which ensures its robust stability regardless of whether the given polynomial is Hurwitz or not. The method is based on introduction and application of the "extended root locus" notion. Polynomial adjustment is performed by setting up each one of its coefficients separately and sequentially and determining permissible values of coefficient variation intervals (intervals of uncertainty). The effect of each coefficient variation upon the polynomial root dynamics (behavior) is considered and analyzed separately, and this influence could be observed in the root locus portraits. Root locus method is thus generalized to the cases when the number of polynomial variable coefficients is arbitrary. The root locus parameter distribution diagram along the asymptotic stability bound is introduced and applied for observing the roots behavior regularities. On this basis, the stability conditions are derived, and analytical and graphic-analytical methods are worked out for calculating intervals of variation for the 4th order polynomial family parameters ensuring its robust stability. It also allows to extract Hurwitz subfamilies from the non-Hurwitz families of interval polynomials and to determine whether there exists at least one stable polynomial in the unstable polynomial family.

Keywords: polynomial, dynamic system, uncertainty, stability, robustness, root locus portrait, extended root locus, root locus parameter function

### 1. Introduction

As it is emphasized in [1, 2], the tasks of analysis and synthesis of control processes occurring in dynamic systems of different physical nature, operating in conditions of substantial plant parametric uncertainty, including the engineering ones, are currently the most urgent and challenging within the framework of the control theory. Among these tasks, one could mention the problem of flux control for the electric motor vector control systems operating in uncertainty because the flux control quality strongly affects the electromagnetic torque and speed control

quality, and thus the drive power efficiency. For this reason, of great importance are the tasks of stability investigation and parametric synthesis of robust control systems (their characteristic polynomials) for the plants which parameters vary within the given or unknown intervals of values.

imaginary roots (MIRs) and their stability analysis, which becomes much more complicated than that in the case with only simple imaginary roots, are treated in [13]. For a class of time-delay systems, it was proved that the invariance between the multiple imaginary roots and the simple imaginary roots holds for any multiplicity as well as for the degenerate cases. In paper [14], monic complex polynomials are identified with the sets of their roots instead of being identified with the vectors of their coefficients. A proof is given that the space of Hurwitz polynomials of degree n with positive (resp. negative) coefficients is contractible and also that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. A computational method to verify the stability of a convex combination of polynomials is considered in [15] and aimed at the robust stability analysis of a linear system. A simple algebraic test (a matrix inequality) for the stability of the segment of polynomials determined by the given two Hurwitz stable polynomials is proposed. Kučera gives a survey [16] where he navigates the area of the polynomial approach in the control system design technique. Such areas as parameterization of stabilizing controllers, called Youla–Kučera parameterization, are explained; the results on reference tracking, disturbance elimination, pole placement, deadbeat

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

Of great interest are the problems of ensuring system stability and quality being solved in the modern statements of the problem [2] as tasks of guaranteeing system robustness, which could be solved by application of the root locus approach. The basic benefit of this approach is that its application itself, by its nature, implies parametric variations (i.e., uncertainty). The root locus approach is a powerful method used for the system synthesis [2] and is notable for its descriptiveness ensuring both calculation of the system robust parameters' values and possibility of detailed overview of the dynamic properties variation changes, the system response to uncertainties that is particularly important when investigating systems with

Root locus approach to the problem is considered in [17–23]. Paper [17] gives

The above analyzed literature covers various approaches to the uncertainty treatment. However, most of the theoretical works are focused on the tasks of robust stability analysis. The methods for synthesis are not that widely represented, often suffer from complexity and in most cases are enough narrow, which means that they certainly provide instruments for system synthesis, but they are mostly "closed on themselves," which means that they do not provide the complete picture in the sense of showing up what is happening "under cover," which is especially important for the qualitative robust system (polynomials) synthesis. The root locus approach is rarely applied even though it represents the dynamic picture of the system response to uncertainties in the most comprehensive way and thus seems to

As for polynomial families, the root locus approach gives us the transparent picture of root dynamics making it possible to see as if from the inside, for example, what subfamilies constitute the whole family of uncertain polynomials in terms of their configuration and stability or some other dynamic indicators bearing significant information about the system behavior and thus leading the way for its

a solution for a compensator synthesis on the basis of the root locus method application. The task of a stable characteristic polynomial synthesis for the interval dynamic system (IDS) by setting up coefficients of the given (initial) unstable one for the case of location of its root locus initial point (where the variable parameter is equal to zero) family within the left half-plane is solved in [21], where the stability is attained via simple setting up the interval of the free

control, robust stabilization, and some others are described.

uncertain and in particular interval parameters.

DOI: http://dx.doi.org/10.5772/intechopen.83705

be the most suitable one to deal with uncertainties.

investigation and synthesis.

111

term variation.

In the area of investigation and synthesis of dynamic system characteristic polynomials, there exists a lot of approaches and methods. For the first time, the necessary and sufficient conditions for systems up to the 3-rd order were formulated by James Maxwell in 1868. Later appeared the stability criteria of Routh– Hurwitz, Mikhajlov, Nyquist, and Bode, which made it possible to check stability of the systems of order n. The frequency Nyquist criterion was the first one that could be used for synthesis by estimation of the system degree of stability. Among the modern methods of synthesis [1, 2] together with the frequency ones, the root locus and state-space methods could be listed. In his book [1], Jurgen Ackermann gives, in particular, the algebraic approach to uncertainty considering different, including the nonlinear, types of the coefficient functions and generating stability regions in the parameter space of real physical parameters of the system (polynomial). The main results in the area of the frequency approach to analysis and synthesis of robust dynamic (control) systems are given in [3], where the stability of uncertain polynomials, including interval ones, is also considered.

The methods for analysis and synthesis of polynomial families represent a separate group. One of the most effective solutions for the task of interval polynomial family investigation within the algebraic approach has been proposed by Kharitonov [4], where in the general case, the task of polynomial stability analysis is reduced to consideration of only four specific polynomials of the whole family with constant coefficients. In [3, 5], the frequency criteria of Hurwitz robust stability are considered, which allow to define the coefficient perturbation sweep for the nominally stable polynomial and various types of uncertainties. Hurwitz robust stability is also investigated in [6–10]. In [6], the maximal deviation intervals of perturbed Hurwitz polynomial coefficients assuring strict Hurwitz property are determined on the basis of the algebraic method worked out using Kharitonov's polynomials [4]. The similar task is solved in [7] but using the Hermite-Biler theorem, which allows to reduce twice the power of investigated polynomial. The way for calculation of perturbed polynomial coefficients' maximal limit values that guarantee sector stability is given in [8]. The linear dependence of coefficient perturbation is considered by Bartlett, et al. for a class of polynomial families generated by convex polytopes in the coefficient space [9]. Here the so-called edge theorem was proved assuring derivation of the stability analysis task to investigation of root location for the finite number of the parametric families. The edge theorem allows to analyze both stability and quality characteristics of the family. A combination of the stochastic and worst-case approaches to the problem of uncertainty is proposed in [10]. It certainly widens the scope of types of treatable uncertainties and reduces conservatism. However, it works properly only in the cases permitting an arbitrarily small probability of specification violation. Thus, to the specific extent, it still bears the drawbacks of the stochastic approach to control, which guarantees only the "average" performance.

An analog of Kharitonov theorem [11] was formulated for the unstable interval polynomials' homogeneous classes of equivalence. Criteria of existence of such classes of equivalence were obtained. Based on the new interval polynomial stability criterion and Lyapunov theorem, a robust optimal proportional-integral-derivative (PID) controller is proposed in paper [12] to carry out design for different plants that contain perturbations of multiple parameters. A new stability criterion of the interval polynomial is presented to determine whether the interval polynomial belongs to Hurwitz polynomial or not. Time-delay systems involving multiple

#### Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705

imaginary roots (MIRs) and their stability analysis, which becomes much more complicated than that in the case with only simple imaginary roots, are treated in [13]. For a class of time-delay systems, it was proved that the invariance between the multiple imaginary roots and the simple imaginary roots holds for any multiplicity as well as for the degenerate cases. In paper [14], monic complex polynomials are identified with the sets of their roots instead of being identified with the vectors of their coefficients. A proof is given that the space of Hurwitz polynomials of degree n with positive (resp. negative) coefficients is contractible and also that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. A computational method to verify the stability of a convex combination of polynomials is considered in [15] and aimed at the robust stability analysis of a linear system. A simple algebraic test (a matrix inequality) for the stability of the segment of polynomials determined by the given two Hurwitz stable polynomials is proposed. Kučera gives a survey [16] where he navigates the area of the polynomial approach in the control system design technique. Such areas as parameterization of stabilizing controllers, called Youla–Kučera parameterization, are explained; the results on reference tracking, disturbance elimination, pole placement, deadbeat control, robust stabilization, and some others are described.

Of great interest are the problems of ensuring system stability and quality being solved in the modern statements of the problem [2] as tasks of guaranteeing system robustness, which could be solved by application of the root locus approach. The basic benefit of this approach is that its application itself, by its nature, implies parametric variations (i.e., uncertainty). The root locus approach is a powerful method used for the system synthesis [2] and is notable for its descriptiveness ensuring both calculation of the system robust parameters' values and possibility of detailed overview of the dynamic properties variation changes, the system response to uncertainties that is particularly important when investigating systems with uncertain and in particular interval parameters.

Root locus approach to the problem is considered in [17–23]. Paper [17] gives a solution for a compensator synthesis on the basis of the root locus method application. The task of a stable characteristic polynomial synthesis for the interval dynamic system (IDS) by setting up coefficients of the given (initial) unstable one for the case of location of its root locus initial point (where the variable parameter is equal to zero) family within the left half-plane is solved in [21], where the stability is attained via simple setting up the interval of the free term variation.

The above analyzed literature covers various approaches to the uncertainty treatment. However, most of the theoretical works are focused on the tasks of robust stability analysis. The methods for synthesis are not that widely represented, often suffer from complexity and in most cases are enough narrow, which means that they certainly provide instruments for system synthesis, but they are mostly "closed on themselves," which means that they do not provide the complete picture in the sense of showing up what is happening "under cover," which is especially important for the qualitative robust system (polynomials) synthesis. The root locus approach is rarely applied even though it represents the dynamic picture of the system response to uncertainties in the most comprehensive way and thus seems to be the most suitable one to deal with uncertainties.

As for polynomial families, the root locus approach gives us the transparent picture of root dynamics making it possible to see as if from the inside, for example, what subfamilies constitute the whole family of uncertain polynomials in terms of their configuration and stability or some other dynamic indicators bearing significant information about the system behavior and thus leading the way for its investigation and synthesis.

quality, and thus the drive power efficiency. For this reason, of great importance are the tasks of stability investigation and parametric synthesis of robust control systems (their characteristic polynomials) for the plants which parameters vary

nomials, there exists a lot of approaches and methods. For the first time, the necessary and sufficient conditions for systems up to the 3-rd order were formulated by James Maxwell in 1868. Later appeared the stability criteria of Routh– Hurwitz, Mikhajlov, Nyquist, and Bode, which made it possible to check stability of the systems of order n. The frequency Nyquist criterion was the first one that could be used for synthesis by estimation of the system degree of stability. Among the modern methods of synthesis [1, 2] together with the frequency ones, the root locus and state-space methods could be listed. In his book [1], Jurgen Ackermann gives, in particular, the algebraic approach to uncertainty considering different, including the nonlinear, types of the coefficient functions and generating stability regions in the parameter space of real physical parameters of the system (polynomial). The main results in the area of the frequency approach to analysis and synthesis of robust dynamic (control) systems are given in [3], where the stability of uncertain

In the area of investigation and synthesis of dynamic system characteristic poly-

The methods for analysis and synthesis of polynomial families represent a separate group. One of the most effective solutions for the task of interval polynomial

Kharitonov [4], where in the general case, the task of polynomial stability analysis is reduced to consideration of only four specific polynomials of the whole family with constant coefficients. In [3, 5], the frequency criteria of Hurwitz robust stability are considered, which allow to define the coefficient perturbation sweep for the nominally stable polynomial and various types of uncertainties. Hurwitz robust stability is also investigated in [6–10]. In [6], the maximal deviation intervals of perturbed Hurwitz polynomial coefficients assuring strict Hurwitz property are determined on the basis of the algebraic method worked out using Kharitonov's polynomials [4]. The similar task is solved in [7] but using the Hermite-Biler theorem, which allows to reduce twice the power of investigated polynomial. The way for calculation of perturbed polynomial coefficients' maximal limit values that guarantee sector stability is given in [8]. The linear dependence of coefficient perturbation is considered by Bartlett, et al. for a class of polynomial families generated by convex polytopes in the coefficient space [9]. Here the so-called edge theorem was proved assuring derivation of the stability analysis task to investigation of root location for the finite number of the parametric families. The edge theorem allows to analyze both stability and quality characteristics of the family. A combination of the stochastic and worst-case approaches to the problem of uncertainty is proposed in [10]. It certainly widens the scope of types of treatable uncertainties and reduces conservatism. However, it works properly only in the cases permitting an arbitrarily small probability of specification violation. Thus, to the specific extent, it still bears the drawbacks of the stochastic approach to

family investigation within the algebraic approach has been proposed by

within the given or unknown intervals of values.

Polynomials - Theory and Application

polynomials, including interval ones, is also considered.

control, which guarantees only the "average" performance.

110

An analog of Kharitonov theorem [11] was formulated for the unstable interval polynomials' homogeneous classes of equivalence. Criteria of existence of such classes of equivalence were obtained. Based on the new interval polynomial stability criterion and Lyapunov theorem, a robust optimal proportional-integral-derivative (PID) controller is proposed in paper [12] to carry out design for different plants that contain perturbations of multiple parameters. A new stability criterion of the interval polynomial is presented to determine whether the interval polynomial belongs to Hurwitz polynomial or not. Time-delay systems involving multiple

In this work, the root locus methods are described for calculating intervals of uncertainty for coefficients of the given (initial) stable or unstable polynomial with coefficients subject to perturbations, which ensure its robust stability. The proposed methods are based on introduction and application of the notions "extended root locus," "diagram of the root locus parameter function values distribution along the stability bound" and can be used for both synthesis of interval stable polynomials by setting up (adjusting) the unstable ones and analysis of the polynomial behavior under coefficient perturbations. The influence of every coefficient upon the polynomial behavior could be observed.

The work further develops results represented in the papers of Anderson [22] and Kharitonov [4] where they consider the issues of analysis and synthesis of robust interval polynomial families.

#### 2. The problem formulation

Define a polynomial like

$$\mathbf{g}\_n(\mathfrak{s}) = \mathfrak{s}^n + a\_1 \mathfrak{s}^{n-1} + \dots + a\_{n-1} \mathfrak{s} + a\_n \mathbf{n} \tag{1}$$

3. Root locus portraits of uncertain polynomials

DOI: http://dx.doi.org/10.5772/intechopen.83705

(coefficient) aj.

the free root locus.

system root locus).

of root locus fields,

root locus field Fk.

free root locus" [18].

4.1 Extended root locus

En ¼

where

113

s

8

>>>>>>>>>>><

>>>>>>>>>>>:

… s <sup>i</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

… s

s <sup>n</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

> gi ðÞ¼ s s

Introduce the following system of polynomials:

Definition 1. The algebraic equation coefficient or the parameter of the dynamic system, described by this algebraic equation, which is being varied in a definite way for generating the root locus, when it is assumed that all the rest coefficients (parameters) are constant, is called the root locus parameter or free parameter. If the root locus parameter is aj, it is named the root locus relative to parameter

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

Definition 2. The root locus relative to the algebraic equation free term is called

Definition 3. Points, where the root locus branches begin and where the root

Definition 4. The family P of root loci of interval polynomial (1) with coefficients varying within (2) name as the interval polynomial root locus portrait (interval polynomial root locus) or interval dynamic system root locus portrait (interval dynamic

Let us along with the parameter an vary also parameter an-1 of (1). Thus, we generate a (free) root locus field Fk (k = 1. 2, …) in the plane s of system roots, which could also be named a two-parameter root locus field or a (interval) root locus subfamily. Parameter an-1 used for the field generation is named a root locus field parameter.

It is evident that the root locus Eq. (3) represents also the equation of level lines of the free root locus field Fk. Root locus portrait P is then represented by the family

that represents the infinite set of root locus fields and therefore possesses their

Hereinafter the term "root locus" is used in the sense of "Teodorchik – Ewans

4. Polynomial analysis and synthesis based on the extended root locus

s þ a<sup>1</sup> ¼ g1ð Þs ð Þ 6:1

<sup>2</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>g</sup>2ð Þ<sup>s</sup> ð Þ <sup>6</sup>:<sup>2</sup>

<sup>n</sup>�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> an�<sup>2</sup><sup>s</sup> <sup>þ</sup> an�<sup>1</sup> <sup>¼</sup> gn�<sup>1</sup>ð Þ<sup>s</sup> ð Þ <sup>6</sup>:ð Þ <sup>n</sup> � <sup>1</sup>

<sup>n</sup>�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> an�<sup>1</sup><sup>s</sup> <sup>þ</sup> an <sup>¼</sup> gnð Þ<sup>s</sup> ð Þ <sup>6</sup>:<sup>n</sup>

<sup>i</sup>�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> ai�<sup>1</sup><sup>s</sup> <sup>þ</sup> ai <sup>¼</sup> gi

<sup>i</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

properties, and from the mathematical point of view, all root locus fields of P feature the same qualities. Therefore, the portrait P can be investigated as a single

P ¼ f g Fk j k ¼ 1; 2; … (5)

ð Þs ð Þ 6:i

<sup>i</sup>�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> ai1<sup>s</sup> <sup>þ</sup> ai, (7)

;

(6)

locus parameter is equal to zero are called the root locus initial points.

where aj are given (initial) values of real polynomial coefficients, j = 1, 2, …, n. In the event of coefficient perturbations, a vector of coefficients of (1),

a = (a1, …, an-1, an), belongs to some connected set A ⊂ R<sup>n</sup> , a ∈ A; n is a degree of the polynomial (integer value); s is a complex variable, s = σ + iω.

Suppose that coefficients of (1) vary within the following intervals:

$$
\underline{a}\_{j} \le a\_{j} \le \overline{a}\_{j}, \quad j = \overline{1, n}. \tag{2}
$$

where aj and aj are the minimal and maximal limit values of closed interval (2) of coefficients aj variation correspondingly. Polynomial (1) can be both, non-Hurwitz or Hurwitz one.

After substituting s = σ + iω, write the root locus and parameter equations [18] correspondingly:

$$\nu(\sigma, \mathfrak{a}) = \mathbf{0}, \text{and} \tag{3}$$

$$\mathfrak{a}\_{\mathfrak{n}} = \mathfrak{u}(\mathfrak{o}, \mathfrak{o}), \tag{4}$$

where u(σ,ω) and v(σ,ω) are the real functions of two independent variables σ and ω.

The root locus method represents a powerful and effective tool for stable and qualitative polynomial synthesis and analysis. However, as it is known, this method allows to consider polynomials with only a single variable coefficient (parameter) and cannot be applied in the cases when all coefficients are uncertain. Therefore, the task is to generalize the root locus method for the cases when the number of variable coefficients is arbitrary and thus to solve the problem of investigation of the uncertain polynomial dynamics and working out methods for synthesis of the robustly stable uncertain (interval) polynomial by setting up the given polynomial (non-Hurwitz or Hurwitz) with constant/variable coefficients and determining intervals of all its coefficients (stability intervals) assuring its robust stability.

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705

#### 3. Root locus portraits of uncertain polynomials

In this work, the root locus methods are described for calculating intervals of uncertainty for coefficients of the given (initial) stable or unstable polynomial with coefficients subject to perturbations, which ensure its robust stability. The proposed methods are based on introduction and application of the notions "extended root locus," "diagram of the root locus parameter function values distribution along the stability bound" and can be used for both synthesis of interval stable polynomials by setting up (adjusting) the unstable ones and analysis of the polynomial behavior under coefficient perturbations. The influence of every coefficient upon the poly-

The work further develops results represented in the papers of Anderson [22] and Kharitonov [4] where they consider the issues of analysis and synthesis of

where aj are given (initial) values of real polynomial coefficients, j = 1, 2, …, n. In the event of coefficient perturbations, a vector of coefficients of (1),

where aj and aj are the minimal and maximal limit values of closed interval (2) of coefficients aj variation correspondingly. Polynomial (1) can be both,

After substituting s = σ + iω, write the root locus and parameter equations [18]

where u(σ,ω) and v(σ,ω) are the real functions of two independent variables

The root locus method represents a powerful and effective tool for stable and

qualitative polynomial synthesis and analysis. However, as it is known, this method allows to consider polynomials with only a single variable coefficient (parameter) and cannot be applied in the cases when all coefficients are uncertain. Therefore, the task is to generalize the root locus method for the cases when the number of variable coefficients is arbitrary and thus to solve the problem of investigation of the uncertain polynomial dynamics and working out methods for synthesis of the robustly stable uncertain (interval) polynomial by setting up the given polynomial (non-Hurwitz or Hurwitz) with constant/variable coefficients and determining intervals of all its coefficients (stability intervals) assuring its

<sup>n</sup>–<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> an–1<sup>s</sup> <sup>þ</sup> an, (1)

aj ≤ aj ≤ aj, j ¼ 1, n: (2)

vð Þ¼ σ; ω 0, and (3)

an ¼ uð Þ σ; ω , (4)

, a ∈ A; n is a degree of the

nomial behavior could be observed.

Polynomials - Theory and Application

robust interval polynomial families.

2. The problem formulation

Define a polynomial like

non-Hurwitz or Hurwitz one.

correspondingly:

σ and ω.

robust stability.

112

gnðÞ¼ s s

a = (a1, …, an-1, an), belongs to some connected set A ⊂ R<sup>n</sup>

polynomial (integer value); s is a complex variable, s = σ + iω.

<sup>n</sup> <sup>þ</sup> a1s

Suppose that coefficients of (1) vary within the following intervals:

Definition 1. The algebraic equation coefficient or the parameter of the dynamic system, described by this algebraic equation, which is being varied in a definite way for generating the root locus, when it is assumed that all the rest coefficients (parameters) are constant, is called the root locus parameter or free parameter.

If the root locus parameter is aj, it is named the root locus relative to parameter (coefficient) aj.

Definition 2. The root locus relative to the algebraic equation free term is called the free root locus.

Definition 3. Points, where the root locus branches begin and where the root locus parameter is equal to zero are called the root locus initial points.

Definition 4. The family P of root loci of interval polynomial (1) with coefficients varying within (2) name as the interval polynomial root locus portrait (interval polynomial root locus) or interval dynamic system root locus portrait (interval dynamic system root locus).

Let us along with the parameter an vary also parameter an-1 of (1). Thus, we generate a (free) root locus field Fk (k = 1. 2, …) in the plane s of system roots, which could also be named a two-parameter root locus field or a (interval) root locus subfamily. Parameter an-1 used for the field generation is named a root locus field parameter.

It is evident that the root locus Eq. (3) represents also the equation of level lines of the free root locus field Fk. Root locus portrait P is then represented by the family of root locus fields,

$$P = \{ F\_k \mid k = 1, 2, \ldots \} \tag{5}$$

that represents the infinite set of root locus fields and therefore possesses their properties, and from the mathematical point of view, all root locus fields of P feature the same qualities. Therefore, the portrait P can be investigated as a single root locus field Fk.

Hereinafter the term "root locus" is used in the sense of "Teodorchik – Ewans free root locus" [18].

#### 4. Polynomial analysis and synthesis based on the extended root locus

#### 4.1 Extended root locus

Introduce the following system of polynomials:

$$\begin{cases} \mathbf{s} + \mathbf{a}\_1 = \mathbf{g}\_1(\mathbf{s}) \\ \mathbf{s}^2 + \mathbf{a}\_1 \mathbf{s} + \mathbf{a}\_2 = \mathbf{g}\_2(\mathbf{s}) \end{cases} \tag{6.1}$$

$$E\_n = \begin{cases} s^2 + a\_1 s + a\_2 s & \text{(6.2)}\\ s^2 + a\_1 s + a\_2 = \mathbf{g}\_2(s) & \text{(6.2)}\\ \cdots & \text{(6.3)}\\ s^i + a\_1 s^{i-1} + \ldots + a\_{i-1} s + a\_i = \mathbf{g}\_i(s) & \text{(6.3)}\\ \cdots & \text{(6.4)}\\ s^{n-1} + \ldots + a\_{n-2} s + a\_{n-1} = \mathbf{g}\_{n-1}(s) & \text{(6.6)}\\ s^n + a\_1 s^{n-1} + \ldots + a\_{n-1} s + a\_n = \mathbf{g}\_n(s) & \text{(6.n)} \end{cases},\tag{6.7}$$

where

$$\mathbf{g}\_{i}(\mathbf{s}) = \mathbf{s}^{i} + a\_{1}\mathbf{s}^{i-1} + \dots + a\_{i1}\mathbf{s} + a\_{i\bullet} \tag{7}$$

$$\mathfrak{g}\_{i-1}(\mathfrak{s}) = \left(\mathfrak{g}\_i(\mathfrak{s})\mathfrak{-}a\_i\right)/\mathfrak{s};\tag{8}$$

Further in the text, polynomial gi-1(s) free root locus is referred to as the originative one relative to that of gi(s) and gi(s) free root locus—as the originated one

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

Statement 1 is illustrated by Figures 1 and 2. Initial points here are designated by signs "x" (crosses) and letters "p" with the lower indexes, designating the point sequential numbers, and upper indexes, designating the sequential numbers of the corresponding root locus. The root locus sequential number is indicated by a digit

Polynomial (1) root locus portrait (field) at n = 3, 5 ≤ а<sup>2</sup> ≤ 45: (a) originated portrait and (b) originated

<sup>4</sup> + 10 s

<sup>3</sup> + 35 s

<sup>2</sup> + a3s + a4, 100 ≤ а<sup>3</sup> ≤ 5 combined with

relative to that of gi-1(s).

Figure 1.

Figure 2.

115

next to its corresponding branch.

DOI: http://dx.doi.org/10.5772/intechopen.83705

portrait combined with its originative root locus (n = 2).

Free root locus portrait (field) for polynomial g4(s) = s

its originative root locus (n = 3).

i–sequential number of the polynomial in (6), which is equal to its degree, i ¼ 1, n; aj—coefficients, j ¼ 1, i.

Every polynomial (8) of (i�1) degree is generated from the i-degree polynomial supposed that ai = 0. Polynomials of (6) have common coefficients, but not common roots.

Definition 5. System of polynomials (6) name as the extension of polynomial (1) or extended polynomial.

Definition 6. Complete set of extension (6) root loci name as the extended root loci of (1).

Extension En of polynomial gn(s) could be represented by the finite set of polynomials,

$$E\_n = \left\{ \mathbf{g}\_i(\mathfrak{s}) \right\}. \tag{9}$$

Statement 1. In case of variation of any coefficient aj,j ¼ 1, ið Þ � 1 , of polynomial gi(s) (7) within the specific interval, aj ≤ aj ≤ aj, every initial point of its free root locus (excluding the point located at the origin) moves along its unique trajectory, representing itself one of the branches of polynomial gi�1(s) (8) root locus, generated relative to this coefficient, and its current position is determined at a point corresponding to the current value of aj.

Proof. As at initial points of polynomial (6) free root locus the free term aj is equal to zero, it is evident that (8) represents the equation of initial points of the free root locus of (6), that is, when varying aj j ¼ 1, ið Þ � 1 � �, the root locus of (8) relative to aj represents the geometric place of initial points of the root locus of (7). Therefore, every initial point of the free root locus of (7) at fixed aj coincides in the complex plane s with one of the polynomial (8) roots at the given value of aj. It is evident, that while varying aj, this root (and hence, this initial point) moves in the complex plane s, generating one of the (i – 1) branches (trajectories) of the root loci of (8) relative to aj. Thus, the statement has been proved.

Definition 7. Name gi�1(s) (8) as the originative polynomial relative to gi(s) (7) and the root locus of (8)—the originative root locus of polynomial (7) free root loci.

Every (i�1)-th polynomial of (6) is the originative one relative to i-th polynomial (6).

Consequence 1. In case of continuous variation of the polynomial gi(s) coefficient aj, j ¼ 1, nð Þ � 1 , every branch of this polynomial root locus, initiated at the specific initial point, migrates continuously along the corresponding branch of the originative root locus relative to aj-1, being the trajectory of this initial point, correspondingly in direction of increase or decrease of the originative root loci parameter aj.

Consequence 2. If polynomial gi-1(s) being the originative one for the polynomial gi(s) is asymptotically stable, all initial points of polynomial gi(s) free root locus, excluding zero one, are located in the left half-plane s:

$$\forall s\_{\mu}^{i-1} \left[ \text{Re } s\_{\mu}^{i-1} \prec \mathbf{0} \to \mathbf{Re } p\_{\mu}^{i} \prec \mathbf{0} \right],\tag{10}$$

$$s^{i-1}\_{\mu} = p^i\_{\mu}.\tag{11}$$

where s i <sup>μ</sup>—roots of gi(s); p<sup>i</sup> <sup>μ</sup>—initial points of polynomial gi(s) free root locus; μ root (initial point) sequential number, μ ¼ 1, i � 1.

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705

Further in the text, polynomial gi-1(s) free root locus is referred to as the originative one relative to that of gi(s) and gi(s) free root locus—as the originated one relative to that of gi-1(s).

Statement 1 is illustrated by Figures 1 and 2. Initial points here are designated by signs "x" (crosses) and letters "p" with the lower indexes, designating the point sequential numbers, and upper indexes, designating the sequential numbers of the corresponding root locus. The root locus sequential number is indicated by a digit next to its corresponding branch.

#### Figure 1.

gi�1ðÞ¼ <sup>s</sup> gi

i ¼ 1, n; aj—coefficients, j ¼ 1, i.

Polynomials - Theory and Application

point corresponding to the current value of aj.

free root locus of (6), that is, when varying aj j ¼ 1, ið Þ � 1

of (8) relative to aj. Thus, the statement has been proved.

locus, excluding zero one, are located in the left half-plane s:

<sup>i</sup>�<sup>1</sup> <sup>μ</sup> Re s

s <sup>i</sup>�<sup>1</sup> <sup>μ</sup> <sup>¼</sup> pi

∀s

root (initial point) sequential number, μ ¼ 1, i � 1.

<sup>μ</sup>—roots of gi(s); p<sup>i</sup>

mon roots.

loci of (1).

nomials,

mial (6).

parameter aj.

where s i

114

or extended polynomial.

i–sequential number of the polynomial in (6), which is equal to its degree,

Every polynomial (8) of (i�1) degree is generated from the i-degree polynomial supposed that ai = 0. Polynomials of (6) have common coefficients, but not com-

Definition 5. System of polynomials (6) name as the extension of polynomial (1)

Definition 6. Complete set of extension (6) root loci name as the extended root

Extension En of polynomial gn(s) could be represented by the finite set of poly-

Statement 1. In case of variation of any coefficient aj,j ¼ 1, ið Þ � 1 , of polynomial gi(s) (7) within the specific interval, aj ≤ aj ≤ aj, every initial point of its free root locus (excluding the point located at the origin) moves along its unique trajectory, representing itself one of the branches of polynomial gi�1(s) (8) root locus, generated relative to this coefficient, and its current position is determined at a

Proof. As at initial points of polynomial (6) free root locus the free term aj is equal to zero, it is evident that (8) represents the equation of initial points of the

relative to aj represents the geometric place of initial points of the root locus of (7). Therefore, every initial point of the free root locus of (7) at fixed aj coincides in the complex plane s with one of the polynomial (8) roots at the given value of aj. It is evident, that while varying aj, this root (and hence, this initial point) moves in the complex plane s, generating one of the (i – 1) branches (trajectories) of the root loci

Definition 7. Name gi�1(s) (8) as the originative polynomial relative to gi(s) (7) and the root locus of (8)—the originative root locus of polynomial (7) free root loci. Every (i�1)-th polynomial of (6) is the originative one relative to i-th polyno-

Consequence 2. If polynomial gi-1(s) being the originative one for the polynomial gi(s) is asymptotically stable, all initial points of polynomial gi(s) free root

<sup>i</sup>�<sup>1</sup> <sup>μ</sup> < 0 ! Re pi

h i

<sup>μ</sup> < 0

<sup>μ</sup>—initial points of polynomial gi(s) free root locus; μ—

Consequence 1. In case of continuous variation of the polynomial gi(s) coefficient aj, j ¼ 1, nð Þ � 1 , every branch of this polynomial root locus, initiated at the specific initial point, migrates continuously along the corresponding branch of the originative root locus relative to aj-1, being the trajectory of this initial point, correspondingly in direction of increase or decrease of the originative root loci

En ¼ gi

ð Þs –ai

� �=s; (8)

ð Þ<sup>s</sup> � �: (9)

, the root locus of (8)

, (10)

<sup>μ</sup>, (11)

� �

Polynomial (1) root locus portrait (field) at n = 3, 5 ≤ а<sup>2</sup> ≤ 45: (a) originated portrait and (b) originated portrait combined with its originative root locus (n = 2).

#### Figure 2.

Free root locus portrait (field) for polynomial g4(s) = s <sup>4</sup> + 10 s <sup>3</sup> + 35 s <sup>2</sup> + a3s + a4, 100 ≤ а<sup>3</sup> ≤ 5 combined with its originative root locus (n = 3).

#### 4.2 Synthesis of stable interval polynomials based on the extended root locus

Consider Eq. (3) in the sense of four following possible cases: n is uneven, (n – 1)/2 is even/uneven, n is even, and n/2 is even/uneven. The root locus parameter equations (as it is in the general form see (4)) are composed in the same way.

Specify the set A<sup>þ</sup> <sup>i</sup> of ai values at the cross points of polynomial (7) root locus positive branches with axis ω:

$$A\_i^+ = \left\{ a\_i^+ l, l = \overline{1, n\_i^+} \right\},\tag{12}$$

An algorithm for the robustly stable regular or interval polynomial synthesis is

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

Step 1. Composing the extension En (6) of the given initial nominal polynomial

Step 2. Sequential check for stability of the extension polynomials, beginning with the polynomial of degree n, until finding the stable polynomial of degree i = k. In case of synthesis of the whole interval polynomial, begin the procedure with

il of cross points of the polynomial gi(s) free

<sup>i</sup> (16)

i :

il (13) values at

<sup>i</sup> , (17)

<sup>i</sup> (12) for polynomial gi(s) by calculating

<sup>i</sup> max solving twice Eq. (4) for polynomial

ai ¼ uð Þ ω , (18)

<sup>i</sup> <sup>¼</sup> min <sup>a</sup><sup>0</sup> ð Þ¼ <sup>a</sup>″ ai, (19)

<sup>i</sup> max of coefficient ai

the 1-st degree polynomial, i = k = 1, specifying interval of a<sup>1</sup> according to the

Step 3. Transfer to the polynomial of the next higher degree, i = k + 1.

ωþ il ∈W<sup>þ</sup>

whole interval family, it is required to calculate the parameter ai ¼ a<sup>þ</sup>

<sup>i</sup>min ∈infW<sup>þ</sup>

root locus equation and thus two different equations should be solved.

<sup>i</sup> min and <sup>a</sup>

infA<sup>þ</sup>

Synthesis of the interval polynomial of the 3-rd degree.

<sup>i</sup> min and ω<sup>þ</sup>

ωþ

Properties of this domain and behavior of the interval root locus portrait at the stability bound iω have been investigated in [18]. On the basis of the fact, that every function of (3) represents continuous differentiable function (steadily increasing/ decreasing function), it has been found in [18] that for ensuring stability of the

<sup>i</sup> , ω<sup>þ</sup>

by solving the corresponding Eq. (3) after substituting preliminarily into this equation, the appropriate combination [18] of the limit values of each coefficient, from a<sup>1</sup> to ai�1, which have been calculated already in this algorithm when generating the originative polynomial gi�1(s). For finding two coordinates (17), two different combinations of coefficients should be substituted into the

> 00 ¼ a<sup>þ</sup>

after substituting previously into (18) the corresponding combinations of

where ai is the upper limit of a<sup>i</sup> variation interval. The required interval (13) is:

Step 6. If the last polynomial of extension (6), that is, that of degree n, has been already processed (i = n), the calculation is considered finished. Otherwise

<sup>i</sup>min ∈ supW<sup>þ</sup>

<sup>i</sup> min ω<sup>þ</sup>

root locus positive branches with the axis iω by solving its appropriate root locus

il generate on the axis iω a so-called "crossing domain" W<sup>þ</sup>

given below.

appropriate requirements or arbitrarily.

DOI: http://dx.doi.org/10.5772/intechopen.83705

Step 4. Calculating coordinates ω<sup>þ</sup>

only two extreme "dominating points":

Step 5. Determining the value of infA<sup>þ</sup>

0 ¼ a 00 <sup>i</sup> min ω<sup>þ</sup>

coefficients (from a<sup>1</sup> to ai�1) [18]. Thus,

correspondingly at points ω<sup>þ</sup>

gi(s) at the stability bound:

gn(s) (1).

Eq. (3).

Cross points ω<sup>þ</sup>

minimal values a

0 < ai < ai:

proceed to step 3.

4.3 Example

117

where n<sup>þ</sup> <sup>i</sup> is a number of cross points.

