Section 2 Applications

**Chapter 6**

**Abstract**

defined by *f*

**1. Introduction**

**91**

**1.1 Basics of graph labelling**

Graphs

*Sudev Naduvath*

Modular Sumset Labelling of

Graph labelling is an assignment of labels or weights to the vertices and/or edges

þð Þ¼ *uv f u*ð Þþ *f v*ð Þ, for all *uv*∈*E G*ð Þ, where *f u*ð Þþ *f v*ð Þ is the sumset

<sup>⊕</sup> : *E G*ð Þ!Pð Þ *<sup>X</sup>* is

<sup>⊕</sup> : *V G*ð Þ!Pð Þ *<sup>X</sup>* defined by

of a graph. For a ground set *X* of integers, a sumset labelling of a graph is an

of the set-label, the vertices *u* and *v*. In this chapter, we discuss a special type of sumset labelling of a graph, called modular sumset labelling and its variations. We also discuss some interesting characteristics and structural properties of the graphs

**Keywords:** sumset labelling, modular sumset labelling, weak modular sumset labelling, strong modular sumset labelling, arithmetic modular sumset labelling

For terminology and results in graph theory, we refer to [1–5]. For further notions and concepts on graph classes, graph operations, graph products and derived graphs, refer to [3, 6–10]. Unless mentioned otherwise, all graphs mentioned in this chapter are simple, finite, connected and undirected.

Labelling of a graph *G* can broadly be considered as an assignment of labels or weights to the elements (vertices and edges) of *G* subject to certain pre-defined conditions. The research on graph labelling has flourished in the second half of twentieth century after the introduction of the notion of *β*-valuations of graphs in [11]. The *β-valuation* of a graph *G* is an injective map *f* : *V G*ð Þ! f g 1, 2, 3, … , j*E*j such that the induced function *<sup>f</sup>* <sup>∗</sup> : *E G*ð Þ! f g 1, 2, 3, … , <sup>j</sup>*E*<sup>j</sup> , defined by *<sup>f</sup>* <sup>∗</sup> ð Þ¼ *uv* ∣ *f u*ð Þ� *f v*ð Þ∣ for all *uv*∈*E G*ð Þ, is also injective. Later, *β*-valuation of graphs was popularly known to be the *graceful labelling* of graphs (see [12]). Many variations of number valuations have been defined in the literature since then and most of those studies were based on the number theory and/or number theoretic properties of

Analogous to the number valuations of graphs, the notion of set-labelling of graphs has been introduced in [17] as follows: Given a non-empty ground set *X*, a *set-labelling* or a *set-valuation* of a graph *G* is an injective function *f* : *V G*ð Þ!Pð Þ *X* ,

sets. For concepts and results in number theory, see [13–16].

the power set of *X*, such that the induced function *f*

injective map *f* : *V G*ð Þ!Pð Þ *X* such that the induced function *f*

which admit these new types of graph labellings.

#### **Chapter 6**

## Modular Sumset Labelling of Graphs

*Sudev Naduvath*

#### **Abstract**

Graph labelling is an assignment of labels or weights to the vertices and/or edges of a graph. For a ground set *X* of integers, a sumset labelling of a graph is an injective map *f* : *V G*ð Þ!Pð Þ *X* such that the induced function *f* <sup>⊕</sup> : *E G*ð Þ!Pð Þ *<sup>X</sup>* is defined by *f* þð Þ¼ *uv f u*ð Þþ *f v*ð Þ, for all *uv*∈*E G*ð Þ, where *f u*ð Þþ *f v*ð Þ is the sumset of the set-label, the vertices *u* and *v*. In this chapter, we discuss a special type of sumset labelling of a graph, called modular sumset labelling and its variations. We also discuss some interesting characteristics and structural properties of the graphs which admit these new types of graph labellings.

**Keywords:** sumset labelling, modular sumset labelling, weak modular sumset labelling, strong modular sumset labelling, arithmetic modular sumset labelling

#### **1. Introduction**

For terminology and results in graph theory, we refer to [1–5]. For further notions and concepts on graph classes, graph operations, graph products and derived graphs, refer to [3, 6–10]. Unless mentioned otherwise, all graphs mentioned in this chapter are simple, finite, connected and undirected.

#### **1.1 Basics of graph labelling**

Labelling of a graph *G* can broadly be considered as an assignment of labels or weights to the elements (vertices and edges) of *G* subject to certain pre-defined conditions. The research on graph labelling has flourished in the second half of twentieth century after the introduction of the notion of *β*-valuations of graphs in [11]. The *β-valuation* of a graph *G* is an injective map *f* : *V G*ð Þ! f g 1, 2, 3, … , j*E*j such that the induced function *<sup>f</sup>* <sup>∗</sup> : *E G*ð Þ! f g 1, 2, 3, … , <sup>j</sup>*E*<sup>j</sup> , defined by *<sup>f</sup>* <sup>∗</sup> ð Þ¼ *uv* ∣ *f u*ð Þ� *f v*ð Þ∣ for all *uv*∈*E G*ð Þ, is also injective. Later, *β*-valuation of graphs was popularly known to be the *graceful labelling* of graphs (see [12]). Many variations of number valuations have been defined in the literature since then and most of those studies were based on the number theory and/or number theoretic properties of sets. For concepts and results in number theory, see [13–16].

Analogous to the number valuations of graphs, the notion of set-labelling of graphs has been introduced in [17] as follows: Given a non-empty ground set *X*, a *set-labelling* or a *set-valuation* of a graph *G* is an injective function *f* : *V G*ð Þ!Pð Þ *X* , the power set of *X*, such that the induced function *f* <sup>⊕</sup> : *V G*ð Þ!Pð Þ *<sup>X</sup>* defined by

*f* <sup>⊕</sup>ð Þ¼ *uv f u*ð Þ⊕*f v*ð Þ, for all *uv*∈*E G*ð Þ, where <sup>⨁</sup> is the symmetric difference of two sets. A graph which admits a set-labelling is called a *set-labelled graph* or a *set-valued graph*. If the induced function *f* <sup>⊕</sup> is also injective, then the set-labelling *f* is called a *set-indexer*.

set-labelling is called an *integer additive set-labelled graph*. It can very easily be verified that every graph *G* admits an integer additive set-labelling, provided the

Following to the above path-breaking study, the structural properties and characteristics of different types of integer additive set-labellings of graphs are studied intensively in accordance with the cardinality of the set-label, nature and pattern of elements in the set-label, nature of the collection of set-label, etc. Some interesting and significant studies in this area can be found in [35–44]. Later, the studies in this area have been extended by including the sets of integers (including negative integers also) for labelling the elements of a graph. Some extensive studies in this

As a specialisation of the sumset labelling of graphs, the notion of modular sumset labelling of graphs and corresponding results are discussed in the following

Recall that *<sup>n</sup>* denotes the set of integers modulo *n*, where *n* is a positive integer.

If we assign the null set Ø to any vertex as the set-label, the set-label of every edge incident at that vertex will also be a null set. To avoid such an embarrassing situation, we do not consider the null set for labelling any vertex of graphs. Thus, the set of all non-empty subsets of a set *X* is denoted by P0ð Þ *X* . That is, P0ð Þ¼P *X* ð Þn *X* f g0 .

In view of the facts stated above, the modular sumset labelling of a graph is

**Definition 2.1.** [47] A modular sumset labelling of a graph *G* is an injective

labels of the vertices *u* and *v*. A graph which admits a modular sumset labelling is

**Definition 2.2.** [47] A modular sumset labelling of a graph *G* is said to be a *uniform modular sumset labelling* of *G* if the set-label of all its edges have the same cardinality. A modular sumset labelling *f* of *G* is said to be a *k-uniform modular*

The proof of the above proposition is immediate from the fact that *f u*ð Þþ

**Proposition 2.1.** [47] *Every graph G admits modular sumset labelling (for a suitable*

An immediate question that arises in this context is about the minimum size of the ground set *<sup>n</sup>* (that is, the minimum value of *n*) required for the existence of a

þð Þ¼ *uv f u*ð Þþ *f v*ð Þ, where *f u*ð Þþ *f v*ð Þ is the modular sumset of the set

<sup>þ</sup> : *E G*ð Þ!P0ð Þ *<sup>n</sup>* is

The *modular sumset* of two subsets *A* and *B* of *<sup>n</sup>* is the set f*k* : *a*∈ *A*, *b*∈*B*, *a* þ *b* � *k* ð Þg mod *n* . Unless mentioned otherwise, throughout this chapter, the notation *A* þ *B* denotes the modular sumset of the sets *A* and *B*. Unlike the ordinary sumsets, the modular sumset *A* þ *B* ⊆*<sup>n</sup>* if and only if *A*, *B*⊆ *n*. This fact will ease many restrictions imposed on the vertex set-label of a sumset graph *G* in order to

ensure that the edge set-label are also subsets of the ground set.

function *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* such that the induced function *f*

þð Þ¼ *uv k*, ∀*uv*∈*E G*ð Þ.

ground set *X* is chosen judiciously.

*Modular Sumset Labelling of Graphs*

*DOI: http://dx.doi.org/10.5772/intechopen.92701*

area, can be seen in [45, 46].

**2.1 Basics of modular sumsets**

**2.2 Modular sumset graphs**

called a *modular sumset graph*.

*f v*ð Þ⊆*<sup>n</sup>* if and only if *f u*ð Þ, *f v*ð Þ⊆*n*.

modular sumset labelling of *G*.

defined as follows:

*sumset labelling* if *f*

defined as *f*

*choice of n).*

**93**

**2. Modular sumset labelling of graphs**

section.

Subsequent to this study, many intensive investigations on set-labelling of graphs and its different types have been taken place. An overview of such studies on set-labelling of graphs can be seen in [17–23]. The study on set-labelling has been extended by replacing the binary operation ⨁ by some other binary operations of sets. For example, two different types of set-labellings—called disjunctive setlabelling and conjunctive set-labelling—of graphs have been studied in [24]. These set-labellings are defined respectively in terms of the union and the intersection of two sets instead of the symmetric difference of two sets.

#### **1.2 Sumsets and integer additive set-labelled graphs**

Integer additive set-labelling or sumset labelling of graphs has been a new addition to the theory of set-labelling of graphs recently. The notion of sumsets of two sets is explained as follows: Let *A* and *B* be two sets of numbers. The *sumset* of *A* and *B* is denoted by *A* þ *B* and is defined by *A* þ *B* ¼ f g *a* þ *b* : *a*∈ *A*, *b*∈*B* (see [25]). Remember that a sumset of two sets can be determined if and only if both of them are number sets. If either *A* or *B* is countably infinite, then their sumset *A* þ *B* is also a countably infinite set and if any one of them is a null set, then the sumset is also a null set. If *C* is the sumset of two sets *A* and *B*, then both *A* and *B* are said to be the *summands* of *C*.

We note that *A* þ f g¼ 0 *A* and hence *A* and 0f g are called the trivial summands of the set *A*. Also, note that *A* þ *B* need not be a subset or a super set of *A* and/or *B*. But, *A* ⊂ *A* þ *B* if 0∈*B*. Furthermore, the sumset of two subsets of a set *X* need not be a subset of the ground set *X*. These observations are clear deviations from the other common binary operations of sets and thus the study of sumsets becomes more interesting. For the terms, concepts and results on sumsets, we refer to [25–34].

Note that if *A* and *B* are two non-empty finite sets of integers, then ∣*A*∣ þ ∣*B*∣ � 1≤ ∣*A* þ *B*∣≤∣*A* � *B*∣ ¼ ∣*A*∣∣*B*∣ (see [25]). The exact cardinality of the sumset *A* þ *B* always depends on the number as well as the pattern of elements in both the summands *A* and *B*. The counting procedure in this case is explained in [35] as follows: Two ordered pairs ð Þ *a*, *b* and ð Þ *c*, *d* in *A* � *B* is said to be *compatible* if *a* þ *b* ¼ *c* þ *d*. If ð Þ *a*, *b* and ð Þ *c*, *d* are compatible, then it is written as ð Þ� *a*, *b* ð Þ *c*, *d* . It can easily be verified that this relation is an equivalence relation. A *compatibility class* of an ordered pair ð Þ *a*, *b* in *A* � *B* with respect to the integer *k* ¼ *a* þ *b* is the subset of *A* � *B* defined by f g ð Þ *c*, *d* ∈ *A* � *B* : ð Þ� *a*, *b* ð Þ *c*, *d* and is denoted by ½ � ð Þ *a*, *b <sup>k</sup>* or C*k*. The cardinality of a compatibility class in *A* � *B* lies between 1 and min f g j*A*j, j*B*j . Note that the sum of coordinates of all elements in a compatibility class is the same and this sum will be an element of the sumset *A* þ *B*. That is, the cardinality of the sumset of two sets is equal to the number of equivalence classes on the Cartesian product of the two sets generated by the compatibility relation defined on it.

Using the concepts of the sumsets of sets, the notion of integer additive setlabelling of graphs has been introduced in [36] as follows: Let *X* be a set of nonnegative integers and P0ð Þ *X* be the collection of the non-empty subsets of *X*. Then, an *integer additive set-labelling* or an *integer additive set-valuation* of a graph *G* is an injective map *f* : *V G*ð Þ!Pð Þ *X* such that the induced function *f* <sup>⊕</sup> : *E G*ð Þ!Pð Þ *<sup>X</sup>* is defined by *f* þð Þ¼ *uv f u*ð Þþ *f v*ð Þ, for all *uv*∈*E G*ð Þ, where *f u*ð Þþ *f v*ð Þ is the sumset of the set-label the vertices *u* and *v* (see [36, 37]). A graph with an integer additive

#### *Modular Sumset Labelling of Graphs DOI: http://dx.doi.org/10.5772/intechopen.92701*

*f*

*set-indexer*.

the *summands* of *C*.

[25–34].

defined by *f*

**92**

*graph*. If the induced function *f*

*Number Theory and Its Applications*

<sup>⊕</sup>ð Þ¼ *uv f u*ð Þ⊕*f v*ð Þ, for all *uv*∈*E G*ð Þ, where <sup>⨁</sup> is the symmetric difference of two sets. A graph which admits a set-labelling is called a *set-labelled graph* or a *set-valued*

Subsequent to this study, many intensive investigations on set-labelling of graphs and its different types have been taken place. An overview of such studies on set-labelling of graphs can be seen in [17–23]. The study on set-labelling has been extended by replacing the binary operation ⨁ by some other binary operations of sets. For example, two different types of set-labellings—called disjunctive setlabelling and conjunctive set-labelling—of graphs have been studied in [24]. These set-labellings are defined respectively in terms of the union and the intersection of

Integer additive set-labelling or sumset labelling of graphs has been a new addition to the theory of set-labelling of graphs recently. The notion of sumsets of two sets is explained as follows: Let *A* and *B* be two sets of numbers. The *sumset* of *A* and *B* is denoted by *A* þ *B* and is defined by *A* þ *B* ¼ f g *a* þ *b* : *a*∈ *A*, *b*∈*B* (see [25]). Remember that a sumset of two sets can be determined if and only if both of them are number sets. If either *A* or *B* is countably infinite, then their sumset *A* þ *B* is also a countably infinite set and if any one of them is a null set, then the sumset is also a null set. If *C* is the sumset of two sets *A* and *B*, then both *A* and *B* are said to be

We note that *A* þ f g¼ 0 *A* and hence *A* and 0f g are called the trivial summands of the set *A*. Also, note that *A* þ *B* need not be a subset or a super set of *A* and/or *B*. But, *A* ⊂ *A* þ *B* if 0∈*B*. Furthermore, the sumset of two subsets of a set *X* need not be a subset of the ground set *X*. These observations are clear deviations from the other common binary operations of sets and thus the study of sumsets becomes more interesting. For the terms, concepts and results on sumsets, we refer to

Note that if *A* and *B* are two non-empty finite sets of integers, then ∣*A*∣ þ ∣*B*∣ � 1≤ ∣*A* þ *B*∣≤∣*A* � *B*∣ ¼ ∣*A*∣∣*B*∣ (see [25]). The exact cardinality of the sumset *A* þ *B* always depends on the number as well as the pattern of elements in both the summands *A* and *B*. The counting procedure in this case is explained in [35] as follows: Two ordered pairs ð Þ *a*, *b* and ð Þ *c*, *d* in *A* � *B* is said to be *compatible* if *a* þ *b* ¼ *c* þ *d*. If ð Þ *a*, *b* and ð Þ *c*, *d* are compatible, then it is written as ð Þ� *a*, *b* ð Þ *c*, *d* . It can easily be verified that this relation is an equivalence relation. A *compatibility class* of an ordered pair ð Þ *a*, *b* in *A* � *B* with respect to the integer *k* ¼ *a* þ *b* is the subset of *A* � *B* defined by f g ð Þ *c*, *d* ∈ *A* � *B* : ð Þ� *a*, *b* ð Þ *c*, *d* and is denoted by ½ � ð Þ *a*, *b <sup>k</sup>* or C*k*. The cardinality of a compatibility class in *A* � *B* lies between 1 and min f g j*A*j, j*B*j . Note that the sum of coordinates of all elements in a compatibility class is the same and this sum will be an element of the sumset *A* þ *B*. That is, the cardinality of the sumset of two sets is equal to the number of equivalence classes on the Cartesian product of the two sets generated by the compatibility relation defined on it. Using the concepts of the sumsets of sets, the notion of integer additive setlabelling of graphs has been introduced in [36] as follows: Let *X* be a set of nonnegative integers and P0ð Þ *X* be the collection of the non-empty subsets of *X*. Then, an *integer additive set-labelling* or an *integer additive set-valuation* of a graph *G* is an

þð Þ¼ *uv f u*ð Þþ *f v*ð Þ, for all *uv*∈*E G*ð Þ, where *f u*ð Þþ *f v*ð Þ is the sumset

of the set-label the vertices *u* and *v* (see [36, 37]). A graph with an integer additive

<sup>⊕</sup> : *E G*ð Þ!Pð Þ *<sup>X</sup>* is

injective map *f* : *V G*ð Þ!Pð Þ *X* such that the induced function *f*

two sets instead of the symmetric difference of two sets.

**1.2 Sumsets and integer additive set-labelled graphs**

<sup>⊕</sup> is also injective, then the set-labelling *f* is called a

set-labelling is called an *integer additive set-labelled graph*. It can very easily be verified that every graph *G* admits an integer additive set-labelling, provided the ground set *X* is chosen judiciously.

Following to the above path-breaking study, the structural properties and characteristics of different types of integer additive set-labellings of graphs are studied intensively in accordance with the cardinality of the set-label, nature and pattern of elements in the set-label, nature of the collection of set-label, etc. Some interesting and significant studies in this area can be found in [35–44]. Later, the studies in this area have been extended by including the sets of integers (including negative integers also) for labelling the elements of a graph. Some extensive studies in this area, can be seen in [45, 46].

As a specialisation of the sumset labelling of graphs, the notion of modular sumset labelling of graphs and corresponding results are discussed in the following section.

#### **2. Modular sumset labelling of graphs**

#### **2.1 Basics of modular sumsets**

Recall that *<sup>n</sup>* denotes the set of integers modulo *n*, where *n* is a positive integer. The *modular sumset* of two subsets *A* and *B* of *<sup>n</sup>* is the set f*k* : *a*∈ *A*, *b*∈*B*, *a* þ *b* � *k* ð Þg mod *n* . Unless mentioned otherwise, throughout this chapter, the notation *A* þ *B* denotes the modular sumset of the sets *A* and *B*. Unlike the ordinary sumsets, the modular sumset *A* þ *B* ⊆*<sup>n</sup>* if and only if *A*, *B*⊆ *n*. This fact will ease many restrictions imposed on the vertex set-label of a sumset graph *G* in order to ensure that the edge set-label are also subsets of the ground set.

If we assign the null set Ø to any vertex as the set-label, the set-label of every edge incident at that vertex will also be a null set. To avoid such an embarrassing situation, we do not consider the null set for labelling any vertex of graphs. Thus, the set of all non-empty subsets of a set *X* is denoted by P0ð Þ *X* . That is, P0ð Þ¼P *X* ð Þn *X* f g0 .

#### **2.2 Modular sumset graphs**

In view of the facts stated above, the modular sumset labelling of a graph is defined as follows:

**Definition 2.1.** [47] A modular sumset labelling of a graph *G* is an injective function *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* such that the induced function *f* <sup>þ</sup> : *E G*ð Þ!P0ð Þ *<sup>n</sup>* is defined as *f* þð Þ¼ *uv f u*ð Þþ *f v*ð Þ, where *f u*ð Þþ *f v*ð Þ is the modular sumset of the set labels of the vertices *u* and *v*. A graph which admits a modular sumset labelling is called a *modular sumset graph*.

**Definition 2.2.** [47] A modular sumset labelling of a graph *G* is said to be a *uniform modular sumset labelling* of *G* if the set-label of all its edges have the same cardinality. A modular sumset labelling *f* of *G* is said to be a *k-uniform modular sumset labelling* if *f* þð Þ¼ *uv k*, ∀*uv*∈*E G*ð Þ.

**Proposition 2.1.** [47] *Every graph G admits modular sumset labelling (for a suitable choice of n).*

The proof of the above proposition is immediate from the fact that *f u*ð Þþ *f v*ð Þ⊆*<sup>n</sup>* if and only if *f u*ð Þ, *f v*ð Þ⊆*n*.

An immediate question that arises in this context is about the minimum size of the ground set *<sup>n</sup>* (that is, the minimum value of *n*) required for the existence of a modular sumset labelling of *G*.

As in the case of sumsets, the cardinality of the modular sumsets also attracted the attention. Hence, we have the bounds for the cardinality of an edge set-label of a modular sumset graph *G* is as follows:

**Theorem 2.2.** [47] *Let f* : *V G*ð Þ!P0ð Þ *<sup>n</sup> be a modular sumset labelling of a given graph G. Then, for any edge uv*∈*E G*ð Þ*, we have*

$$|f(u)| + |f(v)| - \mathbf{1} \le |f^+(uv)| = |f(u) + f(v)| \le |f(u)| \, |f(v)| \le n. \tag{1}$$

*DA*¼ j f g *a* � *b*j: *a*, *b*∈ *A*

*Modular Sumset Labelling of Graphs*

*DOI: http://dx.doi.org/10.5772/intechopen.92701*

¼ *D f u*ð Þ þ *D f v*ð Þ

mind, we define the following notion.

*having smaller common difference.*

adjacent vertices of *G* are the same.

then *f*

**95**

ling is called an *arithmetic modular sumset graph*.

labelling is discussed in the following theorem.

*Proof.* Here, we need to consider the following two cases:

¼ j*ai* þ *br* � *a <sup>j</sup>* þ *bs*

Hence, *D f u*ð Þ þ *D f v*ð Þ is an arithmetic progression.

progression, then by the above remark, we have *A* ¼ *f u*ð Þþ *f v*ð Þ¼ *f*

 <sup>j</sup>: *ai*, *<sup>a</sup> <sup>j</sup>* <sup>∈</sup> *f u*ð Þ, *br*, *bs* <sup>∈</sup>*f v*ð Þ ¼ j*ai* � *<sup>a</sup> <sup>j</sup>*jþj*br* � *bs*j: *ai*, *<sup>a</sup> <sup>j</sup>* <sup>∈</sup>*f u*ð Þ, *br*, *bs* <sup>∈</sup>*f v*ð Þ ¼ j*ai* � *<sup>a</sup> <sup>j</sup>*j: *ai*, *<sup>a</sup> <sup>j</sup>* <sup>∈</sup>*f u*ð Þ þ j*br* � *bs*j: *br* f g , *bs* <sup>∈</sup> *f v*ð Þ

Conversely, assume that *D f u*ð Þ þ *D f v*ð Þ is an arithmetic progression. Then, by previous step, we have *D f u*ð Þ þ *D f v*ð Þ ¼ *DA*, where *A* ¼ *f u*ð Þþ *f v*ð Þ. Then, we have *DA* is an arithmetic progression. Since the difference set *DA* is an arithmetic

arithmetic progression. Hence, the edge *e* ¼ *uv* has an arithmetic progression as its set-label. □ In view of the notions mentioned above, we note that there are some graphs, all whose elements have arithmetic progressions as their set-label and there are some graphs, the set-label of whose edges are not arithmetic progressions. Keeping this in

**Definition 3.2.** An *arithmetic sumset labelling* of a graph *G* is a modular sumset labelling *f* of *G*, with respect to which the set-label of all vertices and edges of *G* are arithmetic progressions. A graph that admits an arithmetic modular sumset label-

Analogous to the condition for an arithmetic sumset graphs (see [44]), a necessary and sufficient condition for a graph to admit an arithmetic modular sumset

**Theorem 3.2.** *A graph G admits an arithmetic modular sumset labelling f if and only if for any two adjacent vertices in G, the deterministic ratio of every edge of G is a positive integer, which is less than or equal to the set-labelling number of its end vertex*

*Case 1*: First note that if the set-label of two adjacent vertices are arithmetic progressions with the same common difference, say *d*, then the set-label of the corresponding edge is also an arithmetic progression with the same common difference *d*. Then, it is clear that a vertex arithmetic modular sumset graph is an arithmetic modular sumset graph if the common differences between any two

*Case 2*: Assume that *u*, *v* be any two adjacent vertices in *G* with common differ-

f g *ar* ¼ *a* þ *rdu* : 0≤*r*< *m* and *f v*ð Þ¼ f g *bs* ¼ *b* þ *sdv* : 0≤*s*<*n* . Then, ∣ *f u*ð Þ∣ ¼ *m* and

follows. For any *bs* ∈*f v*ð Þ, 0≤*s* <*n*, arrange the terms of *A* þ *bs* in ð Þ *s* þ 1 th row in such a way that equal terms of different rows come in the same column of this arrangement. Without loss of generality, assume that *dv* ¼ *kdu* and *k*≤ *m*. If *k*< *m*, then for any *a*∈*f u*ð Þ and *b*∈*f v*ð Þ we have *a* þ ð Þ¼ *b* þ *dv a* þ *b* þ *kdu* <*a* þ *b* þ *mdi*. That is, a few final elements of each row of the above arrangement occur as the initial elements of the succeeding row (or rows) and the difference between two successive elements in each row is *du* itself. If *k* ¼ *m*, then the difference between the final element of each row and the first element of the next row is *du* and the difference between two consecutive elements in each row is *du*. Hence, if *k*≤ *m*,

þð Þ *uv* is an arithmetic progression with common difference *du*.

ences *du* and *dv* respectively such that *du* ≤ *dv*. Also, assume that *f u*ð Þ¼

∣ *f v*ð Þ∣ ¼ *n*. Now, arrange the terms of *f u*ð Þþ *f v*ð Þ¼ *f*

þð Þ *uv* is also an

þð Þ *uv* in rows and columns as

The theorem follows immediately from the theorem on the cardinality of sumsets (see Theorem 2.7, p. 52, [25]).

In this context, it is quite interesting to investigate whether the bounds are sharp. It has also been proved in [25] that the lower bound is sharp when both *f u*ð Þ and *f v*ð Þ are arithmetic progressions (we call set an arithmetic progression if its elements are in arithmetic progression) with the same common difference. We shall discuss the different types of modular sumset graphs based on the set-labelling numbers of its vertices and edges, one by one in the coming discussions.

#### **3. Arithmetic modular sumset graphs**

As mentioned above, the lower bound of the inequality (1) is sharp if both summand set-label are arithmetic progressions with the same common difference. If the context is clear, the common difference of the set-label (if exists) of an element may be called the *common difference* of that element. The *deterministic ratio* of an edge of *G* is the ratio, *k*≥1 between the common differences of its end vertices. In view of this terminology we have the following definition.

**Definition 3.1.** For any vertex *v* of *G*, if *f v*ð Þ is an arithmetic progression, then the modular sumset labelling *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* is called a *vertex arithmetic modular sumset labelling* of *G*. In a similar manner, for any edge *e* of *G*, if *f e*ð Þ is an arithmetic progression, then the modular sumset labelling *f* : *E G*ð Þ!P0ð Þ *<sup>n</sup>* is called an *edge arithmetic modular sumset labelling* of *G*.

The difference set of a non-empty set *A*, denoted by *DA*, is the set defined by *DA* ¼ j f g *a* � *b*j: *a*, *b*∈ *A* . Note that if *A* is an arithmetic progression, then its difference set *DA* is also an arithmetic progression and vice versa. Analogous to the corresponding result of the edge-arithmetic sumset labelling of graphs (see [44, 46]), the following result is a necessary and sufficient condition for a graph *G* to be edge-arithmetic modular sumset graph in terms of the difference sets the setlabel of vertices of *G*.

**Theorem 3.1.** *Let f be a modular sumset labelling defined on a graph G. If the setlabel of an edge of G is an arithmetic progression if and only if the sumset of the difference sets of set-label of its end vertices is an arithmetic progression.*

*Proof.* Let *f* : *V G*ð Þ!Pð Þ ℕ<sup>0</sup> be a modular sumset labelling defined on *G*. Let *ai*, *a <sup>j</sup>* be two arbitrary elements in *f u*ð Þ and let *br*, *bs* be two elements in *f v*ð Þ. Then, <sup>∣</sup>*ai* � *<sup>a</sup> <sup>j</sup>*∣ ∈ *<sup>D</sup> f u*ð Þ and <sup>∣</sup>*ai* � *<sup>a</sup> <sup>j</sup>*∣ ∈ *<sup>D</sup> f u*ð Þ. That is, *<sup>D</sup> f u*ð Þ ¼ j*ai* � *<sup>a</sup> <sup>j</sup>*j: *ai*, *<sup>a</sup> <sup>j</sup>* <sup>∈</sup>*f u*ð Þ and *D f v*ð Þ ¼ j*br* � *bs*j: *br* f g , *bs* ∈ *f v*ð Þ .

Now, assume that *f* þð Þ¼ *e f* þð Þ *uv* is an arithmetic progression for an edge *e* ¼ *uv*∈*E G*ð Þ. That is, *A* ¼ *f u*ð Þþ *f v*ð Þ is an arithmetic progression. Then, the difference set *DA* ¼ j f g *a* � *b*j: *a*, *b*∈ *A* ¼ *f u*ð Þþ *f v*ð Þ is also an arithmetic progression. Since *a*, *b*∈ *A*, we have *a* ¼ *ai* þ *br* and *b* ¼ *a <sup>j</sup>* þ *bs*, where *ai*, *a <sup>j</sup>* ∈*f u*ð Þ and *br*, *bs* ∈*f v*ð Þ. Then,

*Modular Sumset Labelling of Graphs DOI: http://dx.doi.org/10.5772/intechopen.92701*

As in the case of sumsets, the cardinality of the modular sumsets also attracted the attention. Hence, we have the bounds for the cardinality of an edge set-label of a

**Theorem 2.2.** [47] *Let f* : *V G*ð Þ!P0ð Þ *<sup>n</sup> be a modular sumset labelling of a given*

The theorem follows immediately from the theorem on the cardinality of

In this context, it is quite interesting to investigate whether the bounds are sharp. It has also been proved in [25] that the lower bound is sharp when both *f u*ð Þ and *f v*ð Þ are arithmetic progressions (we call set an arithmetic

progression if its elements are in arithmetic progression) with the same common difference. We shall discuss the different types of modular sumset graphs based on the set-labelling numbers of its vertices and edges, one by one in the coming

As mentioned above, the lower bound of the inequality (1) is sharp if both summand set-label are arithmetic progressions with the same common difference. If the context is clear, the common difference of the set-label (if exists) of an element may be called the *common difference* of that element. The *deterministic ratio* of an edge of *G* is the ratio, *k*≥1 between the common differences of its end vertices. In view of this terminology we have the following definition.

**Definition 3.1.** For any vertex *v* of *G*, if *f v*ð Þ is an arithmetic progression, then the modular sumset labelling *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* is called a *vertex arithmetic modular sumset labelling* of *G*. In a similar manner, for any edge *e* of *G*, if *f e*ð Þ is an arithmetic progression, then the modular sumset labelling *f* : *E G*ð Þ!P0ð Þ *<sup>n</sup>* is called an *edge*

The difference set of a non-empty set *A*, denoted by *DA*, is the set defined by *DA* ¼ j f g *a* � *b*j: *a*, *b*∈ *A* . Note that if *A* is an arithmetic progression, then its difference set *DA* is also an arithmetic progression and vice versa. Analogous to the corresponding result of the edge-arithmetic sumset labelling of graphs (see

[44, 46]), the following result is a necessary and sufficient condition for a graph *G* to be edge-arithmetic modular sumset graph in terms of the difference sets the set-

**Theorem 3.1.** *Let f be a modular sumset labelling defined on a graph G. If the setlabel of an edge of G is an arithmetic progression if and only if the sumset of the difference*

*Proof.* Let *f* : *V G*ð Þ!Pð Þ ℕ<sup>0</sup> be a modular sumset labelling defined on *G*. Let *ai*, *a <sup>j</sup>* be two arbitrary elements in *f u*ð Þ and let *br*, *bs* be two elements in *f v*ð Þ. Then, <sup>∣</sup>*ai* � *<sup>a</sup> <sup>j</sup>*∣ ∈ *<sup>D</sup> f u*ð Þ and <sup>∣</sup>*ai* � *<sup>a</sup> <sup>j</sup>*∣ ∈ *<sup>D</sup> f u*ð Þ. That is, *<sup>D</sup> f u*ð Þ ¼ j*ai* � *<sup>a</sup> <sup>j</sup>*j: *ai*, *<sup>a</sup> <sup>j</sup>* <sup>∈</sup>*f u*ð Þ and

*uv*∈*E G*ð Þ. That is, *A* ¼ *f u*ð Þþ *f v*ð Þ is an arithmetic progression. Then, the difference set *DA* ¼ j f g *a* � *b*j: *a*, *b*∈ *A* ¼ *f u*ð Þþ *f v*ð Þ is also an arithmetic progression. Since *a*, *b*∈ *A*, we have *a* ¼ *ai* þ *br* and *b* ¼ *a <sup>j</sup>* þ *bs*, where *ai*, *a <sup>j</sup>* ∈*f u*ð Þ and

þð Þ *uv* is an arithmetic progression for an edge *e* ¼

þð Þ *uv* ∣ ¼ ∣ *f u*ð Þþ *f v*ð Þ∣≤∣ *f u*ð Þ∣ ∣ *f v*ð Þ∣ ≤ *n:* (1)

modular sumset graph *G* is as follows:

*Number Theory and Its Applications*

∣ *f u*ð Þ∣ þ ∣ *f v*ð Þ∣ � 1≤ ∣*f*

sumsets (see Theorem 2.7, p. 52, [25]).

**3. Arithmetic modular sumset graphs**

*arithmetic modular sumset labelling* of *G*.

*sets of set-label of its end vertices is an arithmetic progression.*

þð Þ¼ *e f*

label of vertices of *G*.

*br*, *bs* ∈*f v*ð Þ. Then,

**94**

*D f v*ð Þ ¼ j*br* � *bs*j: *br* f g , *bs* ∈ *f v*ð Þ . Now, assume that *f*

discussions.

*graph G. Then, for any edge uv*∈*E G*ð Þ*, we have*

$$\begin{aligned} D\_A &= \{ |a - b| \colon a, b \in A \} \\ &= \left\{ |a\_i + b\_r - (a\_j + b\_s)| \colon a\_i, a\_j \in f(u), b\_r, b\_s \in f(v) \right\} \\ &= \left\{ |a\_i - a\_j| + |b\_r - b\_s| \colon a\_i, a\_j \in f(u), b\_r, b\_s \in f(v) \right\} \\ &= \left\{ |a\_i - a\_j| \colon a\_i, a\_j \in f(u) \right\} + \left\{ |b\_r - b\_s| \colon b\_r, b\_s \in f(v) \right\} \\ &= D\_{f(u)} + D\_{f(v)} \end{aligned}$$

Hence, *D f u*ð Þ þ *D f v*ð Þ is an arithmetic progression.

Conversely, assume that *D f u*ð Þ þ *D f v*ð Þ is an arithmetic progression. Then, by previous step, we have *D f u*ð Þ þ *D f v*ð Þ ¼ *DA*, where *A* ¼ *f u*ð Þþ *f v*ð Þ. Then, we have *DA* is an arithmetic progression. Since the difference set *DA* is an arithmetic progression, then by the above remark, we have *A* ¼ *f u*ð Þþ *f v*ð Þ¼ *f* þð Þ *uv* is also an arithmetic progression. Hence, the edge *e* ¼ *uv* has an arithmetic progression as its set-label. □

In view of the notions mentioned above, we note that there are some graphs, all whose elements have arithmetic progressions as their set-label and there are some graphs, the set-label of whose edges are not arithmetic progressions. Keeping this in mind, we define the following notion.

**Definition 3.2.** An *arithmetic sumset labelling* of a graph *G* is a modular sumset labelling *f* of *G*, with respect to which the set-label of all vertices and edges of *G* are arithmetic progressions. A graph that admits an arithmetic modular sumset labelling is called an *arithmetic modular sumset graph*.

Analogous to the condition for an arithmetic sumset graphs (see [44]), a necessary and sufficient condition for a graph to admit an arithmetic modular sumset labelling is discussed in the following theorem.

**Theorem 3.2.** *A graph G admits an arithmetic modular sumset labelling f if and only if for any two adjacent vertices in G, the deterministic ratio of every edge of G is a positive integer, which is less than or equal to the set-labelling number of its end vertex having smaller common difference.*

*Proof.* Here, we need to consider the following two cases:

*Case 1*: First note that if the set-label of two adjacent vertices are arithmetic progressions with the same common difference, say *d*, then the set-label of the corresponding edge is also an arithmetic progression with the same common difference *d*. Then, it is clear that a vertex arithmetic modular sumset graph is an arithmetic modular sumset graph if the common differences between any two adjacent vertices of *G* are the same.

*Case 2*: Assume that *u*, *v* be any two adjacent vertices in *G* with common differences *du* and *dv* respectively such that *du* ≤ *dv*. Also, assume that *f u*ð Þ¼ f g *ar* ¼ *a* þ *rdu* : 0≤*r*< *m* and *f v*ð Þ¼ f g *bs* ¼ *b* þ *sdv* : 0≤*s*<*n* . Then, ∣ *f u*ð Þ∣ ¼ *m* and ∣ *f v*ð Þ∣ ¼ *n*. Now, arrange the terms of *f u*ð Þþ *f v*ð Þ¼ *f* þð Þ *uv* in rows and columns as follows. For any *bs* ∈*f v*ð Þ, 0≤*s* <*n*, arrange the terms of *A* þ *bs* in ð Þ *s* þ 1 th row in such a way that equal terms of different rows come in the same column of this arrangement. Without loss of generality, assume that *dv* ¼ *kdu* and *k*≤ *m*. If *k*< *m*, then for any *a*∈*f u*ð Þ and *b*∈*f v*ð Þ we have *a* þ ð Þ¼ *b* þ *dv a* þ *b* þ *kdu* <*a* þ *b* þ *mdi*. That is, a few final elements of each row of the above arrangement occur as the initial elements of the succeeding row (or rows) and the difference between two successive elements in each row is *du* itself. If *k* ¼ *m*, then the difference between the final element of each row and the first element of the next row is *du* and the difference between two consecutive elements in each row is *du*. Hence, if *k*≤ *m*, then *f* þð Þ *uv* is an arithmetic progression with common difference *du*.

In both cases, note that if the given conditions are true, then *f* is an arithmetic modular sumset labelling of *G*.

*Proof.* Let *f* be an arithmetic modular sumset labelling defined on *G*. For any two vertices *vi* and *v <sup>j</sup>* of *G*, let *f v*ð Þ¼*<sup>i</sup> ai*, *ai* þ *di*, *ai* þ 2*di*, *ai* þ 3*di* f g , … , *ai* þ ð Þ *m* � 1 *di*

Let *di* and *d <sup>j</sup>* be the common differences of the vertices *vi* and *v <sup>j</sup>* respectively,

Theorem 3.2, there exists a positive integer *k* such that *d <sup>j</sup>* ¼ *k:di*, where 1≤ *k*≤ ∣ *f v*ð Þ*<sup>i</sup>* ∣.

<sup>¼</sup> *ai* <sup>þ</sup> *<sup>a</sup> <sup>j</sup>*, *ai* <sup>þ</sup> *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> *di*, *ai* <sup>þ</sup> *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>2</sup>*di*, … , *ai* <sup>þ</sup> *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> ½ � ð Þþ *<sup>m</sup>* � <sup>1</sup> *k n*ð Þ � <sup>1</sup> *di*

That is, the set-labelling number of the edge *viv <sup>j</sup>* is *<sup>m</sup>* <sup>þ</sup> *k n*ð Þ � <sup>1</sup> . □

The next type of a modular sumset labelling we are going to discuss is the one with the upper bound in Inequality (1) is sharp (that is, ∣*A* þ *B*∣ ¼ ∣*A*k*B*∣). Thus, we

**Definition 4.1.** [47] A modular sumset labelling *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* defined on a given graph *G* is said to be a *strongly modular sumset labelling* if for the associated

admits a strongly modular sumset labelling is called a *strongly modular sumset graph*. Invoking the notion difference set of a set, a necessary and sufficient condition of a modular sumset labelling of a graph *G* to be a strongly modular sumset labelling

**Theorem 4.1.** *A modular sumset labelling f* : *V G*ð Þ!P0ð Þ *<sup>n</sup> of a given graph G is a strongly modular sumset labelling of G if and only if D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø, ∀*uv*∈*E G*ð Þ*,*

*Proof.* Let *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* be a modular sumset labelling on a given graph *G*.

Let *uv* be an arbitrary edge in *E G*ð Þ. Assume that *f* is a strong modular sumset

*D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø. Therefore, the difference sets of the set-label of any two adjacent

Conversely, assume that the difference *D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø for any edge *uv* in *G*. That is, ∣*ai* � *a <sup>j</sup>*∣ 6¼ ∣*bs* � *br*∣ for any *ai*, *a <sup>j</sup>* ∈ *f u*ð Þ and *br*, *bs* ∈*f v*ð Þ. Then, *ai* � *a <sup>j</sup>* 6¼ *bs* � *br*. That is, *ai* þ *br* 6¼ *a <sup>j</sup>* þ *bs*. Therefore, all elements in *f u*ð Þþ *f v*ð Þ are distinct.

þð Þ *uv* ∣ ¼ ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ for any edge *uv*∈*E G*ð Þ. Hence, *f* is a strongly

Also, note that the maximum possible cardinality in the set-label of any element of *G* is *n*, the product ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ cannot exceed the number *n*. This completes the proof. □ A necessary and sufficient condition for a modular sumset labelling of a graph *G*

**Theorem 4.2.** [47] *For a positive integer k*≤*n, a modular sumset labelling f* : *V G*ð Þ!P0ð Þ *<sup>n</sup> of a given connected graph G is a strongly k-uniform modular sumset*

2

graph by distinct *l*-element sets in such a way that the difference sets of the set-label

For any vertex *<sup>u</sup>* <sup>∈</sup>*V G*ð Þ, define *Df*ð Þ¼ *<sup>u</sup> ai* � *<sup>a</sup> <sup>j</sup>* : *ai*, *<sup>a</sup> <sup>j</sup>* <sup>∈</sup>*f u*ð Þ .

ments *ai*, *a <sup>j</sup>* ∈*f u*ð Þ and *br*, *bs* ∈*f v*ð Þ, we have *ai* þ *br* 6¼ *a <sup>j</sup>* þ *bs* in *f*

to be a strongly *k*-uniform modular sumset labelling is given below:

*labelling of G if and only if either k is a perfect square or G is bipartite.*

*Proof.* If *k* is a perfect square, say *k* ¼ *l*

That is, ∣*ai* � *a <sup>j</sup>*∣ 6¼ ∣*bs* � *br*∣ for any *ai*, *a <sup>j</sup>* ∈ *f u*ð Þ and *br*, *bs* ∈*f v*ð Þ. That is,

þð Þ *uv* ∣ ¼ ∣ *f u*ð Þ∣∣ *f v*ð Þ∣∀*uv*∈*E G*ð Þ. A graph which

þð Þ *uv* ∣ ¼ ∣ *f u*ð Þ∣∣ *f v*ð Þ∣. Therefore, for any ele-

, then we can label all the vertices of a

þð Þ *uv* ∀*uv*∈*E G*ð Þ.

. Here <sup>∣</sup> *f v*ð Þ*<sup>i</sup>* <sup>∣</sup> <sup>¼</sup> *<sup>m</sup>*

. Therefore,

.

<sup>¼</sup> *<sup>a</sup> <sup>j</sup>*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> *<sup>d</sup> <sup>j</sup>*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>2</sup>*<sup>d</sup> <sup>j</sup>*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>3</sup>*<sup>d</sup> <sup>j</sup>*, … , *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>d</sup> <sup>j</sup>*

such that *di* <*d <sup>j</sup>*. Since *f* is an arithmetic modular sumset labelling on *G*, by

<sup>¼</sup> *<sup>a</sup> <sup>j</sup>*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> *kdi*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>2</sup>*kdi*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>3</sup>*kdi*, … , *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *kdi*

and let *f v <sup>j</sup>*

<sup>∣</sup> <sup>¼</sup> *<sup>n</sup>*.

*Modular Sumset Labelling of Graphs*

*DOI: http://dx.doi.org/10.5772/intechopen.92701*

**4. Strongly modular sumset graphs**

<sup>þ</sup> : *E G*ð Þ!P0ð Þ *<sup>n</sup>* , ∣*f*

have the following definition.

and ∣ *f v <sup>j</sup>*

Then, *f v <sup>j</sup>*

function *f*

is given below:

*where* ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ ≤ *n.*

vertices are disjoint.

That is, ∣*f*

**97**

labelling of *G*. Then, by definition ∣*f*

modular sumset labelling of *G*.

*f* <sup>þ</sup> *viv <sup>j</sup>*

We prove the converse part by contradiction method. For this, assume that *f* is an arithmetic modular sumset labelling of *G*. Let us proceed by considering the following two cases.

*Case-1:* Assume that *d <sup>j</sup>* is not a multiple of *di* (or *di* is not a multiple of *d <sup>j</sup>*). Without loss generality, let *di* <*d <sup>j</sup>*. Then, by division algorithm, *d <sup>j</sup>* ¼ *pdi* þ *q*, 0 <*q*< *di*. Then, the differences between any two consecutive terms in *f* <sup>þ</sup> *viv <sup>j</sup>* are not equal. Hence, in this case also, *f* is not an arithmetic modular sumset labelling, contradiction to the hypothesis. Therefore, *di*∣*d <sup>j</sup>*.

*Case 2:* Let *d <sup>j</sup>* ¼ *kdi* where *k*> *m*. Then, the difference between two successive elements in each row is *di*, but the difference between the final element of each row and the first element of the next row is *tdi*, where *t* ¼ *k* � *m* þ 1 6¼ 1. Hence, *f* is not an arithmetic modular sumset labelling, a contradiction to the hypothesis. Hence, we have *d <sup>j</sup>* ¼ *kdi*; *k*≤ *m*.

Therefore, from the above cases it can be noted that if a vertex arithmetic modular sumset labelling of *G* is an arithmetic modular sumset labelling of *G*, then the deterministic ratio of every edge of *G* is a positive integer, which is greater than or equal to the set-labelling number of its end vertex having smaller common difference. This completes the proof. □

In the following theorem, we establish a relation between the common differences of the elements of an arithmetic modular sumset graph *G*.

**Theorem 3.3.** *If G is an arithmetic modular sumset graph, the greatest common divisor of the common differences of vertices of G and the greatest common divisor of the common differences of the edges of G are equal to the smallest among the common differences of the vertices of G.*

*Proof.* Let *f* be an arithmetic modular sumset labelling of *G*. Then, by Theorem 3.2, for any two adjacent vertices *vi* and *v <sup>j</sup>* of *G* with common differences *di* and *d <sup>j</sup>* respectively, either *di* ¼ *d <sup>j</sup>*, or if *d <sup>j</sup>* >*di*, *d <sup>j</sup>* ¼ *kd <sup>j</sup>*, where *k* is a positive integer such that 1<*k*≤ ∣ *f v*ð Þ*<sup>i</sup>* ∣.

If the common differences of the elements of *G* are the same, the result is obvious. Hence, assume that for any two adjacent vertices *vi* and *v <sup>j</sup>* of *G*, *d <sup>j</sup>* ¼ *kd <sup>j</sup>*, *k*≤ ∣ *f v*ð Þ*<sup>i</sup>* ∣, where *di* is the smallest among the common differences of the vertices of *G*. If *vr* is another vertex that is adjacent to *v <sup>j</sup>*, then it has the common difference *dr* which is equal to either *di* or *d <sup>j</sup>* or *ld <sup>j</sup>*. In all the three cases, *dr* is a multiple of *di*. Hence, the greatest common divisor of *di*, *d <sup>j</sup>*, *dr* is *di*. Proceeding like this, we have the greatest common divisor of the common differences of the vertices of *G* is *di*.

Also, by Theorem 3.2, the edge *uiv <sup>j</sup>* has the common difference *di*. The edge *v jvk* has the common difference *di*, if *dk* ¼ *di*, or *d <sup>j</sup>* in the other two cases. Proceeding like this, we observe that the greatest common divisor of the common differences of the edges of *<sup>G</sup>* is also *di*. This completes the proof. □

The study on the set-labelling number of edges of an arithmetic modular sumset graphs arouses much interest. Analogous to the result on set-labelling number of the edges of an arithmetic sumset graph (see [43]), The set-labelling number of an edge of an arithmetic modular sumset graph *G*, in terms of the set-labelling numbers of its end vertices, is determined in the following theorem.

**Theorem 3.4.** *Let G be a graph which admits an arithmetic modular sumset labelling, say f and let di and d <sup>j</sup> be the common differences of two adjacent vertices vi and v <sup>j</sup> in G. If* ∣ *f v*ð Þ*<sup>i</sup>* ∣≥∣ *f v <sup>j</sup>* <sup>∣</sup>*, then for some positive integer* <sup>1</sup><sup>≤</sup> *<sup>k</sup>*≤ ∣ *f v*ð Þ*<sup>i</sup>* <sup>∣</sup>*, the edge viv <sup>j</sup> has the set-labelling number* ∣ *f v*ð Þ*<sup>i</sup>* ∣ þ *k* j*f v <sup>j</sup>* j�<sup>1</sup> *.*

*Modular Sumset Labelling of Graphs DOI: http://dx.doi.org/10.5772/intechopen.92701*

In both cases, note that if the given conditions are true, then *f* is an arithmetic

We prove the converse part by contradiction method. For this, assume that *f* is an arithmetic modular sumset labelling of *G*. Let us proceed by considering the

> <sup>þ</sup> *viv <sup>j</sup>*

*Case-1:* Assume that *d <sup>j</sup>* is not a multiple of *di* (or *di* is not a multiple of *d <sup>j</sup>*). Without loss generality, let *di* <*d <sup>j</sup>*. Then, by division algorithm, *d <sup>j</sup>* ¼ *pdi* þ *q*, 0 <*q*< *di*. Then, the differences between any two consecutive terms in *f*

*Case 2:* Let *d <sup>j</sup>* ¼ *kdi* where *k*> *m*. Then, the difference between two successive elements in each row is *di*, but the difference between the final element of each row and the first element of the next row is *tdi*, where *t* ¼ *k* � *m* þ 1 6¼ 1. Hence, *f* is not an arithmetic modular sumset labelling, a contradiction to the hypothesis. Hence,

Therefore, from the above cases it can be noted that if a vertex arithmetic modular sumset labelling of *G* is an arithmetic modular sumset labelling of *G*, then the deterministic ratio of every edge of *G* is a positive integer, which is greater than or equal to the set-labelling number of its end vertex having smaller common

difference. This completes the proof. □ In the following theorem, we establish a relation between the common differ-

**Theorem 3.3.** *If G is an arithmetic modular sumset graph, the greatest common divisor of the common differences of vertices of G and the greatest common divisor of the common differences of the edges of G are equal to the smallest among the common*

*Proof.* Let *f* be an arithmetic modular sumset labelling of *G*. Then, by Theorem 3.2, for any two adjacent vertices *vi* and *v <sup>j</sup>* of *G* with common differences *di* and *d <sup>j</sup>* respectively, either *di* ¼ *d <sup>j</sup>*, or if *d <sup>j</sup>* >*di*, *d <sup>j</sup>* ¼ *kd <sup>j</sup>*, where *k* is a positive integer such

Also, by Theorem 3.2, the edge *uiv <sup>j</sup>* has the common difference *di*. The edge *v jvk* has the common difference *di*, if *dk* ¼ *di*, or *d <sup>j</sup>* in the other two cases. Proceeding like this, we observe that the greatest common divisor of the common differences of the edges of *<sup>G</sup>* is also *di*. This completes the proof. □ The study on the set-labelling number of edges of an arithmetic modular sumset graphs arouses much interest. Analogous to the result on set-labelling number of the edges of an arithmetic sumset graph (see [43]), The set-labelling number of an edge of an arithmetic modular sumset graph *G*, in terms of the set-labelling numbers of

**Theorem 3.4.** *Let G be a graph which admits an arithmetic modular sumset labelling, say f and let di and d <sup>j</sup> be the common differences of two adjacent vertices vi and v <sup>j</sup> in*

j�<sup>1</sup> *.*

<sup>∣</sup>*, then for some positive integer* <sup>1</sup><sup>≤</sup> *<sup>k</sup>*≤ ∣ *f v*ð Þ*<sup>i</sup>* <sup>∣</sup>*, the edge viv <sup>j</sup> has the*

If the common differences of the elements of *G* are the same, the result is obvious. Hence, assume that for any two adjacent vertices *vi* and *v <sup>j</sup>* of *G*, *d <sup>j</sup>* ¼ *kd <sup>j</sup>*, *k*≤ ∣ *f v*ð Þ*<sup>i</sup>* ∣, where *di* is the smallest among the common differences of the vertices of *G*. If *vr* is another vertex that is adjacent to *v <sup>j</sup>*, then it has the common difference *dr* which is equal to either *di* or *d <sup>j</sup>* or *ld <sup>j</sup>*. In all the three cases, *dr* is a multiple of *di*. Hence, the greatest common divisor of *di*, *d <sup>j</sup>*, *dr* is *di*. Proceeding like this, we have the greatest common divisor of the common differences of the

are not equal. Hence, in this case also, *f* is not an arithmetic modular sumset

labelling, contradiction to the hypothesis. Therefore, *di*∣*d <sup>j</sup>*.

ences of the elements of an arithmetic modular sumset graph *G*.

its end vertices, is determined in the following theorem.

modular sumset labelling of *G*.

*Number Theory and Its Applications*

following two cases.

we have *d <sup>j</sup>* ¼ *kdi*; *k*≤ *m*.

*differences of the vertices of G.*

that 1<*k*≤ ∣ *f v*ð Þ*<sup>i</sup>* ∣.

vertices of *G* is *di*.

*G. If* ∣ *f v*ð Þ*<sup>i</sup>* ∣≥∣ *f v <sup>j</sup>*

**96**

*set-labelling number* ∣ *f v*ð Þ*<sup>i</sup>* ∣ þ *k* j*f v <sup>j</sup>*

*Proof.* Let *f* be an arithmetic modular sumset labelling defined on *G*. For any two vertices *vi* and *v <sup>j</sup>* of *G*, let *f v*ð Þ¼*<sup>i</sup> ai*, *ai* þ *di*, *ai* þ 2*di*, *ai* þ 3*di* f g , … , *ai* þ ð Þ *m* � 1 *di* and let *f v <sup>j</sup>* <sup>¼</sup> *<sup>a</sup> <sup>j</sup>*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> *<sup>d</sup> <sup>j</sup>*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>2</sup>*<sup>d</sup> <sup>j</sup>*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>3</sup>*<sup>d</sup> <sup>j</sup>*, … , *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>d</sup> <sup>j</sup>* . Here <sup>∣</sup> *f v*ð Þ*<sup>i</sup>* <sup>∣</sup> <sup>¼</sup> *<sup>m</sup>* and ∣ *f v <sup>j</sup>* <sup>∣</sup> <sup>¼</sup> *<sup>n</sup>*.

Let *di* and *d <sup>j</sup>* be the common differences of the vertices *vi* and *v <sup>j</sup>* respectively, such that *di* <*d <sup>j</sup>*. Since *f* is an arithmetic modular sumset labelling on *G*, by Theorem 3.2, there exists a positive integer *k* such that *d <sup>j</sup>* ¼ *k:di*, where 1≤ *k*≤ ∣ *f v*ð Þ*<sup>i</sup>* ∣. Then, *f v <sup>j</sup>* <sup>¼</sup> *<sup>a</sup> <sup>j</sup>*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> *kdi*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>2</sup>*kdi*, *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>3</sup>*kdi*, … , *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *kdi* . Therefore, *f* <sup>þ</sup> *viv <sup>j</sup>* <sup>¼</sup> *ai* <sup>þ</sup> *<sup>a</sup> <sup>j</sup>*, *ai* <sup>þ</sup> *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> *di*, *ai* <sup>þ</sup> *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> <sup>2</sup>*di*, … , *ai* <sup>þ</sup> *<sup>a</sup> <sup>j</sup>* <sup>þ</sup> ½ � ð Þþ *<sup>m</sup>* � <sup>1</sup> *k n*ð Þ � <sup>1</sup> *di* . That is, the set-labelling number of the edge *viv <sup>j</sup>* is *<sup>m</sup>* <sup>þ</sup> *k n*ð Þ � <sup>1</sup> . □

#### **4. Strongly modular sumset graphs**

The next type of a modular sumset labelling we are going to discuss is the one with the upper bound in Inequality (1) is sharp (that is, ∣*A* þ *B*∣ ¼ ∣*A*k*B*∣). Thus, we have the following definition.

**Definition 4.1.** [47] A modular sumset labelling *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* defined on a given graph *G* is said to be a *strongly modular sumset labelling* if for the associated function *f* <sup>þ</sup> : *E G*ð Þ!P0ð Þ *<sup>n</sup>* , ∣*f* þð Þ *uv* ∣ ¼ ∣ *f u*ð Þ∣∣ *f v*ð Þ∣∀*uv*∈*E G*ð Þ. A graph which admits a strongly modular sumset labelling is called a *strongly modular sumset graph*.

Invoking the notion difference set of a set, a necessary and sufficient condition of a modular sumset labelling of a graph *G* to be a strongly modular sumset labelling is given below:

**Theorem 4.1.** *A modular sumset labelling f* : *V G*ð Þ!P0ð Þ *<sup>n</sup> of a given graph G is a strongly modular sumset labelling of G if and only if D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø, ∀*uv*∈*E G*ð Þ*, where* ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ ≤ *n.*

*Proof.* Let *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* be a modular sumset labelling on a given graph *G*. For any vertex *<sup>u</sup>* <sup>∈</sup>*V G*ð Þ, define *Df*ð Þ¼ *<sup>u</sup> ai* � *<sup>a</sup> <sup>j</sup>* : *ai*, *<sup>a</sup> <sup>j</sup>* <sup>∈</sup>*f u*ð Þ .

Let *uv* be an arbitrary edge in *E G*ð Þ. Assume that *f* is a strong modular sumset labelling of *G*. Then, by definition ∣*f* þð Þ *uv* ∣ ¼ ∣ *f u*ð Þ∣∣ *f v*ð Þ∣. Therefore, for any elements *ai*, *a <sup>j</sup>* ∈*f u*ð Þ and *br*, *bs* ∈*f v*ð Þ, we have *ai* þ *br* 6¼ *a <sup>j</sup>* þ *bs* in *f* þð Þ *uv* ∀*uv*∈*E G*ð Þ. That is, ∣*ai* � *a <sup>j</sup>*∣ 6¼ ∣*bs* � *br*∣ for any *ai*, *a <sup>j</sup>* ∈ *f u*ð Þ and *br*, *bs* ∈*f v*ð Þ. That is, *D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø. Therefore, the difference sets of the set-label of any two adjacent vertices are disjoint.

Conversely, assume that the difference *D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø for any edge *uv* in *G*. That is, ∣*ai* � *a <sup>j</sup>*∣ 6¼ ∣*bs* � *br*∣ for any *ai*, *a <sup>j</sup>* ∈ *f u*ð Þ and *br*, *bs* ∈*f v*ð Þ. Then, *ai* � *a <sup>j</sup>* 6¼ *bs* � *br*. That is, *ai* þ *br* 6¼ *a <sup>j</sup>* þ *bs*. Therefore, all elements in *f u*ð Þþ *f v*ð Þ are distinct. That is, ∣*f* þð Þ *uv* ∣ ¼ ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ for any edge *uv*∈*E G*ð Þ. Hence, *f* is a strongly modular sumset labelling of *G*.

Also, note that the maximum possible cardinality in the set-label of any element of *G* is *n*, the product ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ cannot exceed the number *n*. This completes the proof. □

A necessary and sufficient condition for a modular sumset labelling of a graph *G* to be a strongly *k*-uniform modular sumset labelling is given below:

**Theorem 4.2.** [47] *For a positive integer k*≤*n, a modular sumset labelling f* : *V G*ð Þ!P0ð Þ *<sup>n</sup> of a given connected graph G is a strongly k-uniform modular sumset labelling of G if and only if either k is a perfect square or G is bipartite.*

*Proof.* If *k* is a perfect square, say *k* ¼ *l* 2 , then we can label all the vertices of a graph by distinct *l*-element sets in such a way that the difference sets of the set-label of every pair of adjacent vertices are disjoint. Hence, assume that *k* is not a perfect square.

columns as follows. For *bs* ∈ *f v <sup>j</sup>*

*Modular Sumset Labelling of Graphs*

*DOI: http://dx.doi.org/10.5772/intechopen.92701*

is <sup>∣</sup> *f v*ð Þ*<sup>i</sup>* ∣∣ *f v <sup>j</sup>*

**5. Supreme modular sumset labelling of** *G*

*f v*ð Þþ*<sup>i</sup> f v <sup>j</sup>*

cardinality *n*.

below:

*labelling* of *G* if and only if *f*

*G* is the ground set *<sup>n</sup>* itself.

elements.

fore, we have ∣*f*

**99**

, 0 ≤*s*<∣ *f v <sup>j</sup>*

ð Þ *s* þ 1 -th row in such a way that equal terms of different rows come in the same column of this arrangement. Then, the common difference between consecutive elements in each row is *di*. Since *k* ¼ ∣ *f v*ð Þ*<sup>i</sup>* ∣, the difference between the final element of any row (other than the last row) and first element of its succeeding row is also *di*. That is, no column in this arrangement contains more than one element. Hence, all elements in this arrangement are distinct. Therefore, total number of elements in

In both types of modular sumset labelling discussed above, it is observed that the cardinality of the edge set-label cannot exceed the value *n*. This fact creates much interest in investigating the case where all the edge set-label have the

**Definition 5.1.** [47] A modular sumset labelling *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* of a given graph *G* is said to be a *supreme modular sumset labelling* or *maximal modular sumset*

Put in a different way, a modular sumset labelling *f* : *V G*ð Þ!Pð Þ *<sup>n</sup>* of a given graph *G* is a supreme modular sumset labelling of *G* if the set-label of every edge of

A necessary and sufficient condition for a modular sumset labelling of a graph *G* to be its supreme modular sumset labelling is discussed in the theorem given

**Theorem 5.1.** [47] *The modular sumset labelling f* : *V G*ð Þ!Pð Þ *<sup>n</sup> of a given graph G is a supreme modular sumset labelling of G if and only if for every pair of*

i. ∣ *f u*ð Þ∣ þ ∣ *f v*ð Þ∣ ≥*n* if *D f u*ð Þ ∩ *D f v*ð Þ 6¼ Ø. The strict inequality hold

*Proof.* For two adjacent vertices *u* and *v* in *G*, let *D f u*ð Þ ¼ *D f v*ð Þ ¼ f g*d* are arithmetic progressions containing the same elements. Then, the elements in *f u*ð Þ and *f v*ð Þ are also in arithmetic progression, with the same common difference *d*. Then, by Theorem 3.4, *n* ¼ ∣ *f u*ð Þþ *f v*ð Þ∣ ¼ ∣ *f u*ð Þ∣ þ ∣ *f v*ð Þ∣ � 1. Therefore, the set-

Now, let *D f u*ð Þ ∩ *D f v*ð Þ 6¼ Ø such that *D f u*ð Þ 6¼ *D f v*ð Þ. Then, clearly ∣ *f u*ð Þþ

<sup>∣</sup> *f v*ð Þ∣ ≥ *<sup>n</sup>*. □ Next assume that *D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø. Then, ∣ *f u*ð Þþ *f v*ð Þ∣ ¼ ∣ *f u*ð Þ∣∣ *f v*ð Þ∣. There-

A necessary and sufficient condition for a strong modular sumset labelling of a

**Theorem 5.2.** [47] *Let f be a strong sumset-labelling of a given graph G. Then, f is a maximal sumset-labelling of G if and only if n is a perfect square or G is bipartite or a*

*Proof.* The proof is an immediate consequence of Theorem 4.2, when *<sup>k</sup>* <sup>¼</sup> *<sup>n</sup>*. □

þð Þ *uv* ∣ ¼ *n* if and only if ∣ *f u*ð Þ∣ þ

when *D f u*ð Þ and *D f v*ð Þ are arithmetic progressions containing the same

*adjacent vertices u and v of G some or all of the following conditions hold.*

labelling number of the edge *uv* is *n* if and only if ∣ *f u*ð Þ∣ þ ∣ *f v*ð Þ∣> *n*.

þð Þ *uv* ∣ ¼ *n* if and only if ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ ≥ *n*.

ii. ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ ≥*n* if *D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø.

*f v*ð Þ∣≥∣ *f u*ð Þ∣ þ ∣ *f v*ð Þ∣. Therefore, we have ∣*f*

*disjoint union of bipartite components.*

graph *G* to be a maximal modular sumset labelling of *G*.

þð Þ¼ *E G*ð Þ f g *<sup>n</sup>* .

<sup>∣</sup>, arrange the terms of *f v*ð Þþ*<sup>i</sup> bs* in

<sup>∣</sup>. Hence, *<sup>f</sup>* is a strongly modular sumset labelling. □

Let *G* be a bipartite graph with bipartition ð Þ *X*, *Y* . Let *r*, *s* be two divisors of *k*. Label all vertices of *X* by distinct *r*-element sets all of whose difference sets are the same, say *DX*. Similarly, label all vertices of *Y* by distinct *s*-element sets all of whose difference sets the same, say *DY*, such that *DX* ∩ *DY* ¼ Ø. Then, all the edges of *G* have the set-labelling number *k* ¼ *rs*. Therefore, *G* is a strongly *k*-uniform modular sumset graph.

Conversely, assume that *G* admits a strongly *k*-uniform modular sumset labelling, say *f*. Then, *f* þð Þ¼ *uv k*∀*uv*∈*E G*ð Þ. Since, *f* is a strong modular sumset labelling, the set-labelling number of every vertex of *G* is a divisor of the set-labelling numbers of the edges incident on that vertex. Let *v* be a vertex of *G* with the setlabelling number *<sup>r</sup>*, where *<sup>r</sup>* is a divisor of *<sup>k</sup>*, but *<sup>r</sup>*<sup>2</sup> 6¼ *<sup>k</sup>*. Since *<sup>f</sup>* is *<sup>k</sup>*-uniform, all the vertices in *N v*ð Þ, must have the set-labelling number *s*, where *rs* ¼ *k*. Again, all vertices, which are adjacent to the vertices of *N v*ð Þ, must have the set-labelling number *r*. Since *G* is a connected graph, all vertices of *G* have the set-labelling number *r* or *s*. Let *X* be the set of all vertices of *G* having the set-labelling number *r* and *<sup>Y</sup>* be the set of all vertices of *<sup>G</sup>* having the set-labelling number *<sup>s</sup>*. Since *<sup>r</sup>*<sup>2</sup> 6¼ *<sup>k</sup>*, no two elements in *X* (and no elements in *Y* also) can be adjacent to each other. Therefore, *<sup>G</sup>* is bipartite. □

The following result is an immediate consequence of the above theorem.