Statement 2. If all initial points of polynomial (7) root locus, excluding a single one at the origin, are located in the left half-plane s, and this polynomial is asymptotically stabile, when the following condition holds:

$$\mathbf{0} \prec a\_i \prec \inf \mathbf{A}\_i^+. \tag{13}$$

Proof. Based on the root locus properties [2, 18] and expressions (10) and (11), it can be stated, that provided all initial points of polynomial (7) root locus are located in the left half-plane s (excluding the initial point at the origin), the specific number ni of root locus branches (ni = i � 2 when i is even and ni = i � 1 when i is uneven), initiating at these points, cross the stability bound iω striving along the asymptotes directed to the right half-plane. As the rest of the root locus branches does not cross the stability bound, they are completely stable. For positive branches, crossing the stability bound, specify the set

$$\mathbb{S}\_{i} = \{\mathbb{S}\_{il}\} = \{\left(\mathbf{0}, a\_{i}^{+}l\right)\}\tag{14}$$

of intervals Sil of values ai within the segments from the initial point pil (where ai = 0) of every branch up to its cross point with axis iω. Thus, the maximal possible interval of ai values, ensuring stability of (6), is equal to

$$\mathbf{S}\_{\text{imax}} = \underset{\mathbf{S}\_{\text{il}} \subset \mathbf{S}\_{i}}{\text{\small{\kern{0.2em}{ $\mathcal{S}\_{i}$ }}}} \mathbf{S}\_{i\text{l}} = \underset{\mathbf{i}}{\text{inf}} \mathbf{S}\_{i} = \left(\mathbf{0}, \underset{\mathbf{i}}{\text{inf}} \mathbf{A}\_{i}^{+}\right), \tag{15}$$

that proofs the statement being considered.

For the 4-th degree polynomial represented in Figure 2, the interval Simax ¼ ð Þ 0; a4ð Þt .

Theorem 1. For ensuring asymptotic stability of regular or interval polynomial (1), it is enough to.


Proof. If polynomial gi(s) = gk(s) is stable, then on the basis of Consequence 2 of Statement 1 (expressions (10) and (11)), the stability of gi+1(s) can be ensured by simple application of condition (13). Thus, stability of all polynomials gi(s) is sequentially ensured beginning with the polynomial of degree i = k + 1 up to the polynomial of degree i = n inclusive, that is, for i ¼ ð Þ k þ 1 , n. Thus, Theorem 1 has been proved.

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705

An algorithm for the robustly stable regular or interval polynomial synthesis is given below.

Step 1. Composing the extension En (6) of the given initial nominal polynomial gn(s) (1).

Step 2. Sequential check for stability of the extension polynomials, beginning with the polynomial of degree n, until finding the stable polynomial of degree i = k.

In case of synthesis of the whole interval polynomial, begin the procedure with the 1-st degree polynomial, i = k = 1, specifying interval of a<sup>1</sup> according to the appropriate requirements or arbitrarily.

Step 3. Transfer to the polynomial of the next higher degree, i = k + 1.

Step 4. Calculating coordinates ω<sup>þ</sup> il of cross points of the polynomial gi(s) free root locus positive branches with the axis iω by solving its appropriate root locus Eq. (3).

Cross points ω<sup>þ</sup> il generate on the axis iω a so-called "crossing domain" W<sup>þ</sup> i :

$$
\boldsymbol{\alpha}\_{i\!\!\! }^+ \in \mathbf{W}\_i^+ \tag{16}
$$

Properties of this domain and behavior of the interval root locus portrait at the stability bound iω have been investigated in [18]. On the basis of the fact, that every function of (3) represents continuous differentiable function (steadily increasing/ decreasing function), it has been found in [18] that for ensuring stability of the whole interval family, it is required to calculate the parameter ai ¼ a<sup>þ</sup> il (13) values at only two extreme "dominating points":

$$o\_{\text{imin}}^{+} \in \text{inf}\,\mathcal{W}\_{i}^{+}\,, o\_{\text{imin}}^{+} \in \text{sup}\,\mathcal{W}\_{i}^{+}\,,\tag{17}$$

by solving the corresponding Eq. (3) after substituting preliminarily into this equation, the appropriate combination [18] of the limit values of each coefficient, from a<sup>1</sup> to ai�1, which have been calculated already in this algorithm when generating the originative polynomial gi�1(s). For finding two coordinates (17), two different combinations of coefficients should be substituted into the root locus equation and thus two different equations should be solved.

Step 5. Determining the value of infA<sup>þ</sup> <sup>i</sup> (12) for polynomial gi(s) by calculating minimal values a 0 ¼ a 00 <sup>i</sup> min ω<sup>þ</sup> <sup>i</sup> min and <sup>a</sup> 00 ¼ a<sup>þ</sup> <sup>i</sup> min ω<sup>þ</sup> <sup>i</sup> max of coefficient ai correspondingly at points ω<sup>þ</sup> <sup>i</sup> min and ω<sup>þ</sup> <sup>i</sup> max solving twice Eq. (4) for polynomial gi(s) at the stability bound:

$$a\_i = u(\alpha),\tag{18}$$

after substituting previously into (18) the corresponding combinations of coefficients (from a<sup>1</sup> to ai�1) [18]. Thus,

$$\inf A\_i^+ = \min(a'a'') = \overline{a}\_i,\tag{19}$$

where ai is the upper limit of a<sup>i</sup> variation interval. The required interval (13) is: 0 < ai < ai:

Step 6. If the last polynomial of extension (6), that is, that of degree n, has been already processed (i = n), the calculation is considered finished. Otherwise proceed to step 3.

#### 4.3 Example

Synthesis of the interval polynomial of the 3-rd degree.

4.2 Synthesis of stable interval polynomials based on the extended root locus

Consider Eq. (3) in the sense of four following possible cases: n is uneven, (n – 1)/2 is even/uneven, n is even, and n/2 is even/uneven. The root locus parameter equations (as it is in the general form see (4)) are composed in the same way.

> Aþ <sup>i</sup> ¼ a<sup>þ</sup>

<sup>i</sup> is a number of cross points.

totically stabile, when the following condition holds:

crossing the stability bound, specify the set

interval of ai values, ensuring stability of (6), is equal to

Simax ¼ ∩

that proofs the statement being considered.

Sil ⊂Si

For the 4-th degree polynomial represented in Figure 2, the interval

polynomial of extension (5) as per condition (13) assuming i = j.

<sup>i</sup> of ai values at the cross points of polynomial (7) root locus

i

, (12)

<sup>i</sup> : (13)

<sup>i</sup> <sup>l</sup> � � � � (14)

� �, (15)

i

<sup>i</sup> l; l ¼ 1, n<sup>þ</sup>

n o

Statement 2. If all initial points of polynomial (7) root locus, excluding a single one at the origin, are located in the left half-plane s, and this polynomial is asymp-

0 < ai < infA<sup>þ</sup>

it can be stated, that provided all initial points of polynomial (7) root locus are located in the left half-plane s (excluding the initial point at the origin), the specific number ni of root locus branches (ni = i � 2 when i is even and ni = i � 1 when i is uneven), initiating at these points, cross the stability bound iω striving along the asymptotes directed to the right half-plane. As the rest of the root locus branches does not cross the stability bound, they are completely stable. For positive branches,

Si ¼ f g Sil ¼ 0; a<sup>þ</sup>

of intervals Sil of values ai within the segments from the initial point pil (where ai = 0) of every branch up to its cross point with axis iω. Thus, the maximal possible

Theorem 1. For ensuring asymptotic stability of regular or interval polynomial

a. find among polynomials of extension (6), the stable polynomial of degree i = k

b.set up sequentially every coefficient aj of (1), beginning with aj = ak + 1, within interval (k + 1) < j ≤ n by setting up the free term ai of the corresponding i-th

Proof. If polynomial gi(s) = gk(s) is stable, then on the basis of Consequence 2 of Statement 1 (expressions (10) and (11)), the stability of gi+1(s) can be ensured by simple application of condition (13). Thus, stability of all polynomials gi(s) is sequentially ensured beginning with the polynomial of degree i = k + 1 up to the polynomial of degree i = n inclusive, that is, for i ¼ ð Þ k þ 1 , n. Thus, Theorem 1 has

Sil ¼ infSi ¼ 0; infA<sup>þ</sup>

Proof. Based on the root locus properties [2, 18] and expressions (10) and (11),

Specify the set A<sup>þ</sup>

where n<sup>þ</sup>

Simax ¼ ð Þ 0; a4ð Þt .

(1), it is enough to.

been proved.

116

being the closest one to n;

positive branches with axis ω:

Polynomials - Theory and Application

Consider polynomial family

$$\mathbf{g}^{0}\_{~3}\ (\mathbf{s}) = \mathbf{s}^{3} + a\_{1}\mathbf{s}^{2} + a\_{2}\mathbf{s} + a\_{3}\tag{20}$$

g4ðÞ¼ s s

DOI: http://dx.doi.org/10.5772/intechopen.83705

Substitute s ¼ σ þ iω ¼ iω (σ ¼ 0) into (26) and rewrite:

<sup>ω</sup><sup>4</sup> � <sup>a</sup>1ω<sup>3</sup>

5.1 Crossing region of the polynomial root locus portrait

<sup>ω</sup> of the root locus portrait P.

5.2 Majorant and minorant of the extremum region

Obtain the extremum parameter function values within D<sup>F</sup>

Take the first-order derivative of (32) and set it to zero:

boundary:

boundary:

region DP

The region D<sup>P</sup>

fied region D<sup>P</sup>

considered.

Eq. (30):

119

spondingly subregion D<sup>F</sup>

ω.

(5), given the condition.

<sup>4</sup> <sup>þ</sup> a1s

<sup>3</sup> <sup>þ</sup> a2s

aj ≤ aj ≤ aj, j ¼ 1, …, 4, a<sup>0</sup> ¼ 1: (27)

<sup>i</sup> � <sup>a</sup>2ω<sup>2</sup> <sup>þ</sup> <sup>a</sup>3<sup>ω</sup> <sup>i</sup> <sup>þ</sup> <sup>a</sup><sup>4</sup> <sup>¼</sup> <sup>0</sup> (28)

�a1ω<sup>3</sup> <sup>þ</sup> <sup>a</sup>3<sup>ω</sup> <sup>¼</sup> <sup>0</sup> (29)

<sup>f</sup>ð Þ¼� <sup>ω</sup> <sup>ω</sup><sup>4</sup> <sup>þ</sup> <sup>a</sup>2ω<sup>2</sup> <sup>¼</sup> <sup>a</sup>4: (30)

0 < а<sup>j</sup> < þ ∞, (31)

ω.

<sup>ω</sup>. To do so, it is

ωi, within the above speci-

<sup>ω</sup>k⊂ D<sup>P</sup>

: (32)

�4ω<sup>3</sup> <sup>þ</sup> <sup>2</sup>a2<sup>ω</sup> <sup>¼</sup> <sup>0</sup>: (33)

Coefficients of Eq. (26) to be real, positive, and variable within the intervals

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

and on the base of (28), write the root locus equation [18, 20] at the stability

and the parameter equation (parameter function) [18, 20] at the stability

Functions (29) and (30) imply properties of analyticity and continuity and, thus, the points where axis iω is crossed by the branches of the root locus family P

constitute on the stability boundary, axis iω, a specific crossing region, DP

branch bki, i ¼ 1, 2, … of the field root loci generate specific subregions, corre-

<sup>ω</sup>k and continuous subregion D<sup>b</sup>

Over the symmetry of the portrait hereinafter, the only upper half-plane s is

necessary to carry out investigation of this function for extremum. It is evident that the majorant parameter function (majorant) can be obtained through rewriting

<sup>a</sup>4max ¼ �ω<sup>4</sup> <sup>þ</sup> <sup>a</sup>2ω<sup>2</sup>

Definition 8. The region at the asymptotic stability boundary iω of the interval system root locus portrait P, described by characteristic polynomial (26), where the given portrait parameter function (30) values family is located, name the crossing

<sup>ω</sup> is a continuous one and, thus, each root locus field Fk (5) and each

<sup>2</sup> <sup>þ</sup> <sup>a</sup>3<sup>s</sup> <sup>þ</sup> <sup>a</sup>4: (26)

where а<sup>j</sup> ∈ [aj , aj], j ∈ {1,2,3}; а<sup>1</sup> ∈ [10, 15], а<sup>2</sup> ∈ [25, 35], а<sup>3</sup> ∈ [350, 450]. Step 1. Compose the extended polynomial (6) for (20):

$$\begin{cases} s + a\_1 = 0 & \quad (21.1) \\ s^2 + a\_1 s + a\_2 = 0 & \quad (21.2) \\ s^3 + a\_1 s^2 + a\_2 s + a\_3 = 0 & \quad (21.3) \end{cases} \quad \text{(21.4)}$$

Step 2. As coefficients of polynomials (21.1) and (21.2) are positive, then both families of these polynomials are asymptotically stable (i = k = 2), and therefore, on the basis of Consequence 2 of Statement 1, the root loci family of (21.3) initial points is located in the left half-pane. Thus, for making stable, the polynomial (21.3) uses Statement 2 and Theorem 1.

Step 3. Transfer to the polynomial of the next higher degree, i = 2 + 1 = 3.

Step 4. Calculating coordinates (16) of the "dominating points" for polynomial g3(s). For this purpose, consider the appropriate root locus (3) and parameter (18) equations:

$$
\alpha^3 \text{--} a\_2 \alpha = 0,\tag{22}
$$

and parameter function (18) at the stability bound:

$$
\mathfrak{a}\_1 \mathfrak{o}^2 = \mathfrak{a}\_3 = f\_p(\mathfrak{o}).\tag{23}
$$

Find the 1-st order derivative of (23) and equate it zero:

$$f\_{\;\;p}^{\;\;\;/}(\bullet) = 2a\_1 \bullet = \mathbf{0}.\tag{24}$$

On the basis of (23) and (24), it can be stated that the character of parameter (23) distribution along the axis σ is steadily increasing and the single extreme point is located at the origin. Thus, there exists the only one extreme point:

$$
\rho\_3^+ \text{min} = \pm \sqrt{\underline{a}\_2} = \pm 5,\tag{25}
$$

where function (23) gets the minimal value of the set A<sup>þ</sup> <sup>3</sup> (see Eq. (12)).

Step 5. Determine infA<sup>þ</sup> <sup>3</sup> (12) for g3(s) using (23), (25):

$$\begin{array}{l} \mathsf{inf} \mathsf{A}\_{3}^{+} = \mathsf{a}\_{3}^{+} \mathsf{min} \left( \mathsf{a}\_{3}^{+} \mathsf{min} \right) = \underline{\mathsf{a}}\_{1} \cdot \left( \mathsf{a}\_{3}^{+} \mathsf{min} \right)^{2} = \mathsf{10} \cdot \mathsf{5}^{2} = \mathsf{250}. \text{ Thus,} \mathsf{0} \cdot \mathsf{a}\_{3} \cdot \mathsf{250}.\\ \mathsf{Step 6. As } i = \mathsf{3} = n \text{, the algorithm is considered finished.} \end{array}$$

Thus, coefficient intervals for the resulting robustly stable polynomial ^g3ð Þs are as follows:

а<sup>1</sup> ∈ [10, 15], а<sup>2</sup> ∈ [25, 35], а<sup>3</sup> ∈ (0, 250).

#### 5. Investigation of behavior at the stability bound and synthesis of interval polynomial families: root locus parameter function distribution diagram

Consider a dynamic system described by the family of interval characteristic polynomials [4, 18, 20, 22] like.

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705

$$\mathbf{g}\_4(\mathbf{s}) = \mathbf{s}^4 + \mathbf{a}\_1 \mathbf{s}^3 + \mathbf{a}\_2 \mathbf{s}^2 + \mathbf{a}\_3 \mathbf{s} + \mathbf{a}\_4. \tag{26}$$

Coefficients of Eq. (26) to be real, positive, and variable within the intervals

$$a\_{\underline{i}} \le a\_{\underline{j}} \le \overline{a}\_{\underline{i}}, \quad j = 1, \dots, 4, \quad a\_0 = 1. \tag{27}$$

Substitute s ¼ σ þ iω ¼ iω (σ ¼ 0) into (26) and rewrite:

$$
\alpha^4 - \mathfrak{a}\_1 \alpha^3 \mathbf{i} - \mathfrak{a}\_2 \alpha^2 + \mathfrak{a}\_3 \alpha \mathbf{i} + \mathfrak{a}\_4 = \mathbf{0} \tag{28}
$$

and on the base of (28), write the root locus equation [18, 20] at the stability boundary:

$$-\mathfrak{a}\_1 \mathfrak{a}^3 + \mathfrak{a}\_3 \mathfrak{a} = \mathbf{0} \tag{29}$$

and the parameter equation (parameter function) [18, 20] at the stability boundary:

$$f(a) = -a^4 + a\_2a^2 = a\_4. \tag{30}$$

#### 5.1 Crossing region of the polynomial root locus portrait

Functions (29) and (30) imply properties of analyticity and continuity and, thus, the points where axis iω is crossed by the branches of the root locus family P (5), given the condition.

$$0 \prec a\_j \prec +\infty,\tag{31}$$

constitute on the stability boundary, axis iω, a specific crossing region, DP ω.

Definition 8. The region at the asymptotic stability boundary iω of the interval system root locus portrait P, described by characteristic polynomial (26), where the given portrait parameter function (30) values family is located, name the crossing region DP <sup>ω</sup> of the root locus portrait P.

The region D<sup>P</sup> <sup>ω</sup> is a continuous one and, thus, each root locus field Fk (5) and each branch bki, i ¼ 1, 2, … of the field root loci generate specific subregions, correspondingly subregion D<sup>F</sup> <sup>ω</sup>k and continuous subregion D<sup>b</sup> ωi, within the above specified region D<sup>P</sup> ω.

Over the symmetry of the portrait hereinafter, the only upper half-plane s is considered.

#### 5.2 Majorant and minorant of the extremum region

Obtain the extremum parameter function values within D<sup>F</sup> <sup>ω</sup>k⊂ D<sup>P</sup> <sup>ω</sup>. To do so, it is necessary to carry out investigation of this function for extremum. It is evident that the majorant parameter function (majorant) can be obtained through rewriting Eq. (30):

$$
\mathfrak{a}\_{4\text{max}} = -\boldsymbol{\alpha}^4 + \overline{\mathfrak{a}}\_2 \boldsymbol{\alpha}^2. \tag{32}
$$

Take the first-order derivative of (32) and set it to zero:

$$-4o^3 + 2\overline{a}\_2 o = 0.\tag{33}$$

Consider polynomial family

Polynomials - Theory and Application

where а<sup>j</sup> ∈ [aj

Statement 2 and Theorem 1.

Step 5. Determine infA<sup>þ</sup>

distribution diagram

polynomials [4, 18, 20, 22] like.

<sup>3</sup> min ω<sup>þ</sup>

<sup>3</sup> ¼ a<sup>þ</sup>

infA<sup>þ</sup>

as follows:

118

equations:

g0

s

s <sup>3</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

8 ><

>:

<sup>3</sup> ðÞ¼ s s

Step 1. Compose the extended polynomial (6) for (20):

<sup>3</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

s þ a<sup>1</sup> ¼ 0 ð Þ 21:1

<sup>2</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>¼</sup> 0 21 ð Þ :<sup>2</sup>

Step 2. As coefficients of polynomials (21.1) and (21.2) are positive, then both families of these polynomials are asymptotically stable (i = k = 2), and therefore, on the basis of Consequence 2 of Statement 1, the root loci family of (21.3) initial points is located in the left half-pane. Thus, for making stable, the polynomial (21.3) uses

Step 3. Transfer to the polynomial of the next higher degree, i = 2 + 1 = 3. Step 4. Calculating coordinates (16) of the "dominating points" for polynomial g3(s). For this purpose, consider the appropriate root locus (3) and parameter (18)

ω3

and parameter function (18) at the stability bound:

Find the 1-st order derivative of (23) and equate it zero:

f p 0

is located at the origin. Thus, there exists the only one extreme point:

ω<sup>þ</sup>

where function (23) gets the minimal value of the set A<sup>þ</sup>

Step 6. As i =3= n, the algorithm is considered finished.

<sup>3</sup> min � � <sup>¼</sup> <sup>a</sup><sup>1</sup> � <sup>ω</sup><sup>þ</sup>

а<sup>1</sup> ∈ [10, 15], а<sup>2</sup> ∈ [25, 35], а<sup>3</sup> ∈ (0, 250).

On the basis of (23) and (24), it can be stated that the character of parameter (23) distribution along the axis σ is steadily increasing and the single extreme point

a2

<sup>3</sup> (12) for g3(s) using (23), (25):

Thus, coefficient intervals for the resulting robustly stable polynomial ^g3ð Þs are

Consider a dynamic system described by the family of interval characteristic

<sup>3</sup> min ¼ � ffiffiffiffiffi

5. Investigation of behavior at the stability bound and synthesis of interval polynomial families: root locus parameter function

<sup>2</sup> <sup>þ</sup> <sup>a</sup>2<sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>3</sup> <sup>¼</sup> 0 21 ð Þ :<sup>3</sup>

, aj], j ∈ {1,2,3}; а<sup>1</sup> ∈ [10, 15], а<sup>2</sup> ∈ [25, 35], а<sup>3</sup> ∈ [350, 450].

<sup>2</sup> <sup>þ</sup> <sup>a</sup>2<sup>s</sup> <sup>þ</sup> <sup>a</sup>3, (20)

:

–a2ω ¼ 0, (22)

<sup>a</sup>1ω<sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>3</sup> <sup>¼</sup> <sup>f</sup> <sup>p</sup>ð Þ <sup>ω</sup> : (23)

ð Þ¼ ω 2a1ω ¼ 0: (24)

<sup>3</sup> min � �<sup>2</sup> <sup>¼</sup> <sup>10</sup> � 52 <sup>¼</sup> <sup>250</sup>: Thus,0 < <sup>a</sup><sup>3</sup> < 250:

<sup>p</sup> ¼ �5, (25)

<sup>3</sup> (see Eq. (12)).

(21)

After solving Eq. (33), obtain three points of extremum for the majorant parameter function for the field when a<sup>2</sup> ¼ a2:

$$\boldsymbol{a}\_{\boldsymbol{\epsilon}\_{\text{max}}} = \mathbf{0}, \qquad \boldsymbol{a}\_{4\boldsymbol{\epsilon}\_{\text{max}}} = \mathbf{0}; \; \boldsymbol{a}\_{\boldsymbol{\epsilon}\_{\text{max}}} = \pm \sqrt{\frac{\overline{a}\_2}{2}}, \quad \boldsymbol{a}\_{4\boldsymbol{\epsilon}\_{\text{max}}} = -\boldsymbol{a}\_{\boldsymbol{\epsilon}\_{\text{max}}}^4 + \overline{a}\_2 \cdot \boldsymbol{a}\_{\boldsymbol{\epsilon}\_{\text{max}}}^2. \tag{34}$$

Rewrite (30) for determination of a minorant parameter function (or a minorant):

$$
\mathfrak{a}\_{4\text{min}} = -\mathfrak{a}^4 + \underline{\mathfrak{a}}\_2 \mathfrak{a}^2. \tag{35}
$$

• D<sup>ω</sup>

(D<sup>ω</sup>

D<sup>ω</sup> �, z 0 ; z � �<sup>00</sup>

121

<sup>þ</sup> ⊂ 0; z � �<sup>0</sup>

, 0; z � �<sup>0</sup>

interval [0, z'] the system stability region.

val [z', z"] the system instability region.

5.4 Real crossing region of the portrait

half-plane, name the real crossing region DR

∩ D<sup>ω</sup> c , z 0 ; z � �<sup>00</sup>

complete instability region.

Specify the region D<sup>R</sup>

values of its roots. When ω>0.

<sup>+</sup> where the parameter function is getting increased (increase region);

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

0 ; z � �<sup>00</sup>

), only the positive branches cross the stability bound-

Z ⊂ L: (37)

�. In this case, axis iω is crossed by combination of

�,

�, (38)

, (40)

Z ⊂R: (39)

<sup>ω</sup> where the branches of the given real root locus portrait

ffiffiffiffiffiffiffi a3 a1 r

<sup>ω</sup> of the system root locus portrait:

<sup>ω</sup>: (41)

⊆ Z<sup>ω</sup> where the initial points of

<sup>+</sup> and partly region D<sup>ω</sup>

<sup>с</sup> and some of the region

c

• Dω� where the parameter function is getting decreased (decrease region);

the root locus portrait migrate through the stability boundary to the right half-

ary, and here the whole family Z of initial points is located in the left half-plane L,

But specific pieces of the positive branches are situated within the right half-plane. For this reason, in some cases, the unstable polynomials could have been found within the whole family (26). However, there certainly could always be found the intervals (27) of stability where the whole family is stable. Name the

both positive and negative branches, and the root locus portrait certainly includes a series of initial points, and thus the whole branches, that have migrated over the boundary to the right half-plane. Therefore, this case always gives us the family (26) that includes combination of stable and unstable polynomials. Name the inter-

½ � <sup>z</sup>″<sup>∞</sup> <sup>⊂</sup> <sup>D</sup><sup>ω</sup>

only the negative branches cross the stability boundary iω, and the family Z together with the corresponding positive branches are located in the right half-pane,

No stable polynomial could be found in (26). This region name the system

cross the stability boundary. To find its limits, consider Eq. (29) and determine the

,ωmin ¼

ffiffiffiffiffiffiffi a3 a1

Definition 10. The region [ωmin,ωmax] at the stability boundary iω, where the polynomial (26) root locus portrait branches migrate through to the right

½ � <sup>ω</sup>min;ωmax <sup>⊆</sup> <sup>D</sup><sup>R</sup>

s

• Dω<sup>с</sup> where increase and decrease regions combine (mixed region).

plane. In the diagram, zero points z', z" are mapped by points z1, z2. Within interval [0, z'], covering completely region D<sup>ω</sup>

The interval [z', z"] covers some piece of the region D<sup>ω</sup>

If the interval [z", ∞] completely belongs to the region D<sup>ω</sup>

ωmax ¼

where ωmax, ωmin represent the real crossing region.

∩ D<sup>ω</sup>

Analyze the region Z<sup>ω</sup> with the interval z

DOI: http://dx.doi.org/10.5772/intechopen.83705

∩ D<sup>ω</sup> c

In the same way obtain three points of extremum for the minorant, when a<sup>2</sup> ¼ a2:

$$a\_{\ell\_{\min}} = 0, \qquad a\_{4\epsilon\_{\min}} = 0; \quad a\_{\ell\_{\min}} = \pm \sqrt{\frac{\underline{a}\_2}{2}}, \quad a\_{4\epsilon\_{\min}} = -a\_{\epsilon\_{\min}}^4 + \underline{a}\_2 \cdot a\_{\epsilon\_{\min}}^2. \tag{36}$$

Evidently, for n = 4, Eqs. (32) and (35) are the majorant and the minorant for the whole portrait.

Definition 9. Extremum region D<sup>e</sup> <sup>ω</sup> of the interval system root locus portrait described by the characteristic polynomial (26) is a region [0, ωemax] at the system asymptotic stability boundary iω where the given portrait parameter function (30) extremum values, a<sup>4</sup>emax (34) and a<sup>4</sup>emin (36), family is located provided all coefficients а<sup>j</sup> vary within limits (31).

#### 5.3 Diagram of the parameter function distribution along the stability boundary

Figure 3 represents the character (diagram) of the parameter function (30) distribution along the boundary of stability by its majorant (32) and minorant (35). For better understanding and descriptiveness, the diagram in Figure 3 is shown by strait lines, although it constitutes curves. Region D<sup>P</sup> <sup>ω</sup> constitutes three subregions (see Figure 3):

#### Figure 3.

A diagram for distribution of the interval system root locus portrait parameter function along the asymptotic stability boundary.

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705


Analyze the region Z<sup>ω</sup> with the interval z 0 ; z � �<sup>00</sup> ⊆ Z<sup>ω</sup> where the initial points of the root locus portrait migrate through the stability boundary to the right halfplane. In the diagram, zero points z', z" are mapped by points z1, z2.

Within interval [0, z'], covering completely region D<sup>ω</sup> <sup>+</sup> and partly region D<sup>ω</sup> c (D<sup>ω</sup> <sup>þ</sup> ⊂ 0; z � �<sup>0</sup> , 0; z � �<sup>0</sup> ∩ D<sup>ω</sup> c ), only the positive branches cross the stability boundary, and here the whole family Z of initial points is located in the left half-plane L,

$$\mathbf{Z} \subset \mathbf{L}.\tag{37}$$

But specific pieces of the positive branches are situated within the right half-plane. For this reason, in some cases, the unstable polynomials could have been found within the whole family (26). However, there certainly could always be found the intervals (27) of stability where the whole family is stable. Name the interval [0, z'] the system stability region.

The interval [z', z"] covers some piece of the region D<sup>ω</sup> <sup>с</sup> and some of the region D<sup>ω</sup> �, z 0 ; z � �<sup>00</sup> ∩ D<sup>ω</sup> c , z 0 ; z � �<sup>00</sup> ∩ D<sup>ω</sup> �. In this case, axis iω is crossed by combination of both positive and negative branches, and the root locus portrait certainly includes a series of initial points, and thus the whole branches, that have migrated over the boundary to the right half-plane. Therefore, this case always gives us the family (26) that includes combination of stable and unstable polynomials. Name the interval [z', z"] the system instability region.

If the interval [z", ∞] completely belongs to the region D<sup>ω</sup> �,

$$[z''\infty] \subset D\_w{}^-,\tag{38}$$

only the negative branches cross the stability boundary iω, and the family Z together with the corresponding positive branches are located in the right half-pane,

$$
\mathbb{Z} \subset \mathbb{R}.\tag{39}
$$

No stable polynomial could be found in (26). This region name the system complete instability region.

#### 5.4 Real crossing region of the portrait

Specify the region D<sup>R</sup> <sup>ω</sup> where the branches of the given real root locus portrait cross the stability boundary. To find its limits, consider Eq. (29) and determine the values of its roots. When ω>0.

$$
\rho\_{\text{max}} = \sqrt{\frac{\overline{a}\_3}{\underline{a}\_1}}, \alpha\_{\text{min}} = \sqrt{\frac{\underline{a}\_3}{\overline{a}\_1}} \tag{40}
$$

where ωmax, ωmin represent the real crossing region.

Definition 10. The region [ωmin,ωmax] at the stability boundary iω, where the polynomial (26) root locus portrait branches migrate through to the right half-plane, name the real crossing region DR <sup>ω</sup> of the system root locus portrait:

$$[\boldsymbol{\alpha}\_{\min}, \boldsymbol{\alpha}\_{\max}] \subseteq \boldsymbol{D}\_{\boldsymbol{\alpha}}^{\boldsymbol{R}}.\tag{41}$$

After solving Eq. (33), obtain three points of extremum for the majorant

Rewrite (30) for determination of a minorant parameter function (or a

<sup>a</sup>4min ¼ �ω<sup>4</sup> <sup>þ</sup> <sup>a</sup>2ω<sup>2</sup>

ffiffiffiffiffi a2 2 r

Evidently, for n = 4, Eqs. (32) and (35) are the majorant and the minorant for the

described by the characteristic polynomial (26) is a region [0, ωemax] at the system asymptotic stability boundary iω where the given portrait parameter function (30) extremum values, a<sup>4</sup>emax (34) and a<sup>4</sup>emin (36), family is located provided all coeffi-

Figure 3 represents the character (diagram) of the parameter function (30) distribution along the boundary of stability by its majorant (32) and minorant (35). For better understanding and descriptiveness, the diagram in Figure 3 is shown by

A diagram for distribution of the interval system root locus portrait parameter function along the asymptotic

5.3 Diagram of the parameter function distribution along the stability

In the same way obtain three points of extremum for the minorant, when

ffiffiffiffiffi a2 2 r

, a4emax ¼ �ω<sup>4</sup>

, a4emin ¼ �ω<sup>4</sup>

<sup>ω</sup> of the interval system root locus portrait

<sup>e</sup>max <sup>þ</sup> <sup>a</sup><sup>2</sup> � <sup>ω</sup><sup>2</sup>

: (35)

<sup>e</sup>min <sup>þ</sup> <sup>a</sup><sup>2</sup> � <sup>ω</sup><sup>2</sup>

<sup>ω</sup> constitutes three subregions

<sup>e</sup>max : (34)

<sup>e</sup>min : (36)

parameter function for the field when a<sup>2</sup> ¼ a2:

ωemax ¼ 0, a4emax ¼ 0; ωemax ¼ �

Polynomials - Theory and Application

ω<sup>e</sup>min ¼ 0, a4emin ¼ 0; ω<sup>e</sup>min ¼ �

Definition 9. Extremum region D<sup>e</sup>

strait lines, although it constitutes curves. Region D<sup>P</sup>

cients а<sup>j</sup> vary within limits (31).

minorant):

a<sup>2</sup> ¼ a2:

whole portrait.

boundary

(see Figure 3):

Figure 3.

120

stability boundary.

#### 5.5 Graphic-analytical stability conditions for interval polynomials

Define below three possible ways of the real crossing region location and the corresponding stability conditions.

5.5.1 Real crossing region belongs to the increase region D<sup>ω</sup> +

$$D\_w^R \subset D\_w^{\phantom{\cdot^{R}}}.\tag{42}$$

5.5.2 Real crossing region belongs to the decrease region Dω

It happens in case if ωmin ≥ωemax .

DOI: http://dx.doi.org/10.5772/intechopen.83705

asymptotically unstable.

ωmin ≥ ωemax. For this case

portrait P�.

where ω(z<sup>0</sup>

following condition holds:

of ωmin.

123

of Z location may take place:

D<sup>R</sup> <sup>ω</sup> ⊂ D<sup>ω</sup>

the whole family Z of its initial points satisfies Eq. (39), and the system is

5.5.3 Real crossing region completely or partially belongs to the mixed region D<sup>ω</sup>

<sup>c</sup> ∨ D<sup>R</sup>

We have already discussed the increase part of (52), when P� = ∅. Hence, this section considers the decrease part, P�. Consider first the family Z of the root locus

Statement 6. If condition (51) holds, family Z of initial points of the dynamic system root locus portrait, described by characteristic polynomial (26), can be located in both left half-plane L and right half-plane R, that is, the following options

> DR <sup>ω</sup> ⊂ D<sup>ω</sup>

or D<sup>R</sup>

<sup>ω</sup> ∩ D<sup>ω</sup>

As options (54)–(57) deliberately indicate instability of the system in whole,

In this case proceed just as in (44)–(47) but only substituting ωmax instead

Statement 7. The asymptotic stability of the dynamic system, described by polynomial family (26) and satisfying expression (51), is ensured when the

From condition (59) follows that the system asymptotic stability for part Р� of

portrait (52), provided that condition (53) holds, is defined by the value of

We have this when the following conditions are not satisfied: ωmax <ω<sup>e</sup>min ,

<sup>ω</sup> ∩ D<sup>ω</sup> c

D<sup>R</sup> <sup>ω</sup> ⊂ D<sup>ω</sup>

Evidently, options (54) and (55) take place when

consider below only option (53) of the system poles location,

) is coordinate ω at point z<sup>0</sup> (Figure 3).

The above made conclusions allow to formulate the following statement. Statement 5. If the interval system root locus portrait P satisfies condition (50),

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

�

�: (50)

: (51)

P ¼ P<sup>þ</sup> þ P�, (52)

Z ⊂L, (53) Z ⊂ð Þ L þ R , (54) Z ⊂R: (55)

� (56)

�: (57)

ωmax < ω z<sup>0</sup> ð Þ, (58)

<sup>a</sup><sup>4</sup> < min <sup>a</sup>4ð Þ <sup>ω</sup>min ; <sup>a</sup>4ð Þ <sup>ω</sup>max : (59)

с

In this case ωmax <ωemin .

Statement 3. When the dynamic system root locus portrait, described by polynomial (26), satisfies relationship (42), the whole family Z of the portrait initial points is located in the left half-plane L,

$$\mathbf{Z} \subset \mathbf{L}.\tag{43}$$

Then, define the set S of the root locus portrait Р branches' intervals si:

$$\mathcal{S} = \{ \mathfrak{s}\_i = [\mathbf{0}, a\_4(a\_i)], \ i = \mathbf{1}, \mathfrak{2}, \ldots \}. \tag{44}$$

a4(ωi) represents the parameter function (30) at points with the coordinates ωi; S ⊂ Р and S ⊂ L (40). Thus, from (42) and (43) obtain:

$$\bigcap\_{i=1}^{\infty} \mathfrak{s}\_i = \inf \ S = \left[ \mathbf{0}, \underline{a}\_{\mathsf{A}}(o\_{\min}) \right], \tag{45}$$

where a4ð Þ ωmin —function (30) minimal value at point ωmin (40). Hence,

$$\forall \; a\_4 \in \left[ \underline{a}\_4, \overline{a}\_4 \right] \quad \left[ a\_4 \in \left[ 0, \underline{a}\_4(a\_{\min}) \right] \quad \rightarrow \; a\_4 \in \mathbb{S} \& P \subset L \right], \tag{46}$$

$$\forall \; a\_4 \in \left[ \underline{a}\_4, \overline{a}\_4 \right] \quad \left[ a\_4 \notin \left[ 0, \underline{a}\_4 (o\_{\min}) \right] \quad \rightarrow \; a\_4 \notin \text{S\&P} \not\subset L \right]. \tag{47}$$

The following statement can be formulated on the basis of expressions (42) and (47).

Statement 4. The dynamic system, described by the interval characteristic polynomial family (26) and satisfying expression (42), is asymptotically stable if

$$
\overline{a}\_{\mathfrak{A}} < \underline{a}\_{\mathfrak{A}}(a\_{\min}).\tag{48}
$$

Definition 11. One or more stable polynomials with constant coefficients within the family (26) that guarantee stability of the whole family name the dominating polynomials.

From Statement 4 and the previous conclusions, the following stability condition goes.

Stability condition 1. The asymptotic stability of the interval system family, described by the root locus portrait Р (5) satisfying expression (42), is guaranteed if polynomial

$$s^4 + \overline{a}\_1 s^3 + \underline{a}\_2 s^2 + \underline{a}\_3 s + \overline{a}\_4 = 0 \tag{49}$$

of the family is stable. Polynomial (49) represents the dominating one.

Stability is verified using the Stability condition 1. The polynomial parameters are calculated with application of the Statement 4.

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705

5.5.2 Real crossing region belongs to the decrease region Dω �

$$D\_{\
u}^{\mathbb{R}} \subset D\_{\
u}^{-}.\tag{50}$$

It happens in case if ωmin ≥ωemax .

The above made conclusions allow to formulate the following statement.

Statement 5. If the interval system root locus portrait P satisfies condition (50), the whole family Z of its initial points satisfies Eq. (39), and the system is asymptotically unstable.

5.5.3 Real crossing region completely or partially belongs to the mixed region D<sup>ω</sup> с

$$D\_{\
u}^{\mathbb{R}} \subset D\_{\
u}^{\quad c} \; \lor \; D\_{\
u}^{\mathbb{R}} \cap D\_{\
u}^{\quad c} \; . \tag{51}$$

We have this when the following conditions are not satisfied: ωmax <ω<sup>e</sup>min , ωmin ≥ ωemax.

For this case

5.5 Graphic-analytical stability conditions for interval polynomials

corresponding stability conditions.

Polynomials - Theory and Application

In this case ωmax <ωemin .

points is located in the left half-plane L,

∀ a<sup>4</sup> ∈ a4; a<sup>4</sup>

∀ a<sup>4</sup> ∈ a4; a<sup>4</sup>

(42) and (47).

polynomials.

polynomial

goes.

122

5.5.1 Real crossing region belongs to the increase region D<sup>ω</sup>

S ⊂ Р and S ⊂ L (40). Thus, from (42) and (43) obtain:

∩ ∞ i¼1

<sup>a</sup><sup>4</sup> <sup>∈</sup> <sup>0</sup>; <sup>a</sup>4ð Þ <sup>ω</sup>min

<sup>a</sup><sup>4</sup> <sup>∉</sup> <sup>0</sup>; <sup>a</sup>4ð Þ <sup>ω</sup>min

s <sup>4</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

are calculated with application of the Statement 4.

Define below three possible ways of the real crossing region location and the

D<sup>R</sup> <sup>ω</sup> ⊂ D<sup>ω</sup>

Statement 3. When the dynamic system root locus portrait, described by polynomial (26), satisfies relationship (42), the whole family Z of the portrait initial

Then, define the set S of the root locus portrait Р branches' intervals si:

a4(ωi) represents the parameter function (30) at points with the coordinates ωi;

si ¼ inf S ¼ 0; a4ð Þ ωmin

where a4ð Þ ωmin —function (30) minimal value at point ωmin (40). Hence,

The following statement can be formulated on the basis of expressions

Statement 4. The dynamic system, described by the interval characteristic polynomial family (26) and satisfying expression (42), is asymptotically stable if

Definition 11. One or more stable polynomials with constant coefficients within the family (26) that guarantee stability of the whole family name the dominating

From Statement 4 and the previous conclusions, the following stability condition

Stability condition 1. The asymptotic stability of the interval system family, described by the root locus portrait Р (5) satisfying expression (42), is guaranteed if

<sup>3</sup> <sup>þ</sup> <sup>a</sup>2<sup>s</sup>

of the family is stable. Polynomial (49) represents the dominating one. Stability is verified using the Stability condition 1. The polynomial parameters

+

<sup>þ</sup>: (42)

Z ⊂ L: (43)

, (45)

S ¼ si ¼ 0; a4ð Þ ω<sup>i</sup> f g ½ �; i ¼ 1; 2; … : (44)

! <sup>a</sup><sup>4</sup> <sup>∈</sup> <sup>S</sup>&P<sup>⊂</sup> <sup>L</sup> , (46)

! <sup>a</sup><sup>4</sup> <sup>∉</sup>S&P<sup>⊄</sup> <sup>L</sup> : (47)

a<sup>4</sup> < a4ð Þ ωmin : (48)

<sup>2</sup> <sup>þ</sup> <sup>a</sup>3<sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>4</sup> <sup>¼</sup> <sup>0</sup> (49)

$$P = P^{+} + P^{-},\tag{52}$$

We have already discussed the increase part of (52), when P� = ∅. Hence, this section considers the decrease part, P�. Consider first the family Z of the root locus portrait P�.

Statement 6. If condition (51) holds, family Z of initial points of the dynamic system root locus portrait, described by characteristic polynomial (26), can be located in both left half-plane L and right half-plane R, that is, the following options of Z location may take place:

$$\mathbf{Z} \subset \mathbf{L},\tag{53}$$

$$Z \subset (L+R),\tag{54}$$

$$Z \subset \mathbb{R}.\tag{55}$$

Evidently, options (54) and (55) take place when

$$D\_{\alpha}^{\mathbb{R}} \subset D\_{\alpha}^{-} \tag{56}$$

$$\text{or } D\_{\alpha}^{\mathbb{R}} \cap D\_{w}^{-}. \tag{57}$$

As options (54)–(57) deliberately indicate instability of the system in whole, consider below only option (53) of the system poles location,

$$
\rho\_{\text{max}} < o(z'), \tag{58}
$$

where ω(z<sup>0</sup> ) is coordinate ω at point z<sup>0</sup> (Figure 3).

In this case proceed just as in (44)–(47) but only substituting ωmax instead of ωmin.

Statement 7. The asymptotic stability of the dynamic system, described by polynomial family (26) and satisfying expression (51), is ensured when the following condition holds:

$$
\overline{a}\_4 \prec \min \{ \underline{a}\_4(o\_{\min}), \underline{a}\_4(o\_{\max}) \}. \tag{59}
$$

From condition (59) follows that the system asymptotic stability for part Р� of portrait (52), provided that condition (53) holds, is defined by the value of

a4ð Þ ωmax . Therefore, for checking stability of Р� (52), it is enough to check the only one following dominating polynomial of (26):

$$s^4 + \underline{a}\_1 s^3 + \underline{a}\_2 s^2 + \overline{a}\_3 s + \overline{a}\_4 = \mathbf{0}.\tag{60}$$

Polynomial stability could be estimated graphically directly from the plots

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

Coefficients of the given polynomial (26): a<sup>1</sup> ∈ ½ � 5; 10 , a<sup>2</sup> ∈ ½ � 5; 10 ,

crossing region in this case completely covers the extremum region,

which is confirmed by the negative value of the parameter a4ð Þ ωmax I. Dominating polynomials of the family are the following:

<sup>3</sup> <sup>þ</sup> <sup>20</sup><sup>s</sup>

<sup>3</sup> <sup>þ</sup> <sup>20</sup><sup>s</sup>

<sup>a</sup><sup>3</sup> <sup>&</sup>lt; <sup>a</sup><sup>1</sup> � <sup>4</sup>, <sup>47</sup><sup>2</sup>

<sup>3</sup> <sup>þ</sup> <sup>20</sup><sup>s</sup>

<sup>3</sup> <sup>þ</sup> <sup>20</sup><sup>s</sup>

ified values of а<sup>3</sup> and а<sup>4</sup> (a<sup>3</sup> ¼ 80,a<sup>4</sup> ¼ 60) is asymptotically stable.

<sup>ω</sup>: ω<sup>e</sup> min = 3,16; ω<sup>e</sup> max = 5,92; a4<sup>e</sup> min = 100,04;

<sup>ω</sup>: ωmin = 2; ωmax = 7,1; a4ð Þ ωmin = 64; a4ð Þ ωmax = � 1532,97.

<sup>2</sup> <sup>þ</sup> <sup>40</sup><sup>s</sup> <sup>þ</sup> <sup>30</sup> <sup>¼</sup> <sup>0</sup>: (64)

<sup>2</sup> <sup>þ</sup> <sup>250</sup><sup>s</sup> <sup>þ</sup> <sup>30</sup> <sup>¼</sup> <sup>0</sup>: (65)

, a<sup>3</sup> < 99, 9: (66)

In Figure 4, the above indicated regions are shown. The points, corresponding to the dominating polynomials (61), (62), are designated by r' and r". The real

It is evident that the given polynomial family in whole is unstable. Within region Z<sup>ω</sup> = [z', z"], there exist poles that have migrated to the right half-plane (see (54)),

Polynomials stability check shows that polynomial (6), which root loci crosses the stability boundary at point a4ð Þ ωmin , is stable, and polynomial (66), which root loci crosses the stability boundary at point a4ð Þ ωmax , has two roots with positive real

Extraction of the stable polynomial subfamily of the given unstable family: The stable root locus family, satisfying conditions (58) and (59), should cross the stability boundary within the region bounded by interval [r', z'] as in this case all initial points of the root locus family are located in the left half-plane (53) (see

To calculate the maximal value of а<sup>3</sup> that defines the stable subfamily within the

Based on (59), accept: a<sup>4</sup> < a4ð Þ ωmin , a<sup>4</sup> ¼ 60 and write the dominating poly-

As per stability condition 2, the root locus portrait subfamily having new mod-

A method has been worked out for synthesis of asymptotically stable regular or interval polynomial from the given Hurwitz or non-Hurwitz source polynomial

<sup>2</sup> <sup>þ</sup> <sup>40</sup><sup>s</sup> <sup>þ</sup> <sup>60</sup> <sup>¼</sup> <sup>0</sup>,

<sup>2</sup> <sup>þ</sup> <sup>80</sup><sup>s</sup> <sup>þ</sup> <sup>60</sup> <sup>¼</sup> <sup>0</sup>:

(see Figures 3 and 4).

a<sup>3</sup> ∈ ½ � 5; 10 , a<sup>4</sup> ∈ ½ � 5; 10 : Extremum region: D<sup>e</sup>

[z', z"]: ω(z') = 4,47; ω(z") = 8,37.

DOI: http://dx.doi.org/10.5772/intechopen.83705

⊆ DR ω.

> s <sup>4</sup> <sup>þ</sup> <sup>10</sup><sup>s</sup>

s <sup>4</sup> <sup>þ</sup> <sup>5</sup><sup>s</sup>

given root locus portrait, apply formula (63):

s <sup>4</sup> <sup>þ</sup> <sup>10</sup><sup>s</sup>

6. Conclusions and future developments

s <sup>4</sup> <sup>þ</sup> <sup>5</sup><sup>s</sup>

Based on (66), accept a<sup>3</sup> ¼ 80.

5.6 Example

a4<sup>e</sup> max = 1223,96. Real region: D<sup>R</sup>

> ω, r<sup>0</sup> ;r <sup>00</sup>

De <sup>ω</sup> ⊂ D<sup>R</sup>

parts.

Section 5.5).

nomials:

125

Because in this case, the portrait represents the compound one (52), check the stability by checking both polynomials, (49) and (60).

Stability condition 2. If the interval dynamic system root locus portrait Р (52), describing the family of characteristic polynomials (26), satisfies expression (51), the system asymptotic stability is ensured when the following dominating polynomials

$$s^4 + \overline{a}\_1 s^3 + \underline{a}\_2 s^2 + \underline{a}\_3 s + \overline{a}\_4 = 0,\tag{61}$$

$$
\mathfrak{s}^4 + \underline{a}\_1 \mathfrak{s}^3 + \underline{a}\_2 \mathfrak{s}^2 + \overline{a}\_3 \mathfrak{s} + \overline{a}\_4 = \mathbf{0} \tag{62}
$$

of family (26) are both stable.

From the results obtained above also goes that in case (51) the system asymptotic stability can be verified by only a single polynomial of (26) having constant coefficients. The equation to choose depends of condition (49) verification results. If the verification shows that min <sup>a</sup>4ð Þ <sup>ω</sup>min ; <sup>a</sup>4ð Þ <sup>ω</sup>max � � <sup>¼</sup> <sup>a</sup>4ð Þ <sup>ω</sup>min , then Eq. (61) is applied for the stability check. If it shows that min <sup>a</sup>4ð Þ <sup>ω</sup>min ; <sup>a</sup>4ð Þ <sup>ω</sup>max � � <sup>¼</sup> <sup>a</sup>4ð Þ <sup>ω</sup>max , then the stability is verified by (62).

To determine the coefficients of (26), ensuring satisfaction of expressions (53) and (58), Eqs. (30) and (31) are applied. Thus, coefficients а<sup>1</sup> and а<sup>3</sup> must satisfy the inequality:

$$
\sqrt{\frac{\overline{\underline{a}\_3}}{\underline{a}\_1}} < \alpha \left( z' \right), \overline{\underline{a}\_3} < \underline{a}\_1 \alpha^2 \left( z' \right). \tag{63}
$$

To verify the system stability, the stability conditions 1 and 2 are used. For calculation of the system (polynomial) parameters, expressions (48), (49) and (63) are used.

Figure 4. Dynamics of the interval system root locus portrait at the asymptotic stability boundary.

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705

Polynomial stability could be estimated graphically directly from the plots (see Figures 3 and 4).

#### 5.6 Example

a4ð Þ ωmax . Therefore, for checking stability of Р� (52), it is enough to check the

Because in this case, the portrait represents the compound one (52), check the

Stability condition 2. If the interval dynamic system root locus portrait Р (52), describing the family of characteristic polynomials (26), satisfies expression (51),

<sup>2</sup> <sup>þ</sup> <sup>a</sup>3<sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>4</sup> <sup>¼</sup> <sup>0</sup>: (60)

<sup>2</sup> <sup>þ</sup> <sup>a</sup>3<sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>4</sup> <sup>¼</sup> <sup>0</sup>, (61)

<sup>2</sup> <sup>þ</sup> <sup>a</sup>3<sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>4</sup> <sup>¼</sup> <sup>0</sup> (62)

� � <sup>¼</sup> <sup>a</sup>4ð Þ <sup>ω</sup>min , then Eq. (61) is

� � <sup>¼</sup> <sup>a</sup>4ð Þ <sup>ω</sup>max ,

: (63)

<sup>3</sup> <sup>þ</sup> <sup>a</sup>2<sup>s</sup>

the system asymptotic stability is ensured when the following dominating

<sup>3</sup> <sup>þ</sup> <sup>a</sup>2<sup>s</sup>

<sup>3</sup> <sup>þ</sup> <sup>a</sup>2<sup>s</sup>

applied for the stability check. If it shows that min a4ð Þ ωmin ; a4ð Þ ωmax

ffiffiffiffiffiffiffi a3 a1

Dynamics of the interval system root locus portrait at the asymptotic stability boundary.

< ω z <sup>0</sup> � �

To verify the system stability, the stability conditions 1 and 2 are used. For calculation of the system (polynomial) parameters, expressions (48), (49) and (63)

s

From the results obtained above also goes that in case (51) the system asymptotic stability can be verified by only a single polynomial of (26) having constant coefficients. The equation to choose depends of condition (49) verification results.

To determine the coefficients of (26), ensuring satisfaction of expressions (53) and (58), Eqs. (30) and (31) are applied. Thus, coefficients а<sup>1</sup> and а<sup>3</sup> must satisfy

, <sup>a</sup><sup>3</sup> <sup>&</sup>lt; <sup>a</sup>1ω<sup>2</sup> <sup>z</sup>

<sup>0</sup> � �

only one following dominating polynomial of (26):

Polynomials - Theory and Application

s <sup>4</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

stability by checking both polynomials, (49) and (60).

s <sup>4</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

s <sup>4</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

If the verification shows that min a4ð Þ ωmin ; a4ð Þ ωmax

of family (26) are both stable.

then the stability is verified by (62).

polynomials

the inequality:

are used.

Figure 4.

124

Coefficients of the given polynomial (26): a<sup>1</sup> ∈ ½ � 5; 10 , a<sup>2</sup> ∈ ½ � 5; 10 , a<sup>3</sup> ∈ ½ � 5; 10 , a<sup>4</sup> ∈ ½ � 5; 10 :

Extremum region: D<sup>e</sup> <sup>ω</sup>: ω<sup>e</sup> min = 3,16; ω<sup>e</sup> max = 5,92; a4<sup>e</sup> min = 100,04; a4<sup>e</sup> max = 1223,96.

Real region: D<sup>R</sup> <sup>ω</sup>: ωmin = 2; ωmax = 7,1; a4ð Þ ωmin = 64; a4ð Þ ωmax = � 1532,97. [z', z"]: ω(z') = 4,47; ω(z") = 8,37.

In Figure 4, the above indicated regions are shown. The points, corresponding to the dominating polynomials (61), (62), are designated by r' and r". The real crossing region in this case completely covers the extremum region, De <sup>ω</sup> ⊂ D<sup>R</sup> ω, r<sup>0</sup> ;r <sup>00</sup> ⊆ DR ω.

It is evident that the given polynomial family in whole is unstable. Within region Z<sup>ω</sup> = [z', z"], there exist poles that have migrated to the right half-plane (see (54)), which is confirmed by the negative value of the parameter a4ð Þ ωmax I.

Dominating polynomials of the family are the following:

$$\mathfrak{s}^4 + \mathfrak{1}\mathfrak{s}^3 + \mathfrak{2}\mathfrak{s}^2 + \mathfrak{4}\mathfrak{s} + \mathfrak{3}\mathfrak{0} = \mathfrak{0}.\tag{64}$$

$$\mathbf{s}^4 + \mathbf{5}\mathbf{s}^3 + \mathbf{20}\mathbf{s}^2 + \mathbf{250}\mathbf{s} + \mathbf{30} = \mathbf{0}.\tag{65}$$

Polynomials stability check shows that polynomial (6), which root loci crosses the stability boundary at point a4ð Þ ωmin , is stable, and polynomial (66), which root loci crosses the stability boundary at point a4ð Þ ωmax , has two roots with positive real parts.

Extraction of the stable polynomial subfamily of the given unstable family:

The stable root locus family, satisfying conditions (58) and (59), should cross the stability boundary within the region bounded by interval [r', z'] as in this case all initial points of the root locus family are located in the left half-plane (53) (see Section 5.5).

To calculate the maximal value of а<sup>3</sup> that defines the stable subfamily within the given root locus portrait, apply formula (63):

$$
\overline{a}\_3 < \underline{a}\_1 \cdot 4, \, 4\overline{\tau}^2, \, \overline{a}\_3 < 9\overline{9}, \, 9. \tag{66}
$$

Based on (66), accept a<sup>3</sup> ¼ 80.

Based on (59), accept: a<sup>4</sup> < a4ð Þ ωmin , a<sup>4</sup> ¼ 60 and write the dominating polynomials:

$$\begin{aligned} \mathfrak{s}^4 + \mathfrak{1}\mathfrak{s}^3 + \mathfrak{2}\mathfrak{s}^2 + \mathfrak{4}\mathfrak{s}\mathfrak{s} + \mathfrak{6}\mathfrak{0} &= \mathbf{0}, \\ \mathfrak{s}^4 + \mathfrak{5}\mathfrak{s}^3 + \mathfrak{2}\mathfrak{s}^2 + \mathfrak{8}\mathfrak{0}\mathfrak{s} + \mathfrak{6}\mathfrak{0} &= \mathbf{0}. \end{aligned}$$

As per stability condition 2, the root locus portrait subfamily having new modified values of а<sup>3</sup> and а<sup>4</sup> (a<sup>3</sup> ¼ 80,a<sup>4</sup> ¼ 60) is asymptotically stable.

#### 6. Conclusions and future developments

A method has been worked out for synthesis of asymptotically stable regular or interval polynomial from the given Hurwitz or non-Hurwitz source polynomial

with constant/interval coefficients by setting up coefficients of the given one. The root locus approach is used. The task is solved by introduction of notions of the "extended polynomial" ("generalized polynomial") and the polynomial "extended root locus," which allows to obtain a descriptive picture of the polynomial root dynamics under coefficient variations and to disclose on this basis the cause of instability. The intervals of uncertainty for each coefficient being set up are specified along the root locus branches.

The above described method based on the "extended root locus" notion is new and allows to extend the application sphere of the root locus method, which is traditionally considered to be the method of system synthesis by only a single parameter (coefficient) variation and with only one variable parameter (coefficient), in both directions: system synthesis by many parameter variations and system synthesis with many parameter variations.

Investigation of the fourth power dynamic system behavior in conditions of the interval parameter variations has also been carried out on the basis of root locus portraits and introduction of the notion of the "diagram of the root locus parameter function values distribution along the stability bound." Behavior regularities for interval system root locus portraits at the stability boundary have been formulated. On this basis, the stability conditions have been derived, and graphic-analytical method has been worked out for calculating intervals of parameter variation ensuring the system robust stability.

In continuation of the results of Anderson [22] and Kharitonov [4] in this work, it is proved that for the 4th power interval system family asymptotic stability analysis, it is enough to use the only one polynomial of this kind. It is also shown, how to find and extract the stable families from the unstable ones.

The above discussed topic is certainly worth further investigation in the light of continuous progress of both theory and technology. When speaking of the practical implementations, it could be noted that most of the control system synthesis tasks, especially those in the area of robust control, are currently still being solved in a somewhat "local domestic" way, when a designer each time tries to invent a solution to be suitable for the specific application experiencing the lack of more generalized methods. Besides this, a great deal of existing robust control methods share and suffer complexity. In this connection, further in-depth investigation of the uncertain polynomials' root locus portraits seems helpful, especially the analysis of its composition in terms of configurations variety, constituting subfamilies, placement of various root domains within the prescribed regions in the complex plane and, of course, dynamics. They also could be distinguished for their undoubted descriptiveness.

Polynomial equation approach in the design technique [16], and root locus technique in particular, is descriptive, clear, and easy to use and computerize and thus could be helpful in many application areas including the areas of industry, biology, medicine, etc. It can be used for proper parameterization of robust drive controllers, for example, in the area of railway traffic control, in particular for the cases of tackling the problems of breaking and skidding.

Author details

Nesenchuk Alla

127

Sciences, Minsk, Belarus

United Institute of Informatics Problems of the Belarusian National Academy of

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus…

DOI: http://dx.doi.org/10.5772/intechopen.83705

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: anes@newman.bas-net.by

provided the original work is properly cited.

#### Acknowledgements

The author acknowledges the support of this work by the Belarusian Republican Foundation for Fundamental Research.

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus… DOI: http://dx.doi.org/10.5772/intechopen.83705

### Author details

with constant/interval coefficients by setting up coefficients of the given one. The root locus approach is used. The task is solved by introduction of notions of the "extended polynomial" ("generalized polynomial") and the polynomial "extended root locus," which allows to obtain a descriptive picture of the polynomial root dynamics under coefficient variations and to disclose on this basis the cause of instability. The intervals of uncertainty for each coefficient being set up are speci-

The above described method based on the "extended root locus" notion is new and allows to extend the application sphere of the root locus method, which is traditionally considered to be the method of system synthesis by only a single parameter (coefficient) variation and with only one variable parameter (coefficient), in both directions: system synthesis by many parameter variations and

Investigation of the fourth power dynamic system behavior in conditions of the interval parameter variations has also been carried out on the basis of root locus portraits and introduction of the notion of the "diagram of the root locus parameter function values distribution along the stability bound." Behavior regularities for interval system root locus portraits at the stability boundary have been formulated. On this basis, the stability conditions have been derived, and graphic-analytical method has been worked out for calculating intervals of parameter variation ensur-

In continuation of the results of Anderson [22] and Kharitonov [4] in this work,

The above discussed topic is certainly worth further investigation in the light of continuous progress of both theory and technology. When speaking of the practical implementations, it could be noted that most of the control system synthesis tasks, especially those in the area of robust control, are currently still being solved in a somewhat "local domestic" way, when a designer each time tries to invent a solution to be suitable for the specific application experiencing the lack of more generalized methods. Besides this, a great deal of existing robust control methods share and suffer complexity. In this connection, further in-depth investigation of the uncertain polynomials' root locus portraits seems helpful, especially the analysis of its composition in terms of configurations variety, constituting subfamilies, placement of various root domains within the prescribed regions in the complex plane and, of course, dynamics. They also could be distinguished for their undoubted

Polynomial equation approach in the design technique [16], and root locus technique in particular, is descriptive, clear, and easy to use and computerize and thus could be helpful in many application areas including the areas of industry, biology, medicine, etc. It can be used for proper parameterization of robust drive controllers, for example, in the area of railway traffic control, in particular for the

The author acknowledges the support of this work by the Belarusian Republican

cases of tackling the problems of breaking and skidding.

it is proved that for the 4th power interval system family asymptotic stability analysis, it is enough to use the only one polynomial of this kind. It is also shown,

how to find and extract the stable families from the unstable ones.

fied along the root locus branches.

Polynomials - Theory and Application

ing the system robust stability.

descriptiveness.

Acknowledgements

126

Foundation for Fundamental Research.

system synthesis with many parameter variations.

Nesenchuk Alla United Institute of Informatics Problems of the Belarusian National Academy of Sciences, Minsk, Belarus

\*Address all correspondence to: anes@newman.bas-net.by

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

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[12] Li X, Yu H, Yuan M, Wang J. Design of robust optimal proportional-integralderivative controller based on new interval polynomial stability criterion and Lyapunov theorem in the multiple parameters' perturbations circumstance. IET Control Theory & Applications. 2010;4:2427-2440. DOI: 10.1049/ietcta.2009.0508

[13] Lia X, Niculescub S, Celac A, Wanga H, Caia T. Invariance properties for a class of quasipolynomials. Automatica. 2014;50:890-895. ISSN 0005-1098

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Polynomials - Theory and Application

[10] Tempo R, Calafiori C, Dabbene F. Randomized Algorithms for Analysis and Control of Uncertain Systems with Applications. London: Springer-Verlag; 2013. 357 p. ISBN 978-1-4471-4609-4

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[12] Li X, Yu H, Yuan M, Wang J. Design of robust optimal proportional-integralderivative controller based on new interval polynomial stability criterion and Lyapunov theorem in the multiple parameters' perturbations circumstance. IET Control Theory & Applications. 2010;4:2427-2440. DOI: 10.1049/iet-

[13] Lia X, Niculescub S, Celac A, Wanga H, Caia T. Invariance properties for a class of quasipolynomials. Automatica. 2014;50:890-895. ISSN 0005-1098

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Sciences. 2012. ID 595076:19 pages. DOI:

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[16] Kučera V. Polynomial control: Past, present, and future. International Journal of Robust and Nonlinear Control. 2007;17:682-705. ISSN

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[2] Dorf R, Bishop R. Modern Control Systems. 12th ed. N.Y.: Prentice Hall; 2011. 1084 p. ISBN-13:978-0-13-6024583

[3] Polyak B, Scherbakov P. Robust stability and control [in Russian]. Nauka. 2002. 303 p. ISBN 5-02-

[4] Kharitonov V. About asymptotic stability of equilibrium of the linear differential equations systems family [in Russian]. Differential Equations. 1978; XIV:2086-2088. ISSN 0374-0641

[5] Tsypkin Y. Robust stability of relay control systems [in Russian]. Doklady Mathematics. 1995;340:751-753. ISSN

[6] Barmish B. Invariance of the strict hurwitz property for polynomials with

Transactions on Automatic Control. 1984;2:935-936. ISSN 0018-9286

[7] Soh Y. Strict hurwitz property of polynomials under coefficient perturbations. IEEE Transactions on Automatic Control. 1989;34:629-632.

[8] Soh Y. Maximal perturbation bounds for perturbed polynomials with roots in the left-sector. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 1994;41:

[9] Bartlett A, Hollot C, Lin H. Root location of an entire polytope of polynomials in suffices to check the edges. Mathematics of Control, Signals,

and Systems. 1987;1:61-71. ISSN

perturbed coefficients. IEEE

[19] Nesenchuk A. Parametric synthesis of qualitative robust control systems using root locus fields. In: In Proceedings of the 15th Triennial World Congress of IFAC; 21–26 July 2003. Barselona, Spain. London: Elsevier Science Ltd.; 2003. pp. 331-335. ISBN 008044220X

[20] Nesenchuk A, Nesenchuk V. Industrial robot control system parametric design on the base of methods for uncertain systems robustness. In: Cubero S, editor. Industrial Robotics: Theory, Modeling and Control. Mammendorf: PlV pro literatur Verlag Robert Mayer-Scholz; 2007. pp. 895-926. ISBN 3-86611-285-8. ch33. ISBN 3-86611-285-8

[21] Nesenchuk A. Parametric synthesis of interval control systems using root loci of Kharitonov's polynomials. In: Proceedings of the European Control Conference (ECC'99); 31 August-03 September 1999; Karlsruhe, Germany. 1999. 1 electronic. opt. disc (CD-ROM). ID 123, 6 p

[22] Anderson B. On robust Hurwitz polynomials. IEEE Transactions on Automatic Control. 1987;32:909-913. ISSN 0018-9286

[23] Nesenchuk А. A method for synthesis of robust interval polynomials using the extended root locus. In: Proceedings of the American Control Conference (ACC'2017); 24–26 May 2017. Seattle, USA: Seattle: IEEE; 2017. pp. 1715-1720. ISBN 978-1-5386-5426

Chapter 7

Abstract

1. Introduction

common jump sizes.

131

Pablo Olivares

cubic splines show the best performance.

options, spread options, jump-diffusion model

Pricing Basket Options by

Polynomial Approximations

In this paper, we use polynomial approximations in terms of Taylor, Chebyshev, and cubic splines to compute the price of basket options. The paper extends the use of a similar pricing technique applied under a multivariate Black-Scholes model to a framework where the dynamic of the underlying assets is described by dependent exponential Levy processes generated by a combination of Brownian motions and compound Poisson processes. This model captures some empirical features of the asset dynamics such as common and idiosyncratic random jumps. The approach is implemented in the context of spread options and a multivariate Merton model, i.e., a jump diffusion with Gaussian jumps. Our findings show that, within the range of parameters analyzed, polynomial approximations are comparable in accuracy to a standard Monte Carlo approach with a considerable reduction in computational effort. Among the three expansions,

Keywords: Taylor approximations, Chebyshev polynomials, cubic splines, basket

We study the pricing of basket contracts under a multivariate jump-diffusion process. The paper extends the use of a similar pricing technique applied under a multivariate Black-Scholes model, see [1], to a framework where the dynamic of the underlying assets is described by dependent exponential Levy processes generated by a combination of Brownian motions and compound Poisson processes. This model captures some empirical features of the asset dynamics such as common and idiosyncratic random jumps. The dependence between assets is reflected in both the covariance structure of the Brownian motion and the joint probability law of the

For such class of models, no pricing closed-form formula is available. In singleasset contracts, well-established numerical methods have proven to be effective, but their extensions to several dimensions reveal important instabilities and a costly computational effort. Our paper introduces a novel approach based on polynomial approximations of the conditional price. It is, in the framework considered, less time demanding than a standard Monte Carlo approach to achieve similar results. Moreover, the use of Chebyshev polynomials and cubic splines improves the con-

vergence over previous attempts based on Taylor expansions.

#### Chapter 7

## Pricing Basket Options by Polynomial Approximations

Pablo Olivares

#### Abstract

In this paper, we use polynomial approximations in terms of Taylor, Chebyshev, and cubic splines to compute the price of basket options. The paper extends the use of a similar pricing technique applied under a multivariate Black-Scholes model to a framework where the dynamic of the underlying assets is described by dependent exponential Levy processes generated by a combination of Brownian motions and compound Poisson processes. This model captures some empirical features of the asset dynamics such as common and idiosyncratic random jumps. The approach is implemented in the context of spread options and a multivariate Merton model, i.e., a jump diffusion with Gaussian jumps. Our findings show that, within the range of parameters analyzed, polynomial approximations are comparable in accuracy to a standard Monte Carlo approach with a considerable reduction in computational effort. Among the three expansions, cubic splines show the best performance.

Keywords: Taylor approximations, Chebyshev polynomials, cubic splines, basket options, spread options, jump-diffusion model

#### 1. Introduction

We study the pricing of basket contracts under a multivariate jump-diffusion process. The paper extends the use of a similar pricing technique applied under a multivariate Black-Scholes model, see [1], to a framework where the dynamic of the underlying assets is described by dependent exponential Levy processes generated by a combination of Brownian motions and compound Poisson processes. This model captures some empirical features of the asset dynamics such as common and idiosyncratic random jumps. The dependence between assets is reflected in both the covariance structure of the Brownian motion and the joint probability law of the common jump sizes.

For such class of models, no pricing closed-form formula is available. In singleasset contracts, well-established numerical methods have proven to be effective, but their extensions to several dimensions reveal important instabilities and a costly computational effort. Our paper introduces a novel approach based on polynomial approximations of the conditional price. It is, in the framework considered, less time demanding than a standard Monte Carlo approach to achieve similar results. Moreover, the use of Chebyshev polynomials and cubic splines improves the convergence over previous attempts based on Taylor expansions.

We consider a pricing methodology consisting in a two-step procedure. First, conditioning on d � 1 out of the total number of d assets, we find the price of a payoff based on a single asset with a more complex conditional distribution.

Secondly, we consider some expansions of the conditional price, given either in terms of Taylor, Chebyshev, or cubic spline polynomials, allowing to write the corresponding price as a linear combination of mixed exponential-power moments. bin nð Þ¼ ;Va; b ∑

the corresponding log-price process. They are related by

S ð Þj <sup>t</sup> ¼ S ð Þj <sup>0</sup> exp <sup>Y</sup>ð Þ<sup>j</sup> t

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

price K, are given by

where wj � �

and <sup>μ</sup> <sup>¼</sup> <sup>r</sup> � <sup>1</sup>

ð Þ Bt <sup>t</sup> <sup>≥</sup> <sup>0</sup> such that

while Σ 1

dynamics under Q given by

<sup>2</sup> is such that Σ

where ð Þ Nt <sup>t</sup> <sup>≥</sup> <sup>0</sup> <sup>¼</sup> <sup>N</sup>ð Þ <sup>0</sup>

The processes Nð Þ<sup>j</sup>

k∈ N that Xk � N μJ; DJ

J � �<sup>2</sup>

jump sizes are Xð Þ<sup>j</sup>

DJð Þ¼ <sup>j</sup>; <sup>l</sup> <sup>δ</sup>jl <sup>σ</sup>ð Þ<sup>j</sup>

<sup>Σ</sup>0,Jð Þ¼ <sup>j</sup>; <sup>l</sup> <sup>σ</sup> j,l

133

processes with respective intensities λj.

t � �

<sup>k</sup> and <sup>X</sup>ð Þ<sup>j</sup>

Poisson process m ¼ log φ<sup>Z</sup><sup>1</sup>

1 <sup>2</sup> Σ 1 2 � �<sup>0</sup>

dom vectors in the first sequence are independent.

Zð Þ<sup>j</sup> <sup>t</sup> ¼ ∑ Nð Þ<sup>j</sup> t k¼1 Xð Þ<sup>j</sup> <sup>k</sup> þ ∑ Nð Þ <sup>0</sup> t k¼1 Xð Þ<sup>j</sup>

t ≥ 0

<sup>0</sup>,k.

<sup>t</sup> ; Nð Þ<sup>1</sup>

� �

and X0,k � N μ0,J; Σ0,J

<sup>t</sup> ; …; <sup>N</sup>ð Þ <sup>d</sup> t

and Nð Þ <sup>0</sup> t � �

cratic and common jumps of the j-th underlying asset on the interval 0½ � ; t . Their

For the sake of concreteness, we assume Gaussian jumps, i.e., we assume for any

<sup>0</sup> . The compensator across each dimension takes the form

� �, where DJ is a diagonal matrix with components

n m¼0

PVa <sup>¼</sup> <sup>1</sup>;Va1; …;Va<sup>n</sup>

Also, for a differentiable function <sup>f</sup>, we set the vector DVf <sup>¼</sup> <sup>f</sup>; <sup>D</sup><sup>1</sup> <sup>f</sup>; …; Dn <sup>f</sup> � �. The d-dimensional process of spot prices is denoted by ð Þ St <sup>t</sup> <sup>≥</sup> <sup>0</sup>, while ð Þ Yt <sup>t</sup> <sup>≥</sup> <sup>0</sup> is

We analyze European basket options whose payoff at maturity T, for a strike

d j¼1 wjS ð Þj <sup>T</sup> � K !

<sup>1</sup> <sup>≤</sup> <sup>j</sup> <sup>≤</sup> <sup>d</sup> are some deterministic weights and x<sup>þ</sup> ¼ max xð Þ ; 0 . Furthermore, for the log-prices, we assume a multidimensional jump-diffusion

1

where ð Þ Bt <sup>t</sup> <sup>≥</sup> <sup>0</sup> is a multivariate Brownian motion with independent components

We define two sequences of independent and identically distributed 1 � ddimensional random vectors ð Þ Xk <sup>k</sup><sup>∈</sup> <sup>N</sup> and ð Þ X0,k <sup>k</sup><sup>∈</sup> <sup>N</sup>. The components of the ran-

The process ð Þ Zt <sup>t</sup> <sup>≥</sup> <sup>0</sup> is a d-variate compound Poisson process, independent of

h Sð Þ¼ <sup>T</sup> ∑

dYt ¼ μdt þ Σ

<sup>2</sup> diagð Þ� <sup>Σ</sup> <sup>m</sup>. The matrix <sup>Σ</sup> <sup>¼</sup> <sup>σ</sup>jl � �

ð Þ �i .

n

1

AVambn�<sup>m</sup>

n

� �, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, d (1)

<sup>2</sup>dBt þ dZt (2)

j,l is symmetric, positive definite,

þ

¼ Σ. The value m is the compensator of a compound

<sup>0</sup>,k, j ¼ 1, …, d

<sup>t</sup> <sup>≥</sup> <sup>0</sup> is a vector of independent Poisson

<sup>t</sup> <sup>≥</sup> <sup>0</sup> correspond, respectively, to idiosyn-

� �, with Σ0,J a matrix of components

0 @

m

� �

This approach is implemented in the context of spread options and a multivariate Merton model, that is, a jump diffusion with Gaussian jumps. Our findings show that, within the range of parameters analyzed, polynomial approximations are comparable in accuracy to a standard Monte Carlo approach with a considerable reduction in computational effort. Among the three expansions, cubic splines show the best performance.

The use of a Taylor expansion to pricing has been considered in the pioneering work of [2] for a vanilla European option and in [3, 4] for spread contracts under a bivariate Black-Scholes model. See also [5]. A Chebyshev expansion has been recently considered in [6]. Applications under a multivariate jump-diffusion model have been less explored. Our paper intends to fill this gap.