**Theorem 4.3.** [47] *For a positive non-square integer k*≤*n, a modular sumset labelling f* : *V G*ð Þ!P0ð Þ *<sup>n</sup> of an arbitrary graph G is a strongly k-uniform modular sumset labelling of G if and only if either G is bipartite or a disjoint union of bipartite components.*

For a positive integer *k*≤*n*, the maximum number of components in a strongly *k*uniform modular sumset graph is as follows.

**Proposition 4.4.** [47] *Let f be a strongly k-uniform modular sumset labelling of a graph G with respect to the ground set n. Then, the maximum number of components in G is the number of distinct pairs of divisors r and s of k such that rs* ¼ *k.*

The following theorem discusses the condition for an arithmetic modular sumset labelling of a graph *G* to be a strongly modular sumset labelling of the graph.

**Theorem 4.5.** *Let G be a graph which admits an arithmetic modular sumset labelling, say f . Then, f is a strongly modular sumset labelling of G if and only if the deterministic ratio of every edge of G is equal to the set-labelling number of its end vertex having smaller common difference.*

*Proof.* Let *f* be an arithmetic modular sumset labelling of *G*. Let *vi* and *v <sup>j</sup>* are two adjacent vertices in *G* and *di* and *d <sup>j</sup>* be their common differences under *f*. Without loss of generality, let *di* < *d <sup>j</sup>*. Then, by Theorem 3.4, the set-labelling number of the edge *viv <sup>j</sup>* is ∣ *f v*ð Þ*<sup>i</sup>* ∣ þ *k* j*f v <sup>j</sup>* j�<sup>1</sup> .

Assume that *f* is a strongly modular sumset labelling. Therefore, *f* <sup>þ</sup> *viv <sup>j</sup>* <sup>¼</sup> *mn*. Then,

$$\begin{aligned} |f(v\_i)| + k \left( |f\left(v\_j\right)| - \mathbf{1} \right) &= |f(v\_i)| |f\left(v\_j\right)| \\ \Rightarrow k \left( |f\left(v\_j\right)| - \mathbf{1} \right) &= |f(v\_i)| \left( |f\left(v\_j\right)| - \mathbf{1} \right) \\ \Rightarrow k = |f(v\_i)|. \end{aligned}$$

Conversely, assume that the common differences *di* and *d <sup>j</sup>* of two adjacent vertices *vi* and *v <sup>j</sup>* respectively in *G*, where *di* < *d <sup>j</sup>* such that *d <sup>j</sup>* ¼ ∣ *f v*ð Þ*<sup>i</sup>* ∣*:di*. Assume that *f v*ð Þ¼*<sup>i</sup> ar* ¼ *a* þ *rdi* f g : 0≤*r*<j *f v*ð Þj *<sup>i</sup>* and *f v <sup>j</sup>* <sup>¼</sup> *bs* <sup>¼</sup> *<sup>b</sup>* <sup>þ</sup> *skdi* : <sup>0</sup>≤*<sup>s</sup>* <sup>&</sup>lt;j*f v <sup>j</sup>* <sup>j</sup> , where *k*≤ ∣ *f v*ð Þ*<sup>i</sup>* ∣. Now, arrange the terms of *f* <sup>þ</sup> *viv <sup>j</sup>* <sup>¼</sup> *f v*ð Þþ*<sup>i</sup> f v <sup>j</sup>* in rows and

of every pair of adjacent vertices are disjoint. Hence, assume that *k* is not a perfect

Let *G* be a bipartite graph with bipartition ð Þ *X*, *Y* . Let *r*, *s* be two divisors of *k*. Label all vertices of *X* by distinct *r*-element sets all of whose difference sets are the same, say *DX*. Similarly, label all vertices of *Y* by distinct *s*-element sets all of whose difference sets the same, say *DY*, such that *DX* ∩ *DY* ¼ Ø. Then, all the edges of *G* have the set-labelling number *k* ¼ *rs*. Therefore, *G* is a strongly *k*-uniform modular

Conversely, assume that *G* admits a strongly *k*-uniform modular sumset label-

For a positive integer *k*≤*n*, the maximum number of components in a strongly *k*-

**Proposition 4.4.** [47] *Let f be a strongly k-uniform modular sumset labelling of a graph G with respect to the ground set n. Then, the maximum number of components in*

The following theorem discusses the condition for an arithmetic modular sumset

*Proof.* Let *f* be an arithmetic modular sumset labelling of *G*. Let *vi* and *v <sup>j</sup>* are two adjacent vertices in *G* and *di* and *d <sup>j</sup>* be their common differences under *f*. Without loss of generality, let *di* < *d <sup>j</sup>*. Then, by Theorem 3.4, the set-labelling number of the

j�<sup>1</sup> <sup>¼</sup> <sup>∣</sup> *f v*ð Þ*<sup>i</sup>* ∣∣ *f v <sup>j</sup>*

j�<sup>1</sup> <sup>¼</sup> <sup>∣</sup> *f v*ð Þ*<sup>i</sup>* <sup>∣</sup> <sup>j</sup>*f v <sup>j</sup>*

Conversely, assume that the common differences *di* and *d <sup>j</sup>* of two adjacent vertices

<sup>þ</sup> *viv <sup>j</sup>*

*vi* and *v <sup>j</sup>* respectively in *G*, where *di* < *d <sup>j</sup>* such that *d <sup>j</sup>* ¼ ∣ *f v*ð Þ*<sup>i</sup>* ∣*:di*. Assume that

∣

<sup>¼</sup> *bs* <sup>¼</sup> *<sup>b</sup>* <sup>þ</sup> *skdi* : <sup>0</sup>≤*<sup>s</sup>* <sup>&</sup>lt;j*f v <sup>j</sup>*

<sup>¼</sup> *f v*ð Þþ*<sup>i</sup> f v <sup>j</sup>*

<sup>j</sup> ,

in rows and

j�<sup>1</sup>

<sup>þ</sup> *viv <sup>j</sup>*

<sup>¼</sup> *mn*.

*G is the number of distinct pairs of divisors r and s of k such that rs* ¼ *k.*

j�<sup>1</sup> .

∣ *f v*ð Þ*<sup>i</sup>* ∣ þ *k* j*f v <sup>j</sup>*

) *k* j*f v <sup>j</sup>*

*f v*ð Þ¼*<sup>i</sup> ar* ¼ *a* þ *rdi* f g : 0≤*r*<j *f v*ð Þj *<sup>i</sup>* and *f v <sup>j</sup>*

where *k*≤ ∣ *f v*ð Þ*<sup>i</sup>* ∣. Now, arrange the terms of *f*

) *k* ¼ ∣ *f v*ð Þ*<sup>i</sup>* ∣*:*

labelling of a graph *G* to be a strongly modular sumset labelling of the graph. **Theorem 4.5.** *Let G be a graph which admits an arithmetic modular sumset labelling, say f . Then, f is a strongly modular sumset labelling of G if and only if the deterministic ratio of every edge of G is equal to the set-labelling number of its end vertex*

Assume that *f* is a strongly modular sumset labelling. Therefore, *f*

ling, the set-labelling number of every vertex of *G* is a divisor of the set-labelling numbers of the edges incident on that vertex. Let *v* be a vertex of *G* with the setlabelling number *<sup>r</sup>*, where *<sup>r</sup>* is a divisor of *<sup>k</sup>*, but *<sup>r</sup>*<sup>2</sup> 6¼ *<sup>k</sup>*. Since *<sup>f</sup>* is *<sup>k</sup>*-uniform, all the vertices in *N v*ð Þ, must have the set-labelling number *s*, where *rs* ¼ *k*. Again, all vertices, which are adjacent to the vertices of *N v*ð Þ, must have the set-labelling number *r*. Since *G* is a connected graph, all vertices of *G* have the set-labelling number *r* or *s*. Let *X* be the set of all vertices of *G* having the set-labelling number *r* and *<sup>Y</sup>* be the set of all vertices of *<sup>G</sup>* having the set-labelling number *<sup>s</sup>*. Since *<sup>r</sup>*<sup>2</sup> 6¼ *<sup>k</sup>*, no two elements in *X* (and no elements in *Y* also) can be adjacent to each other. Therefore, *<sup>G</sup>* is bipartite. □ The following result is an immediate consequence of the above theorem. **Theorem 4.3.** [47] *For a positive non-square integer k*≤*n, a modular sumset labelling f* : *V G*ð Þ!P0ð Þ *<sup>n</sup> of an arbitrary graph G is a strongly k-uniform modular sumset labelling of G if and only if either G is bipartite or a disjoint union of bipartite*

þð Þ¼ *uv k*∀*uv*∈*E G*ð Þ. Since, *f* is a strong modular sumset label-

square.

sumset graph.

*components.*

uniform modular sumset graph is as follows.

*having smaller common difference.*

edge *viv <sup>j</sup>* is ∣ *f v*ð Þ*<sup>i</sup>* ∣ þ *k* j*f v <sup>j</sup>*

Then,

**98**

ling, say *f*. Then, *f*

*Number Theory and Its Applications*

columns as follows. For *bs* ∈ *f v <sup>j</sup>* , 0 ≤*s*<∣ *f v <sup>j</sup>* <sup>∣</sup>, arrange the terms of *f v*ð Þþ*<sup>i</sup> bs* in ð Þ *s* þ 1 -th row in such a way that equal terms of different rows come in the same column of this arrangement. Then, the common difference between consecutive elements in each row is *di*. Since *k* ¼ ∣ *f v*ð Þ*<sup>i</sup>* ∣, the difference between the final element of any row (other than the last row) and first element of its succeeding row is also *di*. That is, no column in this arrangement contains more than one element. Hence, all elements in this arrangement are distinct. Therefore, total number of elements in *f v*ð Þþ*<sup>i</sup> f v <sup>j</sup>* is <sup>∣</sup> *f v*ð Þ*<sup>i</sup>* ∣∣ *f v <sup>j</sup>* <sup>∣</sup>. Hence, *<sup>f</sup>* is a strongly modular sumset labelling. □

#### **5. Supreme modular sumset labelling of** *G*

In both types of modular sumset labelling discussed above, it is observed that the cardinality of the edge set-label cannot exceed the value *n*. This fact creates much interest in investigating the case where all the edge set-label have the cardinality *n*.

**Definition 5.1.** [47] A modular sumset labelling *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* of a given graph *G* is said to be a *supreme modular sumset labelling* or *maximal modular sumset labelling* of *G* if and only if *f* þð Þ¼ *E G*ð Þ f g *<sup>n</sup>* .

Put in a different way, a modular sumset labelling *f* : *V G*ð Þ!Pð Þ *<sup>n</sup>* of a given graph *G* is a supreme modular sumset labelling of *G* if the set-label of every edge of *G* is the ground set *<sup>n</sup>* itself.

A necessary and sufficient condition for a modular sumset labelling of a graph *G* to be its supreme modular sumset labelling is discussed in the theorem given below:

**Theorem 5.1.** [47] *The modular sumset labelling f* : *V G*ð Þ!Pð Þ *<sup>n</sup> of a given graph G is a supreme modular sumset labelling of G if and only if for every pair of adjacent vertices u and v of G some or all of the following conditions hold.*


*Proof.* For two adjacent vertices *u* and *v* in *G*, let *D f u*ð Þ ¼ *D f v*ð Þ ¼ f g*d* are arithmetic progressions containing the same elements. Then, the elements in *f u*ð Þ and *f v*ð Þ are also in arithmetic progression, with the same common difference *d*. Then, by Theorem 3.4, *n* ¼ ∣ *f u*ð Þþ *f v*ð Þ∣ ¼ ∣ *f u*ð Þ∣ þ ∣ *f v*ð Þ∣ � 1. Therefore, the setlabelling number of the edge *uv* is *n* if and only if ∣ *f u*ð Þ∣ þ ∣ *f v*ð Þ∣> *n*.

Now, let *D f u*ð Þ ∩ *D f v*ð Þ 6¼ Ø such that *D f u*ð Þ 6¼ *D f v*ð Þ. Then, clearly ∣ *f u*ð Þþ *f v*ð Þ∣≥∣ *f u*ð Þ∣ þ ∣ *f v*ð Þ∣. Therefore, we have ∣*f* þð Þ *uv* ∣ ¼ *n* if and only if ∣ *f u*ð Þ∣ þ <sup>∣</sup> *f v*ð Þ∣ ≥ *<sup>n</sup>*. □

Next assume that *D f u*ð Þ ∩ *D f v*ð Þ ¼ Ø. Then, ∣ *f u*ð Þþ *f v*ð Þ∣ ¼ ∣ *f u*ð Þ∣∣ *f v*ð Þ∣. Therefore, we have ∣*f* þð Þ *uv* ∣ ¼ *n* if and only if ∣ *f u*ð Þ∣∣ *f v*ð Þ∣ ≥ *n*.

A necessary and sufficient condition for a strong modular sumset labelling of a graph *G* to be a maximal modular sumset labelling of *G*.

**Theorem 5.2.** [47] *Let f be a strong sumset-labelling of a given graph G. Then, f is a maximal sumset-labelling of G if and only if n is a perfect square or G is bipartite or a disjoint union of bipartite components.*

*Proof.* The proof is an immediate consequence of Theorem 4.2, when *<sup>k</sup>* <sup>¼</sup> *<sup>n</sup>*. □

#### **6. Weakly modular sumset graphs**

Another interesting question we address in the beginning of this section is whether the lower bound and the upper bound of the sumset can be equal. Suppose that *A* and *B* be two non-empty subsets of *<sup>n</sup>* such that the bounds of their sumset are equal. Then, we have

A set of vertices *X* of a graph *G* is said to have maximal incidence if the maximum number of edges of incidence at the elements of *X*. Then, analogous to the corresponding result of integer additive set-valued graphs (see [40]), we have. **Theorem 6.4.** *Let G be a weakly modular sumset labelled graph and I be I be the largest independent set of G with maximum incidence. Then, the sparing number of G is*

*Proof.* Recall that the degree of a vertex *v*, denoted by *d v*ð Þ, is equal to the number of edges incident on a vertex. Note that any vertex *vi* ∈*I* can have a nonsingleton set-label which gives non-singleton set labels to *d v*ð Þ*<sup>i</sup>* edges incident on it. Since *I* is an independent set, the edges incident at the vertices in *I* assumes nonsingleton set-label. Therefore, the number of edges having non-singleton set-label

that kind, the above expression counts the maximal non-sparing edges in *G*. Hence,

As a special case of the modular sumset number, the notion of weakly modular

**Definition 6.3.** The *weakly modular sumset number* of a graph *G*, denoted by *σ<sup>w</sup>* is

The following theorem discussed the weak sumset number of an arbitrary graph

Also, the number non-empty, non-singleton subsets of the ground setmust be greater than or equal to *α*. Otherwise, all the vertices in *V*<sup>0</sup> cannot be labelled by non-singleton subsets of this ground set. We know that the number of non-empty, non-singleton subsets of a set *<sup>A</sup>* is 2<sup>∣</sup>*A*<sup>∣</sup> � <sup>∣</sup>*A*<sup>∣</sup> � 1, where *<sup>A</sup>* <sup>⊆</sup>*<sup>n</sup>* is the ground set used for labelling.

Therefore, the weak modular sumset number *<sup>G</sup>* is *<sup>α</sup>* if 2*<sup>α</sup>* � *<sup>α</sup>* � <sup>1</sup>≥*β*. Otherwise, the ground set must have at least *<sup>r</sup>* elements such that 2*<sup>r</sup>* � *<sup>r</sup>* � <sup>1</sup>≥*β*. Therefore, in this case, the weak modular sumset number of *G* is *r*, where *r* is the smallest positive integer such that 2*<sup>r</sup>* � *<sup>r</sup>* � <sup>1</sup>≥*β*. Hence, *<sup>σ</sup>* <sup>∗</sup> ð Þ¼ *<sup>G</sup>* max f g *<sup>α</sup>*,*<sup>r</sup>* . This completes the proof. □ The weakly modular sumset number some fundamental graph classes are given

The following theorem discusses the minimum cardinality of the ground set when the given graph *G* admits a weakly uniform modular sumset labelling. **Theorem 6.6.** [47] *Let G be a weakly k-uniform modular sumset graph with covering number α and independence number β, where k*<*α. Then, the minimum cardinality of the ground set <sup>n</sup> is* max f g *α*,*r , where r is the smallest positive integer such*

**Theorem 6.5.** [47] *Let G be a modular sumset graph and α and β be the covering number and independence number of G respectively. Then, the weak modular sumset number of G is* max f g *<sup>α</sup>*,*<sup>r</sup> , where r is the smallest positive integer such that* <sup>2</sup>*<sup>r</sup>* � *<sup>r</sup>* � <sup>1</sup>≥*β. Proof.* Recall that *α*ð Þþ *G β*ð Þ¼ *G* ∣*V G*ð Þ∣ (see [4]). Since *G* is a modular sumset graph, no two adjacent vertices can have non-singleton set-label simultaneously. Therefore, the maximum number of vertices that have non-singleton set-label is *β*. Let *V*<sup>0</sup> be the set of these independent vertices in *G*. Therefore, the minimum number of sparing vertices is ∣*V G*ð Þ∣ � *β* ¼ *α*. Since all these vertices in *V* � *V*<sup>0</sup> must have distinct singleton set-label, the ground set must have at least *α* elements.

defined to be the minimum value of *n* such that a modular sumset labelling *f* :

*vi* <sup>∈</sup>*<sup>I</sup>d v*ð Þ*<sup>i</sup>* . Since *I* is a maximal independent set of

*vi* <sup>∈</sup>*<sup>I</sup>d v*ð Þ*<sup>i</sup>* . □

<sup>∣</sup>*E G*ð Þ<sup>∣</sup> � <sup>P</sup>

in **Table 1**.

*r k*≥*β* � �*.*

*that*

**101**

*vi* <sup>∈</sup>*<sup>I</sup>d v*ð Þ*<sup>i</sup> .*

*Modular Sumset Labelling of Graphs*

*DOI: http://dx.doi.org/10.5772/intechopen.92701*

incident at the vertices in *I* is P

the number of sparing edges in *<sup>G</sup>* is <sup>∣</sup>*E G*ð Þ<sup>∣</sup> � <sup>P</sup>

**6.2 Weakly modular sumset number of graphs**

sumset number is introduced in [47] as follows:

*V G*ð Þ!P0ð Þ *<sup>n</sup>* is a weakly modular sumset labelling of *G*.

*G* in terms of its covering and independence numbers.

$$\begin{aligned} |A| + |B| - \mathbf{1} &= |A| |B| \\ |A| + |B| - \mathbf{1} - |A| |B| &= \mathbf{0} \\ |A|(\mathbf{1} - |B|) + |B| - \mathbf{1} &= \mathbf{0} \\ (|A| - \mathbf{1})(|B| - \mathbf{1}) &= \mathbf{0} \end{aligned}$$

which is possible only when ∣*A*∣ ¼ 1 or ∣*B*∣ ¼ 1 (or both). Also, note that in this case the cardinality of the sumset is equal to equal to that of one of the summands. This interesting phenomenon leads us to a new type of a modular sumset labelling called weakly modular sumset labelling. This type of labelling is investigated in the following section.

#### **6.1 Weakly modular sumset labelling of graphs**

**Definition 6.1.** A modular sumset labelling *f* of a graph *G* is said to be a *weakly modular sumset labelling* of *G* if the cardinality of the set-label of every edge of *G* is equal to the cardinality of the set-label of at least one of its end vertices. A graph which admits a weakly modular sumset labelling is called a *weakly modular sumset graph*.

From the above definition, it can be observed that for any edge *uv* in weakly modular sumset graph *G*, ∣*f* þð Þ *uv* ∣ ¼ ∣ *f u*ð Þ∣ or ∣*f* þð Þ *uv* ∣ ¼ ∣ *f v*ð Þ∣. Putting it in a different way, the set-labelling number of at least one end vertex of every edge of a weakly modular sumset graph is a singleton. An element (a vertex or an edge) of modular sumset graph *G* with set-labelling number 1 is called a *sparing element* or a *monocardinal elements* of *G*. Hence, analogous to the condition for a sumset graph to be a weak sumset graph (see [39]), we have.

**Theorem 6.1.** *A graph G admits a weak modular sumset labelling if and only if G is bipartite or contains sparing edges.*

*Proof.* Note the fact that at least one end vertex of every edge of *G* is a sparing vertex. Also, we note that no two vertices with non-singleton set-label in weakly modular sumset graph can be adjacent to each other. Thus, if every of edge of *G* has exactly one end vertex with singleton set-label, then we can partition the vertex set of *G* into two subsets *X* with all sparing vertices and *Y* with all non-sparing vertices. Here, no two vertices in the same partition are adjacent and hence *G* is a bipartite graph. If *G* is not a bipartite graph, then obviously *G* should have at least one sparing edge, completing the proof.

Invoking Theorem 6.1, the following two results are immediate.

**Corollary 6.2.** *Every graph G admits a weakly modular sumset labelling.*

**Corollary 6.3.** *A graph G admits a weakly uniform modular sumset labelling if and only if G is bipartite.*

The above results are similar to the corresponding result of integer additive setlabelled graphs (see [39]) and hence the notion of *sparing number* of graphs defined and studied in [48–57] can be extended to our current discussion also. The notion of the sparing number of graphs is defined as follows:

**Definition 6.2.** Let *G* be a weakly modular sumset labelled graph. Then, the *sparing number* of *G* is the number of sparing edges in *G*.

*Modular Sumset Labelling of Graphs DOI: http://dx.doi.org/10.5772/intechopen.92701*

**6. Weakly modular sumset graphs**

*Number Theory and Its Applications*

**6.1 Weakly modular sumset labelling of graphs**

be a weak sumset graph (see [39]), we have.

are equal. Then, we have

following section.

modular sumset graph *G*, ∣*f*

*bipartite or contains sparing edges.*

sparing edge, completing the proof.

the sparing number of graphs is defined as follows:

*sparing number* of *G* is the number of sparing edges in *G*.

*only if G is bipartite.*

**100**

*graph*.

Another interesting question we address in the beginning of this section is whether the lower bound and the upper bound of the sumset can be equal. Suppose that *A* and *B* be two non-empty subsets of *<sup>n</sup>* such that the bounds of their sumset

> ∣*A*∣ þ ∣*B*∣ � 1 ¼ ∣*A*k*B*∣ ∣*A*∣ þ ∣*B*∣ � 1 � ∣*A*k*B*∣ ¼ 0 ∣*A*∣ð Þþ 1�j*B*j ∣*B*∣ � 1 ¼ 0 ð Þj j*A*j�1 ð Þ¼ *B*j�1 0

which is possible only when ∣*A*∣ ¼ 1 or ∣*B*∣ ¼ 1 (or both). Also, note that in this case the cardinality of the sumset is equal to equal to that of one of the summands. This interesting phenomenon leads us to a new type of a modular sumset labelling called weakly modular sumset labelling. This type of labelling is investigated in the

**Definition 6.1.** A modular sumset labelling *f* of a graph *G* is said to be a *weakly modular sumset labelling* of *G* if the cardinality of the set-label of every edge of *G* is equal to the cardinality of the set-label of at least one of its end vertices. A graph which admits a weakly modular sumset labelling is called a *weakly modular sumset*

From the above definition, it can be observed that for any edge *uv* in weakly

different way, the set-labelling number of at least one end vertex of every edge of a weakly modular sumset graph is a singleton. An element (a vertex or an edge) of modular sumset graph *G* with set-labelling number 1 is called a *sparing element* or a *monocardinal elements* of *G*. Hence, analogous to the condition for a sumset graph to

**Theorem 6.1.** *A graph G admits a weak modular sumset labelling if and only if G is*

*Proof.* Note the fact that at least one end vertex of every edge of *G* is a sparing vertex. Also, we note that no two vertices with non-singleton set-label in weakly modular sumset graph can be adjacent to each other. Thus, if every of edge of *G* has exactly one end vertex with singleton set-label, then we can partition the vertex set of *G* into two subsets *X* with all sparing vertices and *Y* with all non-sparing vertices. Here, no two vertices in the same partition are adjacent and hence *G* is a bipartite graph. If *G* is not a bipartite graph, then obviously *G* should have at least one

**Corollary 6.3.** *A graph G admits a weakly uniform modular sumset labelling if and*

The above results are similar to the corresponding result of integer additive setlabelled graphs (see [39]) and hence the notion of *sparing number* of graphs defined and studied in [48–57] can be extended to our current discussion also. The notion of

**Definition 6.2.** Let *G* be a weakly modular sumset labelled graph. Then, the

Invoking Theorem 6.1, the following two results are immediate. **Corollary 6.2.** *Every graph G admits a weakly modular sumset labelling.*

þð Þ *uv* ∣ ¼ ∣ *f v*ð Þ∣. Putting it in a

þð Þ *uv* ∣ ¼ ∣ *f u*ð Þ∣ or ∣*f*

A set of vertices *X* of a graph *G* is said to have maximal incidence if the maximum number of edges of incidence at the elements of *X*. Then, analogous to the corresponding result of integer additive set-valued graphs (see [40]), we have.

**Theorem 6.4.** *Let G be a weakly modular sumset labelled graph and I be I be the largest independent set of G with maximum incidence. Then, the sparing number of G is* <sup>∣</sup>*E G*ð Þ<sup>∣</sup> � <sup>P</sup> *vi* <sup>∈</sup>*<sup>I</sup>d v*ð Þ*<sup>i</sup> .*

*Proof.* Recall that the degree of a vertex *v*, denoted by *d v*ð Þ, is equal to the number of edges incident on a vertex. Note that any vertex *vi* ∈*I* can have a nonsingleton set-label which gives non-singleton set labels to *d v*ð Þ*<sup>i</sup>* edges incident on it. Since *I* is an independent set, the edges incident at the vertices in *I* assumes nonsingleton set-label. Therefore, the number of edges having non-singleton set-label incident at the vertices in *I* is P *vi* <sup>∈</sup>*<sup>I</sup>d v*ð Þ*<sup>i</sup>* . Since *I* is a maximal independent set of that kind, the above expression counts the maximal non-sparing edges in *G*. Hence, the number of sparing edges in *<sup>G</sup>* is <sup>∣</sup>*E G*ð Þ<sup>∣</sup> � <sup>P</sup> *vi* <sup>∈</sup>*<sup>I</sup>d v*ð Þ*<sup>i</sup>* . □

#### **6.2 Weakly modular sumset number of graphs**

As a special case of the modular sumset number, the notion of weakly modular sumset number is introduced in [47] as follows:

**Definition 6.3.** The *weakly modular sumset number* of a graph *G*, denoted by *σ<sup>w</sup>* is defined to be the minimum value of *n* such that a modular sumset labelling *f* : *V G*ð Þ!P0ð Þ *<sup>n</sup>* is a weakly modular sumset labelling of *G*.

The following theorem discussed the weak sumset number of an arbitrary graph *G* in terms of its covering and independence numbers.

**Theorem 6.5.** [47] *Let G be a modular sumset graph and α and β be the covering number and independence number of G respectively. Then, the weak modular sumset number of G is* max f g *<sup>α</sup>*,*<sup>r</sup> , where r is the smallest positive integer such that* <sup>2</sup>*<sup>r</sup>* � *<sup>r</sup>* � <sup>1</sup>≥*β.*

*Proof.* Recall that *α*ð Þþ *G β*ð Þ¼ *G* ∣*V G*ð Þ∣ (see [4]). Since *G* is a modular sumset graph, no two adjacent vertices can have non-singleton set-label simultaneously. Therefore, the maximum number of vertices that have non-singleton set-label is *β*. Let *V*<sup>0</sup> be the set of these independent vertices in *G*. Therefore, the minimum number of sparing vertices is ∣*V G*ð Þ∣ � *β* ¼ *α*. Since all these vertices in *V* � *V*<sup>0</sup> must have distinct singleton set-label, the ground set must have at least *α* elements.

Also, the number non-empty, non-singleton subsets of the ground setmust be greater than or equal to *α*. Otherwise, all the vertices in *V*<sup>0</sup> cannot be labelled by non-singleton subsets of this ground set. We know that the number of non-empty, non-singleton subsets of a set *<sup>A</sup>* is 2<sup>∣</sup>*A*<sup>∣</sup> � <sup>∣</sup>*A*<sup>∣</sup> � 1, where *<sup>A</sup>* <sup>⊆</sup>*<sup>n</sup>* is the ground set used for labelling.

Therefore, the weak modular sumset number *<sup>G</sup>* is *<sup>α</sup>* if 2*<sup>α</sup>* � *<sup>α</sup>* � <sup>1</sup>≥*β*. Otherwise, the ground set must have at least *<sup>r</sup>* elements such that 2*<sup>r</sup>* � *<sup>r</sup>* � <sup>1</sup>≥*β*. Therefore, in this case, the weak modular sumset number of *G* is *r*, where *r* is the smallest positive integer such that 2*<sup>r</sup>* � *<sup>r</sup>* � <sup>1</sup>≥*β*. Hence, *<sup>σ</sup>* <sup>∗</sup> ð Þ¼ *<sup>G</sup>* max f g *<sup>α</sup>*,*<sup>r</sup>* . This completes the proof. □

The weakly modular sumset number some fundamental graph classes are given in **Table 1**.

The following theorem discusses the minimum cardinality of the ground set when the given graph *G* admits a weakly uniform modular sumset labelling.

**Theorem 6.6.** [47] *Let G be a weakly k-uniform modular sumset graph with covering number α and independence number β, where k*<*α. Then, the minimum cardinality of the ground set <sup>n</sup> is* max f g *α*,*r , where r is the smallest positive integer such*

*that r k*≥*β* � �*.*


cardinality of the set-label of the elements of the graphs concerned and the patterns of the elements in these set-label. It is to be noted that several other possibilities can be investigated in this regard. For example, analogous to the topological setvaluations of graphs, the case when the collection of set-label of vertices and/or edges of a graph *G* forms a topology of the ground set *<sup>n</sup>* can be studied in detail. Another possibility for future investigation is to extend the graceful and sequential concepts of set-labelling of graphs to modular sumset labelling also. All these points

The author would like to dedicate the chapter to his doctoral thesis supervisor Prof. Dr. Germina K Augusthy for her constant support and inspiration for working in the area of sumset-labelling. The author would also like to acknowledge the critical and creative suggestions of the editor and referees which improved the

Department of Mathematics, CHRIST (Deemed to be University), Bangalore,

© 2020 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: sudev.nk@christuniversity.in

provided the original work is properly cited.

highlight the wide scope for further studies in this area.

content and the presentation style of the chapter.

Mathematics Subject Classification 2010: 05C75

**Acknowledgements**

*Modular Sumset Labelling of Graphs*

*DOI: http://dx.doi.org/10.5772/intechopen.92701*

**Additional information**

**Author details**

Sudev Naduvath

Karnataka, India

**103**

#### **Table 1.**

*Weakly modular sumset number of some graph classes.*

*Proof.* Let a weakly *k*-uniform modular sumset labelling be defined on a graph *G* over the ground set *A* ⊂*n*. Then, by Corollary 6.3, *G* is bipartite. Let *X*, *Y* be the bipartition of the vertex set *V G*ð Þ. Without loss of generality, let ∣*X*∣≤∣*Y*∣. Then, *α* ¼ ∣*X*∣ and *β* ¼ ∣*Y*∣. Then, distinct elements of *X* must have distinct singleton setlabel. Therefore, *n*≥*α*.

On the other hand, since *f* is *k*-uniform, all the elements in *Y* must have distinct *k*-element set-label. The number of *k*-element subsets of a set *A* (obviously, with more than *<sup>k</sup>* elements) is <sup>∣</sup>*A*<sup>∣</sup> *k* . The ground set *<sup>A</sup>* has *<sup>α</sup>* elements only if *<sup>α</sup> k*≥*β* . Otherwise, the ground set *A* must contain at least *r* elements, where *r*> *α* is the smallest positive integer such that *r k*≥ *β* . Therefore, *<sup>n</sup>* <sup>¼</sup> max f g *<sup>α</sup>*,*<sup>r</sup>* . □

In view of the above theorem, the following result is immediate.

**Corollary 6.7.** *Let G be a weakly k-uniform modular sumset graph, where k*≥*α, where α is the covering number of G. Then, the minimum cardinality of the ground set <sup>n</sup> is the smallest positive integer n such that n k*≥*β , where <sup>β</sup> is the independence*

*number of G.*

The following result explains a necessary and sufficient condition for a weak modular sumset labelling of a given graph *G* to be a maximal modular sumset labelling of *G*.

**Proposition 6.8.** [47] *A weakly modular sumset labelling of a graph G is a supreme modular sumset labelling of G if and only if G is a star graph.*

*Proof.* Let *f* be a weak modular sumset labelling of given graph *G*. First, assume that *f* is a maximal modular sumset labelling of *G*. Then, the set-labelling number of one end vertex of every edge of *G* is 1 and the set-labelling number of the other end vertex is *n*. Therefore, *<sup>n</sup>* be the set-label of one end vertex of every edge of *G*, which is possible only if *G* is a star graph with the central vertex has the set-label *<sup>n</sup>* and the pendant vertices of *G* have distinct singleton set-label.