Although a comparison with alternative approaches is beyond the scope of this paper, it is worth noticing the existence of pricing methods based on Fourier or Hilbert transforms. For example, for spread contracts under a different class of Levy processes, a Fast Fourier transform method can be found in [7]. See also [8] for expansions in terms of Fourier series and [9] for Hilbert transforms.

The organization of the paper is as follows: in Section 2, we introduce the model and obtain the pricing expressions for basket contracts under the approximations. In Section 3, we specialize the three expansions in the case of spreads contracts. In Section 4, we discuss the implementation of the methods and present our numerical findings. Finally in Section 5, we present conclusions. Proofs are deferred to the appendix.

#### 2. Pricing under jump-diffusion models

Let <sup>Ω</sup>; <sup>A</sup>;ð Þ <sup>F</sup><sup>t</sup> <sup>t</sup> <sup>≥</sup> <sup>0</sup>; <sup>P</sup> � � be a filtered probability space. We define the filtration <sup>F</sup>Xt <sup>≔</sup> <sup>σ</sup> Xs ð Þ ; <sup>0</sup> <sup>≤</sup> <sup>s</sup> <sup>≤</sup> <sup>t</sup> as the <sup>σ</sup>-algebra generated by the random variables Xs f g ; 0 ≤ s ≤ t completed in the usual way. Denote by Q an equivalent martingale measure (EMM), respectively, by EQ, φX, and MX the expectation, characteristic, and moment-generating functions of a random variable X under Q. The function f <sup>X</sup> is its probability density function.

By r we denote the (constant) interest rate, A ∘ B is the componentwise product between matrices A and B, and A<sup>0</sup> represents the transpose of matrix <sup>A</sup> <sup>¼</sup> aij � � <sup>1</sup> <sup>≤</sup> i,j <sup>≤</sup> <sup>d</sup>, while diagð Þ A is a vector with components ð Þ aii <sup>1</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>d</sup>. The symbol δij is the usual Kronecker's number. The vector Y~ is created from the vector Y after eliminating the first component. For a function f with domain in R<sup>d</sup> and a vector <sup>L</sup> <sup>¼</sup> ð Þ <sup>l</sup>1; <sup>l</sup>2; …; ld with lk <sup>∈</sup> <sup>N</sup>, the symbol <sup>D</sup><sup>L</sup> <sup>f</sup> represents the mixed partial derivative of the function f differentiated lk times w.r.t. the k-th variable.

For vectors <sup>v</sup> <sup>¼</sup> ð Þ <sup>v</sup>1; <sup>v</sup>2; …; vd and <sup>n</sup> <sup>¼</sup> ð Þ <sup>n</sup>1; <sup>n</sup>2; …; nd , we set <sup>v</sup>! <sup>¼</sup> <sup>Q</sup><sup>d</sup> <sup>k</sup>¼<sup>1</sup> vk and <sup>ν</sup><sup>n</sup> <sup>¼</sup> <sup>Q</sup><sup>d</sup> <sup>k</sup>¼<sup>1</sup> <sup>v</sup> nk k .

We introduce the following convenient notations. For a 1 � ð Þ n þ 1 vector Va, b∈ R, and n∈ N

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

We consider a pricing methodology consisting in a two-step procedure. First, conditioning on d � 1 out of the total number of d assets, we find the price of a payoff based on a single asset with a more complex conditional distribution.

Secondly, we consider some expansions of the conditional price, given either in

terms of Taylor, Chebyshev, or cubic spline polynomials, allowing to write the corresponding price as a linear combination of mixed exponential-power moments. This approach is implemented in the context of spread options and a multivariate Merton model, that is, a jump diffusion with Gaussian jumps. Our find-

approximations are comparable in accuracy to a standard Monte Carlo approach with a considerable reduction in computational effort. Among the three expan-

The use of a Taylor expansion to pricing has been considered in the pioneering work of [2] for a vanilla European option and in [3, 4] for spread contracts under a bivariate Black-Scholes model. See also [5]. A Chebyshev expansion has been recently considered in [6]. Applications under a multivariate jump-diffusion model

Although a comparison with alternative approaches is beyond the scope of this paper, it is worth noticing the existence of pricing methods based on Fourier or Hilbert transforms. For example, for spread contracts under a different class of Levy processes, a Fast Fourier transform method can be found in [7]. See also [8] for expansions in terms of Fourier series and [9] for Hilbert

The organization of the paper is as follows: in Section 2, we introduce the model and obtain the pricing expressions for basket contracts under the approximations. In Section 3, we specialize the three expansions in the case of spreads contracts. In Section 4, we discuss the implementation of the methods and present our numerical findings. Finally in Section 5, we present conclusions. Proofs are deferred to the

Let <sup>Ω</sup>; <sup>A</sup>;ð Þ <sup>F</sup><sup>t</sup> <sup>t</sup> <sup>≥</sup> <sup>0</sup>; <sup>P</sup> � � be a filtered probability space. We define the filtration

Xs f g ; 0 ≤ s ≤ t completed in the usual way. Denote by Q an equivalent martingale measure (EMM), respectively, by EQ, φX, and MX the expectation, characteristic, and moment-generating functions of a random variable X under Q. The function f <sup>X</sup>

By r we denote the (constant) interest rate, A ∘ B is the componentwise product

δij is the usual Kronecker's number. The vector Y~ is created from the vector Y after eliminating the first component. For a function f with domain in R<sup>d</sup> and a vector <sup>L</sup> <sup>¼</sup> ð Þ <sup>l</sup>1; <sup>l</sup>2; …; ld with lk <sup>∈</sup> <sup>N</sup>, the symbol <sup>D</sup><sup>L</sup> <sup>f</sup> represents the mixed partial derivative

We introduce the following convenient notations. For a 1 � ð Þ n þ 1 vector Va,

<sup>1</sup> <sup>≤</sup> i,j <sup>≤</sup> <sup>d</sup>, while diagð Þ A is a vector with components ð Þ aii <sup>1</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>d</sup>. The symbol

<sup>k</sup>¼<sup>1</sup> vk and

<sup>F</sup>Xt <sup>≔</sup> <sup>σ</sup> Xs ð Þ ; <sup>0</sup> <sup>≤</sup> <sup>s</sup> <sup>≤</sup> <sup>t</sup> as the <sup>σ</sup>-algebra generated by the random variables

between matrices A and B, and A<sup>0</sup> represents the transpose of matrix

of the function f differentiated lk times w.r.t. the k-th variable.

For vectors <sup>v</sup> <sup>¼</sup> ð Þ <sup>v</sup>1; <sup>v</sup>2; …; vd and <sup>n</sup> <sup>¼</sup> ð Þ <sup>n</sup>1; <sup>n</sup>2; …; nd , we set <sup>v</sup>! <sup>¼</sup> <sup>Q</sup><sup>d</sup>

ings show that, within the range of parameters analyzed, polynomial

sions, cubic splines show the best performance.

Polynomials - Theory and Application

2. Pricing under jump-diffusion models

is its probability density function.

transforms.

appendix.

A ¼ aij � �

<sup>ν</sup><sup>n</sup> <sup>¼</sup> <sup>Q</sup><sup>d</sup>

132

<sup>k</sup>¼<sup>1</sup> <sup>v</sup> nk k .

b∈ R, and n∈ N

have been less explored. Our paper intends to fill this gap.

$$\begin{array}{rcl} \mathit{bin}(n, \mathrm{Va}, b) &=& \sum\_{m=0}^{n} \binom{n}{m} \mathrm{Va}\_{m} b^{n-m} \\\\ \mathrm{PVa} &=& \left(\mathbf{1}, \mathrm{Va}\_{1}, \dots, \mathrm{Va}\_{n}^{n}\right) \end{array}$$

Also, for a differentiable function <sup>f</sup>, we set the vector DVf <sup>¼</sup> <sup>f</sup>; <sup>D</sup><sup>1</sup> <sup>f</sup>; …; Dn <sup>f</sup> � �.

The d-dimensional process of spot prices is denoted by ð Þ St <sup>t</sup> <sup>≥</sup> <sup>0</sup>, while ð Þ Yt <sup>t</sup> <sup>≥</sup> <sup>0</sup> is the corresponding log-price process. They are related by

$$\mathbf{S}\_t^{(j)} = \mathbf{S}\_0^{(j)} \exp\left(Y\_t^{(j)}\right), \ j = \mathbf{1}, \mathbf{2}, \ldots, d\tag{1}$$

We analyze European basket options whose payoff at maturity T, for a strike price K, are given by

$$h(\mathbf{S}\_T) = \left(\sum\_{j=1}^d \omega\_j \mathbf{S}\_T^{(j)} - K\right)\_+$$

where wj � � <sup>1</sup> <sup>≤</sup> <sup>j</sup> <sup>≤</sup> <sup>d</sup> are some deterministic weights and x<sup>þ</sup> ¼ max xð Þ ; 0 .

Furthermore, for the log-prices, we assume a multidimensional jump-diffusion dynamics under Q given by

$$d\mathbf{Y}\_t = \mu d\mathbf{t} + \Sigma^\ddagger d\mathbf{B}\_t + dZ\_t \tag{2}$$

where ð Þ Bt <sup>t</sup> <sup>≥</sup> <sup>0</sup> is a multivariate Brownian motion with independent components and <sup>μ</sup> <sup>¼</sup> <sup>r</sup> � <sup>1</sup> <sup>2</sup> diagð Þ� <sup>Σ</sup> <sup>m</sup>. The matrix <sup>Σ</sup> <sup>¼</sup> <sup>σ</sup>jl � � j,l is symmetric, positive definite, while Σ 1 <sup>2</sup> is such that Σ 1 <sup>2</sup> Σ 1 2 � �<sup>0</sup> ¼ Σ. The value m is the compensator of a compound Poisson process m ¼ log φ<sup>Z</sup><sup>1</sup> ð Þ �i .

We define two sequences of independent and identically distributed 1 � ddimensional random vectors ð Þ Xk <sup>k</sup><sup>∈</sup> <sup>N</sup> and ð Þ X0,k <sup>k</sup><sup>∈</sup> <sup>N</sup>. The components of the random vectors in the first sequence are independent.

The process ð Þ Zt <sup>t</sup> <sup>≥</sup> <sup>0</sup> is a d-variate compound Poisson process, independent of ð Þ Bt <sup>t</sup> <sup>≥</sup> <sup>0</sup> such that

$$Z\_t^{(j)} = \sum\_{k=1}^{N\_t^{(j)}} X\_k^{(j)} + \sum\_{k=1}^{N\_t^{(0)}} X\_{0,k}^{(j)} \quad j = 1, \dots, d$$

where ð Þ Nt <sup>t</sup> <sup>≥</sup> <sup>0</sup> <sup>¼</sup> <sup>N</sup>ð Þ <sup>0</sup> <sup>t</sup> ; Nð Þ<sup>1</sup> <sup>t</sup> ; …; <sup>N</sup>ð Þ <sup>d</sup> t � � <sup>t</sup> <sup>≥</sup> <sup>0</sup> is a vector of independent Poisson processes with respective intensities λj.

The processes Nð Þ<sup>j</sup> t � � t ≥ 0 and Nð Þ <sup>0</sup> t � � <sup>t</sup> <sup>≥</sup> <sup>0</sup> correspond, respectively, to idiosyncratic and common jumps of the j-th underlying asset on the interval 0½ � ; t . Their jump sizes are Xð Þ<sup>j</sup> <sup>k</sup> and <sup>X</sup>ð Þ<sup>j</sup> <sup>0</sup>,k.

For the sake of concreteness, we assume Gaussian jumps, i.e., we assume for any k∈ N that Xk � N μJ; DJ � �, where DJ is a diagonal matrix with components DJð Þ¼ <sup>j</sup>; <sup>l</sup> <sup>δ</sup>jl <sup>σ</sup>ð Þ<sup>j</sup> J � �<sup>2</sup> and X0,k � N μ0,J; Σ0,J � �, with Σ0,J a matrix of components <sup>Σ</sup>0,Jð Þ¼ <sup>j</sup>; <sup>l</sup> <sup>σ</sup> j,l <sup>0</sup> . The compensator across each dimension takes the form

$$m\_j = \lambda\_j^\circ \left( \exp\left(\mu\_j^{(j)} + \frac{1}{2} \left(\sigma\_j^{(j)}\right)^2\right) - \mathbf{1} \right) + \lambda\_0 \left( \exp\left(\mu\_{0,J}^{(j)} + \frac{1}{2} \left(\sigma\_0^{(j)}\right)^2\right) - \mathbf{1} \right), \ j = 1, 2, \ldots, d$$

Let CJD denote the price of a European basket option with payoff h Sð Þ <sup>T</sup> under the model given by Eqs. (1) and (2).

First, we write the price of the basket contract in terms of its conditional price when the number of jumps and d � 1 underlying assets are fixed. Results are given in Theorem 1 below.

Notice that, for any k∈ Ndþ<sup>1</sup>

$$p\_k = P(N\_T = k) = \frac{\exp\left(-\sum\_{j=0}^d \lambda\_j T\right) \prod\_{j=0}^d \lambda\_j^{k\_j} T^{\sum\_{j=0}^d k\_j}}{k!} \tag{3}$$

We also introduce the vector μð Þk with components

$$
\overline{\mu}\_j(k) = \mu\_j T + k\_j \mu\_f^{(j)} + k\_0 \mu\_{0,f}^{(j)} \quad j = 1, 2, \dots, d.
$$

Theorem 1. Let CJD be the price of a European basket contract with maturity T, strike price K, and payoff h Yð Þ <sup>T</sup> , under a model given by Eqs. (1) and (2). See proof in Appendix A.2.

In addition assume Xk � N μJ; DJ � � and X0,k � <sup>N</sup> <sup>μ</sup>0,J; <sup>Σ</sup>0,J � � for any k∈ N, where DJ is a d � d diagonal matrix with components DJð Þ¼ j; l δjl σ ð Þj J � �<sup>2</sup> and Σð Þ <sup>0</sup> <sup>J</sup> is also a

<sup>d</sup> � d matrix with components <sup>Σ</sup>0,Jð Þ¼ <sup>j</sup>; <sup>l</sup> <sup>σ</sup> j,l 0 . Then, we have

$$\mathbf{C}\_{\text{JD}} = \sum\_{k \in \mathbb{N}^{d+1}} \mathbf{C}(k) p\_k \tag{4}$$

Remark 2. Notice that when K yð Þ ; k is nonnegative, C yð Þ ; k is the well-known Black-

and strike price K yð Þ ; k . A sufficient condition for K yð Þ ; k to be positive is w<sup>1</sup> ≥ 0 while wj ≤ 0, 2 ≤ j ≤ d. It is the case of spreads and crack spreads. When K yð Þ ; k is negative, it

Remark 3. The values μð Þ y; k and σð Þk are, respectively, the mean and variance of the first asset after conditioning on a value y of the remaining assets and the certain

variable y by a suitable polynomial. In particular we consider Taylor, Chebyshev

(i) An order <sup>n</sup> Taylor approximation of C yð Þ ; <sup>k</sup> around <sup>y</sup><sup>∗</sup> <sup>∈</sup> <sup>R</sup>d�<sup>1</sup> is described

Approximations based on the three expansions are discussed below.

n l¼0 ∑ L∈Rl

with L ¼ ð Þ l1; l2; …; ld�<sup>1</sup> , where the second sum is taken on the set

<sup>C</sup><sup>T</sup> <sup>y</sup>; <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>k</sup>; <sup>n</sup> <sup>∑</sup>

1 2

¼ 1 2

Rl <sup>¼</sup> <sup>L</sup><sup>∈</sup> <sup>N</sup><sup>d</sup>�<sup>1</sup>

<sup>C</sup>Chð Þ¼ <sup>y</sup>; <sup>k</sup>; <sup>n</sup>

where the sums are taken over the sets

Bn <sup>¼</sup> <sup>l</sup><sup>∈</sup> <sup>N</sup><sup>d</sup>�<sup>1</sup>

Cl <sup>¼</sup> <sup>m</sup> <sup>∈</sup> <sup>N</sup><sup>d</sup>�<sup>1</sup>

, we approximate the conditional price C yð Þ ; k on the

DLC y<sup>∗</sup> ð Þ ; <sup>k</sup>

, we consider an expansion of order n ¼ ð Þ n1; n2; …; nd�<sup>1</sup> of

^clð Þ<sup>k</sup> <sup>T</sup><sup>D</sup>

∑ m ∈Cl <sup>l</sup> ð Þy 1Dð Þy

� �; <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; …; <sup>d</sup> � <sup>1</sup>

<sup>l</sup> . <sup>W</sup>, where , f,g . <sup>W</sup> is the scalar product of func-

^clð Þ<sup>k</sup> bm,ly<sup>l</sup>�2<sup>m</sup>1Dð Þ<sup>y</sup>

<sup>=</sup>l<sup>1</sup> <sup>þ</sup> <sup>l</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> ld�<sup>1</sup> <sup>¼</sup> <sup>l</sup>; <sup>0</sup> <sup>≤</sup> lj <sup>≤</sup> <sup>l</sup> � �:

l∈Bn

l∈Bn

<sup>=</sup><sup>0</sup> <sup>≤</sup> <sup>l</sup> <sup>≤</sup> nj; <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; …; <sup>d</sup> � <sup>1</sup>: � �

lj 2

� �

<sup>l</sup>∈Bn is a family of d � 1-dimensional Chebyshev polynomials with

=0 ≤ mj ≤

approximations of the corresponding Chebyshev coefficients clð Þk , computed using

Notice that, by the orthogonality of the polynomials, the coefficients in the

tions f and g, conveniently weighted by a function W. See, for example, [10] for a

For convenience, we write the Chebyshev polynomials in terms of powers of

In particular, for a rectangular region <sup>D</sup> <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup> and valued vectors

their variables, where bm,l are the coefficients of this expansion.

a ¼ ð Þ a1; a2; …; ad�<sup>1</sup> and b ¼ ð Þ b1; b2; …; bd�<sup>1</sup> , we write

degrees l ∈Bn defined in the region D, while the quantities ^clð Þk are suitable

Notice the existence of the derivatives of any order in the functions K yð Þ and

(ii) An approximation based on Chebyshev polynomials is given as follows:

^c0ð Þk 1Dð Þþ y ∑

^c0ð Þk 1Dð Þþ y ∑

<sup>σ</sup>ð Þ<sup>k</sup> <sup>p</sup> , spot price Sð Þ<sup>1</sup>

<sup>L</sup>! ð Þ <sup>y</sup> � <sup>y</sup><sup>∗</sup> <sup>L</sup> (7)

0 ,

(8)

Scholes price of a call option with maturity at T . 0, volatility ffiffiffiffiffiffiffiffiffi

does not have the meaning of a strike price anymore.

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

number of jumps k.

by

C y ð Þ ; k .

For any fixed k∈ Ndþ<sup>1</sup>

polynomials and cubic splines.

In a region D ⊂ R<sup>d</sup>�<sup>1</sup>

the function C yð Þ ; k as

Here T<sup>D</sup> l � �

135

the trapezoidal rule.

expansion are clð Þ¼ <sup>k</sup> , C, T<sup>D</sup>

general account on Chebyshev polynomials.

where for any k∈ N<sup>d</sup>þ<sup>1</sup>

$$\mathcal{C}(k) := w\_1 \exp\left(\frac{1}{2}\sigma^2(k)T\right) E\_\mathcal{Q}\left[\exp\left(\mu\left(\bar{Y}\_T, N\_T\right)\right)\mathcal{C}\left(\bar{Y}\_T, N\_T\right)/N\_T = k\right] \tag{5}$$

$$\begin{aligned} \mathbf{C}(\boldsymbol{\eta},k) &= e^{-rT} \mathbf{E}\_{\mathcal{Q}} \Big[ \left( \mathbf{S}\_{0}^{(1)} \exp \left( \left( r - \frac{1}{2} \sigma^{2} (\mathbf{N}\_{T}) \right) T \right) \\ &+ \sigma (\mathbf{N}\_{T}) \sqrt{T} \mathbf{Z} \big) ) - \mathbf{K} (\boldsymbol{\tilde{Y}}\_{T}, \mathbf{N}\_{T}) \Big)\_{+} / \mathbf{N}\_{T} = \mathbf{k}, \boldsymbol{\tilde{Y}}\_{T} = \mathbf{y} \Big] \end{aligned} \tag{6}$$

with Z a standard normal random variable independent of NT and <sup>Y</sup>~T. Also

$$\begin{aligned} K(\boldsymbol{y},k) &= \exp\left( \left(r - \frac{1}{2}\sigma^2(k)\right)T - \mu(\boldsymbol{y},k)\right) \left[\frac{K}{\boldsymbol{w}\_1} - \sum\_{j=2}^d \frac{w\_j}{\omega\_1} \boldsymbol{S}\_0^{(j)} \exp\left(\boldsymbol{y}^{(j)}\right)\right], \text{ for } \boldsymbol{y} \in \mathbb{R}^{d-1} \\ \mu(\boldsymbol{y},k) &= \overline{\mu}\_1(\boldsymbol{k}) + \Sigma\_{1\bar{Y}}(\boldsymbol{k})\Sigma\_{\bar{Y}}^{-1}(\boldsymbol{k})(\boldsymbol{y} - \bar{\mu}(\boldsymbol{k}))' \end{aligned} $$
 
$$ \sigma(\boldsymbol{k}) = \frac{1}{T} \left(\sigma\_{11}(\boldsymbol{k}) - \Sigma\_{1\bar{Y}}(\boldsymbol{k})\Sigma\_{\bar{Y}}^{-1}(\boldsymbol{k})\Sigma\_{1\bar{Y}}'(\boldsymbol{k})\right)$$

Here σjlð Þk is the j ð Þ ; l component of the matrix:

$$\begin{aligned} \Sigma\_Y(k) &= \Sigma T + D\_{\bar{f}} \bullet D\_N + k\_0 \Sigma\_{0, \bar{f}} \text{ and} \\ D\_N(\dot{f}, l) &= \delta\_{\bar{l}l} N\_T^{(\bar{j})} \\ \Sigma\_{1\bar{Y}}(k) &= (\sigma\_{12}(k), \sigma\_{13}(k), \dots, \sigma\_{1, d-1}(k))' \end{aligned}$$

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

mj ¼ λ<sup>j</sup> exp μ

in Theorem 1 below.

Appendix A.2.

Then, we have

where for any k∈ N<sup>d</sup>þ<sup>1</sup>

C kð Þ :¼ w<sup>1</sup> exp

K yð Þ¼ ; <sup>k</sup> exp <sup>r</sup> � <sup>1</sup>

<sup>μ</sup>ð Þ¼ <sup>y</sup>; <sup>k</sup> <sup>μ</sup>1ð Þþ <sup>k</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

Also

<sup>σ</sup>ð Þ¼ <sup>k</sup> <sup>1</sup>

134

ð Þj J þ 1 <sup>2</sup> <sup>σ</sup>ð Þ<sup>j</sup> J � �<sup>2</sup> � �

Polynomials - Theory and Application

the model given by Eqs. (1) and (2).

Notice that, for any k∈ Ndþ<sup>1</sup>

pk ¼ P Nð Þ¼ <sup>T</sup> ¼ k

μjð Þ¼ k μ<sup>j</sup>

In addition assume Xk � N μJ; DJ

<sup>d</sup> � d matrix with components <sup>Σ</sup>0,Jð Þ¼ <sup>j</sup>; <sup>l</sup> <sup>σ</sup> j,l

1 2 σ2 ð Þk T � �

C y ð Þ¼ ; <sup>k</sup> <sup>e</sup>�rTE<sup>Q</sup> <sup>S</sup>

2 σ2 ð Þk � �

<sup>T</sup> <sup>σ</sup>11ð Þ� <sup>k</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

þ σð Þ NT

We also introduce the vector μð Þk with components

DJ is a d � d diagonal matrix with components DJð Þ¼ j; l δjl σ

T þ kjμ

ð Þj <sup>J</sup> þ k0μ

CJD ¼ ∑

ð Þ1

ffiffiffi T

�� �

� �

� 1

þ λ<sup>0</sup> exp μ

Let CJD denote the price of a European basket option with payoff h Sð Þ <sup>T</sup> under

First, we write the price of the basket contract in terms of its conditional price when the number of jumps and d � 1 underlying assets are fixed. Results are given

exp �∑<sup>d</sup>

Theorem 1. Let CJD be the price of a European basket contract with maturity T, strike price K, and payoff h Yð Þ <sup>T</sup> , under a model given by Eqs. (1) and (2). See proof in

<sup>j</sup>¼0λjT � � Q<sup>d</sup>

ð Þj

� � and X0,k � <sup>N</sup> <sup>μ</sup>0,J; <sup>Σ</sup>0,J

0 .

k∈ Ndþ<sup>1</sup>

<sup>0</sup> exp <sup>r</sup> � <sup>1</sup>

<sup>p</sup> <sup>Z</sup>ÞÞ � <sup>K</sup>ðY<sup>~</sup> <sup>T</sup>; NT<sup>Þ</sup>

with Z a standard normal random variable independent of NT and <sup>Y</sup>~T.

T � μð Þ y; k

<sup>Y</sup><sup>~</sup> ð Þk ð Þ y � μ~ð Þk <sup>0</sup>

<sup>Y</sup><sup>~</sup> ð Þk Σ<sup>0</sup>

DNð Þ¼ <sup>j</sup>; <sup>l</sup> <sup>δ</sup>jlNð Þ<sup>j</sup>

<sup>1</sup>Y<sup>~</sup> ð Þk

T

ΣYð Þ¼ k ΣT þ DJ ∘ DN þ k0Σ0,J and

Σ1Y<sup>~</sup> ð Þ¼ k ð Þ σ12ð Þk ; σ13ð Þk ; …; σ1,d�<sup>1</sup>ð Þk <sup>0</sup>

� � K

� �

Here σjlð Þk is the j ð Þ ; l component of the matrix:

2 σ2 ð Þ NT � �

> w1 � ∑ d j¼2

�

þ

wj w1 S ð Þj <sup>0</sup> exp <sup>y</sup>ð Þ<sup>j</sup> � � " #

ð Þj 0,J þ 1 2 σ jj 0

� �<sup>2</sup> � �

<sup>j</sup>¼<sup>0</sup> <sup>λ</sup> kj <sup>j</sup> <sup>T</sup><sup>∑</sup><sup>d</sup> <sup>j</sup>¼<sup>0</sup>kj <sup>k</sup>! (3)

<sup>0</sup>,J j ¼ 1, 2, …, d:

� �

� 1

� � for any k∈ N, where

and Σð Þ <sup>0</sup>

<sup>J</sup> is also a

� (6)

, for y∈ R<sup>d</sup>�<sup>1</sup>

ð Þj J � �<sup>2</sup>

C kð Þpk (4)

<sup>E</sup><sup>Q</sup> exp <sup>μ</sup> <sup>Y</sup>~T; NT<sup>Þ</sup> � �<sup>C</sup> <sup>Y</sup>~T; NTÞ=NT <sup>¼</sup> <sup>k</sup> � � � � (5)

T

<sup>=</sup>NT <sup>¼</sup> k, <sup>Y</sup><sup>~</sup> <sup>T</sup> <sup>¼</sup> <sup>y</sup>

, j ¼ 1, 2, …d

Remark 2. Notice that when K yð Þ ; k is nonnegative, C yð Þ ; k is the well-known Black-Scholes price of a call option with maturity at T . 0, volatility ffiffiffiffiffiffiffiffiffi <sup>σ</sup>ð Þ<sup>k</sup> <sup>p</sup> , spot price Sð Þ<sup>1</sup> 0 , and strike price K yð Þ ; k . A sufficient condition for K yð Þ ; k to be positive is w<sup>1</sup> ≥ 0 while wj ≤ 0, 2 ≤ j ≤ d. It is the case of spreads and crack spreads. When K yð Þ ; k is negative, it does not have the meaning of a strike price anymore.

Remark 3. The values μð Þ y; k and σð Þk are, respectively, the mean and variance of the first asset after conditioning on a value y of the remaining assets and the certain number of jumps k.

For any fixed k∈ Ndþ<sup>1</sup> , we approximate the conditional price C yð Þ ; k on the variable y by a suitable polynomial. In particular we consider Taylor, Chebyshev polynomials and cubic splines.

Approximations based on the three expansions are discussed below.

(i) An order <sup>n</sup> Taylor approximation of C yð Þ ; <sup>k</sup> around <sup>y</sup><sup>∗</sup> <sup>∈</sup> <sup>R</sup>d�<sup>1</sup> is described by

$$C^T(\mathbf{y}, \mathbf{y}^\*, k, n) = \sum\_{l=0}^n \sum\_{L \in R\_l} \frac{D^L C(\mathbf{y}^\*, k)}{L!} (\mathbf{y} - \mathbf{y}\*)^L \tag{7}$$

with L ¼ ð Þ l1; l2; …; ld�<sup>1</sup> , where the second sum is taken on the set

$$R\_l = \left\{ L \in \mathbb{N}^{d-1} / l\_1 + l\_2 + \dots + l\_{d-1} = l, \quad 0 \le l\_j \le l \right\}.$$

Notice the existence of the derivatives of any order in the functions K yð Þ and C y ð Þ ; k .

(ii) An approximation based on Chebyshev polynomials is given as follows:

In a region D ⊂ R<sup>d</sup>�<sup>1</sup> , we consider an expansion of order n ¼ ð Þ n1; n2; …; nd�<sup>1</sup> of the function C yð Þ ; k as

$$\begin{split} \mathbf{C}^{\text{Ch}}(\boldsymbol{\jmath},\boldsymbol{k},\boldsymbol{n}) &= \frac{1}{2}\hat{\boldsymbol{c}}\_{0}(\boldsymbol{k})\mathbf{1}\_{D}(\boldsymbol{\jmath}) + \sum\_{l \in B\_{\text{u}}} \hat{\boldsymbol{c}}\_{l}(\boldsymbol{k}) \mathbf{T}\_{l}^{D}(\boldsymbol{\jmath})\mathbf{1}\_{D}(\boldsymbol{\jmath}) \\ &= \frac{1}{2}\hat{\boldsymbol{c}}\_{0}(\boldsymbol{k})\mathbf{1}\_{D}(\boldsymbol{\jmath}) + \sum\_{l \in B\_{\text{u}}} \sum\_{m \in C\_{l}} \hat{\boldsymbol{c}}\_{l}(\boldsymbol{k})\mathbf{b}\_{m,l} \mathbf{\boldsymbol{y}}^{l-2m}\mathbf{1}\_{D}(\boldsymbol{\jmath}) \end{split} \tag{8}$$

where the sums are taken over the sets

$$\begin{array}{rcl} B\_n &=& \left\{ l \in \mathbb{N}^{d-1} / \mathbf{0} \le l \le n\_j; j = \mathbf{1}, 2, \ldots, d-1. \right\} \\ C\_l &=& \left\{ m \in \mathbb{N}^{d-1} / \mathbf{0} \le m\_j \le \left[ \frac{l\_j}{2} \right], j = \mathbf{1}, 2, \ldots, d-1 \right\} \end{array}$$

Here T<sup>D</sup> l � � <sup>l</sup>∈Bn is a family of d � 1-dimensional Chebyshev polynomials with degrees l ∈Bn defined in the region D, while the quantities ^clð Þk are suitable approximations of the corresponding Chebyshev coefficients clð Þk , computed using the trapezoidal rule.

Notice that, by the orthogonality of the polynomials, the coefficients in the expansion are clð Þ¼ <sup>k</sup> , C, T<sup>D</sup> <sup>l</sup> . <sup>W</sup>, where , f,g . <sup>W</sup> is the scalar product of functions f and g, conveniently weighted by a function W. See, for example, [10] for a general account on Chebyshev polynomials.

For convenience, we write the Chebyshev polynomials in terms of powers of their variables, where bm,l are the coefficients of this expansion.

In particular, for a rectangular region <sup>D</sup> <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup> and valued vectors a ¼ ð Þ a1; a2; …; ad�<sup>1</sup> and b ¼ ð Þ b1; b2; …; bd�<sup>1</sup> , we write

$$T^D(\wp) := T\_l^{a,b}(\wp) = T\_l^{-1,1} \left( -1 + 2 \frac{\wp - a}{b - a} \right)$$

Hence, for d ¼ 2

$$\begin{aligned} \mathbf{C}^{Ch}(\mathbf{y}, \mathbf{k}) &= \frac{1}{2} \hat{c}\_0(\mathbf{k}) \mathbf{1}\_D(\mathbf{y}) \\ &+ \sum\_{l=1}^n \sum\_{m=0}^{\left[\frac{l}{2}\right]} (\mathbf{b} - \mathbf{a})^{2m-l} \hat{c}\_l(\mathbf{k}) b\_{m,l} (\mathbf{2y} - (\mathbf{a} + \mathbf{b}))^{l-2m} \mathbf{1}\_D(\mathbf{y}) \end{aligned} \tag{9}$$

See, for example, [11] for specific expressions of bm,l in one dimension.

(iii) Approximation by cubic splines.

On a rectangular region <sup>D</sup> <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup> , we consider an approximation based on cubic splines given by

$$\mathbf{C}^{pl}(\mathbf{y},k) = \sum\_{j=1}^{N} \sum\_{l \in B\_3} a\_{j,l}(k) \left(\mathbf{y} - b\_{j-1}\right)^l \mathbf{1}\_{D\_j}(\mathbf{y}) \tag{10}$$

Dν

and the set

assume Xk � N μJ; DJ

components Σ0,Jð Þ¼ j; l σ

JD <sup>y</sup><sup>∗</sup> ð Þ¼ <sup>w</sup><sup>1</sup> <sup>∑</sup>

CCh JD <sup>¼</sup> <sup>w</sup><sup>1</sup>

<sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

depends on k.

137

k ∈ Ndþ<sup>1</sup> M

is given by

CT

Ndþ<sup>1</sup>

<sup>A</sup>1ð Þ¼ <sup>k</sup> <sup>1</sup>

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

> 2 σ2

<sup>A</sup>3ð Þ¼ <sup>k</sup> <sup>A</sup>1ð Þþ <sup>k</sup> <sup>1</sup>

� � and X0,k � <sup>N</sup> <sup>μ</sup>0,J; <sup>Σ</sup>0,J

diagonal matrix with components DJð Þ¼ j; l δjl σ

∑ n l¼0

for some truncation vector M ∈ N<sup>d</sup>þ<sup>1</sup>

<sup>2</sup> <sup>∑</sup> k ∈ Ndþ<sup>1</sup> M

þ w<sup>1</sup> ∑ k ∈ Ndþ<sup>1</sup> M

<sup>Y</sup><sup>~</sup> ð Þk ; k; a; b � �.

Dl�<sup>2</sup>mMV<sup>~</sup> <sup>T</sup>

Cspl

∑ L ∈Rl

j,l 0 .

<sup>A</sup>2ð Þ¼ <sup>k</sup> <sup>A</sup>1ð Þþ <sup>k</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

MXð Þ¼ <sup>u</sup>; <sup>k</sup>; <sup>D</sup> <sup>E</sup><sup>Q</sup> exp ð Þ uX <sup>X</sup><sup>ν</sup> ½ � <sup>1</sup>Dð Þ <sup>X</sup> <sup>=</sup>NT <sup>¼</sup> <sup>k</sup> , u<sup>∈</sup> <sup>R</sup>d�<sup>1</sup>

ð Þ<sup>k</sup> <sup>T</sup> <sup>þ</sup> <sup>μ</sup>1ð Þ� <sup>k</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

<sup>M</sup> <sup>¼</sup> <sup>k</sup> <sup>¼</sup> ð Þ <sup>k</sup>0; <sup>k</sup>1; …; kd <sup>=</sup>kj <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; …; Mj; <sup>j</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; …; <sup>d</sup> � �

Theorem 4. Let CJD be the price of a European basket contract with maturity T, strike price K, and payoff h Yð Þ <sup>T</sup> under a model given by Eqs. (1) and (2). In addition

Then, its n-th-order approximation around y<sup>∗</sup> ∈ R<sup>d</sup>�<sup>1</sup> in terms of Taylor polynomials

exp ð Þ <sup>A</sup>2ð Þ<sup>k</sup> <sup>D</sup>LC y<sup>∗</sup> ð Þ ; <sup>k</sup>

. The n-th-order Chebyshev approximation on a region D <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup> is

½^c0ð Þk K1ða; b; kÞ

∑ m ∈Cl

<sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

The n-th-order approximation by cubic splines on the region D <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup> is given by

1 2 σ2 ð Þk T

exp <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

Remark 5. The point y<sup>∗</sup> around which the Taylor expansion is taken, in general,

� � �

exp

DmMY<sup>~</sup> �bj�<sup>1</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

where <sup>V</sup><sup>~</sup> <sup>T</sup> <sup>¼</sup> <sup>2</sup>Y~<sup>T</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>b</sup> and K1ð Þ¼ <sup>a</sup>; <sup>b</sup>; <sup>n</sup> exp ð Þ <sup>A</sup>1ð Þ<sup>k</sup> MY<sup>~</sup><sup>T</sup>

k ∈ Ndþ<sup>1</sup> M

∑ l∈Bn

> ð 1 2

JD ¼ w<sup>1</sup> ∑

∑ N j¼1 ∑ l∈B<sup>3</sup> <sup>Y</sup><sup>~</sup> ð Þ<sup>k</sup> <sup>y</sup><sup>∗</sup>

ð Þj J � �<sup>2</sup>

exp ð Þ A3ð Þk ^clð Þk bm,lð Þ b � a

<sup>Y</sup><sup>~</sup> ð Þk ; k; �ð Þ b � a ; b � aÞ�pk

<sup>Y</sup><sup>~</sup> ð Þk bj�<sup>1</sup> � �αj,lð Þ<sup>k</sup>

<sup>Y</sup><sup>~</sup> ð Þk ; k; DjÞ � ipk

ð Þ <sup>a</sup> <sup>þ</sup> <sup>b</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

<sup>Y</sup><sup>~</sup> ð Þk μ~ð Þk <sup>0</sup>

� � for any k<sup>∈</sup> <sup>N</sup>, where DJ is a d � <sup>d</sup>

<sup>L</sup>! <sup>D</sup>LMY<sup>~</sup>T�y<sup>∗</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

<sup>J</sup> be a d � d matrix with

2m�l

<sup>Y</sup><sup>~</sup> ð Þk ; k � �pk

(11)

(12)

(13)

. Let Σð Þ <sup>0</sup>

<sup>Y</sup><sup>~</sup> ð Þk

In order to simplify notations, we introduce the following quantities:

2

where bj is some point on a (d � 1)-dimensional grid f g b0; b1…; bN with N þ 1 points in D.

The local coefficients αj,lð Þk are determined by imposing the conditions C yj ; k <sup>¼</sup> zjk, j, k <sup>¼</sup> <sup>1</sup>, …, N <sup>þ</sup> 1. The family of sets Dj; <sup>j</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; …; <sup>N</sup> is a partition of D. Notice that the coefficients αj,lð Þk depend on the particular rectangle in the grid. See [12] for a general account on multivariate splines.

In the case of d ¼ 2, splines used to approximate the conditional price become one-dimensional polynomials. Additional conditions on the derivatives to smoothen these curves are imposed, namely, D<sup>l</sup> �C yj ; k <sup>¼</sup> <sup>D</sup><sup>l</sup> <sup>þ</sup>C yj ; k , j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, N, <sup>l</sup> <sup>¼</sup> <sup>1</sup>, 2, where D<sup>l</sup> �C yj ; k and <sup>D</sup><sup>l</sup> <sup>þ</sup>C yj ; k are, respectively, the derivatives from the left and the right of the function C yð Þ ; k at point y ¼ yj . Moreover, for end points in the grid, <sup>D</sup><sup>2</sup> <sup>y</sup>0; <sup>k</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup> yN; <sup>k</sup> <sup>¼</sup> 0.

In order to approximate the prices, we replace the function C yð Þ ; k by its respective expansions. The conditional prices on the event ½ � NT ¼ k are estimated by approximating the corresponding conditional expected values. Substituting the approximations of conditional prices into Eq. (4), we obtain, after truncation, estimates of the price of the basket contract, under the jump-diffusion model described by Eqs. (1) and (2). They are denoted, respectively, by C<sup>T</sup> JD <sup>y</sup><sup>∗</sup> ð Þ, <sup>C</sup>Ch JD , and Cspl JD.

Notice that these estimates depend on the mixing exponential-power moments of the log-prices. The latter can be computed from its conditional momentgenerating function under the selected EMM. Hence, for a vector X and a Borel set D, we define

$$\begin{aligned} M\_X(\boldsymbol{u}, k) &= E\_{\mathcal{Q}}[\exp \left( \boldsymbol{u} \mathbf{X} \right) / N\_T = k] \\ M\_X(\boldsymbol{u}, k, D) &= E\_{\mathcal{Q}}[\exp \left( \boldsymbol{u} \mathbf{X} \right) \mathbf{1}\_D(\mathbf{X}) / N\_T = k] \end{aligned}$$

In particular when <sup>D</sup> <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup> , we write MXð Þ¼ u; k; D MXð Þ u; k; a; b .

Concrete expressions of these approximations under a two-dimensional Gaussian model are shown in Theorem 4.

As it is well known, the conditional mixed exponential-power moments of a random vector X are related to the partial derivatives of the corresponding moment-generating function Indeed, for ν∈ N<sup>d</sup>�<sup>1</sup> , we have

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

> Dν MXð Þ¼ <sup>u</sup>; <sup>k</sup>; <sup>D</sup> <sup>E</sup><sup>Q</sup> exp ð Þ uX <sup>X</sup><sup>ν</sup> ½ � <sup>1</sup>Dð Þ <sup>X</sup> <sup>=</sup>NT <sup>¼</sup> <sup>k</sup> , u<sup>∈</sup> <sup>R</sup>d�<sup>1</sup>

In order to simplify notations, we introduce the following quantities:

$$\begin{aligned} A\_1(k) &= \frac{1}{2}\sigma^2(k)T + \overline{\mu}\_1(k) - \Sigma\_{1\hat{Y}}(k)\Sigma\_{\hat{Y}}^{-1}(k)\bar{\mu}(k)^\* \\ A\_2(k) &= A\_1(k) + \Sigma\_{1\hat{Y}}(k)\Sigma\_{\hat{Y}}^{-1}(k)\mathcal{y}^\* \\ A\_3(k) &= A\_1(k) + \frac{1}{2}(a+b)\Sigma\_{1\hat{Y}}(k)\Sigma\_{\hat{Y}}^{-1}(k) \end{aligned}$$

and the set

<sup>T</sup>Dð Þ<sup>y</sup> <sup>≔</sup> <sup>T</sup>a,b

ð Þ b � a

<sup>C</sup>splð Þ¼ <sup>y</sup>; <sup>k</sup> <sup>∑</sup>

grid. See [12] for a general account on multivariate splines.

<sup>þ</sup>C yj ; k 

and the right of the function C yð Þ ; k at point y ¼ yj

2m�l

N j¼1 ∑ l∈B<sup>3</sup>

Hence, for d ¼ 2

CChð Þ¼ <sup>y</sup>; <sup>k</sup> <sup>1</sup>

cubic splines given by

points in D.

C yj ; k 

where D<sup>l</sup>

D, we define

136

�C yj ; k 

2

Polynomials - Theory and Application

þ∑ n l¼1 ∑ l 2½ � m¼0

these curves are imposed, namely, D<sup>l</sup>

grid, <sup>D</sup><sup>2</sup> <sup>y</sup>0; <sup>k</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup> yN; <sup>k</sup> <sup>¼</sup> 0.

and D<sup>l</sup>

(2). They are denoted, respectively, by C<sup>T</sup>

In particular when <sup>D</sup> <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup>

moment-generating function Indeed, for ν∈ N<sup>d</sup>�<sup>1</sup>

ian model are shown in Theorem 4.

(iii) Approximation by cubic splines. On a rectangular region <sup>D</sup> <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup>

^c0ð Þk 1Dð Þy

<sup>l</sup> ð Þ¼ <sup>y</sup> <sup>T</sup>�1, <sup>1</sup>

See, for example, [11] for specific expressions of bm,l in one dimension.

<sup>l</sup> �<sup>1</sup> <sup>þ</sup> <sup>2</sup> <sup>y</sup> � <sup>a</sup>

^clð Þ<sup>k</sup> bm,lð Þ <sup>2</sup><sup>y</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>b</sup> <sup>l</sup>�2m1Dð Þ<sup>y</sup>

αj,lð Þk y � bj�<sup>1</sup> <sup>l</sup>

<sup>¼</sup> zjk, j, k <sup>¼</sup> <sup>1</sup>, …, N <sup>þ</sup> 1. The family of sets Dj; <sup>j</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; …; <sup>N</sup> is a partition

<sup>¼</sup> <sup>D</sup><sup>l</sup>

<sup>þ</sup>C yj ; k 

JD , and Cspl

, we write MXð Þ¼ u; k; D MXð Þ u; k; a; b .

, we have

JD.

are, respectively, the derivatives from the left

where bj is some point on a (d � 1)-dimensional grid f g b0; b1…; bN with N þ 1

The local coefficients αj,lð Þk are determined by imposing the conditions

of D. Notice that the coefficients αj,lð Þk depend on the particular rectangle in the

�C yj ; k 

In the case of d ¼ 2, splines used to approximate the conditional price become one-dimensional polynomials. Additional conditions on the derivatives to smoothen

In order to approximate the prices, we replace the function C yð Þ ; k by its respective expansions. The conditional prices on the event ½ � NT ¼ k are estimated by approximating the corresponding conditional expected values. Substituting the approximations of conditional prices into Eq. (4), we obtain, after truncation, estimates of the price of the basket contract, under the jump-diffusion model described by Eqs. (1) and

JD <sup>y</sup><sup>∗</sup> ð Þ, <sup>C</sup>Ch

Notice that these estimates depend on the mixing exponential-power moments

Concrete expressions of these approximations under a two-dimensional Gauss-

As it is well known, the conditional mixed exponential-power moments of a

random vector X are related to the partial derivatives of the corresponding

generating function under the selected EMM. Hence, for a vector X and a Borel set

MXð Þ¼ u; k EQ½ � exp ð Þ uX =NT ¼ k MXð Þ¼ u; k; D EQ½ � exp ð Þ uX 1Dð Þ X =NT ¼ k

of the log-prices. The latter can be computed from its conditional moment-

b � a

, we consider an approximation based on

ð Þy (10)

, j ¼ 1, 2, …, N, l ¼ 1, 2,

. Moreover, for end points in the

1Dj

(9)

$$\mathbb{N}\_{M}^{d+1} = \left\{ k = (k\_0, k\_1, \dots, k\_d) / k\_j = \mathbf{0}, \mathbf{1}, \dots, \mathbf{M}\_j, j = \mathbf{0}, \mathbf{1}, \dots, d \right\}$$

Theorem 4. Let CJD be the price of a European basket contract with maturity T, strike price K, and payoff h Yð Þ <sup>T</sup> under a model given by Eqs. (1) and (2). In addition assume Xk � N μJ; DJ � � and X0,k � <sup>N</sup> <sup>μ</sup>0,J; <sup>Σ</sup>0,J � � for any k<sup>∈</sup> <sup>N</sup>, where DJ is a d � <sup>d</sup> diagonal matrix with components DJð Þ¼ j; l δjl σ ð Þj J � �<sup>2</sup> . Let Σð Þ <sup>0</sup> <sup>J</sup> be a d � d matrix with components Σ0,Jð Þ¼ j; l σ j,l 0 .

Then, its n-th-order approximation around y<sup>∗</sup> ∈ R<sup>d</sup>�<sup>1</sup> in terms of Taylor polynomials is given by

$$\mathbf{C}\_{\text{lD}}^{T}(\mathbf{y}^{\*}) = w\_{1} \sum\_{k \in \mathbb{N}\_{M}^{l+1}} \sum\_{l=0}^{n} \sum\_{L \in \mathcal{R}\_{l}} \exp\left(A\_{2}(k)\right) \frac{D^{L}\mathbf{C}(\mathbf{y}^{\*},k)}{L!} D^{L}\mathbf{M}\_{\hat{\mathbf{Y}}\_{T}-\mathbf{y}^{\*}} \left(\Sigma\_{\mathbf{1}\hat{\mathbf{Y}}}(k)\Sigma\_{\hat{\mathbf{Y}}}^{-1}(k),k\right) p\_{k} \tag{11}$$

for some truncation vector M ∈ N<sup>d</sup>þ<sup>1</sup> .

The n-th-order Chebyshev approximation on a region D <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup> is

$$\begin{split} \mathbf{C}\_{\text{fD}}^{\text{Ch}} &= \frac{w\_{1}}{2} \sum\_{k \in \mathbb{N}\_{\text{M}}^{d+1}} [\hat{c}\_{0}(k) K\_{1}(a, b, k) \\ &+ w\_{1} \sum\_{k \in \mathbb{N}\_{\text{M}}^{d+1}} \sum\_{l \in \mathbb{B}\_{n}} \sum\_{m \in \mathbb{C}\_{l}} \exp\left(A\_{3}(k)\right) \hat{c}\_{l}(k) b\_{m, l}(b - a)^{2m - l} \\ &\qquad D^{l - 2m} \mathcal{M}\_{\hat{V}\_{\text{T}}} \frac{1}{2} \Sigma\_{1\hat{Y}}(k) \Sigma\_{\hat{Y}}^{-1}(k), k, -(b - a), b - a) [p\_{k}] \end{split} \tag{12}$$

where <sup>V</sup><sup>~</sup> <sup>T</sup> <sup>¼</sup> <sup>2</sup>Y~<sup>T</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>b</sup> and K1ð Þ¼ <sup>a</sup>; <sup>b</sup>; <sup>n</sup> exp ð Þ <sup>A</sup>1ð Þ<sup>k</sup> MY<sup>~</sup><sup>T</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup> <sup>Y</sup><sup>~</sup> ð Þk ; k; a; b � �.

The n-th-order approximation by cubic splines on the region D <sup>¼</sup> ½ � <sup>a</sup>; <sup>b</sup> <sup>d</sup>�<sup>1</sup> is given by

$$\begin{aligned} \mathbf{C}\_{fD}^{pl} &= w\_1 \sum\_{k \in \mathbb{N}\_M^{d+1}} \left[ \exp\left(\frac{1}{2} \sigma^2(k) T\right) \right. \\ &\sum\_{j=1}^N \sum\_{l \in B\_{\hat{Y}}} \exp\left(\Sigma\_{1\hat{Y}}(k) \Sigma\_{\hat{Y}}^{-1}(k) b\_{j-1}\right) a\_{j,l}(k) \\ &D^m \mathcal{M}\_{\check{Y}-b\_{j-1}} \left(\Sigma\_{1\hat{Y}}(k) \Sigma\_{\hat{Y}}^{-1}(k), k, D\_j\right) p\_k \end{aligned} \tag{13}$$

Remark 5. The point y<sup>∗</sup> around which the Taylor expansion is taken, in general, depends on k.

#### 3. Approximating the price of spread contracts

Spread contracts are the most common basket derivatives. In this case the payoff is written as h Sð Þ¼ <sup>T</sup> S ð Þ1 <sup>T</sup> � S ð Þ2 <sup>T</sup> � K � � þ .

Hence for <sup>d</sup> <sup>¼</sup> 2, conditionally on <sup>Y</sup>ð Þ<sup>2</sup> <sup>T</sup> ¼ y h i <sup>∩</sup> ½ � NT <sup>¼</sup> <sup>k</sup> , the log-prices of the first asset are normally distributed, i.e., Yð Þ<sup>1</sup> <sup>T</sup> � <sup>N</sup> <sup>μ</sup>ð Þ <sup>y</sup>; <sup>k</sup> ; <sup>σ</sup><sup>2</sup> ð Þ ð Þ<sup>k</sup> , with

$$\begin{aligned} \mu(\boldsymbol{y},k) &= \overline{\mu}\_1(k) + \sqrt{\frac{\sigma\_{11}(k)}{\sigma\_{22}(k)}} \overline{\rho}(k) (\boldsymbol{y} - \overline{\mu}\_2(k)) \\ \sigma^2(k) &= \frac{1}{T} \left( \sigma\_{11}(k) - \frac{\sigma\_{12}^2(k)}{\sigma\_{22}(k)} \right) = \frac{1}{T} \left[ 1 - \left( \overline{\rho}(k) \right)^2 \right] \sigma\_{11}(k) \end{aligned}$$

Next, we obtain the Taylor approximations up to third order. By elementary calculation we can compute the derivatives of the function C yð Þ ; k with respect to y.

�rTN d<sup>ð</sup> <sup>2</sup>ð Þ K yð Þ ; <sup>k</sup> �

<sup>σ</sup>ð Þ<sup>k</sup> ffiffiffi T p

<sup>σ</sup>ð Þ<sup>k</sup> <sup>T</sup> <sup>p</sup> <sup>T</sup>1ð Þ <sup>y</sup>; <sup>k</sup> A yð Þ ; <sup>k</sup>

T

<sup>p</sup> <sup>e</sup>�rTK yð Þ ; <sup>k</sup> N dð Þ <sup>2</sup>ð Þ K yð Þ ; <sup>k</sup>

<sup>T</sup>1ð Þ <sup>y</sup>; <sup>k</sup> Dn�l�<sup>1</sup>

A yð Þ ; k

<sup>T</sup> and <sup>V</sup>ð Þ<sup>2</sup> T

(17)

<sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup> (16)

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup>

2

<sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup>

<sup>ρ</sup>ð Þ<sup>k</sup> <sup>μ</sup>2ð Þþ <sup>k</sup> <sup>1</sup>

!

<sup>þ</sup> <sup>r</sup> <sup>þ</sup> <sup>σ</sup>ð Þ<sup>k</sup> 2 � �<sup>T</sup>

> T p

<sup>0</sup> N d1ð Þ� K yð Þ ; k K yð Þ ; k e

log <sup>S</sup>

<sup>d</sup>2ð Þ¼ K yð Þ ; <sup>k</sup> <sup>d</sup>1ð Þ� K yð Þ ; <sup>k</sup> <sup>σ</sup>ð Þ<sup>k</sup> ffiffiffi

C yð Þ¼� ; <sup>k</sup> <sup>1</sup>

<sup>0</sup> <sup>f</sup> <sup>Z</sup>ð Þþ <sup>d</sup>1ð Þ K yð Þ ; <sup>k</sup> <sup>σ</sup>ð Þ<sup>k</sup> ffiffiffi

n�1 l¼0

n � 1 l � �Dl

Concrete expressions for second- and third-order derivatives are shown in the

Regarding the approximation based on Chebyshev polynomials, we first com-

<sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup> , <sup>a</sup><sup>~</sup> <sup>¼</sup> <sup>a</sup> � <sup>μ</sup>2ð Þ<sup>k</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � �

> ffiffiffiffiffiffiffiffiffiffiffiffi σ11ð Þk σ22ð Þk

<sup>b</sup> � ffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> � �

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð ÞÞ <sup>k</sup> ; ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>b</sup> � ffiffiffiffiffiffiffiffiffiffiffiffi

s

1 2

� � �

¼ exp

N ~

�e�rTK yð Þ ; <sup>k</sup> <sup>f</sup> <sup>Z</sup>ð Þ <sup>d</sup>2ð Þ K yð Þ ; <sup>k</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>ð Þ<sup>k</sup> <sup>T</sup> <sup>p</sup> <sup>∑</sup>

pute the moment-generating function of the random variables Yð Þ<sup>2</sup>

<sup>b</sup> <sup>¼</sup> <sup>b</sup> � <sup>μ</sup>2ð Þ<sup>k</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> ; <sup>~</sup>

ð Þ1 0 K yð Þ ; k !

and Nð Þ: is the cumulative distribution function of a standard normal distribution.

ffiffiffiffiffiffiffiffiffiffiffiffi

First, notice that, from the Black-Scholes pricing formula:

ð Þ1

d1ð Þ¼ K yð Þ ; k

D1

K yð Þ ; k K yð Þ ; k

Higher derivatives can be calculated recursively.

constrained to the interval ð Þ a; b . To this end we denote

~

b � � <sup>¼</sup> exp

ρð Þk ; k; �∞; b

Notice that, taking into account Eq. (14),

ffiffiffiffiffiffi <sup>σ</sup><sup>11</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> ; <sup>a</sup>~; <sup>~</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi σ11ð Þk σ22ð Þk

s

bin m; <sup>μ</sup> <sup>m</sup>1; <sup>a</sup><sup>~</sup> � ffiffiffiffiffiffiffiffiffiffiffiffi

!

ð Þ1

<sup>D</sup>nC yð Þ¼� ; <sup>k</sup> <sup>1</sup>

C yð Þ¼ ; k S

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

where

Hence

where

appendix.

DmMZ

Moreover

MYð Þ<sup>2</sup> T

139

<sup>T</sup>1ð Þ¼ <sup>y</sup>; <sup>k</sup> <sup>D</sup><sup>1</sup>

A yð Þ¼ ; k S

where

$$\overline{\rho}(k) = \frac{\sigma\_{12}(k)}{\sqrt{\sigma\_{11}(k)}\sqrt{\sigma\_{22}(k)}}$$

is the conditional correlation coefficient between the two assets.

A result about the derivatives of the moment-generating function of a constrained standard normal random variable Z on the interval ð Þ �∞; b is needed. To this end we have

$$\begin{split} D^{\mathfrak{m}} M\_{\mathbb{Z}} \Big( \sqrt{\sigma\_{11}(k)} \overline{\rho}(k), k, -\infty, b \right) &= \exp \left( \frac{1}{2} \sigma\_{11}(k) (\overline{\rho}(k))^2 \right) \\ \min \Big( m, \mu V \Big( -\infty, b - \sqrt{\sigma\_{11}(k)} \overline{\rho}(k) \big), \sqrt{\sigma\_{11}(k)} \overline{\rho}(k) \Big) \end{split} \tag{14}$$

where <sup>μ</sup>ð Þ¼ <sup>m</sup>; <sup>a</sup>; <sup>b</sup> <sup>μ</sup>ð Þ� <sup>m</sup>; �∞; <sup>b</sup> <sup>μ</sup>ð Þ¼ <sup>m</sup>; �∞; <sup>a</sup> <sup>Ð</sup> <sup>b</sup> <sup>a</sup> <sup>z</sup>mf <sup>Z</sup>ð Þ<sup>z</sup> dz is the <sup>m</sup>-th moment of a standard normal random variable constrained to the interval ð Þ a; b and μV að Þ ; b is a vector with components μð Þ j; a; b , j ¼ 0, 1, …, m .

By integration by parts, the later can be calculated recursively as

$$\begin{aligned} \mu(0, a, b) &= N(b) - N(a) \\ \mu(1, a, b) &= f\_Z(a) - f\_Z(b) \\ \mu(m, a, b) &= (m - 1)\mu(m - 2, a, b) + a^{m - 1} f\_Z(a) - b^{m - 1} f\_Z(b)), \quad m \ge 2 \end{aligned}$$

For a Taylor expansion, derivatives of the moment-generating function and constrained moment-generating function for the second component of the log-prices are computed as follows:

$$D^l M\_{Y\_T^{(2)} - \mathcal{Y}^\*} \left( \sqrt{\frac{\sigma\_{11}(k)}{\sigma\_{22}(k)}} \overline{\rho}(k), k \right) = \exp \left( \sqrt{\frac{\sigma\_{11}(k)}{\sigma\_{22}(k)}} \overline{\rho}(k) (\overline{\mu}\_2(k) - \mathcal{Y}^\*) \right),$$

$$\text{bin} \left( l, PV \left( \sqrt{\sigma\_{22}(k)} \right) \mathbf{1} \circ DV \left( M\_Z \left( \sqrt{\sigma\_{11}(k)} \overline{\rho}(k) \right) \right), \overline{\mu}\_2(k) - \mathcal{Y}^\* \right)$$

Now, combining the expressions above with Eq. (11), we have

$$\begin{split} \mathbf{C}\_{l\mathcal{D}}^{T}(\boldsymbol{y}^{\*}) &= \boldsymbol{w}\_{1} \sum\_{k \in \mathbb{N}\_{\rm d}^{3}} \exp\left(\frac{1}{2}\sigma(k)T + \overline{\mu}\_{1}(k)\right) \sum\_{l=0}^{n} \binom{l}{m} \binom{D^{l}C(\boldsymbol{y}^{\*},k)}{l!} (\overline{\mu}\_{2}(k) - \boldsymbol{y}^{\*})^{l-m} \\ &\quad \cdot \text{bin}\left(l, \text{PV}\left(\sqrt{\sigma\_{22}(k)}\right) \mathbf{1} \circ DV\left(M\_{\mathcal{Z}}\left(\sqrt{\sigma\_{11}(k)} \overline{\rho}(k)\right)\right), \overline{\mu}\_{2}(k) - \boldsymbol{y}^{\*}\right) p\_{k} \end{split} \tag{15}$$

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

Next, we obtain the Taylor approximations up to third order. By elementary

calculation we can compute the derivatives of the function C yð Þ ; k with respect to y. First, notice that, from the Black-Scholes pricing formula:

$$\mathcal{C}(\boldsymbol{y},k) = \mathcal{S}\_0^{(1)}N\left(d\_1(K(\boldsymbol{y},k)) - K(\boldsymbol{y},k)e^{-rT}N(d\_2(K(\boldsymbol{y},k)))\right)$$

where

3. Approximating the price of spread contracts

μð Þ¼ y; k μ1ð Þþ k

<sup>σ</sup>2ð Þ¼ <sup>k</sup> <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi

bin m; <sup>μ</sup><sup>V</sup> �∞; <sup>b</sup> � ffiffiffiffiffiffiffiffiffiffiffiffi

where <sup>μ</sup>ð Þ¼ <sup>m</sup>; <sup>a</sup>; <sup>b</sup> <sup>μ</sup>ð Þ� <sup>m</sup>; �∞; <sup>b</sup> <sup>μ</sup>ð Þ¼ <sup>m</sup>; �∞; <sup>a</sup> <sup>Ð</sup> <sup>b</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi σ11ð Þk σ22ð Þk

!

<sup>σ</sup>22ð Þ<sup>k</sup> � � <sup>p</sup> <sup>1</sup> <sup>∘</sup> DV MZ

1 2

<sup>σ</sup>22ð Þ<sup>k</sup> � � <sup>p</sup> <sup>1</sup> <sup>∘</sup> DV MZ

s

μV að Þ ; b is a vector with components μð Þ j; a; b , j ¼ 0, 1, …, m .

ð Þ1 <sup>T</sup> � S ð Þ2 <sup>T</sup> � K � �

Hence for <sup>d</sup> <sup>¼</sup> 2, conditionally on <sup>Y</sup>ð Þ<sup>2</sup>

asset are normally distributed, i.e., Yð Þ<sup>1</sup>

is written as h Sð Þ¼ <sup>T</sup> S

Polynomials - Theory and Application

where

To this end we have

DmMZ

μð Þ¼ 0; a; b N bð Þ� N að Þ μð Þ¼ 1; a; b f <sup>Z</sup>ð Þ� a f <sup>Z</sup>ð Þ b

log-prices are computed as follows:

bin l; PV ffiffiffiffiffiffiffiffiffiffiffiffi

k ∈ N<sup>3</sup> M exp

bin l; PV ffiffiffiffiffiffiffiffiffiffiffiffi

Dl MYð Þ<sup>2</sup> <sup>T</sup> �y∗

JD <sup>y</sup><sup>∗</sup> ð Þ¼ <sup>w</sup><sup>1</sup> <sup>∑</sup>

CT

138

Spread contracts are the most common basket derivatives. In this case the payoff

<sup>T</sup> ¼ y h i

> ffiffiffiffiffiffiffiffiffiffiffiffi σ11ð Þk σ22ð Þk

> > <sup>12</sup>ð Þk σ22ð Þk

<sup>σ</sup>11ð Þ<sup>k</sup> p ffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup>

<sup>T</sup> � <sup>N</sup> <sup>μ</sup>ð Þ <sup>y</sup>; <sup>k</sup> ; <sup>σ</sup><sup>2</sup> ð Þ ð Þ<sup>k</sup> , with

ρð Þk ð Þ y � μ2ð Þk

¼ 1

1 2

ffiffiffiffiffiffiffiffiffiffiffiffi σ11ð Þk σ22ð Þk

s

ffiffiffiffiffiffiffiffiffiffiffiffi

∑ n l¼0

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> � � � � ; <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> � �pk

l m

ffiffiffiffiffiffiffiffiffiffiffiffi

Dl

C y<sup>∗</sup> ð Þ ; <sup>k</sup> l! � �

!

∩ ½ � NT ¼ k , the log-prices of the first

σ11ð Þk

(14)

<sup>T</sup> <sup>1</sup> � ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> h i

<sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � �

<sup>a</sup> <sup>z</sup>mf <sup>Z</sup>ð Þ<sup>z</sup> dz is the <sup>m</sup>-th

f <sup>Z</sup>ð ÞÞ b , m ≥ 2

<sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ<sup>l</sup>�<sup>m</sup>

(15)

<sup>ρ</sup>ð Þ<sup>k</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ

!

þ .

s

� �

<sup>T</sup> <sup>σ</sup>11ð Þ� <sup>k</sup> <sup>σ</sup><sup>2</sup>

<sup>ρ</sup>ð Þ¼ <sup>k</sup> <sup>σ</sup>12ð Þ<sup>k</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

is the conditional correlation coefficient between the two assets. A result about the derivatives of the moment-generating function of a constrained standard normal random variable Z on the interval ð Þ �∞; b is needed.

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> ; <sup>k</sup>; �∞; <sup>b</sup> � � <sup>¼</sup> exp

By integration by parts, the later can be calculated recursively as

<sup>μ</sup>ð Þ¼ <sup>m</sup>; <sup>a</sup>; <sup>b</sup> ð Þ <sup>m</sup> � <sup>1</sup> <sup>μ</sup>ð Þþ <sup>m</sup> � <sup>2</sup>; <sup>a</sup>; <sup>b</sup> am�<sup>1</sup><sup>f</sup> <sup>Z</sup>ð Þ� <sup>a</sup> <sup>b</sup><sup>m</sup>�<sup>1</sup>

ρð Þk ; k

Now, combining the expressions above with Eq. (11), we have

σð Þk T þ μ1ð Þk � �

constrained moment-generating function for the second component of the

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> � �; ffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> � �

moment of a standard normal random variable constrained to the interval ð Þ a; b and

For a Taylor expansion, derivatives of the moment-generating function and

¼ exp

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> � � � � ; <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> � �

$$\begin{aligned} d\_1(K(\boldsymbol{y},k)) &= \frac{\log\left(\frac{S\_0^{(1)}}{K(\boldsymbol{y},k)}\right) + \left(r + \frac{\sigma(k)}{2}\right)T}{\sigma(k)\sqrt{T}} \\ d\_2(K(\boldsymbol{y},k)) &= d\_1(K(\boldsymbol{y},k)) - \sigma(k)\sqrt{T} \end{aligned}$$

and Nð Þ: is the cumulative distribution function of a standard normal distribution. Hence

$$D^1 \mathcal{C}(\mathcal{y}, k) = -\frac{1}{\sqrt{\sigma(k)T}} T\_1(\mathcal{y}, k) A(\mathcal{y}, k)$$

where

$$\begin{aligned} T\_1(\boldsymbol{y},k) &= \frac{D^1 K(\boldsymbol{y},k)}{K(\boldsymbol{y},k)} \\ A(\boldsymbol{y},k) &= \mathcal{S}\_0^{(1)} f\_Z(d\_1(K(\boldsymbol{y},k))) + \sigma(k)\sqrt{T}e^{-rT}K(\boldsymbol{y},k)N(d\_2(K(\boldsymbol{y},k))) \\ &- e^{-rT}K(\boldsymbol{y},k)f\_Z(d\_2(K(\boldsymbol{y},k))) \end{aligned}$$

Higher derivatives can be calculated recursively.

$$D^n C(\boldsymbol{\jmath}, k) = -\frac{1}{\sqrt{\sigma(k)T}} \sum\_{l=0}^{n-1} \binom{n-1}{l} D^l T\_1(\boldsymbol{\jmath}, k) D^{n-l-1} A(\boldsymbol{\jmath}, k)$$

Concrete expressions for second- and third-order derivatives are shown in the appendix.

Regarding the approximation based on Chebyshev polynomials, we first compute the moment-generating function of the random variables Yð Þ<sup>2</sup> <sup>T</sup> and <sup>V</sup>ð Þ<sup>2</sup> T constrained to the interval ð Þ a; b . To this end we denote

$$\tilde{b} = \frac{b - \overline{\mu}\_2(k)}{\sqrt{\sigma\_{22}(k)}}, \quad \tilde{a} = \frac{a - \overline{\mu}\_2(k)}{\sqrt{\sigma\_{22}(k)}} \tag{16}$$

Notice that, taking into account Eq. (14),

$$\begin{split} D^m M\_Z \left( \sqrt{\sigma\_{11}} \overline{\rho}(k), \tilde{a}, \tilde{b} \right) &= \exp \left( \frac{1}{2} \sigma\_{11}(k) (\overline{\rho}(k))^2 \right) \\ \lim \left( m, \mu \left( m\_1, \tilde{a} - \sqrt{\sigma\_{11}(k)} \overline{\rho}(k), \tilde{b} - \sqrt{\sigma\_{11}(k)} \overline{\rho}(k) \right), \sqrt{\sigma\_{11}(k)} \overline{\rho}(k) \right) \end{split} \tag{17}$$

Moreover

$$\begin{aligned} M\_{Y\_T^{(2)}}\left(\sqrt{\frac{\sigma\_{11}(k)}{\sigma\_{22}(k)}}\overline{\rho}(k), k, -\infty, b\right) &= \exp\left(\sqrt{\frac{\sigma\_{11}(k)}{\sigma\_{22}(k)}}\overline{\rho}(k)\overline{\mu}\_2(k) + \frac{1}{2}\sigma\_{11}(k)\left(\overline{\rho}(k)\right)^2\right) \\ N\left(\bar{b} - \sqrt{\sigma\_{11}(k)}\overline{\rho}(k)\right) \end{aligned}$$

Hence

$$\begin{aligned} D^\nu \mathcal{M}\_{V\_T^{(2)}} & \left( \frac{1}{2} \sqrt{\frac{\sigma\_{11}(k)}{\sigma\_{22}(k)}} \overline{\rho}(k), k, -(b-a), b-a \right) \\ &= \exp \left( \frac{1}{2} \sqrt{\frac{\sigma\_{11}(k)}{\sigma\_{22}(k)}} \overline{\rho}(k) (2\overline{\mu}\_2(k) - a - b) \right) \mathcal{G}(\nu, k), \end{aligned}$$

Although these values are somehow arbitrary, they have been selected to produce reasonable asset prices in connection with contracts based on crude oil prices. It is worth noting that there is not a general agreement about the range of the parameters in a jump-diffusion model. Indeed they may depend on the market into consideration. In Table 1 prices of spread contracts under different methods are shown. Prices are obtained using Taylor and cubic splines approximations and contrasted with a Monte Carlo approach. For the latter we carry 10<sup>7</sup> repetitions to achieve stable results, with a relative average error of 0.1%. In addition, 95% Monte Carlo confidence intervals and running times are provided. Implementation is done on a

Prices obtained using the benchmark parameter set and Monte Carlo, first- and second-order Taylor, and cubic

Price 14.7784 10.2980 14.29068 14.8842 Interval ð Þ 14:7683; 14:7885 — —— Run time 624.312 1.68806 1.68806 54.1720

In row three the average computer time (in seconds) for different pricing methods is shown.

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

MC Taylor (f.o) Taylor (s.o.) Spl.

The efficiency of the Monte Carlo method can be improved by considering only the simulation of a single asset with the corresponding conditional probability and then computing the discounted average of the conditional Black-Scholes price. It reduces the computational time by half, still considerably higher than those based on polynomial expansions. Chebyshev polynomial approximation is discussed in [1]. The expansions also require repetitive evaluations of conditional prices, which

For a Taylor approach of order n, evaluations in the order of nM<sup>3</sup> are needed, where M is the maximum truncation level in the number of jumps. In a Chebyshev

be performed, when a grid of N points is used in a trapezoidal approximation of the

For a theoretical analysis of the error using Taylor and Chebyshev expansions, although in different contexts, see [13] for Taylor and [6] for Chebyshev cases. In Figure 1a, a graph of conditional prices in function of log-price values of the first asset (blue line) with average number of jumps equal to k<sup>0</sup> ¼ 3 and k<sup>1</sup> ¼ k<sup>2</sup> ¼ 2

(a) Conditional price (blue curve) as function of log-price values and its Taylor approximations up to third

order around the average. (b) Conditional price vs. its cubic spline approximation.

number of points in the grid where the polynomial coefficients are adjusted.

, evaluations of the conditional price should

. Here N is also the

Surface Pro 4 i7 computer, using MATLAB language.

approach of the same order about n<sup>2</sup>NM<sup>3</sup>

Table 1.

Figure 1.

141

spline approximations.

turn out to be given by simple Black-Scholes closed formulas.

corresponding integrals. In a cubic splines approximation 3NM<sup>3</sup>

where

$$\mathcal{G}(\nu,k) = \text{bin}\left(\nu, M\_Z\left(\sqrt{\sigma\_{11}(k)}\overline{\rho}(k), k, \vec{a}, \tilde{b}\right) \circ PV\left(2\sigma\_{22}^{\ddagger}(k)\mathbf{1}\right), 2\overline{\mu}\_2(k) - a - b\right),$$

Then, combining Eq. (12) with the results above, we get

$$\begin{aligned} \mathbf{C}^{\text{Ch}}(k,n) &= \frac{w\_1}{2}\hat{c}\_0(k)K\_1(a,b,k) \\ &+ w\_1 \exp\left(\frac{1}{2}\sigma(k)T + \overline{\mu}\_1(k)\right) \sum\_{l=1}^n \sum\_{m=0}^{\left[\frac{l}{2}\right]} \hat{c}\_l(k)b\_{m,l}K(a,b,l,m)G(l-2m,k) \end{aligned}$$

Finally, the n-th-order Chebyshev approximation is given by

$$C\_{jD}^{Ch} = \sum\_{k \in \mathbb{N}\_M^3} C^{Ch}(k, n) p\_k$$

Similarly for a cubic spline approximation, we specialize Eq. (13) with D ¼ ð Þ a; b , Dj ¼ bj�<sup>1</sup>; bj � �, b<sup>0</sup> <sup>¼</sup> a, bNþ<sup>1</sup> <sup>¼</sup> <sup>b</sup>. Therefore, we have

$$\begin{split} C\_{\text{fD}}^{pl} &= w\_1 \sum\_{k \in \mathbb{N}\_M^{d+1}} \left[ \exp\left(\frac{1}{2}\sigma(k)T + \overline{\mu}\_1(k)\right) \right. \\ &\sum\_{j=1}^N \sum\_{l=0}^3 a\_{j,l}(k) (\sigma\_{22}(k))^{\frac{l}{2}} \text{bin}(l, \text{DVM}\_Z(\sqrt{\sigma\_{11}(k)}\overline{\rho}(k), \tilde{b}\_{j-1}, \tilde{b}\_j), -\tilde{b}\_{j-1}) \right] p\_k \end{split} \tag{18}$$

where ~ bj is defined as <sup>~</sup> b in Eq. (16) but replacing b by bj.

#### 4. Numerical results

We implement the results from the previous section to price spread contracts and show that the approximations considered above produce accurate price values when compared with a standard Monte Carlo approach, at a lesser computational effort.