Conversely, assume that *G* is a star graph. Label the central vertex of *G* by the ground set *<sup>n</sup>* and label other vertices of *G* by distinct singleton subsets of *n*. Then, all the edges of *G* has the set-labelling number *n*. That is, this labelling is a supreme modular sumset labelling of *<sup>G</sup>*. □

#### **7. Conclusion**

In this chapter, we have discussed certain types of modular sumset graphs and their structural properties and characterisations. These studies are based on the

*Modular Sumset Labelling of Graphs DOI: http://dx.doi.org/10.5772/intechopen.92701*

cardinality of the set-label of the elements of the graphs concerned and the patterns of the elements in these set-label. It is to be noted that several other possibilities can be investigated in this regard. For example, analogous to the topological setvaluations of graphs, the case when the collection of set-label of vertices and/or edges of a graph *G* forms a topology of the ground set *<sup>n</sup>* can be studied in detail. Another possibility for future investigation is to extend the graceful and sequential concepts of set-labelling of graphs to modular sumset labelling also. All these points highlight the wide scope for further studies in this area.

#### **Acknowledgements**

*Proof.* Let a weakly *k*-uniform modular sumset labelling be defined on a graph *G* over the ground set *A* ⊂*n*. Then, by Corollary 6.3, *G* is bipartite. Let *X*, *Y* be the bipartition of the vertex set *V G*ð Þ. Without loss of generality, let ∣*X*∣≤∣*Y*∣. Then, *α* ¼ ∣*X*∣ and *β* ¼ ∣*Y*∣. Then, distinct elements of *X* must have distinct singleton set-

On the other hand, since *f* is *k*-uniform, all the elements in *Y* must have distinct *k*-element set-label. The number of *k*-element subsets of a set *A* (obviously, with

Otherwise, the ground set *A* must contain at least *r* elements, where *r*> *α* is the

**Corollary 6.7.** *Let G be a weakly k-uniform modular sumset graph, where k*≥*α, where α is the covering number of G. Then, the minimum cardinality of the ground set*

The following result explains a necessary and sufficient condition for a weak modular sumset labelling of a given graph *G* to be a maximal modular sumset

**Proposition 6.8.** [47] *A weakly modular sumset labelling of a graph G is a supreme*

*Proof.* Let *f* be a weak modular sumset labelling of given graph *G*. First, assume that *f* is a maximal modular sumset labelling of *G*. Then, the set-labelling number of one end vertex of every edge of *G* is 1 and the set-labelling number of the other end vertex is *n*. Therefore, *<sup>n</sup>* be the set-label of one end vertex of every edge of *G*, which is possible only if *G* is a star graph with the central vertex has the set-label *<sup>n</sup>*

Conversely, assume that *G* is a star graph. Label the central vertex of *G* by the ground set *<sup>n</sup>* and label other vertices of *G* by distinct singleton subsets of *n*. Then, all the edges of *G* has the set-labelling number *n*. That is, this labelling is a supreme modular sumset labelling of *<sup>G</sup>*. □

In this chapter, we have discussed certain types of modular sumset graphs and their structural properties and characterisations. These studies are based on the

*n k*≥*β* 

*r k*≥ *β* 

In view of the above theorem, the following result is immediate.

. The ground set *<sup>A</sup>* has *<sup>α</sup>* elements only if *<sup>α</sup>*

. Therefore, *<sup>n</sup>* <sup>¼</sup> max f g *<sup>α</sup>*,*<sup>r</sup>* . □

**(***G***)**

2 ⌋

<sup>2</sup> ⌋ if *p*>2

<sup>2</sup> ⌋ if *p*>4

*, where β is the independence*

*k*≥*β* 

.

label. Therefore, *n*≥*α*.

**Table 1.**

*number of G.*

labelling of *G*.

**7. Conclusion**

**102**

more than *<sup>k</sup>* elements) is <sup>∣</sup>*A*<sup>∣</sup>

smallest positive integer such that

*<sup>n</sup> is the smallest positive integer n such that*

*Weakly modular sumset number of some graph classes.*

*Number Theory and Its Applications*

*k* 

**Graph class** *σ***\***

Wheel graph, *W*1,*<sup>p</sup>* 1 + ⌊ *<sup>p</sup>*

Helm graph, *Hp p* Ladder graph, *Lp p* Complete graph, *Kp p* � 1

Path, *Pp* 2 if *p*≤2; ⌊ *<sup>p</sup>*

Cycle, *Cp <sup>p</sup>* � 1 if *<sup>p</sup>* <sup>¼</sup> 3, 4; ⌊ *<sup>p</sup>*

*modular sumset labelling of G if and only if G is a star graph.*

and the pendant vertices of *G* have distinct singleton set-label.

The author would like to dedicate the chapter to his doctoral thesis supervisor Prof. Dr. Germina K Augusthy for her constant support and inspiration for working in the area of sumset-labelling. The author would also like to acknowledge the critical and creative suggestions of the editor and referees which improved the content and the presentation style of the chapter.

#### **Additional information**

Mathematics Subject Classification 2010: 05C75

### **Author details**

Sudev Naduvath Department of Mathematics, CHRIST (Deemed to be University), Bangalore, Karnataka, India

\*Address all correspondence to: sudev.nk@christuniversity.in

© 2020 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] Bondy JA, Murty USR. Graph Theory with Applications. Vol. 290. London: Macmillan; 1976

[2] Bondy JA, Murty USR. Graph Theory. New York: Springer; 2008

[3] Gross JL, Yellen J, Zhang P. Handbook of Graph Theory. Boca Raton: Chapman and Hall/CRC; 2013

[4] Harary F. Graph Theory. New Delhi: Narosa Publ. House; 2001

[5] West DB. Introduction to Graph Theory. Vol. 2. Upper Saddle River: Prentice Hall; 2001

[6] Bača M, MacDougall J, Miller M, Wallis W. Survey of certain valuations of graphs. Discussiones Mathematicae Graph Theory. 2000;**20**(2):219-229

[7] Brandstadt A, Spinrad JP. Graph Classes: A Survey. Vol. 3. Philadelphia: SIAM; 1999

[8] Gallian JA. A dynamic survey of graph labeling. Electronic Journal of Combinatorics 1 (Dynamic Surveys). 2018:DS6

[9] Hammack R, Imrich W, Klavžar S. Handbook of Product Graphs. Bocca Raton: CRC Press; 2011

[10] Imrich W, Klavzar S. Product Graphs: Structure and Recognition. New York: Wiley; 2000

[11] Rosa A. On certain valuations of the vertices of a graph. In: Theory of Graphs International Symposium; July; Rome. 1966. pp. 349-355

[12] Golomb SW. How to number a graph. In: Graph Theory and Computing. Cambridge, UK: Academic Press; 1972. pp. 23-37

[13] Apostol TM. Introduction to Analytic Number Theory. New Delhi: Narosa Publ. House; 2013

[23] Germina KA. Set-valuations of graphs and their applications. Final Technical Report. Vol. 4. DST Grant-in-

*DOI: http://dx.doi.org/10.5772/intechopen.92701*

*Modular Sumset Labelling of Graphs*

[33] Sudev NK, Germina KA,

theory for Shannon entropy. Combinatorics, Probability and Computing. 2010;**19**(4):603-639

additive set-indexers of graphs. International Journal of Mathematical Sciences and Engineering Applications.

2015;**7**(01):1450065

1020-1049

Chithra KP. Arithmetic integer additive set-valued graphs: A creative review. The Journal of Mathematics and Computer Science. 2020;**10**(4):

[34] Tao T. Sumset and inverse sumset

[35] Sudev NK, Germina KA. On integer

[36] Germina KA, Sudev NK. On weakly uniform integer additive set-indexers of graphs. International Mathematical Forum. 2013;**8**(37):1837-1845

[37] Germina KA, Anandavally TMK. Integer additive set-indexers of a graph:

Combinatorics, Information & System

[38] Sudev NK. Set valuations of discrete structures and their applications [PhD thesis]. Kannur University; 2015

Sum square graphs. Journal of

Sciences. 2012;**37**(2-4):345-358

[39] Naduvath S, Germina K. A characterisation of weak integer additive set-indexers of graphs. Journal of Fuzzy Set Valued Analysis. 2014,

[40] Sudev NK, Germina KA,

Chithra KP. Weak set-labeling number of certain integer additive set-labeled graphs. International Journal of Computers and Applications. 2015;

[41] Sudev NK, Germina KA. On weak integer additive set-indexers of certain graph classes. Journal of Discrete Mathematical Sciences & Cryptography.

2014;**1** jfsva-00189

2015;**18**(1–2):117-128

**114**(2):1-6

[24] Naduvath S. On disjunctive and conjunctive set-labelings of graphs.

[25] Nathanson MB. Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Vol. 165. New York: Springer Science & Business

[26] Nathanson MB. Sums of finite sets of integers. American Mathematical Monthly. 1972;**79**(9):1010-1012

[27] Nathanson MB. Additive Number Theory: The Classical Bases. Vol. 164. New York: Springer Science & Business

Additive Number Theory. I. In Additive Combinatorics. Vol. 43. Providence, RI: American Mathematical Society; 2007.

[29] Ruzsa IZ. Generalized arithmetical

[30] Ruzsa IZ. Sumsets and structure. In: Combinatorial Number Theory and Additive Group Theory. 2009.

Chithra KP. Weak integer additive setlabeled graphs: A creative review. Asian-European Journal of Mathematics. 2015;**8**(3):1550052

Chithra KP. Strong integer additive setvalued graphs: A creative review. International Journal of Computer Applications. 2015;**97**(5):8887

progressions and sumsets. Acta Mathematica Hungarica. 1994;**65**(4):

[31] Sudev NK, Germina KA,

[32] Sudev NK, Germina KA,

[28] Nathanson MB. Problems in

Aid Project No. SR/S4/277/05

Southeast Asian Bulletin of Mathematics. 2019;**43**(4):593-600

Media; 1996

Media; 2013

pp. 263-270

379-388

pp. 87-210

**105**

[14] Burton DM. Elementary Number Theory. Noida, India: Tata McGraw-Hill Education; 2006

[15] Cohn H. Advanced Number Theory. N Chelmford: Courier Corporation; 2012

[16] Nathanson MB. Elementary Methods in Number Theory. Vol. 195. New York: Springer Science & Business Media; 2008

[17] Acharya BD. Set valuations of a graph and their applications. In: MRI Lecture Notes in Applied Mathematics, 2. Allahabad: Mehta Research Instt; 1986

[18] Acharya BD. Set-indexers of a graph and set-graceful graphs. Bulletin of the Allahabad Mathematical Society. 2001; **16**:1-23

[19] Acharya BD, Germina KA. Setvaluations of graphs and their applications: A survey. Annals of Pure and Applied Mathematics. 2013;**4**(1):8-42

[20] Acharya BD, Hegde SM. On certain vertex valuations of a graph I. Indian Journal of Pure and Applied Mathematics. 1991;**22**(7):553-560

[21] Acharya BD, Rao SB, Arumugam S. Embeddings and NP-complete problems for graceful graphs. In: Labeling of Discrete Structures and Applications. New Delhi: Narosa Pub. House; 2008. pp. 57-62

[22] Augusthy GK. Set-valuations of graphs and their applications. In: Handbook of Research on Advanced Applications of Graph Theory in Modern Society. Hershey, Pennsylvania: IGI Global; 2020. pp. 171-207

*Modular Sumset Labelling of Graphs DOI: http://dx.doi.org/10.5772/intechopen.92701*

[23] Germina KA. Set-valuations of graphs and their applications. Final Technical Report. Vol. 4. DST Grant-in-Aid Project No. SR/S4/277/05

**References**

Macmillan; 1976

[1] Bondy JA, Murty USR. Graph Theory with Applications. Vol. 290. London:

[13] Apostol TM. Introduction to Analytic Number Theory. New Delhi:

[14] Burton DM. Elementary Number Theory. Noida, India: Tata McGraw-Hill

[15] Cohn H. Advanced Number Theory. N Chelmford: Courier Corporation; 2012

[16] Nathanson MB. Elementary Methods in Number Theory. Vol. 195. New York: Springer Science & Business

[17] Acharya BD. Set valuations of a graph and their applications. In: MRI Lecture Notes in Applied Mathematics, 2. Allahabad: Mehta Research Instt; 1986

[18] Acharya BD. Set-indexers of a graph and set-graceful graphs. Bulletin of the Allahabad Mathematical Society. 2001;

[19] Acharya BD, Germina KA. Setvaluations of graphs and their

applications: A survey. Annals of Pure and Applied Mathematics. 2013;**4**(1):8-42

[20] Acharya BD, Hegde SM. On certain vertex valuations of a graph I. Indian

[21] Acharya BD, Rao SB, Arumugam S. Embeddings and NP-complete problems for graceful graphs. In: Labeling of

Discrete Structures and Applications. New Delhi: Narosa Pub. House; 2008. pp. 57-62

[22] Augusthy GK. Set-valuations of graphs and their applications. In: Handbook of Research on Advanced Applications of Graph Theory in Modern Society. Hershey, Pennsylvania: IGI

Global; 2020. pp. 171-207

Journal of Pure and Applied Mathematics. 1991;**22**(7):553-560

Narosa Publ. House; 2013

Education; 2006

Media; 2008

**16**:1-23

[4] Harary F. Graph Theory. New Delhi:

[5] West DB. Introduction to Graph Theory. Vol. 2. Upper Saddle River:

[6] Bača M, MacDougall J, Miller M, Wallis W. Survey of certain valuations of graphs. Discussiones Mathematicae Graph Theory. 2000;**20**(2):219-229

[7] Brandstadt A, Spinrad JP. Graph Classes: A Survey. Vol. 3. Philadelphia:

[8] Gallian JA. A dynamic survey of graph labeling. Electronic Journal of Combinatorics 1 (Dynamic Surveys).

[9] Hammack R, Imrich W, Klavžar S. Handbook of Product Graphs. Bocca

[10] Imrich W, Klavzar S. Product Graphs: Structure and Recognition. New

[11] Rosa A. On certain valuations of the vertices of a graph. In: Theory of Graphs International Symposium; July; Rome.

[12] Golomb SW. How to number a graph.

In: Graph Theory and Computing. Cambridge, UK: Academic Press; 1972.

Raton: CRC Press; 2011

York: Wiley; 2000

1966. pp. 349-355

pp. 23-37

**104**

[2] Bondy JA, Murty USR. Graph Theory. New York: Springer; 2008

*Number Theory and Its Applications*

[3] Gross JL, Yellen J, Zhang P. Handbook of Graph Theory. Boca Raton: Chapman and Hall/CRC; 2013

Narosa Publ. House; 2001

Prentice Hall; 2001

SIAM; 1999

2018:DS6

[24] Naduvath S. On disjunctive and conjunctive set-labelings of graphs. Southeast Asian Bulletin of Mathematics. 2019;**43**(4):593-600

[25] Nathanson MB. Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Vol. 165. New York: Springer Science & Business Media; 1996

[26] Nathanson MB. Sums of finite sets of integers. American Mathematical Monthly. 1972;**79**(9):1010-1012

[27] Nathanson MB. Additive Number Theory: The Classical Bases. Vol. 164. New York: Springer Science & Business Media; 2013

[28] Nathanson MB. Problems in Additive Number Theory. I. In Additive Combinatorics. Vol. 43. Providence, RI: American Mathematical Society; 2007. pp. 263-270

[29] Ruzsa IZ. Generalized arithmetical progressions and sumsets. Acta Mathematica Hungarica. 1994;**65**(4): 379-388

[30] Ruzsa IZ. Sumsets and structure. In: Combinatorial Number Theory and Additive Group Theory. 2009. pp. 87-210

[31] Sudev NK, Germina KA, Chithra KP. Weak integer additive setlabeled graphs: A creative review. Asian-European Journal of Mathematics. 2015;**8**(3):1550052

[32] Sudev NK, Germina KA, Chithra KP. Strong integer additive setvalued graphs: A creative review. International Journal of Computer Applications. 2015;**97**(5):8887

[33] Sudev NK, Germina KA, Chithra KP. Arithmetic integer additive set-valued graphs: A creative review. The Journal of Mathematics and Computer Science. 2020;**10**(4): 1020-1049

[34] Tao T. Sumset and inverse sumset theory for Shannon entropy. Combinatorics, Probability and Computing. 2010;**19**(4):603-639

[35] Sudev NK, Germina KA. On integer additive set-indexers of graphs. International Journal of Mathematical Sciences and Engineering Applications. 2015;**7**(01):1450065

[36] Germina KA, Sudev NK. On weakly uniform integer additive set-indexers of graphs. International Mathematical Forum. 2013;**8**(37):1837-1845

[37] Germina KA, Anandavally TMK. Integer additive set-indexers of a graph: Sum square graphs. Journal of Combinatorics, Information & System Sciences. 2012;**37**(2-4):345-358

[38] Sudev NK. Set valuations of discrete structures and their applications [PhD thesis]. Kannur University; 2015

[39] Naduvath S, Germina K. A characterisation of weak integer additive set-indexers of graphs. Journal of Fuzzy Set Valued Analysis. 2014, 2014;**1** jfsva-00189

[40] Sudev NK, Germina KA, Chithra KP. Weak set-labeling number of certain integer additive set-labeled graphs. International Journal of Computers and Applications. 2015; **114**(2):1-6

[41] Sudev NK, Germina KA. On weak integer additive set-indexers of certain graph classes. Journal of Discrete Mathematical Sciences & Cryptography. 2015;**18**(1–2):117-128

[42] Sudev NK, Germina KA. Some new results on strong integer additive setindexers of graphs. Discrete Mathematics, Algorithms and Applications. 2015;**7**(01):1450065

[43] Sudev NK, Germina KA. On certain arithmetic integer additive set-indexers of graphs. Discrete Mathematics, Algorithms and Applications. 2015; **7**(03):1550025

[44] Sudev NK, Germina KA. A study on arithmetic integer additive set-indexers of graphs. Journal of Informatics and Mathematical Sciences. 2018;**10**(1–2): 321-332

[45] Naduvath S, Augusthy GK, Kok J. Sumset valuations of graphs and their applications. In: Handbook of Research on Advanced Applications of Graph Theory in Modern Society. Hershey, Pennsylvania: IGI Global; 2020. pp. 208-250

[46] Naduvath S, Germina KA. An Introduction to Sumset Valued Graphs. Mauritius: Lambert Academic Publ; 2018

[47] Naduvath S. A study on the modular sumset labeling of graphs. Discrete Mathematics, Algorithms and Applications. 2017;**9**(03):1750039

[48] Chithra KP, Sudev NK, Germina KA. Sparing number of Cartesian products of certain graphs. Communications in Mathematics and Applications. 2014;**5**(1):23-30

[49] Chithra KP, Sudev NK, Germina KA. A study on the sparing number of corona of certain graphs. Research & Reviews: Discrete Mathematical Structures. 2014;**1**(2):5-15

[50] Naduvath S, Kaithavalappil C, Augustine G. A note on the sparing number of generalised petersen graphs. Journal of Combinatorics, Information & System Sciences. 2017;**42**(1–2):23-31

[51] Sudev NK, Germina KA. A note on the sparing number of graphs. Advances and Applications in Discrete Mathematics. 2014;**14**(1):51-65

**Chapter 7**

*Jung Yoog Kang*

**Abstract**

**1. Introduction**

[1–21]).

**107**

*exponential functions are defined by*

Determination of the Properties of

Nowadays, many mathematicians have great concern about ð Þ *p*, *q* -numbers, which are various applications, and have studied these numbers in many different research areas. We know that ð Þ *p*, *q* -numbers are different to *q*-numbers because of the symmetric property. We find the addition theorem, recurrence formula, and ð Þ *p*, *q* -derivative about sigmoid polynomials including ð Þ *p*, *q* -numbers. Also, we derive the relevant symmetric relations between ð Þ *p*, *q* -sigmoid polynomials and ð Þ *p*, *q* -Euler polynomials. Moreover, we observe the structures of appreciative roots and fixed points about ð Þ *p*, *q* -sigmoid polynomials. By using the fixed points of ð Þ *p*, *q* -sigmoid polynomials and Newton's algorithm, we show self-similarity and

**Keywords:** (*p*,*q*)-sigmoid numbers, (*p*,*q*)-sigmoid polynomials, (*p*,*q*)-Euler

to the quantum group related to mathematics and physics literature.

½ � *<sup>n</sup> <sup>p</sup>*, *<sup>q</sup>* <sup>¼</sup> *pn* � *<sup>q</sup><sup>n</sup>*

For any *n*∈ , the ð Þ *p*, *q* -number is defined by

In 1991, Chakrabarti and Jagannathan [1] introduced the ð Þ *p*, *q* -number in order to unify varied forms of *q*-oscillator algebras in physics literature. Around the same time, Brodimas et al. and Arik et al. independently discovered the ð Þ *p*, *q* -number (see [2, 3]). Contemporarily, Wachs and White [4] introduced the ð Þ *p*, *q* -number in mathematics literature by certain combinatorial problems without any connection

*<sup>p</sup>* � *<sup>q</sup>* , <sup>∣</sup>

Thereby, several physical and mathematical problems lead to the necessity of ð Þ *p*, *q* -calculus. Based on the aforementioned papers, many mathematicians and physicists have developed the ð Þ *p*, *q* -calculus in many different research areas (see

**Definition 1.1.** *Let z be any complex numbers with* ∣*z*∣<1*. The two forms of p*ð Þ , *q -*

*q p*

∣<1*:* (1)

ð Þ *<sup>p</sup>*, *<sup>q</sup>* ‐Sigmoid Polynomials and

the Structure of Their Roots

conjectures about ð Þ *p*, *q* -sigmoid polynomials.

polynomials, roots structure, fixed point

[52] Sudev NK, Germina KA. On the sparing number of certain graph structures. Annals of Pure and Applied Mathematics. 2014;**6**(2):140-149

[53] Sudev NK, Germina KA. Further studies on the sparing number of graphs. TechS Vidya e-Journal of Research. 2014;**2**(2):25-36

[54] Sudev NK, Germina KA. A note on the sparing number on the sieve graphs of certain graphs. Applied Mathematics E-Notes. 2015;**15**(1):29-37

[55] Sudev NK, Germina KA. Some new results on weak integer additive setlabeled graphs. International Journal of Computers and Applications. 2015; **128**(1):1-5

[56] Sudev NK, Chithra KP, Germina KA. Sparing number of the certain graph powers. Acta Universitatis Sapientiae Mathematica. 2019;**11**(1): 186-202

[57] Sudev NK, Chithra KP, Germina KA. Sparing number of the powers of certain Mycielski graphs. Algebra and Discrete Mathematics. 2019;**28**(2):291-307

#### **Chapter 7**

[42] Sudev NK, Germina KA. Some new results on strong integer additive set[51] Sudev NK, Germina KA. A note on the sparing number of graphs. Advances

[52] Sudev NK, Germina KA. On the sparing number of certain graph structures. Annals of Pure and Applied Mathematics. 2014;**6**(2):140-149

[53] Sudev NK, Germina KA. Further studies on the sparing number of graphs. TechS Vidya e-Journal of Research. 2014;**2**(2):25-36

[54] Sudev NK, Germina KA. A note on the sparing number on the sieve graphs of certain graphs. Applied Mathematics

[55] Sudev NK, Germina KA. Some new results on weak integer additive setlabeled graphs. International Journal of Computers and Applications. 2015;

E-Notes. 2015;**15**(1):29-37

[56] Sudev NK, Chithra KP,

[57] Sudev NK, Chithra KP,

2019;**28**(2):291-307

Germina KA. Sparing number of the certain graph powers. Acta Universitatis Sapientiae Mathematica. 2019;**11**(1):

Germina KA. Sparing number of the powers of certain Mycielski graphs. Algebra and Discrete Mathematics.

**128**(1):1-5

186-202

and Applications in Discrete Mathematics. 2014;**14**(1):51-65

[43] Sudev NK, Germina KA. On certain arithmetic integer additive set-indexers of graphs. Discrete Mathematics, Algorithms and Applications. 2015;

[44] Sudev NK, Germina KA. A study on arithmetic integer additive set-indexers of graphs. Journal of Informatics and Mathematical Sciences. 2018;**10**(1–2):

[45] Naduvath S, Augusthy GK, Kok J. Sumset valuations of graphs and their applications. In: Handbook of Research on Advanced Applications of Graph Theory in Modern Society. Hershey, Pennsylvania: IGI Global; 2020.

[46] Naduvath S, Germina KA. An Introduction to Sumset Valued Graphs. Mauritius: Lambert Academic Publ;

[48] Chithra KP, Sudev NK, Germina KA. Sparing number of Cartesian products of certain graphs. Communications in Mathematics and

Applications. 2014;**5**(1):23-30

[49] Chithra KP, Sudev NK,

Germina KA. A study on the sparing number of corona of certain graphs. Research & Reviews: Discrete

[50] Naduvath S, Kaithavalappil C, Augustine G. A note on the sparing number of generalised petersen graphs. Journal of Combinatorics, Information & System Sciences. 2017;**42**(1–2):23-31

Mathematical Structures. 2014;**1**(2):5-15

[47] Naduvath S. A study on the modular sumset labeling of graphs. Discrete Mathematics, Algorithms and Applications. 2017;**9**(03):1750039

indexers of graphs. Discrete Mathematics, Algorithms and Applications. 2015;**7**(01):1450065

*Number Theory and Its Applications*

**7**(03):1550025

321-332

pp. 208-250

2018

**106**

## Determination of the Properties of ð Þ *<sup>p</sup>*, *<sup>q</sup>* ‐Sigmoid Polynomials and the Structure of Their Roots

*Jung Yoog Kang*

#### **Abstract**

Nowadays, many mathematicians have great concern about ð Þ *p*, *q* -numbers, which are various applications, and have studied these numbers in many different research areas. We know that ð Þ *p*, *q* -numbers are different to *q*-numbers because of the symmetric property. We find the addition theorem, recurrence formula, and ð Þ *p*, *q* -derivative about sigmoid polynomials including ð Þ *p*, *q* -numbers. Also, we derive the relevant symmetric relations between ð Þ *p*, *q* -sigmoid polynomials and ð Þ *p*, *q* -Euler polynomials. Moreover, we observe the structures of appreciative roots and fixed points about ð Þ *p*, *q* -sigmoid polynomials. By using the fixed points of ð Þ *p*, *q* -sigmoid polynomials and Newton's algorithm, we show self-similarity and conjectures about ð Þ *p*, *q* -sigmoid polynomials.

**Keywords:** (*p*,*q*)-sigmoid numbers, (*p*,*q*)-sigmoid polynomials, (*p*,*q*)-Euler polynomials, roots structure, fixed point

#### **1. Introduction**

In 1991, Chakrabarti and Jagannathan [1] introduced the ð Þ *p*, *q* -number in order to unify varied forms of *q*-oscillator algebras in physics literature. Around the same time, Brodimas et al. and Arik et al. independently discovered the ð Þ *p*, *q* -number (see [2, 3]). Contemporarily, Wachs and White [4] introduced the ð Þ *p*, *q* -number in mathematics literature by certain combinatorial problems without any connection to the quantum group related to mathematics and physics literature.

For any *n*∈ , the ð Þ *p*, *q* -number is defined by

$$[n]\_{p,q} = \frac{p^n - q^n}{p - q}, \qquad |\frac{q}{p}| < 1. \tag{1}$$

Thereby, several physical and mathematical problems lead to the necessity of ð Þ *p*, *q* -calculus. Based on the aforementioned papers, many mathematicians and physicists have developed the ð Þ *p*, *q* -calculus in many different research areas (see [1–21]).

**Definition 1.1.** *Let z be any complex numbers with* ∣*z*∣<1*. The two forms of p*ð Þ , *q exponential functions are defined by*

$$\begin{aligned} \mathcal{e}\_{p,q}(z) &= \sum\_{n=0}^{\infty} p \binom{n}{2} \frac{z^n}{[n]\_{p,q}!}, \\ \mathcal{e}\_{p^{-1},q^{-1}}(z) &= \sum\_{n=0}^{\infty} q \binom{n}{2} \frac{z^n}{[n]\_{p,q}!}. \end{aligned} \tag{2}$$

In 2016, Araci et al. [6] introduced a new class of Bernoulli, Euler and Genocchi

<sup>¼</sup> <sup>2</sup> *ep*, *<sup>q</sup>*ðÞþ*t* 1

<sup>¼</sup> <sup>1</sup> *e*�*<sup>t</sup>* þ 1 *e*

One of the most widely used methods of solving equations is Newton's method. This method is also based on a linear approximation of the function, but does so using a tangent to the curve. Starting from an initial estimate that is not too far from a root *x*, we extrapolate along the tangent to its intersection with the *x*-axis, and take that as the next approximation. This is continued until either the successive *x*values are sufficiently close, or the value of the function is sufficiently near zero. The calculation scheme follows immediately from the right triangle, which has the angle of inclination of the tangent line to the curve at *x* ¼ *x*<sup>1</sup> as one of its acute

> *f x*ð Þ<sup>1</sup> *x*<sup>1</sup> � *x*<sup>2</sup>

*<sup>x</sup>*<sup>3</sup> <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> � *f x*ð Þ<sup>2</sup> *f* 0 ð Þ *x*<sup>2</sup> ,

Newton's algorithm is widely used because, at least in the near neighborhood of a root, it is more rapidly convergent than any of the methods so far discussed. The method is quadratically convergent, by which we mean that the error of each step approaches a constant *K* times the square of the error of the previous step. The net result of this is that the number of decimal places of accuracy nearly doubles at each iteration. However, offsetting this is the need for two function evaluations at each

, *<sup>x</sup>*<sup>2</sup> <sup>¼</sup> *<sup>x</sup>*<sup>1</sup> � *f x*ð Þ<sup>1</sup>

, *n* ¼ 1, 2, 3, ⋯*:*

ð Þ *xn* . We now use the result to show a criterion for convergence of

*f* 0 ð Þ *x*<sup>1</sup> *:*

Several studies have investigated the sigmoid function for various applications (see [11, 12, 15, 16]). For example, a variant sigmoid function with three parameters has been employed to explain hybrid sigmoidal networks [10] and sigmoid function has been defined using flexible sigmoidal mixed models based on logistic family

*ep*, *<sup>q</sup>*ð Þ *tx :* (9)

*tx:* (10)

polynomials based on the theory of ð Þ *p*, *q* -number and found some properties including difference equations, addition theorem, recurrence relations were derived. We observe some special properties and roots structures of Bernoulli, Euler, and tangent polynomials (see [4, 11, 16–20]). In particular, roots structures and fixed points of tangent polynomials including *q*-numbers are shown in a dif-

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots*

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

**Definition 1.5.** ð Þ *p*, *q -Euler polynomials are defined by*

*En*,*p*, *<sup>q</sup>*ð Þ *x*

**Definition 1.6.** *We define the sigmoid polynomials as follows:*

S*n*ð Þ *x t n n*!

X∞ *n*¼0

X∞ *n*¼0

*DOI: http://dx.doi.org/10.5772/intechopen.91862*

curves for medical applications [11, 12, 15].

tan *θ* ¼ *f*

or, in more general terms,

0

step, *f x*ð Þ*<sup>n</sup>* and *f*

**109**

0 ð Þ¼ *x*<sup>1</sup>

We continue the calculation scheme by computing

*xn*þ<sup>1</sup> <sup>¼</sup> *xn* � *f x*ð Þ*<sup>n</sup>*

*f* 0 ð Þ *xn*

ferent shape by [17].

angles:

The useful relation of two forms of ð Þ *p*, *q* -exponential functions is taken by

$$
\sigma\_{p,q}(z)\sigma\_{p^{-1},q^{-1}}(-z) = \mathbf{1}.\tag{3}
$$

In [9], Corcino created the theorem of ð Þ *p*, *q* -extension of binomials coefficients and found various properties which are related to horizontal function, triangular function, and vertical function.

**Definition 1.2.** *Let n* ≥*k. p*ð Þ , *q -Gauss Binomial coefficients are defined by*

$$
\begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} = \frac{[n]\_{p,q}!}{[n-k]\_{p,q}! [k]\_{p,q}!},\tag{4}
$$

where ½ � *n <sup>p</sup>*, *<sup>q</sup>*! ¼ ½ � *n <sup>p</sup>*, *<sup>q</sup>*½ � *n* � 1 *<sup>p</sup>*, *<sup>q</sup>*⋯½ � 1 *<sup>p</sup>*, *<sup>q</sup>*.

In 2013, Sadjang [21] derived some properties of the ð Þ *p*, *q* -derivative, ð Þ *p*, *q* integration and investigated two ð Þ *p*, *q* -Taylor formulas for polynomials.

**Definition 1.3.** *We define the p*ð Þ , *q -derivative operator of any function f, also referred to as the Jackson derivative, as follows:*

$$D\_{p,q}f(\mathbf{x}) = \frac{f(p\mathbf{x}) - f(q\mathbf{x})}{(p-q)\mathbf{x}}, \qquad \mathbf{x} \neq \mathbf{0},\tag{5}$$

and *Dp*, *qf*ð Þ¼ 0 *f* 0 ð Þ 0 .

If *t x*ð Þ¼ <sup>P</sup>*<sup>n</sup> <sup>k</sup>*¼<sup>0</sup>*akx<sup>k</sup>* then *Dp*, *qt x*ð Þ¼ <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>k</sup>*¼<sup>0</sup>*ak*þ<sup>1</sup>½ � *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>p</sup>*, *<sup>q</sup>xk*, since *Dp*, *qz<sup>n</sup>* <sup>¼</sup> ½ � *<sup>n</sup> <sup>p</sup>*, *<sup>q</sup>zn*�1. This equation is equivalent to the ð Þ *<sup>p</sup>*, *<sup>q</sup>* -difference equation in *<sup>q</sup>* with known *f*, *Dp*, *qg x*ð Þ¼ *f x*ð Þ*:*.