To this end we consider the following benchmark set of parameters:

The contract specifications consist a strike price of K ¼ \$1, maturity T ¼ 1 year, spot prices S ð Þ1 <sup>0</sup> ¼ \$100, S ð Þ2 <sup>0</sup> ¼ \$96, and a fix interest rate of 3%.

Volatilities corresponding to the diffusion part of both assets are σ<sup>1</sup> ¼ 10% and σ<sup>2</sup> ¼ 30%, while the correlation coefficient between the two Brownian noises is ρ ¼ 0:3. Regarding the jump part, we consider an average intensity of the common jumps equal to λ<sup>0</sup> ¼ 3 jumps per year and idiosyncratic intensities λ<sup>1</sup> ¼ λ<sup>2</sup> ¼ 2 jumps per year for the respective assets, while jump sizes have means equal to zero; volatilities of common jump sizes are σ0, <sup>1</sup> ¼ 1%, σ0,<sup>2</sup> ¼ 5%, with a linear correlation ρ<sup>J</sup> ¼ 0:5. Volatilities of the idiosyncratic jumps are taken as σJ, <sup>1</sup> ¼ 10% and σJ, <sup>2</sup> ¼ 20%.

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383


#### Table 1.

Hence

where

Gð Þ¼ ν; k bin ν; MZ

w1

þ w<sup>1</sup> exp

CChð Þ¼ <sup>k</sup>; <sup>n</sup>

D ¼ ð Þ a; b , Dj ¼ bj�<sup>1</sup>; bj

JD ¼ w<sup>1</sup> ∑

∑ N j¼1 ∑ 3 l¼0

4. Numerical results

ð Þ1

<sup>0</sup> ¼ \$100, S

ð Þ2

where ~

spot prices S

and σJ, <sup>2</sup> ¼ 20%.

140

k ∈ Ndþ<sup>1</sup> M

exp

<sup>α</sup>j,lð Þ<sup>k</sup> ð Þ <sup>σ</sup>22ð Þ<sup>k</sup> <sup>l</sup>

bj is defined as <sup>~</sup>

1 2

Cspl

Dν MVð Þ<sup>2</sup> T

Polynomials - Theory and Application

1 2

¼ exp

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> ; <sup>k</sup>; <sup>a</sup>~; <sup>~</sup>

Then, combining Eq. (12) with the results above, we get

σð Þk T þ μ1ð Þk � �

Finally, the n-th-order Chebyshev approximation is given by

σð Þk T þ μ1ð Þk � � �

Similarly for a cubic spline approximation, we specialize Eq. (13) with

� �, b<sup>0</sup> <sup>¼</sup> a, bNþ<sup>1</sup> <sup>¼</sup> <sup>b</sup>. Therefore, we have

<sup>2</sup>binðl; DVMZ<sup>ð</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

b in Eq. (16) but replacing b by bj.

We implement the results from the previous section to price spread contracts and show that the approximations considered above produce accurate price values when compared with a standard Monte Carlo approach, at a lesser computational effort. To this end we consider the following benchmark set of parameters:

The contract specifications consist a strike price of K ¼ \$1, maturity T ¼ 1 year,

volatilities of common jump sizes are σ0, <sup>1</sup> ¼ 1%, σ0,<sup>2</sup> ¼ 5%, with a linear

correlation ρ<sup>J</sup> ¼ 0:5. Volatilities of the idiosyncratic jumps are taken as σJ, <sup>1</sup> ¼ 10%

<sup>0</sup> ¼ \$96, and a fix interest rate of 3%. Volatilities corresponding to the diffusion part of both assets are σ<sup>1</sup> ¼ 10% and σ<sup>2</sup> ¼ 30%, while the correlation coefficient between the two Brownian noises is ρ ¼ 0:3. Regarding the jump part, we consider an average intensity of the common jumps equal to λ<sup>0</sup> ¼ 3 jumps per year and idiosyncratic intensities λ<sup>1</sup> ¼ λ<sup>2</sup> ¼ 2 jumps per year for the respective assets, while jump sizes have means equal to zero;

CCh JD ¼ ∑ k∈ N<sup>3</sup> M

<sup>2</sup> ^c0ð Þ<sup>k</sup> <sup>K</sup>1ð Þ <sup>a</sup>; <sup>b</sup>; <sup>k</sup>

1 2 s

ffiffiffiffiffiffiffiffiffiffiffiffi σ11ð Þk σ22ð Þk

> 1 2

s

ρð Þk ; k; �ð Þ b � a ; b � a

!

bÞ∘ PV 2σ

� � �

∑ n l¼1 ∑ l 2½ � m¼0

<sup>C</sup>Chð Þ <sup>k</sup>; <sup>n</sup> pk

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> ; <sup>~</sup>

bj�<sup>1</sup>; <sup>~</sup> bj <sup>Þ</sup>; �<sup>~</sup> bj�<sup>1</sup>Þ # pk

ρð Þk ð Þ 2μ2ð Þ� k a � b

1 2 <sup>22</sup>ð Þk 1 � � Gð Þ ν; k

; 2μ2ð Þ� k a � b

^clð Þk bm,lK að Þ ; b; l; m G lð Þ � 2m; k

(18)

!

ffiffiffiffiffiffiffiffiffiffiffiffi σ11ð Þk σ22ð Þk

> Prices obtained using the benchmark parameter set and Monte Carlo, first- and second-order Taylor, and cubic spline approximations.

> Although these values are somehow arbitrary, they have been selected to produce reasonable asset prices in connection with contracts based on crude oil prices. It is worth noting that there is not a general agreement about the range of the parameters in a jump-diffusion model. Indeed they may depend on the market into consideration.

> In Table 1 prices of spread contracts under different methods are shown. Prices are obtained using Taylor and cubic splines approximations and contrasted with a Monte Carlo approach. For the latter we carry 10<sup>7</sup> repetitions to achieve stable results, with a relative average error of 0.1%. In addition, 95% Monte Carlo confidence intervals and running times are provided. Implementation is done on a Surface Pro 4 i7 computer, using MATLAB language.

> The efficiency of the Monte Carlo method can be improved by considering only the simulation of a single asset with the corresponding conditional probability and then computing the discounted average of the conditional Black-Scholes price. It reduces the computational time by half, still considerably higher than those based on polynomial expansions. Chebyshev polynomial approximation is discussed in [1].

> The expansions also require repetitive evaluations of conditional prices, which turn out to be given by simple Black-Scholes closed formulas.

> For a Taylor approach of order n, evaluations in the order of nM<sup>3</sup> are needed, where M is the maximum truncation level in the number of jumps. In a Chebyshev approach of the same order about n<sup>2</sup>NM<sup>3</sup> , evaluations of the conditional price should be performed, when a grid of N points is used in a trapezoidal approximation of the corresponding integrals. In a cubic splines approximation 3NM<sup>3</sup> . Here N is also the number of points in the grid where the polynomial coefficients are adjusted.

> For a theoretical analysis of the error using Taylor and Chebyshev expansions, although in different contexts, see [13] for Taylor and [6] for Chebyshev cases.

In Figure 1a, a graph of conditional prices in function of log-price values of the first asset (blue line) with average number of jumps equal to k<sup>0</sup> ¼ 3 and k<sup>1</sup> ¼ k<sup>2</sup> ¼ 2

#### Figure 1.

(a) Conditional price (blue curve) as function of log-price values and its Taylor approximations up to third order around the average. (b) Conditional price vs. its cubic spline approximation.

is shown. The remaining three curves represent the first-order (green), secondorder (red), and third-order (magenta) Taylor polynomials around the average value <sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>E</sup><sup>Q</sup> <sup>Y</sup>ð Þ<sup>2</sup> T . In Figure 1b, conditional prices and its cubic spline approximation are shown. At this scale both are indistinguishable. Notice that, although the Taylor approximation is excellent in a neighborhood of the expansion point, there are significant deviations for values far from the mean. These deviations, under the assumption of normality of the jump sizes, result to be infrequent; therefore, they do not impact the global error, but might be significant when other probability distributions, in particular heavy-tailed ones, or even normal jumps with higher volatilities, are taken into account. Instead of local approximations, as the case of Taylor polynomial expansion, uniform approximations on a given interval may reduce the error. Expansions based on orthogonal basis, e.g., Chebyshev or varying coefficients as in the case of cubic splines, are suggested. Notice that the function C yð Þ ; k is continuous in y for any value of k; therefore, Weierstrass' theorem of uniform convergence applies. Curiously, the convergence of Bernstein polynomials, applied in the original proof of the theorem, is remarkably slow.

Figure 2 shows the differences between the conditional price and the cubic spline for different values of the underlying price. Truncation values were selected as a ¼ �1 and b ¼ 1. Generally speaking the choice of these values depends on the probability distribution of the underlying asset. In practice it requires an exploratory study of the available data. On the other hand, the larger the interval, the more accurate is the approximation but also is the computational effort. Moreover, we have found that the results are sensible to this choice, though rather robust to the number of splines and the truncation values.

where ½ � <sup>x</sup> <sup>þ</sup> represents the maximum of the integer part of <sup>x</sup> and zero, then

Probabilities pk to observe k ¼ ð Þ k0; k<sup>1</sup> jumps when k<sup>2</sup> ¼ 5. Truncation values M<sup>0</sup> ¼ 15, M<sup>1</sup> ¼ 10, M<sup>2</sup> ¼ 10

0; j 1; j 2 over the set

> <sup>=</sup> kl � Ml 2

In Figure 3 we show the probability distribution pk; <sup>k</sup><sup>∈</sup> <sup>N</sup><sup>3</sup> , for <sup>k</sup><sup>2</sup> <sup>¼</sup> 5 varying k<sup>0</sup> and k1. We observe probabilities become negligible after certain values of ð Þ k0; k<sup>1</sup> with a peak around the center of the distribution. For the benchmark parameter set truncation values M<sup>0</sup> ¼ 15, M<sup>1</sup> ¼ 10, M<sup>2</sup> ¼ 10 capture 99.67% of the probability

The paper establishes a methodology over the use of polynomial approximations based on Taylor, Chebyshev, and cubic splines to the price of basket contracts. This approach produces accurate results at a lesser computational effort than a standard Monte Carlo technique. The claim is supported by numerical evidence in the case of spread options, under a bivariate jump-diffusion model with a complex Gaussian

The study needs to be extended to different parameter values to corroborate the results in a wider scope. Moreover, optimal choices in the numerical implementation, for example, the order of the polynomials, the number of points in the grid,

Sensitivities with respect to the parameters in the model and the contract, i.e., maturity, strike, interest rate, correlation, etc., can be easily calculated with a straightforward adaptation of the current method. It is enough to approximate the

A natural question is how to adapt our method when a non-Gaussian joint distribution of the jump sizes is considered. In this setting, the conditional

þ ≤ j

<sup>l</sup> ≤ kl þ

Ml

<sup>2</sup> ; <sup>l</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>

adding expression (18) for points j ¼ j

<sup>0</sup>; k<sup>1</sup> þ j

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

5. Conclusions and future developments

and truncation levels, require a further study.

corresponding derivatives instead.

∈ N<sup>3</sup>

capture 99.67% of the probability distribution in the number of jumps.

<sup>1</sup>; k<sup>2</sup> þ j 2

until ∑kpk ≥ δ, where δ is a predetermined value close to one.

jump structure that allows to capture the dependence between assets.

NMð Þ¼ k k<sup>0</sup> þ j

mass.

143

Figure 3.

Truncation values for the number of jumps, denoted in the paper by M0, M<sup>1</sup> and <sup>M</sup>2, should cover most of the jump probability distribution pk; <sup>k</sup><sup>∈</sup> <sup>N</sup><sup>3</sup> . An efficient way of choosing these values consists in starting to evaluate the sum at a point close to where the maximum value of the pk's is attained, namely,

$$k = (k\_0, k\_1, k\_2) = \left( [\lambda\_0 T - \mathbf{1}]\_+, [\lambda\_1 T - \mathbf{1}]\_+, [\lambda\_2 T - \mathbf{1}]\_+ \right)$$

Figure 2.

Curve representing the difference between conditional price and cubic spline approximation for the benchmark parameters.

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

#### Figure 3.

is shown. The remaining three curves represent the first-order (green), secondorder (red), and third-order (magenta) Taylor polynomials around the average

mation are shown. At this scale both are indistinguishable. Notice that, although the Taylor approximation is excellent in a neighborhood of the expansion point, there are significant deviations for values far from the mean. These deviations, under the assumption of normality of the jump sizes, result to be infrequent; therefore, they do not impact the global error, but might be significant when other probability distributions, in particular heavy-tailed ones, or even normal jumps with higher volatilities, are taken into account. Instead of local approximations, as the case of Taylor polynomial expansion, uniform approximations on a given interval may reduce the error. Expansions based on orthogonal basis, e.g., Chebyshev or varying coefficients as in the case of cubic splines, are suggested. Notice that the function C yð Þ ; k is continuous in y for any value of k; therefore, Weierstrass' theorem of uniform convergence applies. Curiously, the convergence of Bernstein polynomials,

Figure 2 shows the differences between the conditional price and the cubic spline for different values of the underlying price. Truncation values were selected as a ¼ �1 and b ¼ 1. Generally speaking the choice of these values depends on the probability distribution of the underlying asset. In practice it requires an exploratory study of the available data. On the other hand, the larger the interval, the more accurate is the approximation but also is the computational effort. Moreover, we have found that the results are sensible to this choice, though rather robust to

Truncation values for the number of jumps, denoted in the paper by M0, M<sup>1</sup> and <sup>M</sup>2, should cover most of the jump probability distribution pk; <sup>k</sup><sup>∈</sup> <sup>N</sup><sup>3</sup> . An efficient way of choosing these values consists in starting to evaluate the sum at a point

<sup>k</sup> <sup>¼</sup> ð Þ¼ <sup>k</sup>0; <sup>k</sup>1; <sup>k</sup><sup>2</sup> ½ � <sup>λ</sup>0<sup>T</sup> � <sup>1</sup> <sup>þ</sup>; ½ � <sup>λ</sup>1<sup>T</sup> � <sup>1</sup> <sup>þ</sup>; ½ � <sup>λ</sup>2<sup>T</sup> � <sup>1</sup> <sup>þ</sup>

Curve representing the difference between conditional price and cubic spline approximation for the benchmark

applied in the original proof of the theorem, is remarkably slow.

close to where the maximum value of the pk's is attained, namely,

the number of splines and the truncation values.

. In Figure 1b, conditional prices and its cubic spline approxi-

value <sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>E</sup><sup>Q</sup> <sup>Y</sup>ð Þ<sup>2</sup>

Figure 2.

parameters.

142

T 

Polynomials - Theory and Application

Probabilities pk to observe k ¼ ð Þ k0; k<sup>1</sup> jumps when k<sup>2</sup> ¼ 5. Truncation values M<sup>0</sup> ¼ 15, M<sup>1</sup> ¼ 10, M<sup>2</sup> ¼ 10 capture 99.67% of the probability distribution in the number of jumps.

where ½ � <sup>x</sup> <sup>þ</sup> represents the maximum of the integer part of <sup>x</sup> and zero, then adding expression (18) for points j ¼ j 0; j 1; j 2 over the set

$$N\_M(k) = \left\{ \left( k\_0 + j\_0, k\_1 + j\_1, k\_2 + j\_2 \right) \in \mathbb{N}^3 / \left( k\_l - \frac{M\_l}{2} \right)\_+ \le j\_l \le k\_l + \frac{M\_l}{2}, l = 0, 1, 2 \right\}.$$

until ∑kpk ≥ δ, where δ is a predetermined value close to one.

In Figure 3 we show the probability distribution pk; <sup>k</sup><sup>∈</sup> <sup>N</sup><sup>3</sup> , for <sup>k</sup><sup>2</sup> <sup>¼</sup> 5 varying k<sup>0</sup> and k1. We observe probabilities become negligible after certain values of ð Þ k0; k<sup>1</sup> with a peak around the center of the distribution. For the benchmark parameter set truncation values M<sup>0</sup> ¼ 15, M<sup>1</sup> ¼ 10, M<sup>2</sup> ¼ 10 capture 99.67% of the probability mass.

#### 5. Conclusions and future developments

The paper establishes a methodology over the use of polynomial approximations based on Taylor, Chebyshev, and cubic splines to the price of basket contracts. This approach produces accurate results at a lesser computational effort than a standard Monte Carlo technique. The claim is supported by numerical evidence in the case of spread options, under a bivariate jump-diffusion model with a complex Gaussian jump structure that allows to capture the dependence between assets.

The study needs to be extended to different parameter values to corroborate the results in a wider scope. Moreover, optimal choices in the numerical implementation, for example, the order of the polynomials, the number of points in the grid, and truncation levels, require a further study.

Sensitivities with respect to the parameters in the model and the contract, i.e., maturity, strike, interest rate, correlation, etc., can be easily calculated with a straightforward adaptation of the current method. It is enough to approximate the corresponding derivatives instead.

A natural question is how to adapt our method when a non-Gaussian joint distribution of the jump sizes is considered. In this setting, the conditional

probability distribution is generally unknown; nonetheless, the use of a copula approach to capture the dependence may provide some insight.

it is easy to see that

cov Zð Þ<sup>j</sup>

C kð Þ <sup>≔</sup> <sup>e</sup>�rT

E<sup>Q</sup> E<sup>Q</sup> S

¼ w1e

ple, [14].

145

<sup>¼</sup> <sup>w</sup>1e�rT

ð Þ1

where <sup>K</sup>1ð Þ¼ <sup>y</sup> <sup>K</sup>

<sup>0</sup> exp <sup>Y</sup>ð Þ<sup>1</sup> T � � � <sup>K</sup>

�rTE<sup>Q</sup> E<sup>Q</sup> S

h h

and NT <sup>¼</sup> <sup>k</sup>, it is well known that <sup>Y</sup>ð Þ<sup>1</sup>

ð Þ1

<sup>w</sup><sup>1</sup> � <sup>∑</sup><sup>d</sup> j¼2 wj w1 S ð Þj <sup>0</sup> <sup>e</sup><sup>y</sup>ð Þ<sup>j</sup> . Taking into account Eq. (19), again conditioning on the events <sup>Y</sup>~<sup>T</sup> <sup>¼</sup> <sup>y</sup>

<sup>0</sup> exp <sup>Y</sup>ð Þ<sup>1</sup> T

Hence, we can write, on the set <sup>Y</sup><sup>~</sup> <sup>¼</sup> <sup>y</sup> <sup>∩</sup> NT <sup>¼</sup> <sup>k</sup>� � :

Yð Þ<sup>1</sup>

<sup>T</sup> ; <sup>Z</sup>ð Þ<sup>l</sup> T � �=NT

Similarly, for j ¼ l

� � <sup>¼</sup> <sup>E</sup><sup>Q</sup> <sup>∑</sup>

cov Zð Þ<sup>j</sup>

Then, conditionally on NT, we have

Hence, the price is expressed as

CJD ¼ e

where C kð Þ¼ <sup>e</sup>�rTEQ½ � h Sð Þ <sup>T</sup> <sup>=</sup>NT <sup>¼</sup> <sup>k</sup> .

<sup>E</sup>Qð Þ¼ YT=NT <sup>μ</sup><sup>T</sup> <sup>þ</sup> <sup>N</sup><sup>~</sup> <sup>T</sup> <sup>∘</sup> <sup>μ</sup><sup>J</sup> <sup>þ</sup> <sup>N</sup>ð Þ <sup>0</sup>

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

ΣYð Þ NT :¼ Var Yð Þ¼ <sup>T</sup>=NT Σ

From the expression above, in the case of j 6¼ l, we have

0 @

<sup>¼</sup> <sup>N</sup>ð Þ <sup>0</sup>

<sup>T</sup> ; <sup>Z</sup>ð Þ<sup>l</sup> <sup>T</sup> =NT � � <sup>¼</sup> <sup>N</sup>ð Þ<sup>j</sup>

�rT ∑ k∈ Ndþ<sup>1</sup>

On the other hand, conditioning on ½ � NT <sup>¼</sup> <sup>k</sup> <sup>∩</sup> <sup>Y</sup>~T:

<sup>E</sup>Q½h Sð Þ <sup>T</sup> <sup>=</sup>NT <sup>¼</sup> <sup>k</sup>� ¼ <sup>e</sup>�rTE<sup>Q</sup> <sup>E</sup><sup>Q</sup> h Sð Þ <sup>T</sup> <sup>=</sup>NT <sup>¼</sup> <sup>k</sup>; YT<sup>~</sup> � �=NT <sup>¼</sup> <sup>k</sup> � �

<sup>w</sup><sup>1</sup> � ∑ d j¼2

� � ! !

wj w1 S ð Þj <sup>0</sup> exp <sup>Y</sup>ð Þ<sup>j</sup> T

� � � <sup>K</sup><sup>1</sup> <sup>Y</sup>~T<sup>Þ</sup> � �

with mean and variance given, respectively, by <sup>μ</sup>ð Þ <sup>y</sup>; <sup>k</sup> and <sup>σ</sup><sup>2</sup>ð Þ<sup>k</sup> <sup>T</sup>. See, for exam-

<sup>T</sup> <sup>¼</sup> <sup>μ</sup>ð Þþ <sup>y</sup>; <sup>k</sup> <sup>σ</sup>ð Þ<sup>k</sup> ffiffiffi

" " #

� i=NT <sup>¼</sup> <sup>k</sup>�

¼ ΣT þ Var Zð Þ <sup>T</sup>=NT

Nð Þ <sup>0</sup> T k¼1 ∑ Nð Þ <sup>0</sup> T k0 ¼1

<sup>T</sup> cov <sup>X</sup>ð Þ<sup>j</sup>

YT � <sup>N</sup> <sup>μ</sup><sup>T</sup> <sup>þ</sup> NT <sup>∘</sup> <sup>μ</sup><sup>J</sup> <sup>þ</sup> <sup>N</sup>ð Þ <sup>0</sup>

<sup>T</sup> <sup>μ</sup>0,J <sup>¼</sup> <sup>μ</sup><sup>j</sup> <sup>N</sup><sup>~</sup> <sup>T</sup>

<sup>2</sup>Var Bð Þ <sup>T</sup> Σ

<sup>0</sup>,k � <sup>E</sup>QXð Þ<sup>j</sup>

<sup>0</sup>,k=NT � � <sup>¼</sup> <sup>N</sup>ð Þ <sup>0</sup>

> <sup>T</sup> <sup>σ</sup>ð Þ<sup>j</sup> J � �<sup>2</sup>

EQð Þ h Sð Þ <sup>T</sup> =NT ¼ k pk ¼ ∑

0,k � � <sup>X</sup><sup>~</sup> ð Þ<sup>l</sup>

<sup>T</sup> μ0,J; ΣYð Þ NT � � (19)

1

Xð Þ<sup>j</sup>

0,k;Xð Þ<sup>l</sup>

� �

<sup>2</sup> þ Var Zð Þ <sup>T</sup>=NT

<sup>k</sup><sup>0</sup> � <sup>E</sup>QXð Þ<sup>l</sup>

<sup>T</sup> σ j,l 0

<sup>þ</sup> <sup>N</sup>ð Þ <sup>0</sup> <sup>T</sup> σ j,j 0

k∈ Ndþ<sup>1</sup>

þ

<sup>þ</sup>=NT <sup>¼</sup> <sup>k</sup>; <sup>Y</sup>~<sup>T</sup>

<sup>T</sup> has a univariate normal distribution

T <sup>p</sup> <sup>Z</sup> <sup>=</sup>NT <sup>¼</sup> <sup>k</sup>; <sup>Y</sup>~<sup>T</sup>

0,k � �=NT

C kð Þpk (20)

#

=NT ¼ k

(21)

1 A

1

#### Acknowledgements

This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.

#### A. Appendix

#### A.1 Taylor implementation up to third order

After computing the second and third derivatives of C yð Þ ; k and the corresponding derivatives of the moment-generating function of Z, we can compute Taylor approximations up to third order around the point y<sup>∗</sup> as

CT JD <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>1</sup> <sup>w</sup><sup>1</sup> <sup>∑</sup> k∈ N<sup>3</sup> M exp 1 2 <sup>σ</sup>ð Þ<sup>k</sup> <sup>T</sup> <sup>þ</sup> <sup>μ</sup>1ð Þþ <sup>k</sup> <sup>1</sup> 2 <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � � C y<sup>∗</sup> ð Þþ ; <sup>k</sup> <sup>D</sup>ð Þ<sup>1</sup> <sup>C</sup>ðy∗; <sup>k</sup><sup>Þ</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð þ ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>11ð Þ<sup>k</sup> <sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> �ipk h CT JD <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>2</sup> CT JD <sup>y</sup><sup>∗</sup> ð Þþ ; <sup>1</sup> <sup>w</sup><sup>1</sup> <sup>∑</sup> k∈ N<sup>3</sup> M exp 1 2 <sup>σ</sup>ð Þ<sup>k</sup> <sup>T</sup> <sup>þ</sup> <sup>μ</sup>1ð Þþ <sup>k</sup> <sup>1</sup> 2 <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � �Dð Þ<sup>2</sup> C y<sup>∗</sup> ð Þ ; <sup>k</sup> <sup>1</sup> <sup>2</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ<sup>2</sup> <sup>þ</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>11ð Þ<sup>k</sup> <sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> � þ 1 <sup>2</sup> <sup>σ</sup>22ð Þ<sup>k</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � �� pk CT JD <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>3</sup> CT JD <sup>y</sup><sup>∗</sup> ð Þþ ; <sup>2</sup> <sup>w</sup><sup>1</sup> <sup>∑</sup> k ∈ N<sup>3</sup> M exp 1 2 <sup>σ</sup>ð Þ<sup>k</sup> <sup>T</sup> <sup>þ</sup> <sup>μ</sup>1ð Þþ <sup>k</sup> <sup>1</sup> 2 <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � �Dð Þ<sup>3</sup> C y<sup>∗</sup> ð Þ ; <sup>k</sup> 1 <sup>6</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ<sup>3</sup> <sup>þ</sup> 1 <sup>2</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>11ð Þ<sup>k</sup> <sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> � þ 1 <sup>2</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þσ22ð Þ<sup>k</sup> <sup>1</sup> <sup>þ</sup> <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � � þ 1 <sup>6</sup> <sup>σ</sup>22ð Þ<sup>k</sup> <sup>3</sup> 2 ffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>Þ</sup> � �pk

#### A.2 Proof of Theorem 1

From Eq. (2) written in its integral form

$$Y\_T = \mu T + \Sigma^\ddagger B\_T + Z\_T$$

it is easy to see that

probability distribution is generally unknown; nonetheless, the use of a copula

After computing the second and third derivatives of C yð Þ ; k and the corresponding derivatives of the moment-generating function of Z, we can com-

> 1 2

<sup>σ</sup>ð Þ<sup>k</sup> <sup>T</sup> <sup>þ</sup> <sup>μ</sup>1ð Þþ <sup>k</sup> <sup>1</sup>

C y<sup>∗</sup> ð Þþ ; <sup>k</sup> <sup>D</sup>ð Þ<sup>1</sup> <sup>C</sup>ðy∗; <sup>k</sup><sup>Þ</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð þ ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2

<sup>2</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ<sup>2</sup> <sup>þ</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �

1 <sup>2</sup>BT þ ZT

pk

<sup>σ</sup>ð Þ<sup>k</sup> <sup>T</sup> <sup>þ</sup> <sup>μ</sup>1ð Þþ <sup>k</sup> <sup>1</sup>

2

� �

<sup>σ</sup>ð Þ<sup>k</sup> <sup>T</sup> <sup>þ</sup> <sup>μ</sup>1ð Þþ <sup>k</sup> <sup>1</sup>

<sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup>

2

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup>

<sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup>

� �

2

pk

� �

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup>

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>σ</sup>22ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup>

<sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup>

�i pk

Dð Þ<sup>2</sup>

<sup>D</sup>ð Þ<sup>3</sup> C y<sup>∗</sup> ð Þ ; <sup>k</sup>

pute Taylor approximations up to third order around the point y<sup>∗</sup> as

k∈ N<sup>3</sup> M exp

<sup>2</sup> <sup>σ</sup>22ð Þ<sup>k</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � ��

k ∈ N<sup>3</sup> M exp 1 2

1

ffiffiffiffiffiffiffiffiffiffiffiffi

From Eq. (2) written in its integral form

<sup>2</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þσ22ð Þ<sup>k</sup> <sup>1</sup> <sup>þ</sup> <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> � �

<sup>σ</sup>11ð Þ<sup>k</sup> <sup>p</sup> <sup>ρ</sup>ð Þ<sup>k</sup> <sup>σ</sup>11ð Þ<sup>k</sup> ð Þ <sup>ρ</sup>ð Þ<sup>k</sup> <sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>Þ</sup>

YT ¼ μT þ Σ

This research was partially supported by the Natural Sciences and Engineering

approach to capture the dependence may provide some insight.

A.1 Taylor implementation up to third order

k∈ N<sup>3</sup> M exp

JD <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>1</sup> <sup>w</sup><sup>1</sup> <sup>∑</sup>

C y<sup>∗</sup> ð Þ ; <sup>k</sup> <sup>1</sup>

þ 1

1

þ 1

þ 1 <sup>6</sup> <sup>σ</sup>22ð Þ<sup>k</sup> <sup>3</sup> 2

A.2 Proof of Theorem 1

�

h

JD <sup>y</sup><sup>∗</sup> ð Þþ ; <sup>1</sup> <sup>w</sup><sup>1</sup> <sup>∑</sup>

�

JD <sup>y</sup><sup>∗</sup> ð Þþ ; <sup>2</sup> <sup>w</sup><sup>1</sup> <sup>∑</sup>

<sup>6</sup> <sup>μ</sup>2ð Þ� <sup>k</sup> <sup>y</sup><sup>∗</sup> ð Þ<sup>3</sup> <sup>þ</sup>

Acknowledgements

A. Appendix

Research Council of Canada.

Polynomials - Theory and Application

CT

JD <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>2</sup> CT

CT

CT

144

JD <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>3</sup> CT

$$\begin{aligned} E\_{\mathcal{Q}}(Y\_T/N\_T) &= \mu T + \tilde{N}\_T \bullet \mu\_f + N\_T^{(0)} \mu\_{0,f} = \overline{\mu}\_j(\tilde{N}\_T) \\\\ \Sigma\_Y(N\_T) &:= Var(Y\_T/N\_T) = \Sigma^\ddagger Var(B\_T)\Sigma^\ddagger + Var(Z\_T/N\_T) \\\\ &= \Sigma T + Var(Z\_T/N\_T) \end{aligned}$$

From the expression above, in the case of j 6¼ l, we have

$$\begin{split} \text{cov}\left(\left(\mathbf{Z}\_{T}^{(j)},\mathbf{Z}\_{T}^{(l)}\right)/\mathbf{N}\_{T}\right) &= \mathbf{E}\_{\mathcal{Q}}\left(\sum\_{k=1\mathbf{k}'=1}^{N\_{T}^{(0)}} \sum\_{k=1}^{N\_{T}^{(0)}} \left(\mathbf{X}\_{0,k}^{(j)} - \mathbf{E}\_{\mathcal{Q}}\mathbf{X}\_{0,k}^{(j)}\right) \left(\tilde{\mathbf{X}}\_{k'}^{(l)} - \mathbf{E}\_{\mathcal{Q}}\mathbf{X}\_{0,k}^{(l)}\right)/\mathbf{N}\_{T}\right) \\ &= \mathbf{N}\_{T}^{(0)}\text{cov}\left(\mathbf{X}\_{0,k}^{(j)},\mathbf{X}\_{0,k}^{(l)}/\mathbf{N}\_{T}\right) = \mathbf{N}\_{T}^{(0)}\sigma\_{0}^{j,l} \end{split}$$

Similarly, for j ¼ l

$$\mathbf{cov}\left(\mathbf{Z}\_T^{(j)}, \mathbf{Z}\_T^{(l)} / \mathbf{N}\_T\right) \quad = \quad \mathbf{N}\_T^{(j)} \left(\sigma\_J^{(j)}\right)^2 + \mathbf{N}\_T^{(0)} \sigma\_0^{j,j}$$

Then, conditionally on NT, we have

$$Y\_T \sim \mathcal{N}\left(\mu T + \mathcal{N}\_T \bullet \mu\_f + \mathcal{N}\_T^{(0)} \mu\_{0,f}, \Sigma\_Y(\mathcal{N}\_T)\right) \tag{19}$$

Hence, the price is expressed as

$$\mathbf{C}\_{\text{ID}} = \mathbf{e}^{-rT} \sum\_{k \in \mathbb{N}^{d+1}} E\_{\mathcal{Q}}(h(\mathbf{S}\_T)/N\_T = k) p\_k = \sum\_{k \in \mathbb{N}^{d+1}} \mathbf{C}(k) p\_k \tag{20}$$

where C kð Þ¼ <sup>e</sup>�rTEQ½ � h Sð Þ <sup>T</sup> <sup>=</sup>NT <sup>¼</sup> <sup>k</sup> . On the other hand, conditioning on ½ � NT <sup>¼</sup> <sup>k</sup> <sup>∩</sup> <sup>Y</sup>~T:

$$\begin{split} C(k) &:= e^{-rT} \\ E\_{\mathcal{Q}}[h(\mathcal{S}\_{T})/N\_{T} = k] &= e^{-rT} E\_{\mathcal{Q}} [E\_{\mathcal{Q}} \{ h(\mathcal{S}\_{T})/N\_{T} = k, \bar{\mathcal{Y}}T \} / N\_{T} = k] \\ &= w\_{l} e^{-rT} \\ E\_{\mathcal{Q}} \left[ E\_{\mathcal{Q}} \left[ \left( S\_{0}^{(1)} \exp \left( Y\_{T}^{(1)} \right) - \left( \frac{k}{w\_{1}} - \sum\_{j=2}^{d} \frac{w\_{j}}{w\_{1}} S\_{0}^{(j)} \exp \left( Y\_{T}^{(j)} \right) \right) \right)\_{+} / N\_{T} = k, \bar{\mathcal{Y}}\_{T} \right] / N\_{T} = k \right] \\ &= w\_{l} e^{-rT} E\_{\mathcal{Q}} \left[ E\_{\mathcal{Q}} \left[ \left( S\_{0}^{(1)} \exp \left( Y\_{T}^{(1)} \right) - K\_{1} (\bar{\mathcal{Y}}\_{T}) \right)\_{+} / N\_{T} = k, \bar{\mathcal{Y}}\_{T} \right] / N\_{T} = k \right] \\ &\qquad \times \bar{\mathcal{Y}}\_{T} \qquad \text{ and } w\_{l} \neq 0. \end{split} \tag{21}$$

where <sup>K</sup>1ð Þ¼ <sup>y</sup> <sup>K</sup> <sup>w</sup><sup>1</sup> � <sup>∑</sup><sup>d</sup> j¼2 wj w1 S ð Þj <sup>0</sup> <sup>e</sup><sup>y</sup>ð Þ<sup>j</sup> .

Taking into account Eq. (19), again conditioning on the events <sup>Y</sup>~<sup>T</sup> <sup>¼</sup> <sup>y</sup> and NT <sup>¼</sup> <sup>k</sup>, it is well known that <sup>Y</sup>ð Þ<sup>1</sup> <sup>T</sup> has a univariate normal distribution with mean and variance given, respectively, by <sup>μ</sup>ð Þ <sup>y</sup>; <sup>k</sup> and <sup>σ</sup><sup>2</sup>ð Þ<sup>k</sup> <sup>T</sup>. See, for example, [14].

Hence, we can write, on the set <sup>Y</sup><sup>~</sup> <sup>¼</sup> <sup>y</sup> <sup>∩</sup> NT <sup>¼</sup> <sup>k</sup>� � :

$$Y\_T^{(1)} = \mu(\mathcal{y}, k) + \sigma(k)\sqrt{T}Z$$

Then, replacing the expression above in Eq. (21), we have

$$\begin{split} \mathbf{C}(k) &= \boldsymbol{w}\_{l} \boldsymbol{e}^{-rT} \mathbb{E}\_{\mathcal{Q}} \left[ \mathcal{E}\_{\mathcal{Q}} \left[ \left( S\_{0}^{(1)} \exp \left( \mu(\bar{Y}\_{T}, N\_{T}) + \sigma(N\_{T}) \sqrt{T} \mathbf{Z} \right) - K\_{1} (\bar{Y}\_{T}) \right)\_{+} / \mathcal{F}^{\bar{Y}\_{T}}, N\_{T} = k \right] / N\_{T} = k \right] \\ &= \boldsymbol{w}\_{l} \boldsymbol{e}^{-rT} \mathbb{E}\_{\mathcal{Q}} \left[ \exp \left( -\left( r - \frac{1}{2} \sigma(k) \right) T + \mu(\bar{Y}\_{T}, k) \right) \mathbb{E}\_{\mathcal{Q}} \left[ \left( S\_{0}^{(1)} \exp \left( \left( r - \frac{1}{2} \sigma^{2}(N\_{T}) \right) T + \sigma(N\_{T}) \sqrt{T} \mathbf{Z} \right) \right) \right] + \mathbf{C}(k) \right] \\ &\quad - \exp \left( \left( r - \frac{1}{2} \sigma^{2}(N\_{T}) \right) T - \mu(\bar{Y}\_{T}, N\_{T}) \right) \mathbf{K}\_{1} (\bar{Y}\_{T}) \Big] / \mathcal{F}^{\bar{Y}\_{T}}, N\_{T} \Big] / N\_{T} = k \right] \\ &= w\_{l} \exp \left( \frac{1}{2} \sigma^{2}(k) T \right) \mathbb{E}\_{\mathcal{Q}} \left[ \exp \left( \mu(\bar{Y}\_{T}, N\_{T}) \right) \mathbf{C}(\bar{Y}\_{T}, N\_{T}) / N\_{T} = k \right] \end{split} \tag{22}$$

¼ w<sup>1</sup> exp

from which (13) follows.

∑ N j¼1 ∑ l∈B<sup>3</sup>

Author details

Ryerson University, Toronto, Canada

provided the original work is properly cited.

\*Address all correspondence to: pablo.olivares@ryerson.ca

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Pablo Olivares

147

1 2 σ2 ð Þk T 

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

exp <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>Y<sup>~</sup> �<sup>1</sup>

ð Þk cj <sup>α</sup>j,lð Þ<sup>k</sup> <sup>D</sup><sup>l</sup>

MY<sup>~</sup> �cj

<sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>Y<sup>~</sup> �<sup>1</sup>

ð Þk ; k; Dj

Eq. (4) easily follows after replacing Eq. (22) into Eq. (20).

#### A.3 Proof of Theorem 4

In Eq. (6) we replace the function C yð Þ ; k by its Taylor expansion given in Eq. (7).

Then, the Taylor approximation of C kð Þ is

$$\mathbf{C}^{T}(\mathbf{y}^{\*},k) = w\_{1} \exp\left(\frac{1}{2}\sigma^{2}(k)T\right)E\_{\mathcal{Q}}\left[\exp\left(\mu\left(\bar{Y}\_{T},N\_{T}\right)\right)\mathbf{C}^{T}(\bar{Y}\_{T},\mathbf{y}^{\*},N\_{T})/N\_{T} = k\right]$$

$$= w\_{1}\sum\_{l=0}^{n}\sum\_{L\in\mathbb{R}\_{l}}\frac{D^{L}\mathbf{C}(\mathbf{y}^{\*},k)}{L!}\exp\left(\frac{1}{2}\sigma^{2}(k)T + \overline{\mu}\_{1}(k) - \Sigma\_{1\bar{Y}}(k)\Sigma\_{\bar{Y}}^{-1}(k)\bar{\mu}(k)^{\prime}\right)$$

$$E\_{\mathcal{Q}}\left[\exp\left(\Sigma\_{1\bar{Y}}(N\_{T})\Sigma\_{\bar{Y}}^{-1}(N\_{T})\bar{Y}\_{T}\right)\left(\bar{Y}\_{T} - \mathbf{y}^{\*}\right)^{L}/N\_{T} = k\right]$$

$$= w\_{1}\exp\left(A\_{2}(k)\right)\sum\_{l=0}^{n}\sum\_{R\_{l}}\frac{D^{L}\mathbf{C}(\mathbf{y}^{\*},k)}{L!}D^{L}M\_{\bar{Y}\_{T}-\mathbf{y}^{\*}}\left(\Sigma\_{1\bar{Y}}(k)\Sigma\_{\bar{Y}}^{-1}(k),k\right)$$

Eq. (11) follows after replacing C kð Þ in Eq. (20) by the expression above and truncating at point M.

After replacing Eq. (9) into Eq. (22), we have

$$\begin{split} \mathbf{C}^{Ch}(k) &= \frac{w\_{1}}{2} \hat{c}\_{0}\left(k\right) \exp\left(A\_{1}(k)\right) \mathbf{E}\_{\mathcal{Q}}\left[\exp\left(\Sigma\_{1\hat{Y}}(N\_{T})\Sigma\_{\hat{Y}}^{-1}(N\_{T})\bar{Y}\_{T}\right) \mathbf{1}\_{\mathcal{Q}}(\bar{Y}\_{T})/N\_{T} = k\right] \\ &+ w\_{1} \exp\left(A\_{1}(k)\right) \sum\_{l\in\mathcal{B}\_{\mathbf{s}}} \hat{c}\_{l}(k) \mathbf{E}\_{\mathcal{Q}}\left[\exp\left(\Sigma\_{1\hat{Y}}(N\_{T})\Sigma\_{\hat{Y}}^{-1}(N\_{T})\bar{Y}\_{T}\right) \mathbf{T}\_{l}^{D}\left(\bar{Y}\_{T}\right)/N\_{T} = k\right] \\ &= \frac{w\_{1}}{2} \hat{c}\_{0}(k) K\_{1}(k,a,b) + w\_{1} \exp\left(A\_{1}(k)\right) \sum\_{l\in\mathcal{B}\_{\mathbf{s}}} \sum\_{m\in\mathcal{C}\_{l}} \hat{c}\_{l}(k) b\_{m,l} \\ &E\_{\mathcal{Q}}\left[\exp\left(\Sigma\_{1\hat{Y}}(N\_{T})\Sigma\_{\hat{Y}}^{-1}(N\_{T})\bar{Y}\_{T}\right) \left(-1 + 2\frac{\bar{Y}\_{T} - a}{b - a}\right)^{l - 2u} \mathbf{1}\_{\mathcal{D}}(\bar{Y}\_{T})\right] \end{split}$$

Eq. (12) easily follows. Finally, by similar arguments,

$$\begin{aligned} C\_{f\mathbb{D}}^{pl}(\mathbb{k}) &= w\_1 \exp\left(\frac{1}{2}\sigma^2(\mathbb{k})T\right) \\ &\sum\_{j=1}^N \sum\_{l \in B\_{\mathbb{S}}} a\_{j,l}(\mathbb{k}) E\_{\mathbb{Q}}\left(\exp\left(\Sigma\_{1\tilde{Y}}(N\_T)\Sigma\_{\tilde{Y}}^{-1}(N\_T)\tilde{Y}\_T\right)\left(\bar{Y}\_T - c\_{\tilde{f}}\right)^l \mathbf{1}\_{\mathbb{D}\_{\tilde{Y}}}\left(\bar{Y}\_T\right)/N\_T = \mathbf{k}\right) \end{aligned}$$

Pricing Basket Options by Polynomial Approximations DOI: http://dx.doi.org/10.5772/intechopen.82383

$$\begin{aligned} 0 &= w\_1 \exp\left(\frac{1}{2}\sigma^2(k)T\right) \\ &\sum\_{j=1}^N \sum\_{l \in B\_{\triangleright}} \exp\left(\Sigma\_{1\check{Y}}(k)\Sigma\tilde{Y}^{-1}(k)c\_j\right) a\_{j,l}(k)D^l M\_{\check{Y}-c\_j}\left(\Sigma\_{1\check{Y}}(k)\Sigma\tilde{Y}^{-1}(k),k,D\_j\right) \end{aligned}$$

from which (13) follows.

Then, replacing the expression above in Eq. (21), we have

2 σð Þk � �

� �<sup>T</sup> � <sup>μ</sup>ðY~T; NTÞÞK<sup>1</sup> <sup>Y</sup>~T<sup>Þ</sup> � �

Eq. (4) easily follows after replacing Eq. (22) into Eq. (20).

DLC y<sup>∗</sup> ð Þ ; <sup>k</sup>

n l¼0 ∑ Rl

<sup>2</sup> ^c<sup>0</sup> ð Þ<sup>k</sup> exp ð Þ <sup>A</sup>1ð Þ<sup>k</sup> <sup>E</sup><sup>Q</sup> exp <sup>Σ</sup>1Y<sup>~</sup> ð Þ NT <sup>Σ</sup>�<sup>1</sup>

l ∈Bn

<sup>2</sup> ^c0ð Þ<sup>k</sup> <sup>K</sup>1ð Þþ <sup>k</sup>; <sup>a</sup>; <sup>b</sup> <sup>w</sup><sup>1</sup> exp ð Þ <sup>A</sup>1ð Þ<sup>k</sup> <sup>∑</sup>

<sup>α</sup>j,lð Þ<sup>k</sup> <sup>E</sup><sup>Q</sup> exp <sup>Σ</sup>1Y<sup>~</sup> ð Þ NT <sup>Σ</sup>�<sup>1</sup>

<sup>L</sup>! exp

� �

<sup>0</sup> exp <sup>μ</sup> <sup>Y</sup>~T; NTÞ þ <sup>σ</sup>ð Þ NT

ffiffiffi T <sup>p</sup> <sup>Z</sup> � � � <sup>K</sup><sup>1</sup> <sup>Y</sup>~T<sup>Þ</sup> � �

<sup>T</sup> <sup>þ</sup> <sup>μ</sup>ðY~T; <sup>k</sup>ÞÞE<sup>Q</sup> <sup>S</sup>

<sup>E</sup><sup>Q</sup> exp <sup>μ</sup> <sup>Y</sup>~T; NT<sup>Þ</sup> � �CðY~T; NTÞ=NT <sup>¼</sup> <sup>k</sup>� � �

In Eq. (6) we replace the function C yð Þ ; k by its Taylor expansion given in

<sup>E</sup><sup>Q</sup> exp <sup>μ</sup> <sup>Y</sup>~T; NT

1 2 σ2

h i

<sup>Y</sup><sup>~</sup> ð Þ NT <sup>Y</sup>~<sup>T</sup>

<sup>D</sup>LC y<sup>∗</sup> ð Þ ; <sup>k</sup>

Eq. (11) follows after replacing C kð Þ in Eq. (20) by the expression above and

^clð Þ<sup>k</sup> <sup>E</sup><sup>Q</sup> exp <sup>Σ</sup>1Y<sup>~</sup> ð Þ NT <sup>Σ</sup>�<sup>1</sup>

<sup>Y</sup><sup>~</sup> ð Þ NT <sup>Y</sup>~TÞ �<sup>1</sup> <sup>þ</sup> <sup>2</sup>

� � " #

h h� i

� i

� �� � �

<sup>þ</sup>=ℱ<sup>Y</sup>~<sup>T</sup> ; NT

ð Þ1

<sup>0</sup> exp <sup>r</sup> � <sup>1</sup>

� =NT ¼ k �

<sup>þ</sup>=ℱ<sup>Y</sup>~<sup>T</sup> ; NT <sup>¼</sup> <sup>k</sup>

2 σ2 ð Þ NT � �

� � � � <sup>C</sup><sup>T</sup>ðY~T; <sup>y</sup>∗; NTÞ=NT <sup>¼</sup> <sup>k</sup> � �

ð Þ<sup>k</sup> <sup>T</sup> <sup>þ</sup> <sup>μ</sup>1ð Þ� <sup>k</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

<sup>Y</sup>~<sup>T</sup> � <sup>y</sup><sup>∗</sup> � �<sup>L</sup>

<sup>L</sup>! <sup>D</sup>LMY<sup>~</sup>T�y<sup>∗</sup> <sup>Σ</sup>1Y<sup>~</sup> ð Þ<sup>k</sup> <sup>Σ</sup>�<sup>1</sup>

<sup>Y</sup><sup>~</sup> ð Þ NT <sup>Y</sup>~<sup>T</sup>

∑ m ∈Cl

<sup>Y</sup>~<sup>T</sup> � <sup>a</sup> b � a � �<sup>l</sup>�2<sup>m</sup>

h i

� �

l∈Bn

<sup>Y</sup><sup>~</sup> ð Þ NT <sup>Y</sup><sup>~</sup> <sup>T</sup>

� �

�

� �

=NT ¼ k

1<sup>D</sup> Y~ <sup>T</sup>

1<sup>D</sup> Y~ <sup>T</sup>

1Dj Y~ <sup>T</sup>

� �=NT <sup>¼</sup> <sup>k</sup>

<sup>Y</sup><sup>~</sup> ð Þ NT <sup>Y</sup>~TÞT<sup>D</sup>

h � i

^clð Þk bm,l

<sup>Y</sup><sup>~</sup> <sup>T</sup> � cj � �<sup>l</sup>

� �

=NT ¼ k

ffiffiffi T <sup>p</sup> <sup>Z</sup>

(22)

<sup>Y</sup><sup>~</sup> ð Þk μ~ð Þk <sup>0</sup>

<sup>Y</sup><sup>~</sup> ð Þk ; kÞ

� �=NT <sup>¼</sup> <sup>k</sup>

<sup>l</sup> <sup>Y</sup><sup>~</sup> <sup>T</sup>

� �=NT <sup>¼</sup> <sup>k</sup>

T þ σð Þ NT

ð Þ1

<sup>2</sup> <sup>σ</sup><sup>2</sup>ð Þ NT

Then, the Taylor approximation of C kð Þ is

1 2 σ2 ð Þk T � �

<sup>E</sup><sup>Q</sup> exp <sup>Σ</sup>1Y<sup>~</sup> ð Þ NT <sup>Σ</sup>�<sup>1</sup>

¼ w<sup>1</sup> exp ð Þ A2ð Þk ∑

After replacing Eq. (9) into Eq. (22), we have

þ w<sup>1</sup> exp ð Þ A1ð Þk ∑

<sup>E</sup><sup>Q</sup> exp <sup>Σ</sup>1Y<sup>~</sup> ð Þ NT <sup>Σ</sup>�<sup>1</sup>

1 2 σ2 ð Þk T � �

C kð Þ¼ <sup>w</sup>1e�rTE<sup>Q</sup> <sup>E</sup><sup>Q</sup> <sup>S</sup>

Polynomials - Theory and Application

� exp <sup>ð</sup> <sup>r</sup> � <sup>1</sup>

1 2 σ2 ð Þk T � �

¼ w<sup>1</sup> exp

A.3 Proof of Theorem 4

CT <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>k</sup> <sup>w</sup><sup>1</sup> exp

truncating at point M.

<sup>¼</sup> <sup>w</sup><sup>1</sup>

JDð Þ¼ k w<sup>1</sup> exp

∑ N j¼1 ∑ l∈B<sup>3</sup>

Eq. (12) easily follows. Finally, by similar arguments,

CChð Þ¼ <sup>k</sup> <sup>w</sup><sup>1</sup>

Cspl

146

¼ w<sup>1</sup> ∑ n l¼0 ∑ L∈Rl

Eq. (7).

<sup>¼</sup> <sup>w</sup>1e�rTE<sup>Q</sup> exp ð� <sup>r</sup> � <sup>1</sup>

#### Author details

Pablo Olivares Ryerson University, Toronto, Canada

\*Address all correspondence to: pablo.olivares@ryerson.ca

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Olivares P, Alvarez A. Pricing basket options by polynomial approximations. Journal of Applied Mathematics. 2016, ID 9747394. p. 12. http://dx.doi.org/ 10.1155/2016/9747394

[2] Hull JC, White A. The pricing of options on assets with stochastic volatilities. Journal of Finance. 1987;42: 281-300

[3] Li M, Zhou J, Deng SJ. Multi-asset spread option pricing and hedging. Quantitative Finance. 2010;10(3): 305-324

[4] Li M, Deng S, Zhou J. Closed-form approximations for spread options prices and Greeks. Journal of Derivatives. 2008;15(3):58-80

[5] Ju N. Pricing Asian and basket options via Taylor expansion. Journal of Computational Finance. 2002;5:79-103

[6] Gass M, Glau K, Mahlstedt M, Mair M. Chebyshev Interpolation for Parametric Option Pricing, 2016. https://arxiv.org/abs/1505.04648v2

[7] Hurd TR, Zhou Z. A Fourier transform method for spread option pricing. SIAM Journal on Financial Mathematics. 2009;1:142-157

[8] Fang F, Oosterlee CW. Efficient pricing of European-style Asian options under exponential Levy processes based on Fourier cosine expansions. SIAM Journal on Financial Mathematics. 2013; 4:399-426

[9] Phelan CE, Marazzina D, Fusai G, Germano G. Hilbert transform, spectral filtering and option pricing. Annals of Operations Research. 2018;2018. DOI: 10.1007/s10479-018-2881-4

[10] Mason JC, Handscomb DC. Chebyshev Polynomials. Florida: CRC Press Company; 2003

Chapter 8

Abstract

1. Introduction

149

The Orthogonal Expansion in

Paralleling-in-Order Scheme

Maxwell Equations Using

Zheng-Yu Huang, Zheng Sun and Wei He

shielding analysis with the long-time response requirement.

Keywords: associated Hermite, finite-difference time-domain (FDTD), Legendre polynomials, paralleling-in-order, unconditionally stable

To overcome the numerical stability constraints of conventional finitedifference time-domain (FDTD) method [1, 2], many unconditionally stable methods to reduce or eliminate requirements of the stability condition have been proposed and developed, such as alternating-direction implicit method [2, 3] and locally one-dimensional schemes [3], explicit and unconditionally stable FDTD method [4], and orthogonal expansions in time domain [5–8]. For the orthogonal expansions schemes, field-versus-time variations in the FDTD space lattice are expanded using an appropriate set of orthogonal temporal basis and testing functions, such as weighted Laguerre polynomials (WLP) and associated Hermite (AH) functions, which leads to two different solution schemes: marching-on-in-order and paralleling-in-order, respectively. Both of them appear to be promising according to the reported work where the computational time can be reduced to at least 10% of the conventional FDTD scheme [1]. Recently, the Legendre (LD) polynomials are

Time-Domain Method for Solving

The orthogonal expansion in time-domain method is a new kind of unconditionally stable finite-difference time-domain (FDTD) method for solving the Maxwell equation efficiently. Generally, it can be implemented by two schemes: marching-on-in-order and paralleling-in-order, which, respectively, use weighted Laguerre polynomials and associated Hermite functions as temporal expansions and testing functions. This chapter summarized paralleling-in-order-based FDTD method using associated Hermite functions and Legendre polynomials. And a comparison from theoretical analysis to numerical examples is shown. The LD integral transfer matrix can be considered as a "dual" transformation for AH differential matrix, which gives a possible way to find more potential orthogonal basis function to implement a paralleling-in-order scheme. In addition, the differences with these two orthogonal functions are also analyzed. From the numerical results, we can see their agreements in some general cases while differing in some cases such as

[11] Gil A, Segura J, Temme NM. Numerical Methods for Special Functions. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics; 2007

[12] Arcangéli R, López de Silanes MC, Torrens JJ. Multidimensional Minimizing Splines: Theory and Applications. Boston: Kluwer Academic Publishers; 2004

[13] Alvarez A, Escobar M, Olivares P. Spread options under stochastic covariance and jumps. Pricing two dimensional derivatives under stochastic correlation. International Journal of Financial Markets and Derivatives. 2012;2(4/2011):265-287

[14] Tong YL. The Multivariate Normal Distribution. Berlin: Springer; 1989

#### Chapter 8

References

281-300

305-324

10.1155/2016/9747394

[1] Olivares P, Alvarez A. Pricing basket options by polynomial approximations. Journal of Applied Mathematics. 2016, ID 9747394. p. 12. http://dx.doi.org/

Polynomials - Theory and Application

[10] Mason JC, Handscomb DC. Chebyshev Polynomials. Florida: CRC

[11] Gil A, Segura J, Temme NM. Numerical Methods for Special Functions. Philadelphia, PA, USA: Society for Industrial and Applied

[12] Arcangéli R, López de Silanes MC,

Applications. Boston: Kluwer Academic

[13] Alvarez A, Escobar M, Olivares P. Spread options under stochastic covariance and jumps. Pricing two dimensional derivatives under stochastic correlation. International Journal of Financial Markets and Derivatives. 2012;2(4/2011):265-287

[14] Tong YL. The Multivariate Normal Distribution. Berlin: Springer; 1989

Torrens JJ. Multidimensional Minimizing Splines: Theory and

Press Company; 2003

Mathematics; 2007

Publishers; 2004

[2] Hull JC, White A. The pricing of options on assets with stochastic volatilities. Journal of Finance. 1987;42:

[3] Li M, Zhou J, Deng SJ. Multi-asset spread option pricing and hedging. Quantitative Finance. 2010;10(3):

[4] Li M, Deng S, Zhou J. Closed-form approximations for spread options prices and Greeks. Journal of Derivatives. 2008;15(3):58-80

[5] Ju N. Pricing Asian and basket options via Taylor expansion. Journal of Computational Finance. 2002;5:79-103

[7] Hurd TR, Zhou Z. A Fourier transform method for spread option pricing. SIAM Journal on Financial Mathematics. 2009;1:142-157

[8] Fang F, Oosterlee CW. Efficient pricing of European-style Asian options under exponential Levy processes based on Fourier cosine expansions. SIAM Journal on Financial Mathematics. 2013;

[9] Phelan CE, Marazzina D, Fusai G, Germano G. Hilbert transform, spectral filtering and option pricing. Annals of Operations Research. 2018;2018. DOI:

10.1007/s10479-018-2881-4

4:399-426

148

[6] Gass M, Glau K, Mahlstedt M, Mair M. Chebyshev Interpolation for Parametric Option Pricing, 2016. https://arxiv.org/abs/1505.04648v2

## The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using Paralleling-in-Order Scheme

Zheng-Yu Huang, Zheng Sun and Wei He

### Abstract

The orthogonal expansion in time-domain method is a new kind of unconditionally stable finite-difference time-domain (FDTD) method for solving the Maxwell equation efficiently. Generally, it can be implemented by two schemes: marching-on-in-order and paralleling-in-order, which, respectively, use weighted Laguerre polynomials and associated Hermite functions as temporal expansions and testing functions. This chapter summarized paralleling-in-order-based FDTD method using associated Hermite functions and Legendre polynomials. And a comparison from theoretical analysis to numerical examples is shown. The LD integral transfer matrix can be considered as a "dual" transformation for AH differential matrix, which gives a possible way to find more potential orthogonal basis function to implement a paralleling-in-order scheme. In addition, the differences with these two orthogonal functions are also analyzed. From the numerical results, we can see their agreements in some general cases while differing in some cases such as shielding analysis with the long-time response requirement.

Keywords: associated Hermite, finite-difference time-domain (FDTD), Legendre polynomials, paralleling-in-order, unconditionally stable

#### 1. Introduction

To overcome the numerical stability constraints of conventional finitedifference time-domain (FDTD) method [1, 2], many unconditionally stable methods to reduce or eliminate requirements of the stability condition have been proposed and developed, such as alternating-direction implicit method [2, 3] and locally one-dimensional schemes [3], explicit and unconditionally stable FDTD method [4], and orthogonal expansions in time domain [5–8]. For the orthogonal expansions schemes, field-versus-time variations in the FDTD space lattice are expanded using an appropriate set of orthogonal temporal basis and testing functions, such as weighted Laguerre polynomials (WLP) and associated Hermite (AH) functions, which leads to two different solution schemes: marching-on-in-order and paralleling-in-order, respectively. Both of them appear to be promising according to the reported work where the computational time can be reduced to at least 10% of the conventional FDTD scheme [1]. Recently, the Legendre (LD) polynomials are

explored as another possible orthogonal expansion incorporated with FDTD to form a paralleling-in-order-based unconditionally stable FDTD method. Based on it, in this chapter, we made a comparison investigation for these two new methods, which are AH FDTD method and LD FDTD method, especially focused on their differences. Through a numerical example, we validate their effectiveness when compared with the conventional FDTD method and summarized the characteristics of the two methods.

#### 2. Formulation for paralleling-in-order scheme: AH and LD functions

#### 2.1 2D Maxwell's equations in time domain

The 2D time-domain Maxwell's equations with the TEz wave case in lossy medium are considered:

$$
\varepsilon \frac{\partial E\_{\mathbf{x}}(r,t)}{\partial t} + \sigma\_{\varepsilon} E\_{\mathbf{x}}(r,t) = \frac{\partial H\_{\mathbf{x}}(r,t)}{\partial \mathbf{y}} - J\_{\mathbf{x}}(r,t) \tag{1}
$$

u rð Þ¼ ; t ∑

unþ1ð Þr

ffiffi 1 p

> � ffiffi 2 <sup>p</sup> <sup>⋱</sup>

� ffiffi 1 p ffiffi 2 p

we can deduce the first derivative of u xð Þ ; t with respect to

∂ ∂t

where

be obtained [9].

polynomial given by [10]:

expanded by (9) as

151

the following recurrence relation:

u rð Þ¼ ; t

DOI: http://dx.doi.org/10.5772/intechopen.83387

obtain the relationship between Uand U\_ as

α ¼

2.2.2 The associated Legendre polynomial

PqðÞ¼ t

Lqþ<sup>1</sup>ðÞ¼ t

ffiffi 2 p 2λ

1 <sup>σ</sup> <sup>∑</sup> ∞ n¼0

can be obtained as <sup>U</sup> <sup>¼</sup> <sup>U</sup><sup>0</sup>⋯UQ�<sup>1</sup> � �<sup>T</sup> and <sup>U</sup>\_ <sup>¼</sup> <sup>U</sup>\_ <sup>0</sup>⋯U\_ <sup>Q</sup>�<sup>1</sup> � �<sup>T</sup>

∞ n¼0

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using…

r

Then, the Q-tuple AH domain coefficients for u rð Þ ; t and u r \_ð Þ ; t from (5) and (6)

ffiffiffiffiffiffiffiffiffiffiffi n þ 1 2

! r

⋱ ffiffiffiffiffiffiffiffiffiffiffiffi <sup>Q</sup> � <sup>1</sup> <sup>p</sup>

� ffiffiffiffiffiffiffiffiffiffiffiffi <sup>Q</sup> � <sup>1</sup> <sup>p</sup>

By using (8), the partial differential term in Maxwell's equations can readily be dealt with, and finally, a five-point banded matrix equation for Hz component can

We expand all the temporal quantities in terms of the associated Legendre

Lq 2 t l � 1

where l is the time support for analyzing a causal response and Lq is the Legendre polynomial with order q, which are orthogonal in the interval [�1,1] satisfying

tLqðÞ�<sup>t</sup> <sup>q</sup>

and L0ðÞ¼ t 0, L1ðÞ¼ t t. Given a time-support field function u rð Þ ; t , it can be

∞ q¼0

where uqð Þr is the q-th expanding coefficients, and it can be calculated by

þ ð∞

�∞

q þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffi 2q þ 1 2

2q þ 1 q þ 1

u rð Þ¼ ; t ∑

uqð Þ¼ r

r

� un�1ð Þr

unð Þ<sup>r</sup> <sup>ϕ</sup>nð Þ~<sup>t</sup> (5)

ffiffiffi n 2

<sup>U</sup>\_ <sup>¼</sup> <sup>α</sup><sup>U</sup> (7)

� �, t∈½ � <sup>0</sup>; <sup>l</sup> (9)

uqð Þr Pqð Þt (11)

u rð Þ ; t Pqð Þt dt (12)

Lq�<sup>1</sup>ð Þt , (10)

<sup>ϕ</sup>nð Þ~<sup>t</sup> (6)

. And, we can readily

(8)

$$
\mu \frac{\partial H\_x(r, t)}{\partial t} + \sigma\_m H\_x(r, t) = \frac{\partial E\_x(r, t)}{\partial y} - \frac{\partial E\_y(r, t)}{\partial \mathbf{x}} - M\_x(r, t) \tag{2}
$$

$$
\varepsilon \frac{\partial E\_{\text{y}}(r, t)}{\partial t} + \sigma\_{\text{t}} E\_{\text{y}}(r, t) = -\frac{\partial H\_{\text{x}}(r, t)}{\partial \mathbf{x}} - J\_{\text{y}}(r, t) \tag{3}
$$

where ε, μ, σe, and σ<sup>m</sup> are the permittivity, the permeability, the electric conductivity, and the magnetic loss of the medium, respectively. Eξð Þ r; t and Jξð Þ r; t (ξ ¼ x, y) are the electric field component and the electric current densities, respectively. Hzð Þ r; t and Mzð Þ r; t are the magnetic field component and magnetic current densities, respectively.

#### 2.2 The differential and integral transfer matrices to deal with the partial differential term in Maxwell's equations

#### 2.2.1 The associated Hermite function

Associated Hermite function is defined as

$$\left\{\phi\_n(t) = \left(2^n n! \pi^{1/2}\right)^{-1/2} e^{-t^2/2} H\_n(t)\right\}, (\mathbf{n} = \mathbf{0}, \ \mathbf{1}...) \tag{4}$$

where HnðÞ¼ � <sup>t</sup> ð Þ<sup>1</sup> <sup>n</sup> et <sup>2</sup> d<sup>n</sup> dt<sup>n</sup> e�<sup>t</sup> <sup>2</sup> � � is Hermite polynomials. Although it is not causal, it can be transformed into causal form by virtue of a proper translating and scaling parameters and then used to span the causal electromagnetic responses. The transformed basis function is <sup>ϕ</sup>nðÞ¼ <sup>~</sup><sup>t</sup> <sup>2</sup>nn!σπ<sup>1</sup>=<sup>2</sup> � ��1=<sup>2</sup> <sup>e</sup>�~t2=<sup>2</sup>Hnð Þ~<sup>t</sup> n o, where transformed time variable <sup>~</sup><sup>t</sup> <sup>¼</sup> <sup>t</sup> � Tf � �=σ. And Tf is a translating parameter and σ is a scaling parameter. By controlling these two parameters, the time-frequency support of the AH functions <sup>ϕ</sup>nð Þ~<sup>t</sup> � � space can be changed flexibly. So, arbitrary locally time-supported functions can be spanned by these transformed basis functions, including the causal electromagnetic responses.

From [7], if a causal function u rð Þ ; t , such as the electric or magnetic field function, can be expanded by

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using… DOI: http://dx.doi.org/10.5772/intechopen.83387

$$u(r,t) = \sum\_{n=0}^{\infty} u\_n(r) \overline{\phi}\_n(\overline{t}) \tag{5}$$

we can deduce the first derivative of u xð Þ ; t with respect to

$$\frac{\partial}{\partial t}u(r,t) = \frac{1}{\sigma} \sum\_{n=0}^{\infty} \left( u\_{n+1}(r) \sqrt{\frac{n+1}{2}} - u\_{n-1}(r) \sqrt{\frac{n}{2}} \right) \overline{\phi}\_n(\bar{t}) \tag{6}$$

Then, the Q-tuple AH domain coefficients for u rð Þ ; t and u r \_ð Þ ; t from (5) and (6) can be obtained as <sup>U</sup> <sup>¼</sup> <sup>U</sup><sup>0</sup>⋯UQ�<sup>1</sup> � �<sup>T</sup> and <sup>U</sup>\_ <sup>¼</sup> <sup>U</sup>\_ <sup>0</sup>⋯U\_ <sup>Q</sup>�<sup>1</sup> � �<sup>T</sup> . And, we can readily obtain the relationship between Uand U\_ as

$$
\dot{U} = aU\tag{7}
$$

where

explored as another possible orthogonal expansion incorporated with FDTD to form a paralleling-in-order-based unconditionally stable FDTD method. Based on it, in this chapter, we made a comparison investigation for these two new methods, which are AH FDTD method and LD FDTD method, especially focused on their differences. Through a numerical example, we validate their effectiveness when compared with the conventional FDTD method and summarized the characteristics

2. Formulation for paralleling-in-order scheme: AH and LD functions

The 2D time-domain Maxwell's equations with the TEz wave case in lossy

<sup>∂</sup>Hzð Þ <sup>r</sup>; <sup>t</sup>

<sup>∂</sup><sup>y</sup> � <sup>∂</sup>Eyð Þ <sup>r</sup>; <sup>t</sup>

<sup>∂</sup>Hzð Þ <sup>r</sup>; <sup>t</sup>

<sup>∂</sup>Exð Þ <sup>r</sup>; <sup>t</sup>

where ε, μ, σe, and σ<sup>m</sup> are the permittivity, the permeability, the electric conductivity, and the magnetic loss of the medium, respectively. Eξð Þ r; t and Jξð Þ r; t (ξ ¼ x, y) are the electric field component and the electric current densities, respectively. Hzð Þ r; t and Mzð Þ r; t are the magnetic field component and magnetic

<sup>∂</sup><sup>y</sup> � Jxð Þ <sup>r</sup>; <sup>t</sup> (1)

<sup>∂</sup><sup>x</sup> � Jyð Þ <sup>r</sup>; <sup>t</sup> (3)

<sup>∂</sup><sup>x</sup> � Mzð Þ <sup>r</sup>; <sup>t</sup> (2)

,ð Þ n ¼ 0; 1… (4)

, where

is Hermite polynomials. Although it is not

� �=σ. And Tf is a translating parameter and σ is

<sup>e</sup>�~t2=<sup>2</sup>Hnð Þ~<sup>t</sup>

þ σeExð Þ¼ r; t

þ σeEyð Þ¼� r; t

2.2 The differential and integral transfer matrices to deal with the partial

e �t <sup>2</sup>=2 Hnð Þt

causal, it can be transformed into causal form by virtue of a proper translating and scaling parameters and then used to span the causal electromagnetic responses. The

a scaling parameter. By controlling these two parameters, the time-frequency support of the AH functions <sup>ϕ</sup>nð Þ~<sup>t</sup> � � space can be changed flexibly. So, arbitrary locally time-supported functions can be spanned by these transformed basis functions,

From [7], if a causal function u rð Þ ; t , such as the electric or magnetic field

n o

þ σmHzð Þ¼ r; t

of the two methods.

Polynomials - Theory and Application

medium are considered:

μ

current densities, respectively.

2.2.1 The associated Hermite function

transformed time variable <sup>~</sup><sup>t</sup> <sup>¼</sup> <sup>t</sup> � Tf

function, can be expanded by

150

where HnðÞ¼ � <sup>t</sup> ð Þ<sup>1</sup> <sup>n</sup>

2.1 2D Maxwell's equations in time domain

ε

<sup>∂</sup>Hzð Þ <sup>r</sup>; <sup>t</sup> ∂t

ε

<sup>∂</sup>Exð Þ <sup>r</sup>; <sup>t</sup> ∂t

<sup>∂</sup>Eyð Þ <sup>r</sup>; <sup>t</sup> ∂t

differential term in Maxwell's equations

Associated Hermite function is defined as

et <sup>2</sup> d<sup>n</sup> dt<sup>n</sup> e�<sup>t</sup> <sup>2</sup> � �

including the causal electromagnetic responses.

transformed basis function is <sup>ϕ</sup>nðÞ¼ <sup>~</sup><sup>t</sup> <sup>2</sup>nn!σπ<sup>1</sup>=<sup>2</sup> � ��1=<sup>2</sup>

<sup>ϕ</sup>nðÞ¼ <sup>t</sup> <sup>2</sup>nn!π<sup>1</sup>=<sup>2</sup> � ��1=<sup>2</sup>

� �

$$a = \frac{\sqrt{2}}{2\lambda} \begin{bmatrix} \sqrt{1} & & & & \\ -\sqrt{1} & \sqrt{2} & & & \\ & -\sqrt{2} & \ddots & & \\ & & \ddots & \sqrt{Q-1} \\ & & -\sqrt{Q-1} & \\ \end{bmatrix}\_{Q \times Q} \tag{8}$$

By using (8), the partial differential term in Maxwell's equations can readily be dealt with, and finally, a five-point banded matrix equation for Hz component can be obtained [9].

#### 2.2.2 The associated Legendre polynomial

We expand all the temporal quantities in terms of the associated Legendre polynomial given by [10]:

$$P\_q(t) = \sqrt{\frac{2q+1}{2}} L\_q \left( 2\frac{t}{l} - 1 \right), \quad t \in [0, l] \tag{9}$$

where l is the time support for analyzing a causal response and Lq is the Legendre polynomial with order q, which are orthogonal in the interval [�1,1] satisfying the following recurrence relation:

$$L\_{q+1}(t) = \frac{2q+1}{q+1} t L\_q(t) - \frac{q}{q+1} L\_{q-1}(t),\tag{10}$$

and L0ðÞ¼ t 0, L1ðÞ¼ t t. Given a time-support field function u rð Þ ; t , it can be expanded by (9) as

$$u(r,t) = \sum\_{q=0}^{\infty} u\_q(r) P\_q(t) \tag{11}$$

where uqð Þr is the q-th expanding coefficients, and it can be calculated by

$$u\_q(r) = \int\_{-\infty}^{+\infty} u(r, t) P\_q(t) dt \tag{12}$$

From the intrinsic features of Legendre function, the differential relationship can be described as

$$P\_q(t) = \frac{1}{2\sqrt{(2q+3)(2q+1)}} P\_{q+1}'(t) - \frac{1}{2\sqrt{(2q+1)(2q-1)}} P\_{q-1}'(t) \tag{13}$$

If the field derivative of u rð Þ ; t to t is expanded as

$$\boldsymbol{u}'(r,t) = \sum\_{q=0}^{\infty} \boldsymbol{u}\_q^{(1)}(r) P\_q(t) \tag{14}$$

αe ið Þ ; <sup>j</sup> Ey 

DOI: http://dx.doi.org/10.5772/intechopen.83387

i,j ¼ Exji,jþ<sup>1</sup> � Exj

αm ið Þ ; <sup>j</sup> Hz 

where Exji,j

al ið Þ ; <sup>j</sup> Hzj

where

bi,j ¼ � <sup>α</sup>�<sup>1</sup>

method, and b<sup>∗</sup>

from (11).