**Theorem 1.4.** *This operator, Dp*, *q, has the following basic properties:*

$$\begin{aligned} (i) \text{Derivative of a product} \qquad D\_{p,q}(f(\mathbf{x})\mathbf{g}(\mathbf{x})) &= f(p\mathbf{x})D\_{p,q}\mathbf{g}(\mathbf{x}) + \mathbf{g}(q\mathbf{x})D\_{p,q}f(\mathbf{x})\\ &= \mathbf{g}(p\mathbf{x})D\_{p,q}f(\mathbf{x}) + f(q\mathbf{x})D\_{p,q}\mathbf{g}(\mathbf{x}). \end{aligned} \tag{6}$$

$$\begin{split} \left( \left( \mathrm{ii} \right) \mathrm{Derivative} \right) \mathrm{a \ ratio} \qquad & D\_{p,q} \left( \frac{f(\mathbf{x})}{\mathbf{g}(\mathbf{x})} \right) = \frac{\mathrm{g}(q\mathbf{x}) D\_{p,q} f(\mathbf{x}) - f(q\mathbf{x}) D\_{p,q} \mathbf{g}(\mathbf{x})}{\mathrm{g}(p\mathbf{x}) \mathrm{g}(q\mathbf{x})} \\ &= \frac{\mathrm{g}(p\mathbf{x}) D\_{p,q} f(\mathbf{x}) - f(p\mathbf{x}) D\_{p,q} \mathbf{g}(\mathbf{x})}{\mathrm{g}(p\mathbf{x}) \mathrm{g}(q\mathbf{x})} . \end{split} \tag{7}$$

Let *f* be an arbitrary function. In [7], we note that the definition of ð Þ *p*, *q* integral is

$$\int f(\mathbf{x})d\_{p,q}\mathbf{x} = (p-q)\mathbf{x}\sum\_{k=0}^{\infty} \frac{q^k}{p^{k+1}} f\left(\frac{q^k}{p^{k+1}}\mathbf{x}\right). \tag{8}$$

**108**

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots DOI: http://dx.doi.org/10.5772/intechopen.91862*

In 2016, Araci et al. [6] introduced a new class of Bernoulli, Euler and Genocchi polynomials based on the theory of ð Þ *p*, *q* -number and found some properties including difference equations, addition theorem, recurrence relations were derived. We observe some special properties and roots structures of Bernoulli, Euler, and tangent polynomials (see [4, 11, 16–20]). In particular, roots structures and fixed points of tangent polynomials including *q*-numbers are shown in a different shape by [17].

**Definition 1.5.** ð Þ *p*, *q -Euler polynomials are defined by*

$$\sum\_{n=0}^{\infty} E\_{n,p,q}(\boldsymbol{\omega}) \frac{t^n}{[n]\_{p,q}!} = \frac{2}{\mathfrak{e}\_{p,q}(t) + \mathbf{1}} \mathfrak{e}\_{p,q}(t\boldsymbol{\omega}).\tag{9}$$

Several studies have investigated the sigmoid function for various applications (see [11, 12, 15, 16]). For example, a variant sigmoid function with three parameters has been employed to explain hybrid sigmoidal networks [10] and sigmoid function has been defined using flexible sigmoidal mixed models based on logistic family curves for medical applications [11, 12, 15].

**Definition 1.6.** *We define the sigmoid polynomials as follows:*

$$\sum\_{n=0}^{\infty} \mathcal{S}\_n(\mathbf{x}) \frac{t^n}{n!} = \frac{1}{e^{-t} + 1} e^{t\mathbf{x}}.\tag{10}$$

One of the most widely used methods of solving equations is Newton's method. This method is also based on a linear approximation of the function, but does so using a tangent to the curve. Starting from an initial estimate that is not too far from a root *x*, we extrapolate along the tangent to its intersection with the *x*-axis, and take that as the next approximation. This is continued until either the successive *x*values are sufficiently close, or the value of the function is sufficiently near zero.

The calculation scheme follows immediately from the right triangle, which has the angle of inclination of the tangent line to the curve at *x* ¼ *x*<sup>1</sup> as one of its acute angles:

$$\tan \theta = f'(\mathbf{x}\_1) = \frac{f(\mathbf{x}\_1)}{\mathbf{x}\_1 - \mathbf{x}\_2}, \qquad \mathbf{x}\_2 = \mathbf{x}\_1 - \frac{f(\mathbf{x}\_1)}{f'(\mathbf{x}\_1)}.$$

We continue the calculation scheme by computing

$$
\mathfrak{x}\_3 = \mathfrak{x}\_2 - \frac{f(\mathfrak{x}\_2)}{f'(\mathfrak{x}\_2)},
$$

or, in more general terms,

$$\mathbf{x}\_{n+1} = \mathbf{x}\_n - \frac{f(\mathbf{x}\_n)}{f'(\mathbf{x}\_n)}, \qquad n = 1, 2, 3, \dots, 4$$

Newton's algorithm is widely used because, at least in the near neighborhood of a root, it is more rapidly convergent than any of the methods so far discussed. The method is quadratically convergent, by which we mean that the error of each step approaches a constant *K* times the square of the error of the previous step. The net result of this is that the number of decimal places of accuracy nearly doubles at each iteration. However, offsetting this is the need for two function evaluations at each step, *f x*ð Þ*<sup>n</sup>* and *f* 0 ð Þ *xn* . We now use the result to show a criterion for convergence of

*ep*, *<sup>q</sup>*ð Þ¼ *<sup>z</sup>* <sup>X</sup><sup>∞</sup>

*ep*�1,*q*�<sup>1</sup> ð Þ¼ *<sup>z</sup>* <sup>X</sup><sup>∞</sup>

function, and vertical function.

*Number Theory and Its Applications*

and *Dp*, *qf*ð Þ¼ 0 *f*

known *f*, *Dp*, *qg x*ð Þ¼ *f x*ð Þ*:*.

If *t x*ð Þ¼ <sup>P</sup>*<sup>n</sup>*

integral is

**108**

where ½ � *n <sup>p</sup>*, *<sup>q</sup>*! ¼ ½ � *n <sup>p</sup>*, *<sup>q</sup>*½ � *n* � 1 *<sup>p</sup>*, *<sup>q</sup>*⋯½ � 1 *<sup>p</sup>*, *<sup>q</sup>*.

*referred to as the Jackson derivative, as follows:*

0 ð Þ 0 .

ð Þ *ii Derivative of a ratio Dp*, *<sup>q</sup>*

ð

*n*¼0 *p*

> *n*¼0 *q*

The useful relation of two forms of ð Þ *p*, *q* -exponential functions is taken by

**Definition 1.2.** *Let n* ≥*k. p*ð Þ , *q -Gauss Binomial coefficients are defined by*

*n k* � �

*p*, *q*

integration and investigated two ð Þ *p*, *q* -Taylor formulas for polynomials.

*Dp*, *qf x*ð Þ¼ *f px* ð Þ� *f qx* ð Þ

*<sup>k</sup>*¼<sup>0</sup>*akx<sup>k</sup>* then *Dp*, *qt x*ð Þ¼ <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

In [9], Corcino created the theorem of ð Þ *p*, *q* -extension of binomials coefficients and found various properties which are related to horizontal function, triangular

> <sup>¼</sup> ½ � *<sup>n</sup> <sup>p</sup>*, *<sup>q</sup>*! ½ � *n* � *k <sup>p</sup>*, *<sup>q</sup>*!½ � *k <sup>p</sup>*, *<sup>q</sup>*!

In 2013, Sadjang [21] derived some properties of the ð Þ *p*, *q* -derivative, ð Þ *p*, *q* -

**Definition 1.3.** *We define the p*ð Þ , *q -derivative operator of any function f, also*

½ � *<sup>n</sup> <sup>p</sup>*, *<sup>q</sup>zn*�1. This equation is equivalent to the ð Þ *<sup>p</sup>*, *<sup>q</sup>* -difference equation in *<sup>q</sup>* with

ð Þ*i Derivative of a product Dp*, *<sup>q</sup>*ð Þ¼ *f x*ð Þ*g x*ð Þ *f px* ð Þ*Dp*, *qg x*ð Þþ *g qx* ð Þ*Dp*, *qf x*ð Þ

Let *f* be an arbitrary function. In [7], we note that the definition of ð Þ *p*, *q* -

*f x*ð Þ*dp*, *qx* ¼ ð Þ *p* � *q x*

*f x*ð Þ *g x*ð Þ � �

> X∞ *k*¼0

*qk pk*þ<sup>1</sup> *<sup>f</sup> <sup>q</sup><sup>k</sup>*

**Theorem 1.4.** *This operator, Dp*, *q, has the following basic properties:*

*n* 2

*zn* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! ,

> *zn* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! *:*

*ep*, *<sup>q</sup>*ð Þ*z ep*�1,*q*�<sup>1</sup> ð Þ¼ �*z* 1*:* (3)

ð Þ *<sup>p</sup>* � *<sup>q</sup> <sup>x</sup>* , *<sup>x</sup>* 6¼ 0, (5)

*<sup>k</sup>*¼<sup>0</sup>*ak*þ<sup>1</sup>½ � *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>p</sup>*, *<sup>q</sup>xk*, since *Dp*, *qz<sup>n</sup>* <sup>¼</sup>

¼ *g px* ð Þ*Dp*, *qf x*ð Þþ *f qx* ð Þ*Dp*, *qg x*ð Þ*:*

<sup>¼</sup> *g px* ð Þ*Dp*, *qf x*ð Þ� *f px* ð Þ*Dp*, *qg x*ð Þ

*pk*þ<sup>1</sup> *<sup>x</sup>* � �

<sup>¼</sup> *g qx* ð Þ*Dp*, *qf x*ð Þ� *f qx* ð Þ*Dp*, *qg x*ð Þ *g px* ð Þ*g qx* ð Þ

*g px* ð Þ*g qx* ð Þ *:*

*:* (8)

(6)

(7)

, (4)

(2)

*n* 2

!

!

Newton's method. Consider the form *xn*þ<sup>1</sup> ¼ *g x*ð Þ*<sup>n</sup>* . Successive iterations converge if ∣*g*0 ð Þ *x* ∣<1. Since

Newton's method to obtain a iterative function of ð Þ *p*, *q* -sigmoid polynomials to

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots*

This section introduces about ð Þ *p*, *q* -sigmoid numbers and polynomials. From the generating function of these polynomials, we can observe some of the basic properties and identities of this polynomials. In particular, we can show the forms of ð Þ *p*, *q* -derivative, symmetric properties, and relations of ð Þ *p*, *q* -Euler polynomials

> <sup>¼</sup> <sup>1</sup> *ep*, *<sup>q</sup>*ð Þþ �*t* 1

S*<sup>n</sup>*,*p*, *<sup>q</sup>*

so we can be called S*<sup>n</sup>*,*p*, *<sup>q</sup>* is ð Þ *p*, *q* -sigmoid numbers. We note that *ep*, *<sup>q</sup>*ð Þ¼ 0 *p*

because of the property for ð Þ *p*, *q* -exponential function. If *p* ¼ 1 in the Definition 2.1,

S*<sup>n</sup>*,*<sup>q</sup>*ð Þ *x*

where S*<sup>n</sup>*,*<sup>q</sup>*ð Þ *x* is *q*-sigmoid polynomials. Moreover, if *p* ¼ 1, *q* ! 1 in the gener-

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

*t n* ½ � *n <sup>q</sup>*!

S*n*ð Þ *x t n n*!

*n* � *k* 2

!

<sup>¼</sup> <sup>1</sup> *eq*ð Þþ �*t* 1

> <sup>¼</sup> <sup>1</sup> *e*�*<sup>t</sup>* þ 1 *e*

S*<sup>k</sup>*,*p*, *<sup>q</sup>*ð ÞþS *x <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x* ,

0 *if n* 6¼ 0*:*

<sup>S</sup>*<sup>k</sup>*,*p*, *<sup>q</sup>* þ S*<sup>n</sup>*,*p*, *<sup>q</sup>* <sup>¼</sup> <sup>1</sup> *if n* <sup>¼</sup> 0,

(

½ � *<sup>n</sup> <sup>p</sup>*, *<sup>q</sup>*! *ep*, *<sup>q</sup>*ð Þþ �*<sup>t</sup>* <sup>1</sup> � � <sup>¼</sup> *ep*, *<sup>q</sup>*ð Þ *tx :* (16)

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! <sup>¼</sup> <sup>1</sup> *ep*, *<sup>q</sup>*ð Þþ �*t* 1

*ep*, *<sup>q</sup>*ð Þ *tx :* (11)

, (12)

*eq*ð Þ *tx* , (13)

*tx*, (14)

(15)

*n* 2 � �

**2. Some properties and identities of** *p***,** *q* � �**-sigmoid polynomials**

**Definition 2.1.** *We define p*ð Þ , *q -sigmoid polynomials as following:*

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

> <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

identify the domain leading to the fixed points.

*DOI: http://dx.doi.org/10.5772/intechopen.91862*

X∞ *n*¼0

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

*n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

In the Definition 2.1, if *x* ¼ 0 we can see that

<sup>S</sup>*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ <sup>0</sup> *<sup>t</sup>*

for ð Þ *p*, *q* -sigmoid polynomials.

X∞ *n*¼0

X∞ *n*¼0

S*<sup>n</sup>*,1,*<sup>q</sup>*ð Þ *x*

X∞ *n*¼0

lim *q*!1

*xn* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> k*¼0

*p*, *q*

X∞ *n*¼0

*t n* ½ � *n* 1,*<sup>q</sup>*!

ation function of ð Þ *p*, *q* -sigmoid polynomials, we have

where S*n*ð Þ *x* is sigmoid polynomials (see [16]). **Theorem 2.2.** *Let be* ∣*q=p*∣< 1*. Then we get*

> *n k*

ð Þ �<sup>1</sup> *<sup>n</sup>*�*<sup>k</sup> p*

*p*, *q*

*Proof. i*ð Þ Consider that *ep*, *<sup>q</sup>*ð Þ �*t* 6¼ �1. Then we can see

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

" #

S*<sup>n</sup>*,1,*<sup>q</sup>*ð Þ *x*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

> *t n* ½ � *n* 1,*<sup>q</sup>*!

ð Þ �<sup>1</sup> *<sup>n</sup>*�*<sup>k</sup> p*

*n* � *k* 2

> *t n*

!

then one holds

ð Þ*i p*

ð Þ *ii* <sup>X</sup>*<sup>n</sup>*

**111**

*k*¼0

*n k*

" #

*n* 2

!

$$\begin{aligned} \mathbf{g}(\mathbf{x}) &= \mathbf{x} - \frac{f(\mathbf{x})}{f'(\mathbf{x})},\\ \mathbf{g'}(\mathbf{x}) &= \mathbf{1} - \frac{f'(\mathbf{x})f'(\mathbf{x}) - f(\mathbf{x})f'(\mathbf{x})}{\left(f'(\mathbf{x})\right)^2} = \frac{f(\mathbf{x})f'(\mathbf{x})}{\left(f'(\mathbf{x})\right)^2}. \end{aligned}$$

Hence if

$$\left| \frac{f(\mathbf{x})f'(\mathbf{x})}{\left(f'(\mathbf{x})\right)^2} \right| < 1$$

on an interval about the root *r*, the method will converge for any initial value *x*<sup>1</sup> in the interval. The condition is sufficient only, and requires the unusual continuity and existence of *f x*ð Þ and its derivatives. Note that *f* 0 ð Þ *x* must not zero. In addition, Newton's method is quadratically convergent and we can apply this method to polynomials.

Let *f* : *D* ! *D* be a complex function, with *D* as a subset of . We define the iterated maps of the complex function as the following:

$$f\_r: z\_0 \to \underbrace{f(f(\cdots(f(z\_0)\cdots)))}\_{r}$$

The iterates of *<sup>f</sup>* are the functions *<sup>f</sup>*, *<sup>f</sup>* ◦ *<sup>f</sup>*, *<sup>f</sup>* ◦ *<sup>f</sup>* ◦ *<sup>f</sup>*, … , which are denoted *f* 1 , *f* 2 , *f* 3 , *:* … If *z*∈ , and then the orbit of *z*<sup>0</sup> under *f* is the sequence < *z*0, *f z*ð Þ<sup>0</sup> , *f fz* ð Þ ð Þ<sup>0</sup> , ⋯ >*:*

**Definition 1.7.** *The orbit of the point z*<sup>0</sup> ∈ *under the action of the function f is said to be bounded if there exists M* ∈ *such that* ∣*f n* ð Þ *z*<sup>0</sup> ∣< *M for all n*∈ *. If the orbit is not bounded, it is said to be unbounded.*

**Definition 1.8.** *Let f* : *D* ! *be a transformation on a metric space. A point z*<sup>0</sup> ∈ *D such that f z*ð Þ¼ <sup>0</sup> *z*<sup>0</sup> *is called a fixed point of the transformation.*

*We know that the fixed point is divided as follows. Suppose that the complex function f is analytic in a region D of , and f has a fixed point at z*<sup>0</sup> ∈ *D. Then z*<sup>0</sup> *is said to be: 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*.*

If *z*<sup>0</sup> is an attracting fixed point of *f*, then there exists a neighborhood of *A* such that if *b*∈ *A* the orbit *b* converges to *z*0. Attractive fixed points of a function have a basin of attraction, which may be disconnected. The component which contains the fixed point is called the immediate basin of attraction. If *z*<sup>0</sup> is a repelling periodic point of *f*, then there is a neighborhood of *N* such that if *b*∈ *N*, there are points in the orbit of *b* which are not in *N*. In the case of polynomials of degree greater than 0 and some rational functions, ∞ is also called an attracting fixed point, as, for each such function, *f*, there exist *R*> 0 such that if ∣*z*∣>*R* then *f n* ð Þ!*z* ∞ as *n* ! ∞.

Based on the above, the contents of the paper are as follows. Section 2 checks the properties of ð Þ *p*, *q* -sigmoid polynomials. For example, we look for addition theorem, recurrence relation, differential, etc. and find the properties associated with the symmetric property and ð Þ *p*, *q* -Euler polynomials. Section 3 identifies the structure and accumulation of roots of ð Þ *p*, *q* -sigmoid polynomials based on the contents of Section 2 and checks the contents related to the fixed points. Also, we use

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots DOI: http://dx.doi.org/10.5772/intechopen.91862*

Newton's method to obtain a iterative function of ð Þ *p*, *q* -sigmoid polynomials to identify the domain leading to the fixed points.

### **2. Some properties and identities of** *p***,** *q* � �**-sigmoid polynomials**

This section introduces about ð Þ *p*, *q* -sigmoid numbers and polynomials. From the generating function of these polynomials, we can observe some of the basic properties and identities of this polynomials. In particular, we can show the forms of ð Þ *p*, *q* -derivative, symmetric properties, and relations of ð Þ *p*, *q* -Euler polynomials for ð Þ *p*, *q* -sigmoid polynomials.

**Definition 2.1.** *We define p*ð Þ , *q -sigmoid polynomials as following:*

$$\sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q}(\boldsymbol{\omega}) \frac{t^n}{[n]\_{p,q}!} = \frac{1}{e\_{p,q}(-t) + 1} e\_{p,q}(t\boldsymbol{\omega}).\tag{11}$$

In the Definition 2.1, if *x* ¼ 0 we can see that

$$\sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q}(\mathbf{0}) \frac{t^n}{[n]\_{p,q}!} = \sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} = \frac{1}{\varepsilon\_{p,q}(-t) + \mathbf{1}},\tag{12}$$

� �

so we can be called S*<sup>n</sup>*,*p*, *<sup>q</sup>* is ð Þ *p*, *q* -sigmoid numbers. We note that *ep*, *<sup>q</sup>*ð Þ¼ 0 *p n* 2 because of the property for ð Þ *p*, *q* -exponential function. If *p* ¼ 1 in the Definition 2.1, then one holds

$$\sum\_{n=0}^{\infty} \mathcal{S}\_{n,1,q}(\boldsymbol{\kappa}) \frac{t^n}{[n]\_{1,q}!} = \sum\_{n=0}^{\infty} \mathcal{S}\_{n,q}(\boldsymbol{\kappa}) \frac{t^n}{[n]\_q!} = \frac{1}{e\_q(-t) + 1} e\_q(t\boldsymbol{\kappa}), \tag{13}$$

where S*<sup>n</sup>*,*<sup>q</sup>*ð Þ *x* is *q*-sigmoid polynomials. Moreover, if *p* ¼ 1, *q* ! 1 in the generation function of ð Þ *p*, *q* -sigmoid polynomials, we have

$$\lim\_{q \to 1} \sum\_{n=0}^{\infty} \mathcal{S}\_{n, 1, q}(\boldsymbol{\kappa}) \frac{t^n}{[n]\_{1, q}!} = \sum\_{n=0}^{\infty} \mathcal{S}\_n(\boldsymbol{\kappa}) \frac{t^n}{n!} = \frac{1}{e^{-t} + 1} e^{t\boldsymbol{\kappa}},\tag{14}$$

where S*n*ð Þ *x* is sigmoid polynomials (see [16]). **Theorem 2.2.** *Let be* ∣*q=p*∣< 1*. Then we get*

$$
\begin{aligned}
\mathbf{p}(i) \quad &p \begin{pmatrix} n \\ 2 \end{pmatrix}\_{\mathbf{x}^n} = \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (-1)^{n-k} p \binom{n-k}{2} \mathcal{S}\_{k,p,q}(\mathbf{x}) + \mathcal{S}\_{n,p,q}(\mathbf{x}), \\ &p \begin{pmatrix} n-k \\ n \end{pmatrix}\_{p,q} \end{aligned} \tag{15}
$$

$$
\mathcal{S}\_n(\vec{u}) \quad \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (-1)^{n-k} p^{\binom{n-k}{2}} \mathcal{S}\_{k,p,q} + \mathcal{S}\_{n,p,q} = \begin{cases} 1 & \text{if } \quad n=0, \\ 0 & \text{if } \quad n \neq 0. \end{cases}
$$

*Proof. i*ð Þ Consider that *ep*, *<sup>q</sup>*ð Þ �*t* 6¼ �1. Then we can see

$$\sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q}(\infty) \frac{t^n}{[n]\_{p,q}!} \left(\mathfrak{e}\_{p,q}(-t) + \mathbf{1}\right) = \mathfrak{e}\_{p,q}(t\infty). \tag{16}$$

Newton's method. Consider the form *xn*þ<sup>1</sup> ¼ *g x*ð Þ*<sup>n</sup>* . Successive iterations converge if

ð Þ� *x f x*ð Þ *f*

*f* 0

*f x*ð Þ *f* 00 ð Þ *x*

� � � � �

*f* 0 ð Þ *<sup>x</sup>* � �<sup>2</sup>

Newton's method is quadratically convergent and we can apply this method to

*fr* : *z*<sup>0</sup> ! *f*ð *f*ð⋯ð *f*

The iterates of *<sup>f</sup>* are the functions *<sup>f</sup>*, *<sup>f</sup>* ◦ *<sup>f</sup>*, *<sup>f</sup>* ◦ *<sup>f</sup>* ◦ *<sup>f</sup>*, … , which are denoted

, *:* … If *z*∈ , and then the orbit of *z*<sup>0</sup> under *f* is the sequence

Let *f* : *D* ! *D* be a complex function, with *D* as a subset of . We define the


**Definition 1.7.** *The orbit of the point z*<sup>0</sup> ∈ *under the action of the function f is said*

**Definition 1.8.** *Let f* : *D* ! *be a transformation on a metric space. A point z*<sup>0</sup> ∈ *D*

*We know that the fixed point is divided as follows. Suppose that the complex function f is analytic in a region D of , and f has a fixed point at z*<sup>0</sup> ∈ *D. Then z*<sup>0</sup> *is said to be:*

If *z*<sup>0</sup> is an attracting fixed point of *f*, then there exists a neighborhood of *A* such that if *b*∈ *A* the orbit *b* converges to *z*0. Attractive fixed points of a function have a basin of attraction, which may be disconnected. The component which contains the fixed point is called the immediate basin of attraction. If *z*<sup>0</sup> is a repelling periodic point of *f*, then there is a neighborhood of *N* such that if *b*∈ *N*, there are points in the orbit of *b* which are not in *N*. In the case of polynomials of degree greater than 0 and some rational functions, ∞ is also called an attracting fixed point, as, for each

Based on the above, the contents of the paper are as follows. Section 2 checks the properties of ð Þ *p*, *q* -sigmoid polynomials. For example, we look for addition theorem, recurrence relation, differential, etc. and find the properties associated with the symmetric property and ð Þ *p*, *q* -Euler polynomials. Section 3 identifies the structure and accumulation of roots of ð Þ *p*, *q* -sigmoid polynomials based on the contents of Section 2 and checks the contents related to the fixed points. Also, we use

*n*

00 ð Þ *x* 00 ð Þ *x*

ð Þ *x* must not zero. In addition,

ð Þ *z*<sup>0</sup> ∣< *M for all n*∈ *. If the orbit is not*

*n*

ð Þ!*z* ∞ as *n* ! ∞.

*f* 0 ð Þ *<sup>x</sup>* � �<sup>2</sup> *:*

ð Þ *<sup>x</sup>* � �<sup>2</sup> <sup>¼</sup> *f x*ð Þ *<sup>f</sup>*

0

ð Þ *z*<sup>0</sup> ⋯ÞÞ

� � � � � <1

on an interval about the root *r*, the method will converge for any initial value *x*<sup>1</sup> in the interval. The condition is sufficient only, and requires the unusual continuity

∣*g*0

ð Þ *x* ∣<1. Since

Hence if

polynomials.

< *z*0, *f z*ð Þ<sup>0</sup> , *f fz* ð Þ ð Þ<sup>0</sup> , ⋯ >*:*

*bounded, it is said to be unbounded.*

*an attracting fixed point if* ∣*f*

*a repelling fixed point if* ∣*f*

*a neutral fixed point if* ∣*f*

*to be bounded if there exists M* ∈ *such that* ∣*f*

*f* 1 , *f* 2 , *f* 3

**110**

*g x*ð Þ¼ *<sup>x</sup>* � *f x*ð Þ *f* 0 ð Þ *x* ,

ð Þ¼ *<sup>x</sup>* <sup>1</sup> � *<sup>f</sup>*

and existence of *f x*ð Þ and its derivatives. Note that *f*

iterated maps of the complex function as the following:

*such that f z*ð Þ¼ <sup>0</sup> *z*<sup>0</sup> *is called a fixed point of the transformation.*

0

0

such function, *f*, there exist *R*> 0 such that if ∣*z*∣>*R* then *f*

0

ð Þ *z*<sup>0</sup> ∣<1*;*

ð Þ *z*<sup>0</sup> ∣>1*;*

ð Þ *z*<sup>0</sup> ∣ ¼ 1*.*

0 ð Þ *x f* 0

*g*0

*Number Theory and Its Applications*

Using power series of ð Þ *p*, *q* -exponential function in the equation above (16) and Cauchy product, we can compare both-sides as following:

$$\sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (-1)^{n-k} p \begin{pmatrix} n-k \\ 2 \end{pmatrix} \mathcal{S}\_{k,p,q}(\mathbf{x}) + \mathcal{S}\_{n,p,q}(\mathbf{x}) \right) \frac{t^n}{[n]\_{p,q}!} $$
 
$$= \sum\_{n=0}^{\infty} p \binom{n}{2} \mathcal{X}^n \frac{t^n}{[n]\_{p,q}!}, \tag{17}$$

X∞ *n*¼0

derive

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

*DOI: http://dx.doi.org/10.5772/intechopen.91862*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

X∞ *n*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

ðÞ S *<sup>i</sup> <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ¼ *<sup>x</sup>* <sup>X</sup>*<sup>n</sup>*

ðÞ S *ii <sup>n</sup>*,*p*, *<sup>q</sup>* ¼ �ð Þ<sup>1</sup> *<sup>n</sup>*

nomials, we can investigate

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

<sup>S</sup>*<sup>n</sup>*,*p*�1,*q*�<sup>1</sup> ð Þ �<sup>1</sup> ð Þ �*<sup>t</sup> <sup>n</sup>*

*p*, *q*

*n k*

" #

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

1 þ *ep*�1,*q*�<sup>1</sup> ð Þ*t*

X*n k*¼0

0

BBB@

<sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

**113**

X∞ *n*¼0

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

0

BBB@

X*n k*¼0

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*, *y*

*n*

" #

*k*

*p*, *q p*

X*n k*¼0

0

BBB@

*k*¼0

*p*

*n k*

*p*, *q*

*q*

*n* 2

*ep*�1*q*�<sup>1</sup> ð Þ*t ep*, *<sup>q</sup>*ð Þ *tx*

½ � *n <sup>p</sup>*�1,*q*�<sup>1</sup> !

ð Þ �<sup>1</sup> *<sup>k</sup> p* X∞ *n*¼0 *p*

*n* � *k* 2

!

!

ð Þ �<sup>1</sup> *<sup>k</sup> p*

> *n* 2

!

" #

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

*n*

" #

*k*

*p*, *q p*

Therefore, we complete the proof of the Theorem 2.4 ð Þ*i* at once.

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

S*<sup>n</sup>*,*p*, *<sup>q</sup>*

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots*

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*n* � *k* 2

ð Þ *ii* We also consider ð Þ *p*, *q* -sigmoid polynomials in two parameters. Then we

<sup>¼</sup> <sup>1</sup> *ep*, *<sup>q</sup>*ð Þþ �*t* 1

where the required result ð Þ *ii* is completed immediately. □

*n* � *k* 2

S*<sup>n</sup>*,*p*�1,*q*�<sup>1</sup> ð Þ �1 *:*

*Proof. i*ð Þ Multiplying *ep*�1,*q*�<sup>1</sup> ð Þ*t* in generating function of ð Þ *p*, *q* -sigmoid poly-

*n* 2

þ *k* 2

*<sup>x</sup><sup>n</sup> <sup>t</sup> n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*q*

*k* 2

<sup>S</sup>*<sup>n</sup>*,*p*�1,*q*�<sup>1</sup> ð Þ �<sup>1</sup> *xn*�*<sup>k</sup>*

1

CCCA *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! *:*

(23)

!

!

!

þ *k* 2

!

**Theorem 2.5.** *Let* ∣*q=p*∣<1 *and k be a nonnegative integer. Then we have*

*n* � *k* 2

!

!

X∞ *n*¼0 *p* *n*

!

2

<sup>S</sup>*<sup>k</sup>*,*p*, *qxn*�*<sup>k</sup>*

<sup>S</sup>*<sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ *<sup>x</sup> <sup>y</sup><sup>n</sup>*�*<sup>k</sup>*

!

*q*

*xn <sup>t</sup> n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

(20)

(21)

,

(22)

1

CCCA *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! *:*

*ep*, *<sup>q</sup>*ð Þ *tx ep*, *<sup>q</sup>*ð Þ *ty*

1

CCCA *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! ,

*k* 2

<sup>S</sup>*<sup>n</sup>*,*p*�1,*q*�<sup>1</sup> ð Þ �<sup>1</sup> *<sup>x</sup><sup>n</sup>*�*<sup>k</sup>*

!

that is shown the required result of Theorem 2.2 ð Þ*i* .

ð Þ *ii* This equation is a recurrence formulae of ð Þ *p*, *q* -sigmoid numbers. We omit the proof of Theorem 2.2 ð Þ *ii* since we can find the result for ð Þ *p*, *q* -sigmoid numbers to calculating the same method ð Þ*<sup>i</sup>* . □

Based on the results from Definition 2.1 and Theorem 2.2, can be taken a few ð Þ *p*, *q* -sigmoid numbers and polynomials can be calculated by using computer. We can observe that some of the ð Þ *p*, *q* -sigmoid numbers are S0,*p*, *<sup>q</sup>* ¼ 1*=*2, S1,*p*, *<sup>q</sup>* ¼ <sup>1</sup>*=*4ð Þ �*<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* , <sup>S</sup>2,*p*, *<sup>q</sup>* <sup>¼</sup> <sup>1</sup>*=*8ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup> <sup>p</sup>*<sup>2</sup> � <sup>3</sup>*pq* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> ð Þ, <sup>S</sup>3,*p*, *<sup>q</sup>* <sup>¼</sup> �1*=*16ð Þ *<sup>p</sup>* � *<sup>q</sup>* ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup> <sup>p</sup>*<sup>2</sup> � <sup>4</sup>*pq* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> ð Þ *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *pq* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> ð Þ, <sup>S</sup>4,*p*, *<sup>q</sup>* <sup>¼</sup> <sup>1</sup>*=*<sup>32</sup> *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> ð Þ *<sup>p</sup>*<sup>8</sup> � <sup>4</sup>*p*<sup>7</sup>*<sup>q</sup>* � <sup>4</sup>*p*<sup>6</sup>*q*<sup>2</sup> <sup>þ</sup> <sup>7</sup>*p*<sup>5</sup>*q*<sup>3</sup> <sup>þ</sup> <sup>8</sup>*p*<sup>4</sup>*q*<sup>4</sup> <sup>þ</sup> <sup>7</sup>*p*<sup>3</sup>*q*<sup>5</sup> � <sup>4</sup>*p*<sup>2</sup>*q*<sup>6</sup> � <sup>4</sup>*pq*<sup>7</sup> <sup>þ</sup> *<sup>q</sup>*<sup>8</sup> � �, <sup>⋯</sup>. **Example 2.3.** *Some of the p*ð Þ , *q -sigmoid polynomials are:*

$$\begin{aligned} \mathcal{S}\_{0,p,q}(\mathbf{x}) &= \frac{1}{2} \\ \mathcal{S}\_{1p,q}(\mathbf{x}) &= \frac{1}{8}(1+2\mathbf{x}) \\ \mathcal{S}\_{2p,q}(\mathbf{x}) &= \frac{1}{8} \left(-p+q+2(p+q)\mathbf{x}+4p\mathbf{x}^2\right) \\ \mathcal{S}\_{3p,q}(\mathbf{x}) &= \frac{1}{16} \left(q^3(1+2\mathbf{x}) + 2p^2q(-1+2\mathbf{x}^2) + 2pq^2(-1+2\mathbf{x}^2) + p^3(1-2\mathbf{x}+4\mathbf{x}^2+8\mathbf{x}^3)\right) \\ \mathcal{S}\_{4p,q}(\mathbf{x}) &= \frac{1}{32} \left(-3p^2q^4(1+2\mathbf{x}) + q^6(1+2\mathbf{x}) + 4p^3q^3\mathbf{x}(-1+\mathbf{x}+2\mathbf{x}^2)\right) \\ &+ \frac{1}{32} \left(p^4q^2(-1+2\mathbf{x})(-3+4\mathbf{x}^2) + pq^5(-3-2\mathbf{x}+4\mathbf{x}^2) + p^5q(3-2\mathbf{x}+8\mathbf{x}^3)\right) \\ &+ \frac{1}{32} \left(p^6(-1+2\mathbf{x}(1-2\mathbf{x}+4\mathbf{x}^2+8\mathbf{x}^3))\right) \\ &\dots \end{aligned} \tag{18}$$

**Theorem 2.4.** *Let k be a nonnegative integer. Then we obtain*

$$\begin{aligned} (i) \quad \mathcal{S}\_{n,p,q}(\mathbf{x}) &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} p^{\binom{n-k}{2}} \mathcal{S}\_{k,p,q} \mathbf{x}^{n-k}, \\\ (ii) \quad \mathcal{S}\_{n,p,q}(\mathbf{x}, \mathbf{y}) &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} p^{\binom{n-k}{2}} \mathcal{S}\_{k,p,q} \mathbf{y}^{n-k}. \end{aligned} \tag{19}$$

*Proof. i*ð Þ Using the definition of ð Þ *p*, *q* -exponential function, we can transform the Definition 2.1 as the follows:

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots DOI: http://dx.doi.org/10.5772/intechopen.91862*

$$\begin{split} \sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q}(\mathbf{x}) \frac{t^n}{[n]\_{p,q}!} &= \sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} \sum\_{n=0}^{\infty} p \binom{n}{2}\_{\mathcal{X}^n} \frac{t^n}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \binom{n}{k}\_{p,q} p \binom{n-k}{2}\_{\mathcal{S}\_{k,p,q}\mathcal{X}^{n-k}} \right) \frac{t^n}{[n]\_{p,q}!} . \end{split} \tag{20}$$

Therefore, we complete the proof of the Theorem 2.4 ð Þ*i* at once.

ð Þ *ii* We also consider ð Þ *p*, *q* -sigmoid polynomials in two parameters. Then we derive

$$\begin{split} &\sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q}(\mathbf{x},\mathbf{y}) \frac{t^n}{[n]\_{p,q}!} = \frac{1}{e\_{p,q}(-t) + \mathbf{1}} e\_{p,q}(t\mathbf{x}) e\_{p,q}(t\mathbf{y}) \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} p \binom{n-k}{2} \mathbf{S}\_{k,p,q}(\mathbf{x}) y^{n-k} \right) \frac{t^n}{[n]\_{p,q}!}, \end{split} \tag{21}$$

where the required result ð Þ *ii* is completed immediately. □ **Theorem 2.5.** *Let* ∣*q=p*∣<1 *and k be a nonnegative integer. Then we have*

$$\begin{aligned} (i) \quad \mathcal{S}\_{n,p,q}(\mathbf{x}) &= \sum\_{k=0}^{n} \binom{n}{k}\_{p,q} (-\mathbf{1})^k p \binom{n-k}{2}\_q \binom{k}{2}\_q \binom{k}{2}\_{\mathcal{S}\_{n,p^{-1},q^{-1}}(-\mathbf{1}) \mathbf{x}^{n-k}}, \\ (ii) \quad \mathcal{S}\_{n,p,q} &= (-\mathbf{1})^n p \binom{n}{2}\_q \binom{n}{2}\_q \end{aligned} \tag{21}$$

*Proof. i*ð Þ Multiplying *ep*�1,*q*�<sup>1</sup> ð Þ*t* in generating function of ð Þ *p*, *q* -sigmoid polynomials, we can investigate

$$\begin{split} &\sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q}(\mathbf{x}) \frac{t^n}{[n]\_{p,q}!} \\ &= \frac{1}{\mathbf{1} + e\_{p^{-1},q^{-1}}(t)} e\_{p^{-1}q^{-1}}(t) e\_{p,q}(t\mathbf{x}) \\ &= \sum\_{n=0}^{\infty} \mathcal{S}\_{n,p^{-1},q^{-1}}(-1) \frac{(-t)^n}{[n]\_{p^{-1},q^{-1}}!} \sum\_{n=0}^{\infty} p \binom{n}{2}\_{\mathbf{x}^n} \mathbf{x}^n \frac{t^n}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \binom{n}{k}\_{p,q} (-\mathbf{1})^k p \binom{n-k}{2}\_q + \binom{k}{2}\_q \binom{k}{2}\_q \binom{k}{2}\_{\mathbf{S}\_{n,p^{-1},q^{-1}}(-1) \mathbf{x}^{n-k}} \right) \frac{t^n}{[n]\_{p,q}!}. \end{split} \tag{23}$$

Using power series of ð Þ *p*, *q* -exponential function in the equation above (16) and

ð Þ *ii* This equation is a recurrence formulae of ð Þ *p*, *q* -sigmoid numbers. We omit the proof of Theorem 2.2 ð Þ *ii* since we can find the result for ð Þ *p*, *q* -sigmoid numbers to calculating the same method ð Þ*<sup>i</sup>* . □ Based on the results from Definition 2.1 and Theorem 2.2, can be taken a few ð Þ *p*, *q* -sigmoid numbers and polynomials can be calculated by using computer. We can observe that some of the ð Þ *p*, *q* -sigmoid numbers are S0,*p*, *<sup>q</sup>* ¼ 1*=*2, S1,*p*, *<sup>q</sup>* ¼

<sup>1</sup>*=*<sup>32</sup> *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> ð Þ *<sup>p</sup>*<sup>8</sup> � <sup>4</sup>*p*<sup>7</sup>*<sup>q</sup>* � <sup>4</sup>*p*<sup>6</sup>*q*<sup>2</sup> <sup>þ</sup> <sup>7</sup>*p*<sup>5</sup>*q*<sup>3</sup> <sup>þ</sup> <sup>8</sup>*p*<sup>4</sup>*q*<sup>4</sup> <sup>þ</sup> <sup>7</sup>*p*<sup>3</sup>*q*<sup>5</sup> � <sup>4</sup>*p*<sup>2</sup>*q*<sup>6</sup> � <sup>4</sup>*pq*<sup>7</sup> <sup>þ</sup> *<sup>q</sup>*<sup>8</sup> � �, <sup>⋯</sup>.

*<sup>q</sup>* �<sup>1</sup> <sup>þ</sup> <sup>2</sup>*x*<sup>2</sup> � � <sup>þ</sup> <sup>2</sup>*pq*<sup>2</sup> �<sup>1</sup> <sup>þ</sup> <sup>2</sup>*x*<sup>2</sup> � � <sup>þ</sup> *<sup>p</sup>*<sup>3</sup> <sup>1</sup> � <sup>2</sup>*<sup>x</sup>* <sup>þ</sup> <sup>4</sup>*x*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*x*<sup>3</sup> � � � �

*q*3

*<sup>q</sup>* <sup>3</sup> � <sup>2</sup>*<sup>x</sup>* <sup>þ</sup> <sup>8</sup>*x*<sup>3</sup> � � � �

*n* � *k* 2

> *n* � *k* 2

!

<sup>S</sup>*<sup>k</sup>*,*p*, *qx<sup>n</sup>*�*<sup>k</sup>*,

<sup>S</sup>*<sup>k</sup>*,*p*, *qy<sup>n</sup>*�*<sup>k</sup>*

*:*

!

(18)

(19)

S*<sup>k</sup>*,*p*, *<sup>q</sup>*ð ÞþS *x <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

, (17)

1

CCA *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*n* � *k* 2 � �

Cauchy product, we can compare both-sides as following:

ð Þ �<sup>1</sup> *<sup>n</sup>*�*<sup>k</sup> p*

*xn <sup>t</sup> n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

that is shown the required result of Theorem 2.2 ð Þ*i* .

<sup>1</sup>*=*4ð Þ �*<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* , <sup>S</sup>2,*p*, *<sup>q</sup>* <sup>¼</sup> <sup>1</sup>*=*8ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup> <sup>p</sup>*<sup>2</sup> � <sup>3</sup>*pq* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> ð Þ, <sup>S</sup>3,*p*, *<sup>q</sup>* <sup>¼</sup> �1*=*16ð Þ *<sup>p</sup>* � *<sup>q</sup>* ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup> <sup>p</sup>*<sup>2</sup> � <sup>4</sup>*pq* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> ð Þ *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *pq* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> ð Þ, <sup>S</sup>4,*p*, *<sup>q</sup>* <sup>¼</sup>

<sup>8</sup> �*<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* <sup>þ</sup> <sup>2</sup>ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup> <sup>x</sup>* <sup>þ</sup> <sup>4</sup>*px*<sup>2</sup> � �

ð Þþ <sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>x</sup>* <sup>2</sup>*p*<sup>2</sup>

**Example 2.3.** *Some of the p*ð Þ , *q -sigmoid polynomials are:*

*<sup>q</sup>*<sup>4</sup>ð Þþ <sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>x</sup> <sup>q</sup>*<sup>6</sup>ð Þþ <sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>x</sup>* <sup>4</sup>*p*<sup>3</sup>

<sup>32</sup> *<sup>p</sup>*<sup>6</sup> �<sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>x</sup>* <sup>1</sup> � <sup>2</sup>*<sup>x</sup>* <sup>þ</sup> <sup>4</sup>*x*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*x*<sup>3</sup> � � � � � �

**Theorem 2.4.** *Let k be a nonnegative integer. Then we obtain*

*k*¼0

*n k*

*k*¼0

*p*, *q p*

*n k*

*p*, *q p*

*Proof. i*ð Þ Using the definition of ð Þ *p*, *q* -exponential function, we can transform

" #

" #

ðÞ S *<sup>i</sup> <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ¼ *<sup>x</sup>* <sup>X</sup>*<sup>n</sup>*

ðÞ S *ii <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>* <sup>X</sup>*<sup>n</sup>*

*<sup>x</sup>* �<sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> <sup>2</sup>*x*<sup>2</sup> � � � �

ð Þ� �<sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>x</sup>* <sup>3</sup> <sup>þ</sup> <sup>4</sup>*x*<sup>2</sup> � � <sup>þ</sup> *pq*<sup>5</sup> �<sup>3</sup> � <sup>2</sup>*<sup>x</sup>* <sup>þ</sup> <sup>4</sup>*x*<sup>2</sup> � � <sup>þ</sup> *<sup>p</sup>*<sup>5</sup>

X∞ *n*¼0

<sup>S</sup>0,*p*, *<sup>q</sup>*ð Þ¼ *<sup>x</sup>* <sup>1</sup>

<sup>S</sup>1,*p*, *<sup>q</sup>*ð Þ¼ *<sup>x</sup>* <sup>1</sup>

<sup>S</sup>2,*p*, *<sup>q</sup>*ð Þ¼ *<sup>x</sup>* <sup>1</sup>

<sup>S</sup>3,*p*, *<sup>q</sup>*ð Þ¼ *<sup>x</sup>* <sup>1</sup>

<sup>S</sup>4,*p*, *<sup>q</sup>*ð Þ¼ *<sup>x</sup>* <sup>1</sup>

⋯*:*

**112**

2

<sup>4</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>x</sup>*

<sup>16</sup> *<sup>q</sup>*<sup>3</sup>

þ 1 <sup>32</sup> *<sup>p</sup>*<sup>4</sup>*q*<sup>2</sup>

þ 1

<sup>32</sup> �3*p*<sup>2</sup>

the Definition 2.1 as the follows:

0

BB@

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0 *p*

X*n k*¼0

*Number Theory and Its Applications*

*n k* � �

> *n* 2 � �

*p*, *q*

Comparing the both-side in the equation above, (23), we find the required results ð Þ*i* .

ð Þ *ii* Using the same method ð Þ*i* , we make the equation ð Þ *ii* , so we omit the proof of Theorem 2.5 ð Þ *ii* . □

**Corollary 2.6.** *From Theorem 2.5, one holds*

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} p^{\binom{n-k}{2}} \mathcal{S}\_{k,p,q} = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (-1)^k p^{\binom{n-k}{2} + \binom{k}{2}} q^{\binom{k}{2}} \mathcal{S}\_{n,p^{-1},q^{-1}} (-1). \tag{24}$$

**Theorem 2.7.** *For* ∣*q=p*∣<1*, p*ð Þ , *q -derivative of p*ð Þ , *q -sigmoid polynomials is as the follows:*

$$\frac{\mathcal{D}\_{p,q}}{\mathcal{D}\_{p,q}\mathcal{X}}\mathcal{S}\_{n,p,q}(\mathbf{x}) = [n]\_{p,q}\mathcal{S}\_{n-1,p,q}(p\mathbf{x}).\tag{25}$$

D*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ¼ *x*

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

<sup>¼</sup> <sup>1</sup> *ep*, *<sup>q</sup>*ð Þþ �*t* 1

> X∞ *n*¼0

D*<sup>p</sup>*, *<sup>q</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.91862*

½ � *<sup>n</sup> <sup>p</sup>*, *<sup>q</sup>*ð Þ *<sup>p</sup>* � *<sup>q</sup>* <sup>X</sup>*<sup>n</sup>*�<sup>1</sup>

X∞ *n*¼0

and

**115**

Dð Þ<sup>2</sup>

*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

ðÞ D *i <sup>p</sup>*, *<sup>q</sup>*

ðÞ D *ii <sup>p</sup>*, *<sup>q</sup>*

Here, we can obtain that

*k*¼0

X∞ *n*¼0

<sup>¼</sup> <sup>1</sup> ð Þ *p* � *q x*

*n* � 1 *k* � �

*Proof.* Using the Theorem 2.8 above, we have

*<sup>x</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>x</sup>* <sup>¼</sup> <sup>1</sup>

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*ep*, *<sup>q</sup>*ð Þ *ptx*

*ep*, *<sup>q</sup>*ð Þ *qtx*

*p*, *q p* S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ�S *px <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *qx*

D*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

*ep*, *<sup>q</sup>*ð Þ� *ptx ep*, *<sup>q</sup>*ð Þ *qtx* ð Þ *p* � *q x* � �

*Proof.* Applying ð Þ *p*, *q* -derivative in the ð Þ *p*, *q* -exponential function, we have

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

which is the required result. □

<sup>S</sup>*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ�S *px <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *qx* � � *<sup>t</sup>*

½ � *n <sup>p</sup>*, *<sup>q</sup>*ð Þ *p* � *q x*S*<sup>n</sup>*�1,*p*, *<sup>q</sup>*ð ÞþS *px <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ¼S *qx <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *px :* (31)

þ*n*�1�*k*

*pqx*<sup>2</sup> *qx*D*<sup>p</sup>*, *qep*, *<sup>q</sup>*ð Þ� *ptx ep*, *<sup>q</sup>*ð ÞD *pqtx <sup>p</sup>*, *qx* � �

*tx* � �D*<sup>p</sup>*, *qx* � �

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

**Corollary 2.9.** *Comparing the Theorem 2.7 and Theorem 2.8, one holds*

**Corollary 2.10.** *Putting x* ¼ 1 *in the Theorem 2.7 and Theorem 2.8, one holds*

*n* � 1 � *k* 2 � �

**Theorem 2.11.** *Let* ∣*q=p*∣<1*, p* 6¼ *q, and p*, *q* 6¼ 0*. Then we obtain*

*p pqxD*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð ÞþD *<sup>x</sup> <sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *qx* � � <sup>¼</sup> *<sup>q</sup>* <sup>D</sup>*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þþ *<sup>x</sup> pqx*Dð Þ<sup>2</sup>

<sup>¼</sup> <sup>1</sup>

ð Þ *<sup>p</sup>* � *<sup>q</sup> ep*, *<sup>q</sup>*ð Þþ �*<sup>t</sup>* <sup>1</sup> � �

<sup>¼</sup> *qep*, *<sup>q</sup> <sup>p</sup>*<sup>2</sup> ð Þ� *tx pep*, *<sup>q</sup>*ð Þ *pqtx*

*pq p*ð Þ � *<sup>q</sup> <sup>x</sup>*<sup>2</sup> ,

*pqx*<sup>2</sup> *qx*D*<sup>p</sup>*, *qep*, *<sup>q</sup>*ð Þ� *qtx ep*, *<sup>q</sup> <sup>q</sup>*<sup>2</sup>

*pq p*ð Þ � *<sup>q</sup> <sup>x</sup>*<sup>2</sup> *:*

Applying Eqs. (35) and (36) in the Eq. (34), we can catch the following equation:

<sup>¼</sup> *qep*, *<sup>q</sup>*ð Þ� *pqtx pep*, *<sup>q</sup> <sup>q</sup>*<sup>2</sup> ð Þ *tx*

ð Þ *<sup>p</sup>* � *<sup>q</sup> <sup>x</sup> :* (29)

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! ,

S*<sup>k</sup>*,*p*, *<sup>q</sup>* ¼ S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ�S *p <sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ*q :*

*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

� �*:* (33)

*ep*, *<sup>q</sup>*ð Þ� *ptx ep*, *<sup>q</sup>*ð Þ *qtx x*

� �*:* (34)

(30)

(32)

(35)

(36)

*Proof.* Using ð Þ *p*, *q* -derivative for S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x* , we have

$$\begin{split} \sum\_{n=0}^{\infty} \frac{\mathcal{D}\_{p,q}}{\mathcal{D}\_{p,q} \mathfrak{x}} \mathcal{S}\_{n,p,q}(\mathbf{x}) \frac{t^n}{[n]\_{p,q}!} &= \frac{1}{\mathfrak{e}\_{p,q}(-t) + \mathbf{1}} \mathcal{D}\_{p,q} \mathfrak{e}\_{p,q}(t\mathbf{x}) \\ &= \sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} \sum\_{n=0}^{\infty} p \binom{n}{2} \mathcal{D}\_{p,q} \mathfrak{x}^n \frac{t^n}{[n]\_{p,q}!} . \end{split} \tag{26}$$

Here, we can note that <sup>D</sup>*<sup>p</sup>*, *qx<sup>n</sup>* <sup>¼</sup> ð Þ *px <sup>n</sup>* �ð Þ *qx <sup>n</sup>* ð Þ *<sup>p</sup>*�*<sup>q</sup> <sup>x</sup>* <sup>¼</sup> ½ � *<sup>n</sup> <sup>p</sup>*, *<sup>q</sup>xn*�<sup>1</sup> (see [7]). From the equation above (26), we can transform the equation to

$$\begin{split} &\sum\_{n=0}^{\infty} \frac{\mathcal{D}\_{p,q}}{\mathcal{D}\_{p,q} \mathbf{x}} \mathcal{S}\_{n,p,q}(\mathbf{x}) \frac{t^{n}}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \binom{n}{k}\_{p,q} [n-k]\_{p,q} p^{n} \binom{n-k}{2}\_{\mathcal{S}\_{k,p,q}} \mathbf{x}^{n-1-k} \right) \frac{t^{n}}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n-1} [n]\_{p,q} \binom{n-1}{k}\_{p,q} p^{n} \binom{n-1-k}{2}\_{\mathcal{S}\_{k,p,q}} \mathbf{x}^{n} \right) \frac{t^{n}}{[n]\_{p,q}!} .\end{split} \tag{27}$$

Using the comparison of coefficients in the both-sides, we can find

$$\frac{\mathcal{D}\_{p,q}}{\mathcal{D}\_{p,q}\varkappa} \mathcal{S}\_{n,p,q}(\varkappa) = [n]\_{p,q} \sum\_{k=0}^{n-1} \begin{bmatrix} n-1 \\ k \end{bmatrix}\_{p,q} p^{\binom{n-1-k}{2}} \mathcal{S}\_{k,p,q}(p\varkappa)^{n-1-k}.\tag{28}$$

Applying the Theorem 2.4 ð Þ*i* in the equation above, (28), we complete the proof of Theorem 2.7. □

**Theorem 2.8.** *Let* ∣*q=p*∣<1 *and p* 6¼ *q. Then we investigate*

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots DOI: http://dx.doi.org/10.5772/intechopen.91862*

$$\mathcal{D}\_{p,q}\mathcal{S}\_{n,p,q}(\mathbf{x}) = \frac{\mathcal{S}\_{n,p,q}(p\mathbf{x}) - \mathcal{S}\_{n,p,q}(q\mathbf{x})}{(p-q)\mathbf{x}}.\tag{29}$$

*Proof.* Applying ð Þ *p*, *q* -derivative in the ð Þ *p*, *q* -exponential function, we have

$$\begin{split} \mathcal{D}\_{p,q} \sum\_{n=0}^{\infty} \mathcal{S}\_{n,p,q}(\mathbf{x}) \frac{t^n}{[n]\_{p,q}!} &= \sum\_{n=0}^{\infty} \mathcal{D}\_{p,q} \mathcal{S}\_{n,p,q}(\mathbf{x}) \frac{t^n}{[n]\_{p,q}!} \\ &= \frac{1}{e\_{p,q}(-t) + 1} \left( \frac{e\_{p,q}(p\mathbf{x}) - e\_{p,q}(q\mathbf{x})}{(p-q)\mathbf{x}} \right) \\ &= \frac{1}{(p-q)\mathbf{x}} \sum\_{n=0}^{\infty} \left( \mathcal{S}\_{n,p,q}(p\mathbf{x}) - \mathcal{S}\_{n,p,q}(q\mathbf{x}) \right) \frac{t^n}{[n]\_{p,q}!}, \end{split} \tag{30}$$

Comparing the both-side in the equation above, (23), we find the required

**Corollary 2.6.** *From Theorem 2.5, one holds*

<sup>S</sup>*<sup>k</sup>*,*p*, *<sup>q</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

D*<sup>p</sup>*, *<sup>q</sup>* D*<sup>p</sup>*, *qx*

*Proof.* Using ð Þ *p*, *q* -derivative for S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x* , we have

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

S*<sup>n</sup>*,*p*, *<sup>q</sup>*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

> *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*n* � *k* 2

*n* � 1 � *k* 2

> *p*, *q p*

Applying the Theorem 2.4 ð Þ*i* in the equation above, (28), we complete the proof of Theorem 2.7. □

!

!

*k*¼0

*n k* � �

*p*, *q*

ð Þ *ii* Using the same method ð Þ*i* , we make the equation ð Þ *ii* , so we omit the proof of Theorem 2.5 ð Þ *ii* . □

> ð Þ �<sup>1</sup> *<sup>k</sup> p*

**Theorem 2.7.** *For* ∣*q=p*∣<1*, p*ð Þ , *q -derivative of p*ð Þ , *q -sigmoid polynomials is as the*

<sup>¼</sup> <sup>1</sup> *ep*, *<sup>q</sup>*ð Þþ �*t* 1

*n*

!

2

<sup>S</sup>*<sup>k</sup>*,*p*, *qx<sup>n</sup>*�1�*<sup>k</sup>*

*n* � 1 � *k* 2 � �

X∞ *n*¼0 *p*

�ð Þ *qx <sup>n</sup>*

*n* � *k* 2 � �

þ *k* 2 � �

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ¼ *x* ½ � *n <sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*�1,*p*, *<sup>q</sup>*ð Þ *px :* (25)

D*<sup>p</sup>*, *qep*, *<sup>q</sup>*ð Þ *tx*

*n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! *:*

1

CCCA *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! *:*

<sup>S</sup>*<sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ *px <sup>n</sup>*�1�*<sup>k</sup>*

<sup>D</sup>*<sup>p</sup>*, *qx<sup>n</sup> <sup>t</sup>*

1

CCCA *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

<sup>S</sup>*<sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ *px <sup>n</sup>*�1�*<sup>k</sup>*

ð Þ *<sup>p</sup>*�*<sup>q</sup> <sup>x</sup>* <sup>¼</sup> ½ � *<sup>n</sup> <sup>p</sup>*, *<sup>q</sup>xn*�<sup>1</sup> (see [7]). From the equa-

*q*

*k* 2 � �

S*<sup>n</sup>*,*p*�1,*q*�<sup>1</sup> ð Þ �1 *:*

(24)

(26)

(27)

*:* (28)

results ð Þ*i* .

*n k* � �

*p*, *q p*

*n* � *k* 2 � �

*Number Theory and Its Applications*

X∞ *n*¼0

D*<sup>p</sup>*, *<sup>q</sup>* D*<sup>p</sup>*, *qx*

Here, we can note that <sup>D</sup>*<sup>p</sup>*, *qx<sup>n</sup>* <sup>¼</sup> ð Þ *px <sup>n</sup>*

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

*n k*

½ � *n <sup>p</sup>*, *<sup>q</sup>*

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ¼ *x* ½ � *n <sup>p</sup>*, *<sup>q</sup>*

*p*, *q*

" #

tion above (26), we can transform the equation to

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*n* � 1 *k*

" #

½ � *n* � *k <sup>p</sup>*, *<sup>q</sup>p*

*p*, *q p*

X*n*�1 *k*¼0

**Theorem 2.8.** *Let* ∣*q=p*∣<1 *and p* 6¼ *q. Then we investigate*

Using the comparison of coefficients in the both-sides, we can find

*n* � 1 *k* � �

X*n k*¼0

*follows:*

X∞ *n*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

D*<sup>p</sup>*, *<sup>q</sup>* D*<sup>p</sup>*, *qx*

**114**

D*<sup>p</sup>*, *<sup>q</sup>* D*<sup>p</sup>*, *qx*

0

BBB@

0

BBB@

X*n k*¼0

X*n*�1 *k*¼0

which is the required result. □ **Corollary 2.9.** *Comparing the Theorem 2.7 and Theorem 2.8, one holds*

$$
\hat{\mathfrak{a}}[n]\_{p,q}(p-q) \mathfrak{x} \mathcal{S}\_{n-1,p,q}(p\mathbf{x}) + \mathcal{S}\_{n,p,q}(q\mathbf{x}) = \mathcal{S}\_{n,p,q}(p\mathbf{x}).\tag{31}
$$

**Corollary 2.10.** *Putting x* ¼ 1 *in the Theorem 2.7 and Theorem 2.8, one holds*

$$
\begin{split}
\boldsymbol{\varepsilon}[\boldsymbol{n}]\_{p,q}(\boldsymbol{p}-\boldsymbol{q}) \sum\_{k=0}^{n-1} \begin{bmatrix} \boldsymbol{n}-\boldsymbol{1} \\ \boldsymbol{k} \end{bmatrix}\_{p,q} \boldsymbol{p} \binom{\boldsymbol{n}-\boldsymbol{1}-\boldsymbol{k}}{\boldsymbol{2}}\_{\boldsymbol{p},\boldsymbol{q}} \boldsymbol{\varepsilon}\_{\boldsymbol{p},\boldsymbol{q}} &= \boldsymbol{\mathcal{S}}\_{\boldsymbol{n},p,q}(\boldsymbol{p}) - \boldsymbol{\mathcal{S}}\_{\boldsymbol{n},p,q}(\boldsymbol{q}).
\end{split}
\tag{32}
$$

**Theorem 2.11.** *Let* ∣*q=p*∣<1*, p* 6¼ *q, and p*, *q* 6¼ 0*. Then we obtain*

$$p\left(pq\mathbf{x}D\mathcal{S}\_{\boldsymbol{n},\boldsymbol{p},\boldsymbol{q}}(\boldsymbol{\chi}) + \mathcal{D}\_{\boldsymbol{p},\boldsymbol{q}}\mathcal{S}\_{\boldsymbol{n},\boldsymbol{p},\boldsymbol{q}}(q\boldsymbol{\chi})\right) = q\left(\mathcal{D}\_{\boldsymbol{p},\boldsymbol{q}}\mathcal{S}\_{\boldsymbol{n},\boldsymbol{p},\boldsymbol{q}}(\boldsymbol{\chi}) + pq\mathbf{x}\mathcal{D}\_{\boldsymbol{p},\boldsymbol{q}}^{(2)}\mathcal{S}\_{\boldsymbol{n},\boldsymbol{p},\boldsymbol{q}}(\boldsymbol{\chi})\right). \tag{33}$$

*Proof.* Using the Theorem 2.8 above, we have

$$\sum\_{n=0}^{\infty} \mathcal{D}\_{p,q}^{(2)} \mathcal{S}\_{n,p,q}(\infty) \frac{t^n}{[n]\_{p,q}!} = \frac{1}{(p-q)\left(e\_{p,q}(-t) + 1\right)} \left(\frac{e\_{p,q}(pt\infty) - e\_{p,q}(qt\infty)}{\varkappa}\right). \tag{34}$$

Here, we can obtain that

$$\begin{split} (i) \quad \mathcal{D}\_{p,q} \frac{e\_{p,q}(p\text{tx})}{\mathfrak{x}} &= \frac{\mathbf{1}}{pq\mathfrak{x}^2} \big( q\mathbf{x}\mathcal{D}\_{p,q}e\_{p,q}(p\text{tx}) - e\_{p,q}(pq\text{tx})\mathcal{D}\_{p,q}\mathfrak{x} \big) \\ &= \frac{qe\_{p,q}(p^2\text{tx}) - pe\_{p,q}(pq\text{tx})}{pq(p-q)\mathfrak{x}^2}, \end{split} \tag{35}$$

and

$$\begin{split} \mathbf{r}(\text{ii}) \quad \mathcal{D}\_{p,q} \frac{e\_{p,q}(q\text{tx})}{\text{x}} &= \frac{\mathbf{1}}{pq\text{x}^2} \big( q\mathbf{x}\mathcal{D}\_{p,q}\mathbf{e}\_{p,q}(q\text{tx}) - \mathbf{e}\_{p,q}\left(q^2\text{tx}\right)\mathcal{D}\_{p,q}\mathbf{x} \big) \\ &= \frac{q e\_{p,q}(pq\text{tx}) - p e\_{p,q}(q^2\text{tx})}{pq(p-q)\text{x}^2} . \end{split} \tag{36}$$

Applying Eqs. (35) and (36) in the Eq. (34), we can catch the following equation:

$$\begin{split} &\sum\_{n=0}^{\infty} \mathcal{D}\_{p,q}^{(2)} \mathcal{S}\_{n,p,q}(\boldsymbol{\kappa}) \frac{t^{n}}{[n]\_{p,q}!} \\ &= \frac{1}{(p-q)pq\boldsymbol{\kappa}} \left( \sum\_{n=0}^{\infty} \left( q \mathcal{D}\_{p,q} \mathcal{S}\_{n,p,q}(p\boldsymbol{\kappa}) - p \mathcal{D}\_{p,q} \mathcal{S}\_{n,p,q}(q\boldsymbol{\kappa}) \right) \frac{t^{n}}{[n]\_{p,q}!} \right) . \end{split} \tag{37}$$

**Theorem 2.16.** *Let a*, *b be any integers without* 0*. Then we obtain*

*k*¼0

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots*

*<sup>B</sup>*<sup>≔</sup> *ep*, *<sup>q</sup>*ð Þ *tx ep*, *<sup>q</sup>*ð Þ *ty ep*, *<sup>q</sup>* � *<sup>t</sup> a* � � <sup>þ</sup> <sup>1</sup> � � *ep*, *<sup>q</sup> <sup>t</sup>*

> X∞ *n*¼0

*n k* � �

*p*, *q*

*b*

*n*

X∞ *n*¼0

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *ax t*

*an*½ � *n <sup>p</sup>*, *<sup>q</sup>*!

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð Þ *ax Ek*,*p*, *<sup>q</sup>*ð Þ *by an*�*kb<sup>k</sup>*

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð Þ �*bx Ek*,*p*, *<sup>q</sup>*ð Þ �*ay*

*n k* � �

*n k* � � *p*, *q*

*p*, *q*

*b*

½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*bk*

*n*

X∞ *n*¼0

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ �*ax t*

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð Þ �*ax Ek*,*p*, *<sup>q</sup>*ð Þ *by* ð Þ �*<sup>a</sup> <sup>n</sup>*�*<sup>k</sup>*

ð Þ �*<sup>a</sup> <sup>n</sup>*

*b<sup>n</sup>*�*<sup>k</sup> ak*

ð Þ �<sup>1</sup> *<sup>n</sup>*

Theorem 2.16. □.

*k*¼0

*k*¼0

*<sup>C</sup>*<sup>≔</sup> *ep*, *<sup>q</sup>*ð Þ *tx ep*, *<sup>q</sup>*ð Þ *ty*

*ep*, *<sup>q</sup> <sup>t</sup> a* � � <sup>þ</sup> <sup>1</sup> � � *ep*, *<sup>q</sup> <sup>t</sup>*

> X∞ *n*¼0

In the form C of the equation above (52), we can find that

*<sup>C</sup>*<sup>≔</sup> <sup>1</sup> ½ � 2 *<sup>p</sup>*, *<sup>q</sup>*

*n*

" #

*k*

*p*, *q*

Comparing the equation above (48) and (49), we derive the result of

! *<sup>t</sup>*

S*n*�*k*,*p*, *<sup>q</sup>*ð Þ �*bx Ek*,*p*, *<sup>q</sup>*ð Þ �*ay*

� � <sup>þ</sup> <sup>1</sup> � � *:* (47)

*En*,*p*, *<sup>q</sup>*ð Þ *by t*

*bn* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

1

CA *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! *:*

*n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

S*n*�*k*,*p*, *<sup>q</sup>*ð Þ �*bx Ek*,*p*, *<sup>q</sup>*ð Þ �*y*

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð Þ �*bx Ek*,*p*, *<sup>q</sup>*ð Þ *ay*

� � <sup>þ</sup> <sup>1</sup> � � *:* (52)

*En*,*p*, *<sup>q</sup>*ð Þ *by t*

*bn* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

1 A *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! ,

*n*

(53)

*n*

(48)

*:* (49)

*<sup>b</sup>n*�*<sup>k</sup>* , (50)

*akb<sup>n</sup>*�*<sup>k</sup>* , (51)

*akb<sup>n</sup>*�*<sup>k</sup>* , (46)

*an*�*kb<sup>k</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

where *En*,*p*, *<sup>q</sup>*ð Þ *x* is the ð Þ *p*, *q* -Euler polynomials (see [10]).