153

m ið Þ ; <sup>j</sup>þ<sup>1</sup> Jx i,jþ1

decoupled equation, we have

j q

3. Comparison for the two methods

, Ey i,j , Hzji,j

<sup>i</sup>�1,j þ ar ið Þ <sup>þ</sup>1; <sup>j</sup> Hzj

where

i,j ¼ � Hzj

i,j <sup>=</sup>Δyj � Ey

αe ið Þ ; <sup>j</sup> ¼ εj

αm ið Þ ; <sup>j</sup> ¼ μmj

<sup>i</sup>þ1,j þ am ið Þ ; <sup>j</sup> Hzj

au ið Þ ; <sup>j</sup>þ<sup>1</sup> ¼ �α�<sup>1</sup>

ad ið Þ ; <sup>j</sup> ¼ �α�<sup>1</sup>

al ið Þ ; <sup>j</sup> ¼ �α�<sup>1</sup>

ar ið Þ <sup>þ</sup>1; <sup>j</sup> ¼ �α�<sup>1</sup>

eigenvector matrix and diagonal matrix composed of eigenvalues λ<sup>q</sup>

A 1=λ<sup>q</sup> H<sup>∗</sup> z <sup>q</sup> <sup>¼</sup> <sup>b</sup><sup>∗</sup> j

is the transformed variables from bj

� <sup>α</sup>�<sup>1</sup> m ið Þ ; <sup>j</sup> Jx i,j

=Δ<sup>y</sup> <sup>þ</sup> <sup>α</sup>�<sup>1</sup>

, Jxj i,j , Jy i,j

banded matrix equation for Hz component can be obtained:

i,j α�<sup>1</sup> <sup>L</sup> þ σej i,j

i,j α�<sup>1</sup> <sup>L</sup> þ σmji,j

fields and sources, respectively. And, I is the Q-dimensional identity matrix. By assembling (20)–(22) and eliminating the electric field components, a five-diagonal

i,j � Hzj

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using…

i�1,j <sup>=</sup>Δxi � Jy

i,j þ ad ið Þ ; <sup>j</sup> Hzj

e ið Þ ; <sup>j</sup> =Δyj

am ið Þ ; <sup>j</sup> ¼ � ar ið Þ <sup>þ</sup>1; <sup>j</sup> þ al ið Þ ; <sup>j</sup> þ au ið Þ ; <sup>j</sup>þ<sup>1</sup> þ ad ið Þ ; <sup>j</sup> þ αm ið Þ ; <sup>j</sup>

By using eigenvalue transformation from αLX ¼ XV, where X and V are the

where Að Þ� is a banded sparse matrix, with the similar form as from AH FDTD

obtain a paralleling-in-order scheme to calculate all of the expanding coefficients of electromagnetic fields, and then the time-domain responses can be reconstructed

The above formula can be regarded and classified as a uniform OF differential transfer matrix transformation. Therefore, as long as the LD differential matrix is replaced by the AH domain differential transfer matrix, the FDTD algorithm based

tively, Eq. (25) can be changed to the paralleling-in-order solution. For the q-th

<sup>i</sup>þ1,j � Ey

 i,j

, and Mzji,j are Q-tuple representations of

i,j�<sup>1</sup> þ au ið Þ ; <sup>j</sup>þ<sup>1</sup> Hzj

e ið Þ ; <sup>j</sup>þ<sup>1</sup> <sup>=</sup>Δyjþ<sup>1</sup>=Δyj (26)

e ið Þ ; <sup>j</sup> =Δxi=Δxi (28)

e ið Þ <sup>þ</sup>1; <sup>j</sup> =Δxiþ<sup>1</sup>=Δxi (29)

� <sup>α</sup>�<sup>1</sup> m ið Þ ; <sup>j</sup> Jy i,j

i,j <sup>¼</sup> Xb<sup>∗</sup>

j i,j

=Δx�Mzji,j

<sup>q</sup> (32)

(30)

m ið Þ <sup>þ</sup>1; <sup>j</sup> Jy iþ1,j

=Δyj (27)

   i,j

<sup>=</sup>Δxi � Mzji,j (22)

I (23)

I (24)

(21)

i,jþ<sup>1</sup> ¼ bi,j (25)

(31)

, respec-

. Finally, we can

where uð Þ<sup>1</sup> <sup>q</sup> ð Þr is q-th expanding coefficients for u 0 ð Þ r; t , then incorporated with (13), it can be deduced as

$$u'(r,t) = \left(\sum\_{q=0}^{\infty} \left(\frac{l}{2\sqrt{(2q+1)(2q-1)}} u\_{q-1}^{(1)}(r) - \frac{l}{2\sqrt{(2q+3)(2q+1)}} u\_{q+1}^{(1)}(r)\right) P\_q(t)\right)^{\frac{1}{2}} \tag{15}$$

Connecting (15) and (11), we can get

$$u\_q(r) = \frac{l}{2\sqrt{(2q+1)(2q-1)}} u\_{q-1}^{(1)}(r) - \frac{l}{2\sqrt{(2q+3)(2q+1)}} u\_{q+1}^{(1)}(r) \tag{16}$$

When assembling uqð Þ<sup>r</sup> � � <sup>q</sup>¼0, <sup>1</sup>⋯Q�<sup>1</sup> as a Q-tuple <sup>U</sup> and <sup>u</sup>ð Þ<sup>1</sup> <sup>q</sup> ð Þr n o q¼0, 1⋯Q�1 as Uð Þ<sup>1</sup> , a matrix-multiply relationship can be obtained from (16) as the following:

$$U = a\_L U^{(1)} \tag{17}$$

where α<sup>L</sup> is integral matrix.

$$a\_L = \frac{l}{2} \begin{bmatrix} \cdot \text{1/} \sqrt{1 \cdot 3} \\ \mathbf{1}/\sqrt{1 \cdot 3} & \cdot \text{1/} \sqrt{3 \cdot 5} \\ \mathbf{1}/\sqrt{3 \cdot 5} & \ddots \\ & \ddots & \mathbf{1}/\sqrt{(2Q-3)(2Q-1)} \\ & & \mathbf{1}/\sqrt{(2Q-3)(2Q-1)} \end{bmatrix}\_{Q \times Q} \tag{18}$$

Alternatively, Eq. (17) can be rewritten as.

$$U^{(1)} = a\_L^{-1} U \tag{19}$$

#### 2.3 From time domain to orthogonal domain and reconstruction

When the differential or integral transfer matrices are obtained, the timedomain Maxwell equation can be transformed directly into AH or LD domain. Here, let us set LD as an example to illustrate the later formulation.

Similar to the paralleling-in-order-based AH FDTD method, we can apply a Qtuple-domain transformation for LD FDTD method to (1)–(3) and discretize them as the following:

$$a\_{\epsilon(i,j)} E\_{\mathbf{x}}|\_{i,j} = \left( H\_x|\_{i,j} - H\_x|\_{i,j-1} \right) / \Delta \overline{y}\_j - J\_x|\_{i,j} \tag{20}$$

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using… DOI: http://dx.doi.org/10.5772/intechopen.83387

$$\left.a\_{\varepsilon(i,j)}E\_{\mathcal{Y}}\right|\_{i,j} = -\left(H\_x|\_{i,j} - H\_x|\_{i-1,j}\right) / \Delta \overline{x\_i} - J\_{\mathcal{Y}}\Big|\_{i,j} \tag{21}$$

$$\left.a\_{m(i,j)}H\_{\mathbf{x}}\right|\_{i,j} = \left(E\_{\mathbf{x}}|\_{i,j+1} - E\_{\mathbf{x}}|\_{i,j}\right) / \Delta \mathbf{y}\_j - \left(E\_{\mathbf{y}}|\_{i+1,j} - E\_{\mathbf{y}}|\_{i,j}\right) / \Delta \mathbf{x}\_i - M\_{\mathbf{z}}|\_{i,j} \tag{22}$$

where

From the intrinsic features of Legendre function, the differential relationship

1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup><sup>q</sup> � <sup>1</sup> <sup>p</sup> <sup>P</sup>

0

l 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>3</sup> ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> <sup>u</sup>ð Þ<sup>1</sup>

l 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>3</sup> ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> <sup>u</sup>ð Þ<sup>1</sup>

0

<sup>q</sup> ð Þr Pqð Þt (14)

<sup>q</sup> ð Þr n o

<sup>L</sup> U (19)

=Δyj � Jxji,j (20)

<sup>U</sup> <sup>¼</sup> <sup>α</sup>LUð Þ<sup>1</sup> (17)

ð Þ r; t , then incorporated with

<sup>q</sup>þ<sup>1</sup>ð Þ<sup>r</sup>

<sup>q</sup>þ<sup>1</sup>ð Þr (16)

q¼0, 1⋯Q�1

Pqð Þt

(15)

as

(18)

<sup>q</sup>�1ð Þt (13)

0 <sup>q</sup>þ1ðÞ�t

ð Þ¼ r; t ∑

∞ q¼0 uð Þ<sup>1</sup>

<sup>q</sup>�<sup>1</sup>ð Þ� <sup>r</sup>

<sup>q</sup>�<sup>1</sup>ð Þ� r

Uð Þ<sup>1</sup> , a matrix-multiply relationship can be obtained from (16) as the following:

<sup>3</sup> � <sup>5</sup> <sup>p</sup>

1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>Q</sup> � <sup>3</sup> ð Þ <sup>2</sup><sup>Q</sup> � <sup>1</sup> <sup>p</sup>

<sup>U</sup>ð Þ<sup>1</sup> <sup>¼</sup> <sup>α</sup>�<sup>1</sup>

When the differential or integral transfer matrices are obtained, the timedomain Maxwell equation can be transformed directly into AH or LD domain. Here,

Similar to the paralleling-in-order-based AH FDTD method, we can apply a Qtuple-domain transformation for LD FDTD method to (1)–(3) and discretize them

> i,j � Hzji,j�<sup>1</sup> � �

!

!<sup>0</sup>

<sup>q</sup>¼0, <sup>1</sup>⋯Q�<sup>1</sup> as a Q-tuple <sup>U</sup> and <sup>u</sup>ð Þ<sup>1</sup>

<sup>⋱</sup> ‐1<sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ <sup>2</sup><sup>Q</sup> � <sup>3</sup> ð Þ <sup>2</sup><sup>Q</sup> � <sup>1</sup> <sup>p</sup>

can be described as

Polynomials - Theory and Application

PqðÞ¼ t

where uð Þ<sup>1</sup>

ð Þ¼ r; t ∑

uqð Þ¼ r

When assembling uqð Þ<sup>r</sup> � �

where α<sup>L</sup> is integral matrix.

1= ffiffiffiffiffiffiffiffi

1= ffiffiffiffiffiffiffiffi

Alternatively, Eq. (17) can be rewritten as.

<sup>α</sup><sup>L</sup> <sup>¼</sup> <sup>l</sup> 2

as the following:

152

u 0

(13), it can be deduced as

∞ q¼0

1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>3</sup> ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> <sup>P</sup>

If the field derivative of u rð Þ ; t to t is expanded as

l 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup><sup>q</sup> � <sup>1</sup> <sup>p</sup> <sup>u</sup>ð Þ<sup>1</sup>

Connecting (15) and (11), we can get

l 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>q</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup><sup>q</sup> � <sup>1</sup> <sup>p</sup> <sup>u</sup>ð Þ<sup>1</sup>

> ‐1<sup>=</sup> ffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>3</sup> <sup>p</sup>

<sup>1</sup> � <sup>3</sup> <sup>p</sup> ‐1<sup>=</sup> ffiffiffiffiffiffiffiffi

<sup>3</sup> � <sup>5</sup> <sup>p</sup> <sup>⋱</sup>

2.3 From time domain to orthogonal domain and reconstruction

let us set LD as an example to illustrate the later formulation.

αe ið Þ ; <sup>j</sup> Exji,j ¼ Hzj

u 0

<sup>q</sup> ð Þr is q-th expanding coefficients for u

$$a\_{\varepsilon(i,j)} = \varepsilon|\_{i,j} a\_L^{-1} + \sigma\_{\varepsilon}|\_{i,j} I \tag{23}$$

$$
\alpha\_{m(i,j)} = \mu\_m|\_{i,j} a\_L^{-1} + \sigma\_m|\_{i,j} I \tag{24}
$$

where Exji,j , Ey i,j , Hzji,j , Jxj i,j , Jy i,j , and Mzji,j are Q-tuple representations of fields and sources, respectively. And, I is the Q-dimensional identity matrix. By assembling (20)–(22) and eliminating the electric field components, a five-diagonal banded matrix equation for Hz component can be obtained:

$$a\_{l(i,j)}H\_x|\_{i=1,j} + a\_{r(i+1,j)}H\_x|\_{i+1,j} + a\_{m(i,j)}H\_x|\_{i,j} + a\_{d(i,j)}H\_x|\_{i,j-1} + a\_{u(i,j+1)}H\_x|\_{i,j+1} = b\_{i,j} \tag{25}$$

where

$$\mathfrak{a}\_{u(i,j+1)} = -\mathfrak{a}\_{e(i,j+1)}^{-1} / \Delta \overline{y}\_{j+1} / \Delta y\_j \tag{26}$$

$$\mathfrak{a}\_{d(i,j)} = -a\_{\mathfrak{e}(i,j)}^{-1} / \Delta \mathfrak{y}\_j / \Delta \mathfrak{y}\_j \tag{27}$$

$$\mathfrak{a}\_{l(i,j)} = -a\_{\mathfrak{e}(i,j)}^{-1} / \Delta \overline{\mathfrak{x}}\_{i} / \Delta \mathfrak{x}\_{i} \tag{28}$$

$$a\_{r(i+1,j)} = -a\_{e(i+1,j)}^{-1} / \Delta \overline{x}\_{i+1} / \Delta x\_i \tag{29}$$

$$\mathfrak{a}\_{m(i,j)} = -\left(\mathfrak{a}\_{r(i+1,j)} + \mathfrak{a}\_{l(i,j)} + \mathfrak{a}\_{u(i,j+1)} + \mathfrak{a}\_{d(i,j)} + \mathfrak{a}\_{m(i,j)}\right) \tag{30}$$

$$b\_{i,j} = -\left(a\_{m(i,j+1)}^{-1} J\_x\Big|\_{i,j+1} - a\_{m(i,j)}^{-1} J\_x\Big|\_{i,j}\right) / \Delta y + \left(a\_{m(i+1,j)}^{-1} J\_y\Big|\_{i+1,j} - a\_{m(i,j)}^{-1} J\_y\Big|\_{i,j}\right) / \Delta x - \mathcal{M}\_{\overline{x}|\_{i,j}}\tag{31}$$

By using eigenvalue transformation from αLX ¼ XV, where X and V are the eigenvector matrix and diagonal matrix composed of eigenvalues λ<sup>q</sup> , respectively, Eq. (25) can be changed to the paralleling-in-order solution. For the q-th decoupled equation, we have

$$\left| A \left( \mathbf{1}/\lambda\_{\mathbf{q}} \right) H\_{\mathbf{z}}^{\*} \right|^{q} = b^{\*} \left|^{q} \right. \tag{32}$$

where Að Þ� is a banded sparse matrix, with the similar form as from AH FDTD method, and b<sup>∗</sup> j q is the transformed variables from bj i,j <sup>¼</sup> Xb<sup>∗</sup> j i,j . Finally, we can obtain a paralleling-in-order scheme to calculate all of the expanding coefficients of electromagnetic fields, and then the time-domain responses can be reconstructed from (11).

#### 3. Comparison for the two methods

The above formula can be regarded and classified as a uniform OF differential transfer matrix transformation. Therefore, as long as the LD differential matrix is replaced by the AH domain differential transfer matrix, the FDTD algorithm based


#### Table 1.

LD comparison of LD FDTD method and AH FDTD method.

on the LD orthogonal basis function, LD FDTD, including the parallel solution AH FDTD algorithm [9], and the alternate direction efficient calculation [11] can be easily realized. The implementation of the program only requires a simple modification.

only differing in the initial part. Therefore, in general, when the order of the two basic functions is the same and the parameters are selected reasonably, the accuracy

Comparison of calculation results between AH FDTD method and HR FDTD method when simulating an

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using…

DOI: http://dx.doi.org/10.5772/intechopen.83387

However, the two methods also have the differences when simulating the longtime response applications, such as the example in [12]. The numerical example is set as a TEz wave propagation in a parallel plate waveguide, as shown in Figure 2. It is with a PEC slot of the thickness 0.2 mm and the distance 0.2 mm and a partly filled dielectric material of the thickness 0.8 mm with the dielectric medium parameters given as two cases: case I, ε = 11 ε0, μ = μ0, σ<sup>e</sup> = 0.003 S/m, and σ<sup>m</sup> = 0 Ω/m; case II, ε = 2 ε0, μ = μ0, σ<sup>e</sup> = 30,000 S/m, and σ<sup>m</sup> = 0 Ω/m. There are 140 � 8 uniform cells (Δ<sup>x</sup> ¼ Δy=0.1 mm) in the computational domain. A Gaussian

<sup>2</sup> sin 2π<sup>f</sup> <sup>c</sup>ð Þ <sup>t</sup> � tc

, tc= 4td, and <sup>f</sup> <sup>c</sup>= 12 GHz. And the total simulation time is set as

(33)

pulse sinusoidally modulated is used as the electric current source profile:

l = 1.28 ns for case I and l = 12.8 ns for case II; then it leads to the marching-in-ontime steps for N = 6000 and N = 60,000, respectively. And the number of orders for LD functions is chosen as 80 and 300, respectively, to obtain a good approximation

The Ey electric field responses at measurement point p1 and p2, located at the center of the slot and behind the medium, respectively, are calculated, which are

Figures 2 and 3. For comparison, the AH FDTD method is also used in these two cases. One can find the good results in Figure 3, but the errors come out in Figure 4 for AH FDTD method when the same number of orthogonal functions (Q = 80 for case I or 300 for case II) is used as LD FDTD method. However, when Q reaches 800, the results from AH FDTD method can achieve a comparable accuracy with the ones from LD FDTD method. One should note that for case II the waveform at point p2 has larger amplitude attenuation and longer delay than the result at point

JyðÞ¼ t exp �ð Þ ð Þ t � tc =td

both in agreement with the conventional FDTD method as shown in

p1 due to the high dielectric medium located between them.

where td= 1= 2fc

Figure 1.

of field components.

155

is basically the same, and the efficiency is almost the same.

infinitely large lossy dielectric plate. (a) Time-domain waveform. (b) Relative error.

4.2 An nonuniform parallel plate waveguide with a slot

Table 1 gives a comparison of the relevant properties of the LD FDTD method and the AH FDTD method. It can be seen that the two methods can be considered as a "dual" system, because the AH differential matrix is the basic element of the AH FDTD method and the LD integration matrix is also the basic element of the LD FDTD method. This gives us a revelation that is it possible that any orthogonal basis function can construct a differential or integral transfer matrix and then easily implement a paralleling-in-order scheme similar like AH FDTD algorithm? The answer might be NOT. Such as the Laguerre FDTD method, as introduced before, cannot be calculated in parallel. However, it is undeniable that there may be more basis functions that can implement the paralleling-in-order scheme. If any, we can collectively call these methods as the AH series unconditionally stable FDTD method.

#### 4. Numerical verification

#### 4.1 An infinitely large lossy dielectric plate

As AH or LD FDTD method shares with almost the same program, a 1-D program is set for a general verification. Figure 1 shows the simulation results when a uniform plane wave penetrates an infinitely large lossy dielectric plate. The figure includes the electric field waveforms calculated by the AH FDTD method and the LD FDTD method and their relative errors with respect to the conventional FDTD method. It can be seen that the time-domain waveforms of both can be consistent with the results of the FDTD method and the relative errors are basically the same,

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using… DOI: http://dx.doi.org/10.5772/intechopen.83387

Figure 1.

on the LD orthogonal basis function, LD FDTD, including the parallel solution AH FDTD algorithm [9], and the alternate direction efficient calculation [11] can be easily realized. The implementation of the program only requires a simple modifi-

Table 1 gives a comparison of the relevant properties of the LD FDTD method and the AH FDTD method. It can be seen that the two methods can be considered as a "dual" system, because the AH differential matrix is the basic element of the AH FDTD method and the LD integration matrix is also the basic element of the LD FDTD method. This gives us a revelation that is it possible that any orthogonal basis function can construct a differential or integral transfer matrix and then easily implement a paralleling-in-order scheme similar like AH FDTD algorithm? The answer might be NOT. Such as the Laguerre FDTD method, as introduced before, cannot be calculated in parallel. However, it is undeniable that there may be more basis functions that can implement the paralleling-in-order scheme. If any, we can collectively call these methods as the AH series unconditionally stable FDTD

As AH or LD FDTD method shares with almost the same program, a 1-D program is set for a general verification. Figure 1 shows the simulation results when a uniform plane wave penetrates an infinitely large lossy dielectric plate. The figure includes the electric field waveforms calculated by the AH FDTD method and the LD FDTD method and their relative errors with respect to the conventional FDTD method. It can be seen that the time-domain waveforms of both can be consistent with the results of the FDTD method and the relative errors are basically the same,

cation.

<sup>α</sup>ð Þ<sup>l</sup> <sup>¼</sup> ffiffi 2 p 2l

FQ ≈

Antisymmetry

A λ<sup>q</sup> � �H<sup>q</sup> <sup>¼</sup> <sup>J</sup>

Table 1.

∂

8 >< >:

method.

154

4. Numerical verification

4.1 An infinitely large lossy dielectric plate

AH FDTD LD FDTD

<sup>U</sup>ð Þ<sup>1</sup> <sup>¼</sup> <sup>α</sup>U U <sup>¼</sup> <sup>α</sup>LUð Þ<sup>1</sup>

⋱ ffiffiffiffiffiffiffiffiffiffiffiffi <sup>Q</sup> � <sup>1</sup> <sup>p</sup>

! ð Þ l; Q

<sup>q</sup> A 1=λ<sup>q</sup>

LD comparison of LD FDTD method and AH FDTD method.

With time-frequency Homomorphism Without time-frequency

� ffiffiffiffiffiffiffiffiffiffiffiffi <sup>Q</sup> � <sup>1</sup> <sup>p</sup>

<sup>∂</sup><sup>t</sup> <sup>α</sup> ! <sup>j</sup><sup>ω</sup> <sup>Ð</sup>

<sup>π</sup>Q=1:<sup>7</sup> <sup>p</sup> <sup>þ</sup> <sup>1</sup>:<sup>8</sup> � �

Eigenvalue conjugate symmetry

ffiffi 1 p � ffiffi 1 p ffiffi 2 p � ffiffi 2 <sup>p</sup> <sup>⋱</sup>

Polynomials - Theory and Application

TQ ≈2l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>π</sup>Q=1:<sup>7</sup> <sup>p</sup> <sup>þ</sup> <sup>1</sup>:<sup>8</sup> 2πl

Differential transfer matrix Integral transfer matrix

<sup>α</sup>L lð Þ <sup>¼</sup> <sup>l</sup> 2 ‐1<sup>=</sup> ffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>3</sup> <sup>p</sup>

<sup>1</sup> � <sup>3</sup> <sup>p</sup> ‐1<sup>=</sup> ffiffiffiffiffiffiffiffi

<sup>3</sup> � <sup>5</sup> <sup>p</sup>

1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>Q</sup> � <sup>3</sup> ð Þ <sup>2</sup><sup>Q</sup> � <sup>1</sup> <sup>p</sup>

<sup>⋱</sup> ‐1<sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ <sup>2</sup><sup>Q</sup> � <sup>3</sup> ð Þ <sup>2</sup><sup>Q</sup> � <sup>1</sup> <sup>p</sup>

1= ffiffiffiffiffiffiffiffi

dt <sup>α</sup><sup>L</sup> ! <sup>1</sup> jω

Scale factor l = TQ Finite order of Q

Homomorphism

Eigenvalue conjugate symmetry

q

Antisymmetry

� �H<sup>q</sup> <sup>¼</sup> <sup>J</sup>

1= ffiffiffiffiffiffiffiffi <sup>3</sup> � <sup>5</sup> <sup>p</sup> <sup>⋱</sup>

Comparison of calculation results between AH FDTD method and HR FDTD method when simulating an infinitely large lossy dielectric plate. (a) Time-domain waveform. (b) Relative error.

only differing in the initial part. Therefore, in general, when the order of the two basic functions is the same and the parameters are selected reasonably, the accuracy is basically the same, and the efficiency is almost the same.

#### 4.2 An nonuniform parallel plate waveguide with a slot

However, the two methods also have the differences when simulating the longtime response applications, such as the example in [12]. The numerical example is set as a TEz wave propagation in a parallel plate waveguide, as shown in Figure 2. It is with a PEC slot of the thickness 0.2 mm and the distance 0.2 mm and a partly filled dielectric material of the thickness 0.8 mm with the dielectric medium parameters given as two cases: case I, ε = 11 ε0, μ = μ0, σ<sup>e</sup> = 0.003 S/m, and σ<sup>m</sup> = 0 Ω/m; case II, ε = 2 ε0, μ = μ0, σ<sup>e</sup> = 30,000 S/m, and σ<sup>m</sup> = 0 Ω/m. There are 140 � 8 uniform cells (Δ<sup>x</sup> ¼ Δy=0.1 mm) in the computational domain. A Gaussian pulse sinusoidally modulated is used as the electric current source profile:

$$J\_{\mathcal{I}}(t) = \exp\left(-\left((t-t\_c)/t\_d\right)^2\right)\sin\left(2\pi f\_c(t-t\_c)\right) \tag{33}$$

where td= 1= 2fc , tc= 4td, and <sup>f</sup> <sup>c</sup>= 12 GHz. And the total simulation time is set as l = 1.28 ns for case I and l = 12.8 ns for case II; then it leads to the marching-in-ontime steps for N = 6000 and N = 60,000, respectively. And the number of orders for LD functions is chosen as 80 and 300, respectively, to obtain a good approximation of field components.

The Ey electric field responses at measurement point p1 and p2, located at the center of the slot and behind the medium, respectively, are calculated, which are both in agreement with the conventional FDTD method as shown in Figures 2 and 3. For comparison, the AH FDTD method is also used in these two cases. One can find the good results in Figure 3, but the errors come out in Figure 4 for AH FDTD method when the same number of orthogonal functions (Q = 80 for case I or 300 for case II) is used as LD FDTD method. However, when Q reaches 800, the results from AH FDTD method can achieve a comparable accuracy with the ones from LD FDTD method. One should note that for case II the waveform at point p2 has larger amplitude attenuation and longer delay than the result at point p1 due to the high dielectric medium located between them.

#### Figure 2.

The geometry configuration for a 2D parallel plate waveguide with a PEC slot and a partly filled dielectric medium [12].

Tables 2 and 3 show the comparison of the computational resources. We can see that the simulation takes much more time for the FDTD method compared with proposed method, especially for the case of II, while the trade-off for the proposed method is that it consumes more memory than conventional FDTD method, which is similar to the AH FDTD method. In addition, from Table 3, we can find the advantages compared with AH FDTD method that the proposed method can use relative smaller memory storage and slightly fewer CPU times to get a readily results.

FDTD (N = 60,000) 0.21 1.8 30.8 AH FDTD (Q = 300) 21 11.8 1.55 AH FDTD (Q = 800) 21 28.9 1.95 LD FDTD (Q = 300) 21 11.8 1.55

FDTD (N = 6000) 0.21 1.8 2.97 AH FDTD (Q = 80) 21 2.9 1.32 LD FDTD (Q = 80) 21 2.9 1.32

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using…

Δt (ps) Memory (MB) CPU time (s)

Δt (ps) Memory (MB) CPU time (s)

The paralleling-in-order-based unconditionally stable FDTD methods are introduced using associated Hermite and Legendre polynomials in this chapter. The direct Q-tuple-domain transformation for time-domain Maxwell equation is guaranteed by using the integral matrix and differential matrix for Legendre function and associated Hermite functions that are introduced from the intrinsic integral or differential features for these orthogonal functions. Normally, the integral matrix of Legendre function can be considered as an inverse relationship from the differential operator, similar to the AH differential matrix. From this view, we can consider them as a uniform algorithm organized from the paralleling-in-order solution scheme. In addition, this chapter also detailed the different properties and the formula with these two methods theoretically and tested by numerical examples. Numerical examples for 1D and 2D cases validate their effectiveness and show LD FDTD with a better performance than AH FDTD method, in long-time simulation applications. In the next step, the more general paralleling-in-order scheme should be summarized, and then find or construct other possible orthogonal functions for

This work is supported by the National Natural Science Foundation of China under Grants 61801217 and 51477183 and Natural Science Foundation of Jiangsu Province under Grant BK20180422. This support is gratefully acknowledged.

5. Conclusions and future developments

The comparison of computational resources for the case of II [12].

The comparison of computational resources for the case of I [12].

DOI: http://dx.doi.org/10.5772/intechopen.83387

their specific applications.

Table 3.

Table 2.

Acknowledgements

157

Figure 3.

The calculated results of transient electric field Ey for the case of I [12].

Figure 4. The calculated results of transient electric field Ey for the case of II [12].


The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using… DOI: http://dx.doi.org/10.5772/intechopen.83387

#### Table 2.

Figure 2.

Figure 3.

Figure 4.

156

The calculated results of transient electric field Ey for the case of I [12].

The calculated results of transient electric field Ey for the case of II [12].

medium [12].

Polynomials - Theory and Application

The geometry configuration for a 2D parallel plate waveguide with a PEC slot and a partly filled dielectric

The comparison of computational resources for the case of I [12].


#### Table 3.

The comparison of computational resources for the case of II [12].

Tables 2 and 3 show the comparison of the computational resources. We can see that the simulation takes much more time for the FDTD method compared with proposed method, especially for the case of II, while the trade-off for the proposed method is that it consumes more memory than conventional FDTD method, which is similar to the AH FDTD method. In addition, from Table 3, we can find the advantages compared with AH FDTD method that the proposed method can use relative smaller memory storage and slightly fewer CPU times to get a readily results.

#### 5. Conclusions and future developments

The paralleling-in-order-based unconditionally stable FDTD methods are introduced using associated Hermite and Legendre polynomials in this chapter. The direct Q-tuple-domain transformation for time-domain Maxwell equation is guaranteed by using the integral matrix and differential matrix for Legendre function and associated Hermite functions that are introduced from the intrinsic integral or differential features for these orthogonal functions. Normally, the integral matrix of Legendre function can be considered as an inverse relationship from the differential operator, similar to the AH differential matrix. From this view, we can consider them as a uniform algorithm organized from the paralleling-in-order solution scheme. In addition, this chapter also detailed the different properties and the formula with these two methods theoretically and tested by numerical examples. Numerical examples for 1D and 2D cases validate their effectiveness and show LD FDTD with a better performance than AH FDTD method, in long-time simulation applications. In the next step, the more general paralleling-in-order scheme should be summarized, and then find or construct other possible orthogonal functions for their specific applications.

#### Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants 61801217 and 51477183 and Natural Science Foundation of Jiangsu Province under Grant BK20180422. This support is gratefully acknowledged.

Polynomials - Theory and Application

### Author details

Zheng-Yu Huang<sup>1</sup> \*, Zheng Sun<sup>2</sup> and Wei He3

1 Key Laboratory of Radar Imaging and Microwave Photonics, Nanjing University of Aeronautics and Astronautics, Nanjing, China

References

[1] Taflove A, Hagness SC. Finite-Difference Time-Domain Solution of Maxwell's Equations. 3rd ed. Vol. 14. Hoboken, NJ, USA: John Wiley & Sons,

DOI: http://dx.doi.org/10.5772/intechopen.83387

and applications. In: Presented at the 2018 IEEE International Symposium on Electromagnetic Compatibility and 2018 IEEE Asia-Pacific Symposium on Electromagnetic Compatibility (EMC/ APEMC 2018); 2018. pp. 119-122

[9] Huang Z-Y, Shi L-H, Zhou Y-H, Chen B. An improved paralleling-inorder solving scheme for AH-FDTD

transformation. IEEE Transactions on Antennas and Propagation. 2015;63(5):

[10] Tohidi E. Legendre approximation

comparison with Taylor and Bernoulli matrix methods. AM. 2012;3(5):410-416

[12] He W, Huang Z-Y, Li Y-B, Sun Z, Zhou Y-H. A new paralleling-in-order based unconditionally stable FDTD method using Legendre polynomials.

[11] Huang Z-Y, Shi L-H, Chen B. Efficient implementation for the AH FDTD method with iterative procedure and CFS-PML. IEEE Transactions on Antennas and Propagation. 2017;65(5):

for solving linear HPDEs and

method using eigenvalue

2135-2140

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using…

2728-2733

JAE. 2017;40(4):1-7

[2] Feng N, Zhang Y, Sun Q, Zhu J, Joines WT, Liu QH. An accurate 3-D CFS-PML based crank–Nicolson FDTD method and its applications in lowfrequency subsurface sensing. IEEE Transactions on Antennas and Propagation. 2018;66(6):2967-2975

[3] Wakabayashi Y, Shibayama J, Yamauchi J, Nakano H. A locally onedimensional finite difference time domain method for the analysis of a periodic structure at oblique incidence.

Radio Science. 2011;46(5):1-9

[4] Yan J, Jiao D. Fast explicit and unconditionally stable FDTD method for electromagnetic analysis. IEEE Transactions on Microwave Theory and Techniques. 2017;65(8):2698-2710

[5] Chung Y-S, Sarkar TK, Jung BH, Salazar-Palma M. An unconditionally stable scheme for the finite-difference

Transactions on Microwave Theory and

[7] Huang Z-Y, Shi L-H, Bin C, Zhou Y-H. A new unconditionally stable scheme for FDTD method using associated Hermite orthogonal functions. IEEE Transactions on Antennas and Propagation. 2014;62(9):4804-4809

[8] Huang Z-Y, Shi L-H. The associated Hermite FDTD method: Developments

time-domain method. IEEE

Techniques. 2003;51(3):697-704

[6] He G-Q , Stiens JH, Shao W, Wang B-Z. Recursively convolutional CFS-PML in 3-D Laguerre-FDTD scheme for arbitrary media. IEEE Transactions on Microwave Theory and

Techniques. 2018;66(5):1-10

159

Inc.; 2016. pp. 1-33. No. 4

2 National Key Laboratory on Electromagnetic Environmental Effects and Electro-Optical Engineering, Army Engineering University, Nanjing, China

3 Luoyang Hydraulic Engineering Technical Institute, Luoyang, China

\*Address all correspondence to: huangzynj@nuaa.edu.cn

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using… DOI: http://dx.doi.org/10.5772/intechopen.83387

### References

[1] Taflove A, Hagness SC. Finite-Difference Time-Domain Solution of Maxwell's Equations. 3rd ed. Vol. 14. Hoboken, NJ, USA: John Wiley & Sons, Inc.; 2016. pp. 1-33. No. 4

[2] Feng N, Zhang Y, Sun Q, Zhu J, Joines WT, Liu QH. An accurate 3-D CFS-PML based crank–Nicolson FDTD method and its applications in lowfrequency subsurface sensing. IEEE Transactions on Antennas and Propagation. 2018;66(6):2967-2975

[3] Wakabayashi Y, Shibayama J, Yamauchi J, Nakano H. A locally onedimensional finite difference time domain method for the analysis of a periodic structure at oblique incidence. Radio Science. 2011;46(5):1-9

[4] Yan J, Jiao D. Fast explicit and unconditionally stable FDTD method for electromagnetic analysis. IEEE Transactions on Microwave Theory and Techniques. 2017;65(8):2698-2710

[5] Chung Y-S, Sarkar TK, Jung BH, Salazar-Palma M. An unconditionally stable scheme for the finite-difference time-domain method. IEEE Transactions on Microwave Theory and Techniques. 2003;51(3):697-704

[6] He G-Q , Stiens JH, Shao W, Wang B-Z. Recursively convolutional CFS-PML in 3-D Laguerre-FDTD scheme for arbitrary media. IEEE Transactions on Microwave Theory and Techniques. 2018;66(5):1-10

[7] Huang Z-Y, Shi L-H, Bin C, Zhou Y-H. A new unconditionally stable scheme for FDTD method using associated Hermite orthogonal functions. IEEE Transactions on Antennas and Propagation. 2014;62(9):4804-4809

[8] Huang Z-Y, Shi L-H. The associated Hermite FDTD method: Developments

and applications. In: Presented at the 2018 IEEE International Symposium on Electromagnetic Compatibility and 2018 IEEE Asia-Pacific Symposium on Electromagnetic Compatibility (EMC/ APEMC 2018); 2018. pp. 119-122

[9] Huang Z-Y, Shi L-H, Zhou Y-H, Chen B. An improved paralleling-inorder solving scheme for AH-FDTD method using eigenvalue transformation. IEEE Transactions on Antennas and Propagation. 2015;63(5): 2135-2140

[10] Tohidi E. Legendre approximation for solving linear HPDEs and comparison with Taylor and Bernoulli matrix methods. AM. 2012;3(5):410-416

[11] Huang Z-Y, Shi L-H, Chen B. Efficient implementation for the AH FDTD method with iterative procedure and CFS-PML. IEEE Transactions on Antennas and Propagation. 2017;65(5): 2728-2733

[12] He W, Huang Z-Y, Li Y-B, Sun Z, Zhou Y-H. A new paralleling-in-order based unconditionally stable FDTD method using Legendre polynomials. JAE. 2017;40(4):1-7

Author details

Polynomials - Theory and Application

Zheng-Yu Huang<sup>1</sup>

158

\*, Zheng Sun<sup>2</sup> and Wei He3

of Aeronautics and Astronautics, Nanjing, China

provided the original work is properly cited.

1 Key Laboratory of Radar Imaging and Microwave Photonics, Nanjing University

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 National Key Laboratory on Electromagnetic Environmental Effects and Electro-Optical Engineering, Army Engineering University, Nanjing, China

3 Luoyang Hydraulic Engineering Technical Institute, Luoyang, China

\*Address all correspondence to: huangzynj@nuaa.edu.cn

## *Edited by Cheon Seoung Ryoo*

Polynomials are well known for their ability to improve their properties and for their applicability in the interdisciplinary fields of engineering and science. Many problems arising in engineering and physics are mathematically constructed by differential equations. Most of these problems can only be solved using special polynomials. Special polynomials and orthonormal polynomials provide a new way to analyze solutions of various equations often encountered in engineering and physical problems. In particular, special polynomials play a fundamental and important role in mathematics and applied mathematics. Until now, research on polynomials has been done in mathematics and applied mathematics only. This book is based on recent results in all areas related to polynomials. Divided into sections on theory and application, this book provides an overview of the current research in the field of polynomials. Topics include cyclotomic and Littlewood polynomials; Descartes' rule of signs; obtaining explicit formulas and identities for polynomials defined by generating functions; polynomials with symmetric zeros; numerical investigation on the structure of the zeros of the q-tangent polynomials; investigation and synthesis of robust polynomials in uncertainty on the basis of the root locus theory; pricing basket options by polynomial approximations; and orthogonal expansion in time domain method for solving Maxwell's equations using paralleling-in-order scheme.

Published in London, UK © 2019 IntechOpen © Sergei Pivovarov / iStock

Polynomials - Theory and Application

Polynomials

Theory and Application

*Edited by Cheon Seoung Ryoo*