The form B of the equation above (47) can be transformed as

*<sup>B</sup>*<sup>≔</sup> <sup>1</sup> ½ � 2 *<sup>p</sup>*, *<sup>q</sup>*

*n*

3 5 *p*, *q*

2 4

*k*

S*n*�*k*,*p*, *<sup>q</sup>*ð Þ *ax Ek*,*p*, *<sup>q</sup>*ð Þ *by*

X*n k*¼0

*n k* � �

*p*, *q*

ð Þ �<sup>1</sup> *<sup>n</sup>*

*Proof.* We consider the form B as

*DOI: http://dx.doi.org/10.5772/intechopen.91862*

<sup>¼</sup> <sup>1</sup> ½ � 2 *<sup>p</sup>*, *<sup>q</sup>*

> X∞ *n*¼0

*<sup>B</sup>*<sup>≔</sup> <sup>1</sup> ½ � 2 *<sup>p</sup>*, *<sup>q</sup>*

X*n k*¼0

X*n k*¼0

**117**

*n k* � �

*n k* � �

*p*, *q*

*p*, *q*

<sup>¼</sup> <sup>1</sup> ½ � 2 *<sup>p</sup>*, *<sup>q</sup>* X∞ *n*¼0

0 @

X*n k*¼0

*Proof.* We set the form C as

ð Þ �<sup>1</sup> *<sup>n</sup>*

X∞ *n*¼0

Also, we can transform the form B such as

X*n k*¼0 X*n k*¼0

*n k* � �

S*n*�*k*,*p*, *<sup>q</sup>*ð Þ *x Ek*,*p*, *<sup>q</sup>*ð Þ *by*

*where En*,*p*, *<sup>q</sup>*ð Þ *x is the p*ð Þ , *q -Euler polynomials*.

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð Þ �*ax Ek*,*p*, *<sup>q</sup>*ð Þ *by*

*where En*,*p*, *<sup>q</sup>*ð Þ *x is the p*ð Þ , *q -Euler polynomials*.

*p*, *q*

**Corollary 2.17.** *Setting a* ¼ 1 *in the Theorem 2.16, one holds*

*bk* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

**Theorem 2.18.** *Let a*, *b be any integers without* 0*. Then we have*

*an*�*kbk* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

0

B@

Therefore, we can see that

$$q(p-q)pq\mathbb{1}\mathcal{D}\_{p,q}^{(2)}\mathcal{S}\_{n,p,q}(\mathbf{x}) = q\mathcal{D}\_{p,q}\mathcal{S}\_{n,p,q}(p\mathbf{x}) - p\mathcal{D}\_{p,q}\mathcal{S}\_{n,p,q}(q\mathbf{x}),\tag{38}$$

and this shows the required result at once. □ **Corollary 2.12.** *From the Theorem 2.11, one holds*

$$\begin{split} &p^2 q \mathfrak{x} \mathcal{D}\_{p,q}^{(2)} \mathcal{S}\_{n,p,q}(\mathfrak{x}) + p[n]\_{p,q} \mathcal{S}\_{n-1,p,q}(pq\mathfrak{x}) \\ &= q[n]\_{p,q} \mathcal{S}\_{n-1,p,q}(p^2 \mathfrak{x}) + pq^2 \mathfrak{x} \mathcal{D}\_{p,q}^{(2)} \mathcal{S}\_{n,p,q}(\mathfrak{x}). \end{split} \tag{39}$$

**Theorem 2.13.** *Let a*, *b be any positive integers. Then we find*

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(a\mathbf{x})\mathcal{S}\_{k,p,q}(b\mathbf{y})}{a^{n-k}b^k} = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(b\mathbf{x})\mathcal{S}\_{k,p,q}(a\mathbf{y})}{a^k b^{n-k}}.\tag{40}$$

*Proof.* Suppose the form A is as the following.

$$A \coloneqq \frac{e\_{p,q}(t\infty)e\_{p,q}(t\flat)}{\left(e\_{p,q}\left(-\frac{t}{a}\right) + \mathbf{1}\right)\left(e\_{p,q}\left(-\frac{t}{b}\right) + \mathbf{1}\right)}.\tag{41}$$

The form A of the equation above (41) can be transformed as

$$A \coloneqq \sum\_{n=0}^{\infty} \frac{\mathcal{S}\_{n,p,q}(a\infty)t^n}{a^n [n]\_{p,q}!} \sum\_{n=0}^{\infty} \frac{\mathcal{S}\_{n,p,q}(b\mathcal{y})t^n}{b^n [n]\_{p,q}!}$$

$$= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(a\infty)\mathcal{S}\_{k,p,q}(b\mathcal{y})}{a^{n-k}b^k} \right) \frac{t^n}{[n]\_{p,q}!},\tag{42}$$

or, equivalently,

$$A \coloneqq \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(b\infty)\mathcal{S}\_{k,p,q}(ay)}{b^{n-k}a^k} \right) \frac{t^n}{[n]\_{p,q}!}. \tag{43}$$

Comparing the coefficients of *t <sup>n</sup>* in both sides, we find the required result. □ **Corollary 2.14.** *Setting a* ¼ 1 *in the Theorem 2.13, one holds*

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(\mathbf{x}) \mathcal{S}\_{k,p,q}(by)}{b^k} = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(bx) \mathcal{S}\_{k,p,q}(y)}{b^{n-k}}.\tag{44}$$

**Corollary 2.15.** *When p* ¼ 1 *and q* ! 1 *in the Theorem 2.13, one holds*

$$\sum\_{k=0}^{n} \binom{n}{k} \frac{\mathbb{S}\_{n-k}(a\boldsymbol{\omega})\mathbb{S}\_{k}(b\boldsymbol{\nu})}{a^{n-k}b^{k}} = \sum\_{k=0}^{n} \binom{n}{k} \frac{\mathbb{S}\_{n-k}(b\boldsymbol{\nu})l\mathbb{S}\_{k}(a\boldsymbol{\nu})}{a^{k}b^{n-k}},\tag{45}$$

*where Sn*ð Þ *x is the sigmoid polynomials (see [16])*.

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots DOI: http://dx.doi.org/10.5772/intechopen.91862*

**Theorem 2.16.** *Let a*, *b be any integers without* 0*. Then we obtain*

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{(-1)^{n} \mathcal{S}\_{n-k,p,q}(a\boldsymbol{\omega}) E\_{k,p,q}(b\boldsymbol{\eta})}{a^{n-k} b^{k}} = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(-b\boldsymbol{\omega}) E\_{k,p,q}(-a\boldsymbol{\eta})}{a^{k} b^{n-k}},\tag{46}$$

where *En*,*p*, *<sup>q</sup>*ð Þ *x* is the ð Þ *p*, *q* -Euler polynomials (see [10]). *Proof.* We consider the form B as

$$B \coloneqq \frac{e\_{p,q}(t\infty)e\_{p,q}(t\flat)}{\left(e\_{p,q}\left(-\frac{t}{a}\right) + 1\right)\left(e\_{p,q}\left(\frac{t}{b}\right) + 1\right)}.\tag{47}$$

The form B of the equation above (47) can be transformed as

$$B \coloneqq \frac{1}{[2]\_{p,q}} \sum\_{n=0}^{\infty} \frac{\mathcal{S}\_{n,p,q}(a\boldsymbol{\alpha})t^n}{a^n [n]\_{p,q}!} \sum\_{n=0}^{\infty} \frac{E\_{n,p,q}(b\boldsymbol{y})t^n}{b^n [n]\_{p,q}!}$$

$$= \frac{1}{[2]\_{p,q}} \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n\\k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(a\boldsymbol{\alpha})E\_{k,p,q}(b\boldsymbol{y})}{a^{n-k}b^k} \right) \frac{t^n}{[n]\_{p,q}!}.\tag{48}$$

Also, we can transform the form B such as

$$B \coloneqq \frac{1}{[2]\_{p,q}} \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(-b\boldsymbol{\varpi})E\_{k,p,q}(-a\boldsymbol{\eta})}{(-1)^n b^{n-k} a^k} \right) \frac{t^n}{[n]\_{p,q}!}.\tag{49}$$

Comparing the equation above (48) and (49), we derive the result of Theorem 2.16. □.

**Corollary 2.17.** *Setting a* ¼ 1 *in the Theorem 2.16, one holds*

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{(-1)^{n} \mathcal{S}\_{n-k,p,q}(\boldsymbol{x}) E\_{k,p,q}(b\boldsymbol{y})}{b^{k}} = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(-b\boldsymbol{x}) E\_{k,p,q}(-\boldsymbol{y})}{b^{n-k}},\tag{50}$$

*where En*,*p*, *<sup>q</sup>*ð Þ *x is the p*ð Þ , *q -Euler polynomials*. **Theorem 2.18.** *Let a*, *b be any integers without* 0*. Then we have*

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(-a\mathbf{x})E\_{k,p,q}(by)}{a^{n-k}b^k} = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(-b\mathbf{x})E\_{k,p,q}(ay)}{a^k b^{n-k}},\tag{51}$$

*where En*,*p*, *<sup>q</sup>*ð Þ *x is the p*ð Þ , *q -Euler polynomials*. *Proof.* We set the form C as

$$\mathbf{C} \coloneqq \frac{e\_{p,q}(t\infty)e\_{p,q}(t\flat)}{\left(e\_{p,q}\left(\frac{t}{a}\right) + \mathbf{1}\right)\left(e\_{p,q}\left(\frac{t}{b}\right) + \mathbf{1}\right)}.\tag{52}$$

In the form C of the equation above (52), we can find that

$$\mathbf{C} \coloneqq \frac{1}{[2]\_{p,q}} \sum\_{n=0}^{\infty} \frac{\mathcal{S}\_{n,p,q}(-a\mathbf{x})t^n}{(-a)^n [n]\_{p,q}!} \sum\_{n=0}^{\infty} \frac{E\_{n,p,q}(by)t^n}{b^n [n]\_{p,q}!}$$

$$= \frac{1}{[2]\_{p,q}} \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(-a\mathbf{x})E\_{k,p,q}(by)}{(-a)^{n-k}b^k} \right) \frac{t^n}{[n]\_{p,q}!},\tag{53}$$

X∞ *n*¼0

X*n k*¼0

*n k* � �

*p*, *q*

Dð Þ<sup>2</sup>

<sup>¼</sup> <sup>1</sup> ð Þ *p* � *q pqx*

*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *x*

*Number Theory and Its Applications*

Therefore, we can see that

ð Þ *<sup>p</sup>* � *<sup>q</sup> pqx*Dð Þ<sup>2</sup>

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

**Corollary 2.12.** *From the Theorem 2.11, one holds*

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð ÞS *ax <sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ *by*

*Proof.* Suppose the form A is as the following.

X*n k*¼0

> X*n k*¼0

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð ÞS *x <sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ *by*

� � *Sn*�*<sup>k</sup>*ð Þ *ax Sk*ð Þ *by*

*where Sn*ð Þ *x is the sigmoid polynomials (see [16])*.

0 @

*n*

" #

*k*

*n k* � �

**Corollary 2.14.** *Setting a* ¼ 1 *in the Theorem 2.13, one holds*

*bk* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*p*, *q*

*p*, *q*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

*A*≔X<sup>∞</sup> *n*¼0

Comparing the coefficients of *t*

or, equivalently,

X*n k*¼0

**116**

*n k* � �

*p*, *q*

X*n k*¼0

*n k*

<sup>¼</sup> *q n*½ �*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*�1,*p*, *<sup>q</sup> <sup>p</sup>*<sup>2</sup>

**Theorem 2.13.** *Let a*, *b be any positive integers. Then we find*

*an*�*kb<sup>k</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*ep*, *<sup>q</sup>* � *<sup>t</sup> a* � � <sup>þ</sup> <sup>1</sup> � � *ep*, *<sup>q</sup>* � *<sup>t</sup>*

The form A of the equation above (41) can be transformed as

*A*≔X<sup>∞</sup> *n*¼0

*p*2 *qx*Dð Þ<sup>2</sup>

*<sup>q</sup>*D*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ� *px <sup>p</sup>*D*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *qx* � � *<sup>t</sup>*

and this shows the required result at once. □

*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þþ *x p n*½ �*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*�1,*p*, *<sup>q</sup>*ð Þ *pqx*

*n k* � �

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *ax t*

*an*½ � *n <sup>p</sup>*, *<sup>q</sup>*!

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð ÞS *ax <sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ *by an*�*kbk*

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð ÞS *bx <sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ *ay bn*�*<sup>k</sup> ak*

!

*k*¼0

**Corollary 2.15.** *When p* ¼ 1 *and q* ! 1 *in the Theorem 2.13, one holds*

*an*�*kbk* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*n k* � �

*k*¼0

*p*, *q*

*n k*

*p*, *q*

*b*

*n*

X∞ *n*¼0

*<sup>x</sup>* � � <sup>þ</sup> *pq*<sup>2</sup>

*k*¼0

*<sup>A</sup>*<sup>≔</sup> *ep*, *<sup>q</sup>*ð Þ *tx ep*, *<sup>q</sup>*ð Þ *ty*

*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ¼ *x q*D*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ� *px p*D*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *qx* , (38)

*<sup>x</sup>*Dð Þ<sup>2</sup>

!

*n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*<sup>p</sup>*, *<sup>q</sup>*S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *<sup>x</sup> :* (39)

*akbn*�*<sup>k</sup> :* (40)

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð ÞS *bx <sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ *ay*

� � <sup>þ</sup> <sup>1</sup> � � *:* (41)

S*<sup>n</sup>*,*p*, *<sup>q</sup>*ð Þ *by t*

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

*<sup>n</sup>* in both sides, we find the required result. □

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð ÞS *bx <sup>k</sup>*,*p*, *<sup>q</sup>*ð Þ*y*

� � *Sn*�*<sup>k</sup>*ð Þ *bx lSk*ð Þ *ay*

*bn* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

1 A *t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*! ,

*n*

(42)

*:* (43)

*<sup>b</sup><sup>n</sup>*�*<sup>k</sup> :* (44)

*akb<sup>n</sup>*�*<sup>k</sup>* , (45)

*:*

(37)

X∞ *n*¼0

$$\mathtt{and}$$

$$\mathbf{C} \coloneqq \frac{1}{[2]\_{p,q}} \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(-b\mathbf{x}) E\_{k,p,q}(ay)}{(-b)^{n-k} a^k} \right) \frac{t^n}{[n]\_{p,q}!}.\tag{54}$$

First, let us find an approximation of the root of ð Þ *p*, *q* -sigmoid polynomials. At this time, *p* and *q* should not be the same value. If they have the same value, the denominator of ð Þ *p*, *q* -sigmoid polynomials will be 0. Consider the roots of S20,*p*, *<sup>q</sup>*. Once we fix the value of *q* to 0*:*1 and switch *p* into 0*:*9, 0*:*5, and 0*:*2, the following

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots*

Here we can see that the approximate values of the roots change as the value of *p* changes, as the roots always contain two real roots. Therefore, we can make the

**Conjecture 3.1.** *Roots of* S20,*p*, *<sup>q</sup> when* ∣*p*∣<1*, n* ¼ 20 *and q* ¼ 0*:*1*, always have two*

In the same way, we can find an approximation of the roots when the values of *q* are changed in order of 0*:*9, 0*:*5, and 0*:*2 and *p* is fixed at 0*:*1. In this case, when *p* ¼ 0*:*1 and *q* ¼ 0*:*2, the approximate roots of S20,*p*, *<sup>q</sup>* all possess real roots which can be confirmed through Mathematica (*x* ¼ �395824, � 136973, � 47951, � 16837,

**Table 1** can be illustrated as **Figure 1**. The figure on the left is when *p* ¼ 0*:*9 and the figure on the right is when *p* ¼ 0*:*2. Here we can see that the structure of the roots is changing. The following will identify the structure and the build-up of the roots of S*<sup>n</sup>*,*p*, *<sup>q</sup>*. The structures of approximations roots in polynomials that combine

�5912, � 2076, � 728, � 255, � 89, �31, � 10, � 3, � 1, � 0, 2, 19, 157,

the existing ð Þ *p*, *q* -number can be found becoming closer to a circle as the *n* increases and can be seen that a single root is continuously stacked at a certain point. Also, as the roots continue to pile up near at some point and the larger the *n* becomes, it can be assumed that S20,*p*, *<sup>q</sup>* becomes closer to the circle. Actually, a

picture of S20,*p*, *<sup>q</sup>* can be made using Mathematica.

*Approximate value of zeros for* S20,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5, 0*:*2*.*

*Zeros structures for* S*<sup>n</sup>*,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5, 0*:*2 *and* 0 ≤ *n*≤50*.*

**Table 1** can be found.

*DOI: http://dx.doi.org/10.5772/intechopen.91862*

following assumptions.

1231, 9610, 71387).

*real roots.*

**Figure 2.**

**119**

**Figure 1.**

Comparing the both sides in the equation above (53) and (54), we complete the required result of Theorem 2.18.

**Corollary 2.19.** *Putting a* ¼ �1 *in the Theorem 2.18, one holds*

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(\mathbf{x}) E\_{k,p,q}(b\mathbf{y})}{b^k} = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \frac{\mathcal{S}\_{n-k,p,q}(-b\mathbf{x}) E\_{k,p,q}(-\mathbf{y})}{(-1)^n b^{n-k}},\tag{55}$$

*where En*,*p*, *<sup>q</sup>*ð Þ *x is the p*ð Þ , *q -Euler polynomials*.

#### **3. Structure and various phenomena of roots of** S*n***,***p***,** *<sup>q</sup>* **using the computer**

This section mentions the structure of roots of S*<sup>n</sup>*,*p*, *<sup>q</sup>*. Furthermore, based on the previous content, an example of ð Þ *p*, *q* -sigmoid polynomials is taken to identify the shape of the fixed points and the iterative function. And by applying it, we can find properties of self-similarity by using Newton's method.


#### **Table 1.** *Approximate zeros of* S20,*p*,0*:*1ð Þ *x .*

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots DOI: http://dx.doi.org/10.5772/intechopen.91862*

First, let us find an approximation of the root of ð Þ *p*, *q* -sigmoid polynomials. At this time, *p* and *q* should not be the same value. If they have the same value, the denominator of ð Þ *p*, *q* -sigmoid polynomials will be 0. Consider the roots of S20,*p*, *<sup>q</sup>*. Once we fix the value of *q* to 0*:*1 and switch *p* into 0*:*9, 0*:*5, and 0*:*2, the following **Table 1** can be found.

Here we can see that the approximate values of the roots change as the value of *p* changes, as the roots always contain two real roots. Therefore, we can make the following assumptions.

**Conjecture 3.1.** *Roots of* S20,*p*, *<sup>q</sup> when* ∣*p*∣<1*, n* ¼ 20 *and q* ¼ 0*:*1*, always have two real roots.*

In the same way, we can find an approximation of the roots when the values of *q* are changed in order of 0*:*9, 0*:*5, and 0*:*2 and *p* is fixed at 0*:*1. In this case, when *p* ¼ 0*:*1 and *q* ¼ 0*:*2, the approximate roots of S20,*p*, *<sup>q</sup>* all possess real roots which can be confirmed through Mathematica (*x* ¼ �395824, � 136973, � 47951, � 16837, �5912, � 2076, � 728, � 255, � 89, �31, � 10, � 3, � 1, � 0, 2, 19, 157, 1231, 9610, 71387).

**Table 1** can be illustrated as **Figure 1**. The figure on the left is when *p* ¼ 0*:*9 and the figure on the right is when *p* ¼ 0*:*2. Here we can see that the structure of the roots is changing. The following will identify the structure and the build-up of the roots of S*<sup>n</sup>*,*p*, *<sup>q</sup>*. The structures of approximations roots in polynomials that combine the existing ð Þ *p*, *q* -number can be found becoming closer to a circle as the *n* increases and can be seen that a single root is continuously stacked at a certain point. Also, as the roots continue to pile up near at some point and the larger the *n* becomes, it can be assumed that S20,*p*, *<sup>q</sup>* becomes closer to the circle. Actually, a picture of S20,*p*, *<sup>q</sup>* can be made using Mathematica.

**Figure 1.** *Approximate value of zeros for* S20,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5, 0*:*2*.*

**Figure 2.** *Zeros structures for* S*<sup>n</sup>*,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5, 0*:*2 *and* 0 ≤ *n*≤50*.*

and

X*n k*¼0

*n k* � �

**computer**

**Table 1.**

**118**

*Approximate zeros of* S20,*p*,0*:*1ð Þ *x .*

*p*, *q*

*<sup>C</sup>*<sup>≔</sup> <sup>1</sup> ½ � 2 *<sup>p</sup>*, *<sup>q</sup>*

*Number Theory and Its Applications*

required result of Theorem 2.18.

X∞ *n*¼0 X*n k*¼0

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð Þ *x Ek*,*p*, *<sup>q</sup>*ð Þ *by*

*where En*,*p*, *<sup>q</sup>*ð Þ *x is the p*ð Þ , *q -Euler polynomials*.

properties of self-similarity by using Newton's method.

*n k* � �

*p*, *q*

**Corollary 2.19.** *Putting a* ¼ �1 *in the Theorem 2.18, one holds*

*bk* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð Þ �*bx Ek*,*p*, *<sup>q</sup>*ð Þ *ay* ð Þ �*<sup>b</sup> <sup>n</sup>*�*<sup>k</sup>*

!

*k*¼0

**3. Structure and various phenomena of roots of** S*n***,***p***,** *<sup>q</sup>* **using the**

*p* ¼ **0***:***9** *p* ¼ **0***:***5** *p* ¼ **0***:***2** �1 �1 �0.99999 �0.390901 � 0.133409i �0.319498 � 0.0466997i �0.632357 � 0.0806528i �0.390901 + 0.133409i �0.319498 + 0.0466997i �0.632357 + 0.0806528i �0.325772 � 0.249143i �0.22888 � 0.212259i �0.574469 � 0.251005i �0.325772 + 0.249143i �0.22888 + 0.212259i �0.574469 + 0.251005i �0.229438 � 0.33589i �0.149885 � 0.266689i �0.458705 � 0.409513i �0.229438 + 0.33589i �0.149885 + 0.266689i �0.458705 + 0.409513i �0.113312 � 0.387358i �0.0593879 � 0.293992i �0.102928 � 0.574072i �0.113312 + 0.387358i �0.0593879 + 0.293992i �0.102928 + 0.574072i 0.0110113 � 0.400647i 0.0334027 � 0.292274i 0.083275 � 0.559033i 0.0110113 + 0.400647i 0.0334027 + 0.292274i 0.083275 + 0.559033i 0.132056 � 0.37596i 0.119568 � 0.262599i 0.136077 0.132056 + 0.37596i 0.119568 + 0.262599i 0.24766 � 0.490886i 0.239081 � 0.316515i 0.18937 0.24766 + 0.490886i 0.239081 + 0.316515i 0.191091 � 0.208427i 0.379531 � 0.381018i 0.322803 � 0.228278i 0.191091 + 0.208427i 0.379531 + 0.381018i 0.322803 + 0.228278i 0.241247 � 0.134952i 0.471422 � 0.24088i 0.376059 � 0.119488i 0.241247 + 0.134952i 0.471422 + 0.24088i 0.376059 + 0.119488i 0.265157 � 0.0476321i 0.518528 � 0.0823508i 0.394325 0.265157 + 0.0476321i 0.518528 + 0.0823508i

Comparing the both sides in the equation above (53) and (54), we complete the

*n k* � �

This section mentions the structure of roots of S*<sup>n</sup>*,*p*, *<sup>q</sup>*. Furthermore, based on the previous content, an example of ð Þ *p*, *q* -sigmoid polynomials is taken to identify the shape of the fixed points and the iterative function. And by applying it, we can find

*p*, *q*

*ak*

*t n* ½ � *n <sup>p</sup>*, *<sup>q</sup>*!

S*<sup>n</sup>*�*k*,*p*, *<sup>q</sup>*ð Þ �*bx Ek*,*p*, *<sup>q</sup>*ð Þ �*y* ð Þ �<sup>1</sup> *<sup>n</sup>*

*:* (54)

*bn*�*<sup>k</sup>* , (55)

times and getting the number of real roots, the value of fixed points will vary depending on the value of *p*. For *p* ¼ 0*:*9 and *p* ¼ 0*:*5, the number of the real roots of each of the five iterated function is 1, 1, 1, 1, 1, 1, but for *p* ¼ 0*:*2, 3, 3, 3, 3, 3 appears. The structure of the approximate value of the actual fixed point is shown in **Figure 6**. The top part of **Figure 6** is when *p* ¼ 0*:*9 and the bottom figures represent *p* ¼ 0*:*2. Typically, most of the roots structures of general polynomials using *q*numbers appear a circular shape, but it is difficult to find constant regularity in fixed points. However, sigmoid function including ð Þ *p*, *q* -numbers might have a special property for fixed points. In other words, we can guess from **Figure 6** that the structure of fixed points for ð Þ *p*, *q* -sigmoid polynomials will become a circular

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots*

Let us look at the following by observing an application of an iterated S3,*p*,0*:*<sup>1</sup> using **Figure 6**. Let us try using the Newton's method that we know well. Let us divide the values that go to the root of the tertiary function. First, fix *p* at 0*:*9 and limit the range of values of *x* and *y* from �4 to 4. Then the approximate root of S3,0*:*9,0*:*<sup>1</sup> becomes �0*:*936067, 0*:*187169 � 0*:*256352*i*, 0*:*187169 þ 0*:*256352*i*. Also, if the values going to �0*:*93606 are shown in red, 0*:*187169 � 0*:*256352*i* in blue, 0*:*187169 þ 0*:*256352*i* in yellow, it is shown as the left of **Figure 7**. The right side of

The structure of the roots of S*<sup>n</sup>*,*p*, *<sup>q</sup>* appears to have one value near �1 and become a circular form as the *n* increases. Also, as the value of *p* increases, the

**Figure 7** is the picture that comes when S3,0*:*9,0*:*<sup>1</sup> are iterated twice.

shape if *n* increases tremendously.

*DOI: http://dx.doi.org/10.5772/intechopen.91862*

**Figure 6.**

**121**

*Scattering of fixed points for* S3,*p*,0*:*1ð Þ *x iterated 5-times.*

**Figure 3.** *Zeros scattering for* S*<sup>n</sup>*,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5, 0*:*2 *and* 0 ≤ *n* ≤ 50*.*

The following **Figure 2** shows the structure of the roots when *n* is 0 to 50. When fixed at *q* ¼ 0*:*1, the figure on the left is when *p* ¼ 0*:*9 and the right is when *p* ¼ 0*:*2.

Three-dimensional identification of **Figure 2** shows the following **Figure 3**. Given the speculation, the roots are piling up near a point (*x* ¼ �1) and as the *p* approaches 1, the rest of the roots become closer to a circle as the value of *n* increases.

Based on the content above, we will now look at fixed points of S*<sup>n</sup>*,*p*, *<sup>q</sup>*. At this point, the value of *q* is fixed at 0*:*1 and *p* is changed to 0*:*9, 0*:*5, and 0*:*2 respectively. This can be found as shown in the following **Figure 4**. **Figure 4** shows a nearly circular appearance and always has the origin. Also, as the value of *p* decreases, it can be seen that the distance from the origin and the roots increases.

Similarly, the fixed points in the 3D structure can be checked as shown in **Figure 5**.

**Conjecture 3.2.** S*<sup>n</sup>*,*p*, *<sup>q</sup> may have one fixed point which is the origin and the rest of the fixed points appear in the form of a circle.*

The following is a third polynomial of S3,*p*, *<sup>q</sup>*, using a iterative function to find the approximate value of the fixed point. First, by iterating this third function five

**Figure 4.** *Fixed points for* S50,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5, 0*:*2*.*

**Figure 5.** *Scattering of fixed points for* S*<sup>n</sup>*,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5 *and* 0≤ *n*≤50*.*

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots DOI: http://dx.doi.org/10.5772/intechopen.91862*

times and getting the number of real roots, the value of fixed points will vary depending on the value of *p*. For *p* ¼ 0*:*9 and *p* ¼ 0*:*5, the number of the real roots of each of the five iterated function is 1, 1, 1, 1, 1, 1, but for *p* ¼ 0*:*2, 3, 3, 3, 3, 3 appears. The structure of the approximate value of the actual fixed point is shown in **Figure 6**. The top part of **Figure 6** is when *p* ¼ 0*:*9 and the bottom figures represent *p* ¼ 0*:*2. Typically, most of the roots structures of general polynomials using *q*numbers appear a circular shape, but it is difficult to find constant regularity in fixed points. However, sigmoid function including ð Þ *p*, *q* -numbers might have a special property for fixed points. In other words, we can guess from **Figure 6** that the structure of fixed points for ð Þ *p*, *q* -sigmoid polynomials will become a circular shape if *n* increases tremendously.

Let us look at the following by observing an application of an iterated S3,*p*,0*:*<sup>1</sup> using **Figure 6**. Let us try using the Newton's method that we know well. Let us divide the values that go to the root of the tertiary function. First, fix *p* at 0*:*9 and limit the range of values of *x* and *y* from �4 to 4. Then the approximate root of S3,0*:*9,0*:*<sup>1</sup> becomes �0*:*936067, 0*:*187169 � 0*:*256352*i*, 0*:*187169 þ 0*:*256352*i*. Also, if the values going to �0*:*93606 are shown in red, 0*:*187169 � 0*:*256352*i* in blue, 0*:*187169 þ 0*:*256352*i* in yellow, it is shown as the left of **Figure 7**. The right side of **Figure 7** is the picture that comes when S3,0*:*9,0*:*<sup>1</sup> are iterated twice.

The structure of the roots of S*<sup>n</sup>*,*p*, *<sup>q</sup>* appears to have one value near �1 and become a circular form as the *n* increases. Also, as the value of *p* increases, the

**Figure 6.** *Scattering of fixed points for* S3,*p*,0*:*1ð Þ *x iterated 5-times.*

The following **Figure 2** shows the structure of the roots when *n* is 0 to 50. When fixed at *q* ¼ 0*:*1, the figure on the left is when *p* ¼ 0*:*9 and the right is when *p* ¼ 0*:*2. Three-dimensional identification of **Figure 2** shows the following **Figure 3**. Given the speculation, the roots are piling up near a point (*x* ¼ �1) and as the *p* approaches 1, the rest of the roots become closer to a circle as the value of *n*

Based on the content above, we will now look at fixed points of S*<sup>n</sup>*,*p*, *<sup>q</sup>*. At this point, the value of *q* is fixed at 0*:*1 and *p* is changed to 0*:*9, 0*:*5, and 0*:*2 respectively. This can be found as shown in the following **Figure 4**. **Figure 4** shows a nearly circular appearance and always has the origin. Also, as the value of *p* decreases, it

Similarly, the fixed points in the 3D structure can be checked as shown in

approximate value of the fixed point. First, by iterating this third function five

**Conjecture 3.2.** S*<sup>n</sup>*,*p*, *<sup>q</sup> may have one fixed point which is the origin and the rest of the*

The following is a third polynomial of S3,*p*, *<sup>q</sup>*, using a iterative function to find the

can be seen that the distance from the origin and the roots increases.

increases.

**Figure 3.**

**Figure 5**.

**Figure 5.**

**120**

**Figure 4.**

*fixed points appear in the form of a circle.*

*Fixed points for* S50,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5, 0*:*2*.*

*Scattering of fixed points for* S*<sup>n</sup>*,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5 *and* 0≤ *n*≤50*.*

*Zeros scattering for* S*<sup>n</sup>*,*p*,0*:*1ð Þ *x for p* ¼ 0*:*9, 0*:*5, 0*:*2 *and* 0 ≤ *n* ≤50*.*

*Number Theory and Its Applications*

**References**

1991;**24**:L711

net/literature/319903

Series A. 1991;**56**:27-46

Press; 1999

**301**:1-8

2008;**8**:A29

**123**

[1] Chakrabarti R, Jagannathan R. A ð Þ *p*, *q* -oscillator realization of twoparameter quantum algebras. Journal of Physics A: Mathematical and General.

*DOI: http://dx.doi.org/10.5772/intechopen.91862*

Computational and Theoretical

[11] Han J, Moraga C. The influence of the sigmoid function parameters on the speed of backpropagation learning. International Conference on Artificial Neural Networks. 2005;**930**:195-201

[12] Han J, Wilson RS, Leurgans SE. Sigmoidal mixed models for longitudinal data. Statistical Methods in Medical Research. March 2018;**27**(3):863-875. DOI: 10.1177/0962280216645632

[13] Jagannathan R, Rao KS. Twoparameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. In: Proceeding of

the International Conference on Number Theory and Mathematical Physics; 20–21 December 2005. Kumbakonam, India: Srinivasa Ramanujan Centre; 2005

[14] Jagannathan R. ð Þ *P*, *Q* -special functions, special functions and

[15] Kwan HK. Simple sigmoid-like activation function suitable for digital hardware implementation. Statistical Methods in Medical Research. 1992;**28**: 1379-1380. DOI: 10.1049/el:19920877

[16] Kang JY. Some relationships

[17] Kang JY, Ryoo CS. A numerical investigation on the structure of the zeros

Polynomials - Theory and Application. 2019:1-18. Available from: https://www. intechopen.com/books/polynomials-

of the *q*-tangent polynomials.

theory-and-application

between sigmoid polynomials and other polynomials. Journal of Pure and Applied Mathematics. 2019;**1**(1–2):57-67

January. 1997. pp. 13-24

differential equations. In: Proceedings of a Workshop Held at the Institute of Mathematical Sciences, Madras, India,

Nanoscience. 2016

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots*

[2] Arik M, Demircan E, Turgut T, Ekinci L, Mungan M. Fibonacci oscillators. Zeitschrift für Physik C Particles and Fields. 1991;**55**:89-95

[3] Brodimas G, Jannussis A, Mignani R. Two-Parameter Quantum Groups. Universita di Roma; Preprint, Nr. 820; 1991. Available from: https://inspirehep.

[4] Wachs M, White D. ð Þ *p*, *q -*Stirling numbers and set partition statistics. Journal of Combinatorial Theory,

[5] Andrews GE, Askey R, Roy R. Special Functions. Cambridge, UK: Cambridge

[6] Araci S, Duran U, Acikgoz M, Srivastava HM. A certain ð Þ *p*, *q* derivative operator and associated divided differences. Journal of Inequalities and Applications. 2016;

[7] Burban M, Klimyk AU. ð Þ *P*, *Q* differentiation, ð Þ *P*, *Q* -integration and ð Þ *P*, *Q* -hypergeometric functions related to quantum groups. Integral Transforms and Special Functions. 1994;**2**:15-36

[8] Cieslinski JL. Improved *q*-exponential and *q*-trigonometric functions. 2010. arXiv:1006.5652v1 [math.CA]

[9] Corcino RB. On ð Þ *P*, *Q* -Binomial coefficients. Integers Electronic Journal of Combinatorial Number Theory.

[10] Duran U, Acikgoz M, Araci S. On ð Þ *p*, *q* -Bernoulli, ð Þ *p*, *q* -Euler and ð Þ *p*, *q* - Genocchi polynomials. Journal of

**Figure 7.** *Classification of values that go near the fixed points of* S3,0*:*9,0*:*1*.*

diameter of the circle increases. A fixed point of S*<sup>n</sup>*,*p*, *<sup>q</sup>* can be seen to have a nearly constant form as *p* is reduced, which can also confirm an increase in the radius.

### **4. Conclusion**

Sigmoid function is a very important function in deep learning. In the current situation of artificial intelligence development, the properties and speculations of the sigmoid polynomials revealed in this paper in the area of using ð Þ *p*, *q* -number could be an useful data in deep learning using activation functions. Through iterating S*<sup>n</sup>*,*p*, *<sup>q</sup>* with these properties, it can be assumed to have self-similarity and can be studied further to confirm new properties.

### **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).

#### **Additional information**

Mathematics Subject Classification: 11B68, 11B75, 12D10

#### **Author details**

Jung Yoog Kang Department of Mathematics Education, Silla University, Busan, Republic of Korea

\*Address all correspondence to: jykang@silla.ac.kr

© 2020 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.

*Determination of the Properties of (*p, q*)‐Sigmoid Polynomials and the Structure of Their Roots DOI: http://dx.doi.org/10.5772/intechopen.91862*

#### **References**

diameter of the circle increases. A fixed point of S*<sup>n</sup>*,*p*, *<sup>q</sup>* can be seen to have a nearly constant form as *p* is reduced, which can also confirm an increase in the radius.

Sigmoid function is a very important function in deep learning. In the current situation of artificial intelligence development, the properties and speculations of the sigmoid polynomials revealed in this paper in the area of using ð Þ *p*, *q* -number could be an useful data in deep learning using activation functions. Through iterating S*<sup>n</sup>*,*p*, *<sup>q</sup>* with these properties, it can be assumed to have self-similarity and can be

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science,

Department of Mathematics Education, Silla University, Busan, Republic of Korea

© 2020 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,

**4. Conclusion**

**Figure 7.**

**Acknowledgements**

**Additional information**

**Author details**

Jung Yoog Kang

**122**

studied further to confirm new properties.

*Classification of values that go near the fixed points of* S3,0*:*9,0*:*1*.*

*Number Theory and Its Applications*

ICT and Future Planning (No. 2017R1E1A1A03070483).

\*Address all correspondence to: jykang@silla.ac.kr

provided the original work is properly cited.

Mathematics Subject Classification: 11B68, 11B75, 12D10

[1] Chakrabarti R, Jagannathan R. A ð Þ *p*, *q* -oscillator realization of twoparameter quantum algebras. Journal of Physics A: Mathematical and General. 1991;**24**:L711

[2] Arik M, Demircan E, Turgut T, Ekinci L, Mungan M. Fibonacci oscillators. Zeitschrift für Physik C Particles and Fields. 1991;**55**:89-95

[3] Brodimas G, Jannussis A, Mignani R. Two-Parameter Quantum Groups. Universita di Roma; Preprint, Nr. 820; 1991. Available from: https://inspirehep. net/literature/319903

[4] Wachs M, White D. ð Þ *p*, *q -*Stirling numbers and set partition statistics. Journal of Combinatorial Theory, Series A. 1991;**56**:27-46

[5] Andrews GE, Askey R, Roy R. Special Functions. Cambridge, UK: Cambridge Press; 1999

[6] Araci S, Duran U, Acikgoz M, Srivastava HM. A certain ð Þ *p*, *q* derivative operator and associated divided differences. Journal of Inequalities and Applications. 2016; **301**:1-8

[7] Burban M, Klimyk AU. ð Þ *P*, *Q* differentiation, ð Þ *P*, *Q* -integration and ð Þ *P*, *Q* -hypergeometric functions related to quantum groups. Integral Transforms and Special Functions. 1994;**2**:15-36

[8] Cieslinski JL. Improved *q*-exponential and *q*-trigonometric functions. 2010. arXiv:1006.5652v1 [math.CA]

[9] Corcino RB. On ð Þ *P*, *Q* -Binomial coefficients. Integers Electronic Journal of Combinatorial Number Theory. 2008;**8**:A29

[10] Duran U, Acikgoz M, Araci S. On ð Þ *p*, *q* -Bernoulli, ð Þ *p*, *q* -Euler and ð Þ *p*, *q* - Genocchi polynomials. Journal of

Computational and Theoretical Nanoscience. 2016

[11] Han J, Moraga C. The influence of the sigmoid function parameters on the speed of backpropagation learning. International Conference on Artificial Neural Networks. 2005;**930**:195-201

[12] Han J, Wilson RS, Leurgans SE. Sigmoidal mixed models for longitudinal data. Statistical Methods in Medical Research. March 2018;**27**(3):863-875. DOI: 10.1177/0962280216645632

[13] Jagannathan R, Rao KS. Twoparameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. In: Proceeding of the International Conference on Number Theory and Mathematical Physics; 20–21 December 2005. Kumbakonam, India: Srinivasa Ramanujan Centre; 2005

[14] Jagannathan R. ð Þ *P*, *Q* -special functions, special functions and differential equations. In: Proceedings of a Workshop Held at the Institute of Mathematical Sciences, Madras, India, January. 1997. pp. 13-24

[15] Kwan HK. Simple sigmoid-like activation function suitable for digital hardware implementation. Statistical Methods in Medical Research. 1992;**28**: 1379-1380. DOI: 10.1049/el:19920877

[16] Kang JY. Some relationships between sigmoid polynomials and other polynomials. Journal of Pure and Applied Mathematics. 2019;**1**(1–2):57-67

[17] Kang JY, Ryoo CS. A numerical investigation on the structure of the zeros of the *q*-tangent polynomials. Polynomials - Theory and Application. 2019:1-18. Available from: https://www. intechopen.com/books/polynomialstheory-and-application

[18] Ryoo CS, Kang JY. A numerical investigation on the structure of the zeros of the Euler polynomials. Discrete Dynamics in Nature and Society. 2015: 1-9. DOI: 10.1155/2015/174173

**Chapter 8**

**Abstract**

*f n*ð Þ¼ <sup>Q</sup>

<sup>I</sup>ð Þ*<sup>q</sup>*

binomial coefficients

**1. Introduction**

sequences, mainly by <sup>I</sup>ð Þ*<sup>q</sup>*

the sequences <sup>Ω</sup>ð Þ *<sup>n</sup>*

**125**

log log *<sup>n</sup>* and *<sup>ω</sup>*ð Þ *<sup>n</sup>*

log log *<sup>n</sup>* are <sup>I</sup>ð Þ*<sup>q</sup>*

Functions

*Vladimír Baláž and Tomáš Visnyai*

*<sup>d</sup>*∣*<sup>n</sup><sup>d</sup>* and *<sup>f</sup>* <sup>∗</sup> ð Þ¼ *<sup>n</sup> f n*ð Þ

well-known arithmetical functions, where I¼Ið Þ*<sup>q</sup>*

an admissible ideal on such that for *<sup>q</sup>*∈ð0, 1<sup>i</sup> we have <sup>I</sup>ð Þ*<sup>q</sup>*

I–Convergence of Arithmetical

Let *<sup>n</sup>*<sup>&</sup>gt; 1 be an integer with its canonical representation, *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup><sup>α</sup>*<sup>1</sup>

*H n*ð Þ¼ max f g *α*1, … , *α<sup>k</sup>* , *h n*ð Þ¼ min f g *α*1, … , *α<sup>k</sup>* , *ω*ð Þ¼ *n k*, Ωð Þ¼ *n α*<sup>1</sup> þ ⋯ þ *αk*,

of these arithmetical functions. For instance, the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with I*d*–convergence, where I*<sup>d</sup>* is the ideal of all subsets of positive integers having the asymptotic density zero. In this part, we will study I–convergence of the

*<sup>c</sup>* –convergence is stronger than the statistical convergence (I*d*–convergence).

The notion of statistical convergence was introduced independently by Fast and Schoenberg in [1, 2], and the notion of I–convergence introduced by Kostyrko et al. in the paper [3] coresponds to the natural generalization of statistical convergence (see also [4] where I–convergence is defined by means of filter – the dual notion to ideal). These notions have been developed in several directions in [5–18] and have been used in various parts of mathematics, in particular in Number Theory and Ergodic Theory, for example [15, 19–28] also in Economic Theory [29, 30] and Political Science [31]. Many authors deal with average and normal order of the wellknown arithmetical functions (see [20, 21, 23, 24, 26, 28, 32, 33] and the monograph [34] for basic properties of the well-known arithmetical functions). In what follows, we shall strengthen these results from the standpoint of I–convergence of

that we can obtain a good information about behaviour and properties of the wellknown arithmetical functions by investigating I–convergence of these functions or some sequences connected with these functions. Specifically in [28] by means of I*d*–convergence, there is recalled the result that normal order of Ωð Þ *n* or *ω*ð Þ *n* respectively is log log *n*. We managed to completely determine for which *q*∈ ð0, 1i

we have that the above sequences are I*d*–convergent to 1, what is equivalent that

**Keywords:** sequences, I–convergence, arithmetical functions, normal order,

<sup>1</sup> *<sup>p</sup><sup>α</sup>*<sup>2</sup> <sup>2</sup> <sup>⋯</sup>*p<sup>α</sup><sup>k</sup>*

P *<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>a*�*<sup>q</sup>* <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup> � � is

*<sup>c</sup>* ⊊ I*d*, thus

*<sup>n</sup>* . Many authors deal with the statistical convergence

*<sup>c</sup>* ¼ *A* ⊂ :

*<sup>c</sup>* –convergence and I*u*–convergence. On connection with

*<sup>c</sup>* –convergent. As consequence of our results,

*<sup>k</sup>* . Put

[19] Ryoo CS. A numerical investigation on the zeros of the tangent polynomials. Journal of Applied Mathematics and Informatics. 2014;**32**:315-322

[20] Kurt B. Relations on the Apostol type ð Þ *p*, *q* -Frobenius-Euler polynomials and generalizations of the Srivastava-Pinter addition theorems. Turkish Journal of Analysis and Number Theory. 2017;**5**:126131

[21] Sadjang PN. On the fundamental theorem of ð Þ *p*, *q* -calculus and some ð Þ *p*, *q* -Taylor formulas. 2013. arXiv: 1309.3934 [math.QA]

**Chapter 8**

[18] Ryoo CS, Kang JY. A numerical investigation on the structure of the zeros of the Euler polynomials. Discrete Dynamics in Nature and Society. 2015:

*Number Theory and Its Applications*

[19] Ryoo CS. A numerical investigation on the zeros of the tangent polynomials. Journal of Applied Mathematics and

[20] Kurt B. Relations on the Apostol type ð Þ *p*, *q* -Frobenius-Euler polynomials and generalizations of the Srivastava-Pinter addition theorems. Turkish Journal of Analysis and Number Theory.

[21] Sadjang PN. On the fundamental theorem of ð Þ *p*, *q* -calculus and some ð Þ *p*, *q* -Taylor formulas. 2013. arXiv:

1-9. DOI: 10.1155/2015/174173

Informatics. 2014;**32**:315-322

2017;**5**:126131

**124**

1309.3934 [math.QA]

## I–Convergence of Arithmetical Functions

*Vladimír Baláž and Tomáš Visnyai*

#### **Abstract**

Let *<sup>n</sup>*<sup>&</sup>gt; 1 be an integer with its canonical representation, *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup><sup>α</sup>*<sup>1</sup> <sup>1</sup> *<sup>p</sup><sup>α</sup>*<sup>2</sup> <sup>2</sup> <sup>⋯</sup>*p<sup>α</sup><sup>k</sup> <sup>k</sup>* . Put *H n*ð Þ¼ max f g *α*1, … , *α<sup>k</sup>* , *h n*ð Þ¼ min f g *α*1, … , *α<sup>k</sup>* , *ω*ð Þ¼ *n k*, Ωð Þ¼ *n α*<sup>1</sup> þ ⋯ þ *αk*, *f n*ð Þ¼ <sup>Q</sup> *<sup>d</sup>*∣*<sup>n</sup><sup>d</sup>* and *<sup>f</sup>* <sup>∗</sup> ð Þ¼ *<sup>n</sup> f n*ð Þ *<sup>n</sup>* . Many authors deal with the statistical convergence of these arithmetical functions. For instance, the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with I*d*–convergence, where I*<sup>d</sup>* is the ideal of all subsets of positive integers having the asymptotic density zero. In this part, we will study I–convergence of the well-known arithmetical functions, where I¼Ið Þ*<sup>q</sup> <sup>c</sup>* ¼ *A* ⊂ : P *<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>a*�*<sup>q</sup>* <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup> � � is an admissible ideal on such that for *<sup>q</sup>*∈ð0, 1<sup>i</sup> we have <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* ⊊ I*d*, thus <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence is stronger than the statistical convergence (I*d*–convergence).

**Keywords:** sequences, I–convergence, arithmetical functions, normal order, binomial coefficients

#### **1. Introduction**

The notion of statistical convergence was introduced independently by Fast and Schoenberg in [1, 2], and the notion of I–convergence introduced by Kostyrko et al. in the paper [3] coresponds to the natural generalization of statistical convergence (see also [4] where I–convergence is defined by means of filter – the dual notion to ideal). These notions have been developed in several directions in [5–18] and have been used in various parts of mathematics, in particular in Number Theory and Ergodic Theory, for example [15, 19–28] also in Economic Theory [29, 30] and Political Science [31]. Many authors deal with average and normal order of the wellknown arithmetical functions (see [20, 21, 23, 24, 26, 28, 32, 33] and the monograph [34] for basic properties of the well-known arithmetical functions). In what follows, we shall strengthen these results from the standpoint of I–convergence of sequences, mainly by <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence and I*u*–convergence. On connection with that we can obtain a good information about behaviour and properties of the wellknown arithmetical functions by investigating I–convergence of these functions or some sequences connected with these functions. Specifically in [28] by means of I*d*–convergence, there is recalled the result that normal order of Ωð Þ *n* or *ω*ð Þ *n* respectively is log log *n*. We managed to completely determine for which *q*∈ ð0, 1i the sequences <sup>Ω</sup>ð Þ *<sup>n</sup>* log log *<sup>n</sup>* and *<sup>ω</sup>*ð Þ *<sup>n</sup>* log log *<sup>n</sup>* are <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergent. As consequence of our results, we have that the above sequences are I*d*–convergent to 1, what is equivalent that

normal order of Ωð Þ *n* or *ω*ð Þ *n* respectively is log log *n*. Further in [26], there is proved that the sequence log *p* � *ap*ð Þ *n* log *n* � � is <sup>I</sup>*d*–convergent to 0 (see also [21]). We shall extend this result by means of I*u*–convergence of the sequence log *p* � *ap*ð Þ *n* log *n* � �. So we can get a better view of the structure of the set *B*ð Þ¼ *ε n*∈ : log *p* � *ap*ð Þ *n* log *<sup>n</sup>* <*ε* n o, *<sup>ε</sup>*>0. We also want to investigate the <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence of further arithmetical functions.

4.define *h(n)* and *H(n)*, put *h*ð Þ¼ 1 1, *H*ð Þ¼ 1 1 and for *n*>1 denote

*<sup>n</sup>* , where *n* ¼ 1, 2, … ,

6.let *p* be a prime number, *ap*ð Þ *n* is defined as follows: *ap*ð Þ¼ 1 0 and if *n*> 0,

7.*γ*ð Þ *n* and *τ*ð Þ *n* – were introduced in connection with representation of natural

<sup>2</sup> ¼ ⋯ ¼ *a*

be all such representations of given natural number *n*, where *ai*, *bi* ∈ . Denote by

*τ*ð Þ¼ *n b*<sup>1</sup> þ *b*<sup>2</sup> þ ⋯ þ *bγ*ð Þ *<sup>n</sup>* , ð Þ *n* >1 *:*

It is clear that *γ*ð Þ *n* ≥1, because for any *n* >1 there exist representation in the

A lot of mathematical disciplines use the term small (large) set from different point of view. For instance a final set, a set having the measure zero and nowhere dense set is a small set from point of view of cardinality, measure (probability) and topology, respectively. The notion of ideal <sup>I</sup> <sup>⊆</sup> <sup>2</sup>*<sup>X</sup>* is the unifying principle how to express that a subset of *X* 6¼ Ø is small. We say a set *A* ⊆ *X* is a small set if *A* ∈I.

Let <sup>I</sup> <sup>⊆</sup> <sup>2</sup>. <sup>I</sup> is said to be an *ideal* in , if <sup>I</sup> is additive (if *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup><sup>I</sup> then *<sup>A</sup>* <sup>∪</sup> *<sup>B</sup>*∈I) and hereditary (if *A* ∈I and *B* ⊂ *A* then *B*∈ I). An ideal I is said to be *non-trivial ideal* if I 6¼ Ø and ∉ I. A non-trivial ideal I is said to be *admissible ideal* if it contains all finite subsets of . The dual notion to the ideal is the notion filter. A non-empty family of sets <sup>F</sup> <sup>⊂</sup> <sup>2</sup> is a *filter* if and only if Ø <sup>∉</sup> <sup>F</sup>, for each *<sup>A</sup>*, *<sup>B</sup>*<sup>∈</sup> <sup>F</sup> we have *A* ∩ *B* ∈ F and for each *A* ∈ F and each *B*⊃ *A* we have *B* ∈ F (for definitions see e.g. [4, 41, 42]). Let I be a proper ideal in (i.e. ∉ I). Then a family of sets F Ið Þ¼ f g *B* ⊆ : *there exists A* ∈I *such that B* ¼ n*A* is a filter in , so called

The following example shows the most commonly used admissible ideals in

a. The class of all finite subsets of forms an admissible ideal usually denoted

b. Let *ϱ* be a density function on , the set I*<sup>ϱ</sup>* ¼ f g *A* ⊂ : *ϱ*ð Þ¼ *A* 0 is an admissible ideal. We will use namely the ideals I*d*, I*<sup>δ</sup>* and I*<sup>u</sup>* related to

c. A wide class of ideals I can be obtained by means of regular non negative

asymptotic, logarithmic and uniform density, respectively.

matrixes **<sup>T</sup>** <sup>¼</sup> ð Þ *tn*,*<sup>k</sup> <sup>n</sup>*,*k*<sup>∈</sup> (see [43]). For *<sup>A</sup>* <sup>⊂</sup> , we put *<sup>d</sup>*ð Þ *<sup>n</sup>*

numbers of the form *<sup>n</sup>* <sup>¼</sup> *ab*, where *<sup>a</sup>*, *<sup>b</sup>* are positive integers. Let

<sup>1</sup> <sup>¼</sup> *<sup>a</sup>b*<sup>2</sup>

*<sup>n</sup>* <sup>¼</sup> *ab*<sup>1</sup>

*α <sup>j</sup>*, *H n*ð Þ¼ max

1≤*j*≤*k*

*bγ*ð Þ *<sup>n</sup> γ*ð Þ *n* *α <sup>j</sup>*,

∣*n*, but *p <sup>j</sup>*þ<sup>1</sup>∤*n* i.e., *pap*ð Þ *<sup>n</sup>* ∥*n*,

**<sup>T</sup>** ð Þ¼ *<sup>A</sup>* <sup>P</sup><sup>∞</sup>

*<sup>k</sup>*¼<sup>1</sup>*tn*,*<sup>k</sup>χA*ð Þ*<sup>k</sup>*

*h n*ð Þ¼ min 1≤*j* ≤*k*

then *ap*ð Þ *<sup>n</sup>* is a unique integer *<sup>j</sup>*≥0 satisfying *<sup>p</sup> <sup>j</sup>*

*<sup>d</sup>*∣*nd*, *<sup>f</sup>* <sup>∗</sup> ð Þ¼ *<sup>n</sup> f n*ð Þ

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

Recall the notion of an ideal I of subsets of .

the *associated filter* with the ideal I.

different areas of mathematics.

**Example 1.2.**

by I *<sup>f</sup>* .

**127**

5.*f n*ð Þ¼ <sup>Q</sup>

form *n*1.

**3. Ideals**

#### **2. Basic notions**

Let be the set of positive integers. Let *A* ⊆ . If *m*, *n*∈ , *m* ≤*n*, we denote by *A m*ð Þ , *n* the cardinality of the set *A* ∩ ½ � *m*, *n* . *A*ð Þ 1, *n* is abbreviated by *A n*ð Þ. We recall the concept of asymthotic, logarithmic and uniform density of the set *A* ⊆ (see [35–38]).

**Definition 1.1.** *Let A* ⊆ *.*


$$
\underline{u}(A) \le \underline{d}(A) \le \underline{\delta}(A) \le \overline{\delta}(A) \le \overline{d}(A) \le \overline{u}(A). \tag{1}
$$

Further densities can be found in papers [11, 12].

Let *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup><sup>α</sup>*<sup>1</sup> <sup>1</sup> *<sup>p</sup><sup>α</sup>*<sup>2</sup> <sup>2</sup> <sup>⋯</sup>*p<sup>α</sup><sup>k</sup> <sup>k</sup>* be the canonical representation of the integer *n*∈ . Recall some arithmetical functions, which belong to our interest.

1.*ω*ð Þ *n* – the number of distinct prime factors of *n* ð Þ *ω*ð Þ¼ *n k* ,

2.Ωð Þ *n* – the number of prime factors of *n* counted with multiplicities ð Þ Ωð Þ¼ *n α*<sup>1</sup> þ ⋯ þ *α<sup>k</sup>* ,

3.*d n*ð Þ – the number of divisors of *n dn*ð Þ¼ <sup>P</sup> *<sup>d</sup>*∣*<sup>n</sup>*1 � �, 4.define *h(n)* and *H(n)*, put *h*ð Þ¼ 1 1, *H*ð Þ¼ 1 1 and for *n*>1 denote

$$h(n) = \min\_{1 \le j \le k} a\_j, \quad H(n) = \max\_{1 \le j \le k} a\_j,$$

5.*f n*ð Þ¼ <sup>Q</sup> *<sup>d</sup>*∣*nd*, *<sup>f</sup>* <sup>∗</sup> ð Þ¼ *<sup>n</sup> f n*ð Þ *<sup>n</sup>* , where *n* ¼ 1, 2, … ,

6.let *p* be a prime number, *ap*ð Þ *n* is defined as follows: *ap*ð Þ¼ 1 0 and if *n*> 0, then *ap*ð Þ *<sup>n</sup>* is a unique integer *<sup>j</sup>*≥0 satisfying *<sup>p</sup> <sup>j</sup>* ∣*n*, but *p <sup>j</sup>*þ<sup>1</sup>∤*n* i.e., *pap*ð Þ *<sup>n</sup>* ∥*n*,

7.*γ*ð Þ *n* and *τ*ð Þ *n* – were introduced in connection with representation of natural numbers of the form *<sup>n</sup>* <sup>¼</sup> *ab*, where *<sup>a</sup>*, *<sup>b</sup>* are positive integers. Let

$$n = a\_1^{b\_1} = a\_2^{b\_2} = \dots = a\_{r(n)}^{b\_{r(n)}}$$

be all such representations of given natural number *n*, where *ai*, *bi* ∈ . Denote by

$$
\pi(n) = b\_1 + b\_2 + \dots + b\_{\gamma(n)}, \quad (n > 1).
$$

It is clear that *γ*ð Þ *n* ≥1, because for any *n* >1 there exist representation in the form *n*1.

#### **3. Ideals**

normal order of Ωð Þ *n* or *ω*ð Þ *n* respectively is log log *n*. Further in [26], there is

shall extend this result by means of I*u*–convergence of the sequence log *p* �

Let be the set of positive integers. Let *A* ⊆ . If *m*, *n*∈ , *m* ≤*n*, we denote by *A m*ð Þ , *n* the cardinality of the set *A* ∩ ½ � *m*, *n* . *A*ð Þ 1, *n* is abbreviated by *A n*ð Þ. We recall the concept of asymthotic, logarithmic and uniform density of the set *A* ⊆ (see

limsup*<sup>n</sup>*!<sup>∞</sup>*dn*ð Þ *<sup>A</sup>* are called the *lower* and *upper asymptotic density* of the set *<sup>A</sup>*, respectively. If there exists lim *<sup>n</sup>*!<sup>∞</sup> *dn*ð Þ *A* , then *d A*ð Þ¼ *d A*ð Þ¼ *d A*ð Þ is said to

> *k*¼1 1

*<sup>s</sup>* , *u A*ð Þ¼ lim *<sup>s</sup>*!<sup>∞</sup> *<sup>α</sup><sup>s</sup>*

*u A*ð Þ≤ *d A*ð Þ≤*δ*ð Þ *A* ≤*δ*ð Þ *A* ≤*d A*ð Þ≤ *u A*ð Þ*:* (1)

*<sup>k</sup>* be the canonical representation of the integer *n*∈ . Recall

*<sup>d</sup>*∣*<sup>n</sup>*1

,

� �

lim *<sup>n</sup>*!<sup>∞</sup> *δn*ð Þ *A* , then *δ*ð Þ¼ *A δ*ð Þ¼ *A δ*ð Þ *A* is said to be the *logarithmic density* of

lim inf *<sup>n</sup>*!<sup>∞</sup> *<sup>δ</sup>n*ð Þ *<sup>A</sup>* and *<sup>δ</sup>*ð Þ¼ *<sup>A</sup>* lim sup*<sup>n</sup>*!<sup>∞</sup> *<sup>δ</sup>n*ð Þ *<sup>A</sup>* are called the *lower* and *upper logarithmic density* of *A*, respectively. Similarly, if there exists

c. Put *<sup>α</sup><sup>s</sup>* <sup>¼</sup> min *<sup>n</sup>*≥<sup>0</sup> *A n*ð Þ <sup>þ</sup> 1, *<sup>n</sup>* <sup>þ</sup> *<sup>s</sup>* and *<sup>α</sup><sup>s</sup>* <sup>¼</sup> max *<sup>n</sup>*≥<sup>0</sup> *A n*ð Þ <sup>þ</sup> 1, *<sup>n</sup>* <sup>þ</sup> *<sup>s</sup>* . The

39, 40]) and they are called *lower* and *upper uniform density* of the set *A*, respectively. If *u A*ð Þ¼ *u A*ð Þ, then we denote it by *u A*ð Þ and it is called the

So we can get a better view of the structure of the set *B*ð Þ¼ *ε n*∈ : log *p* �

set *A*. Then the numbers *d A*ð Þ¼ lim inf *<sup>n</sup>*!<sup>∞</sup>*dn*ð Þ *A* and *d A*ð Þ¼

*<sup>k</sup>* , where *sn* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

*k*

*uniform density* of *A*. It is clear that for each *A* ⊆ we have

1.*ω*ð Þ *n* – the number of distinct prime factors of *n* ð Þ *ω*ð Þ¼ *n k* ,

2.Ωð Þ *n* – the number of prime factors of *n* counted with multiplicities

replaced by log *n* in the definition of *δn*ð Þ *A* .

Further densities can be found in papers [11, 12].

3.*d n*ð Þ – the number of divisors of *n dn*ð Þ¼ <sup>P</sup>

some arithmetical functions, which belong to our interest.

is I*d*–convergent to 0 (see also [21]). We

*<sup>c</sup>* –convergence of further arithmetical

*<sup>k</sup>*¼<sup>1</sup>*χA*ð Þ*<sup>k</sup>* , where *<sup>χ</sup><sup>A</sup>* is the characteristic function of the

*<sup>k</sup>*. Then the numbers *δ*ð Þ¼ *A*

*<sup>s</sup>* exist (see [17, 37,

� � for *<sup>n</sup>* ! <sup>∞</sup> and *<sup>γ</sup>* is the Euler constant, *sn* can be

*ap*ð Þ *n* log *n* � �

> *ap*ð Þ *n* log *<sup>n</sup>* <*ε*

n o

.

,

*ap*ð Þ *n* log *n* � �

proved that the sequence log *p* �

*Number Theory and Its Applications*

**Definition 1.1.** *Let A* ⊆ *.*

*<sup>n</sup>* <sup>¼</sup> <sup>1</sup> *n* P*<sup>n</sup>*

be the *asymptotic density* of *A*.

*sn* P*<sup>n</sup> k*¼1 *χA*ð Þ*k*

*<sup>A</sup>*. Since *sn* <sup>¼</sup> log *<sup>n</sup>* <sup>þ</sup> *<sup>γ</sup>* <sup>þ</sup> *<sup>O</sup>* <sup>1</sup>

following limits *u A*ð Þ¼ lim *<sup>s</sup>*!<sup>∞</sup> *<sup>α</sup><sup>s</sup>*

a. Put *dn*ð Þ¼ *<sup>A</sup> A n*ð Þ

b. Put *<sup>δ</sup>n*ð Þ¼ *<sup>A</sup>* <sup>1</sup>

Let *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup><sup>α</sup>*<sup>1</sup>

**126**

<sup>1</sup> *<sup>p</sup><sup>α</sup>*<sup>2</sup> <sup>2</sup> <sup>⋯</sup>*p<sup>α</sup><sup>k</sup>*

ð Þ Ωð Þ¼ *n α*<sup>1</sup> þ ⋯ þ *α<sup>k</sup>* ,

functions.

[35–38]).

**2. Basic notions**

*<sup>ε</sup>*>0. We also want to investigate the <sup>I</sup>ð Þ*<sup>q</sup>*

A lot of mathematical disciplines use the term small (large) set from different point of view. For instance a final set, a set having the measure zero and nowhere dense set is a small set from point of view of cardinality, measure (probability) and topology, respectively. The notion of ideal <sup>I</sup> <sup>⊆</sup> <sup>2</sup>*<sup>X</sup>* is the unifying principle how to express that a subset of *X* 6¼ Ø is small. We say a set *A* ⊆ *X* is a small set if *A* ∈I. Recall the notion of an ideal I of subsets of .

Let <sup>I</sup> <sup>⊆</sup> <sup>2</sup>. <sup>I</sup> is said to be an *ideal* in , if <sup>I</sup> is additive (if *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup><sup>I</sup> then *<sup>A</sup>* <sup>∪</sup> *<sup>B</sup>*∈I) and hereditary (if *A* ∈I and *B* ⊂ *A* then *B*∈ I). An ideal I is said to be *non-trivial ideal* if I 6¼ Ø and ∉ I. A non-trivial ideal I is said to be *admissible ideal* if it contains all finite subsets of . The dual notion to the ideal is the notion filter. A non-empty family of sets <sup>F</sup> <sup>⊂</sup> <sup>2</sup> is a *filter* if and only if Ø <sup>∉</sup> <sup>F</sup>, for each *<sup>A</sup>*, *<sup>B</sup>*<sup>∈</sup> <sup>F</sup> we have *A* ∩ *B* ∈ F and for each *A* ∈ F and each *B*⊃ *A* we have *B* ∈ F (for definitions see e.g. [4, 41, 42]). Let I be a proper ideal in (i.e. ∉ I). Then a family of sets F Ið Þ¼ f g *B* ⊆ : *there exists A* ∈I *such that B* ¼ n*A* is a filter in , so called the *associated filter* with the ideal I.

The following example shows the most commonly used admissible ideals in different areas of mathematics.

#### **Example 1.2.**


for *<sup>n</sup>* <sup>∈</sup> . If lim *<sup>n</sup>*!<sup>∞</sup> *<sup>d</sup>*ð Þ *<sup>n</sup>* **<sup>T</sup>** ð Þ¼ *A d***T**ð Þ *A* exists, then *d***T**ð Þ *A* is called **T***–density* of *A* (see [3, 44]). Put I*d***<sup>T</sup>** ¼ f g *A* ⊂ : *d***T**ð Þ¼ *A* 0 . Then I*d***<sup>T</sup>** is a non-trivial ideal and I*d***<sup>T</sup>** contains both I*<sup>d</sup>* and I*<sup>δ</sup>* ideals as a special case. Indeed I*<sup>d</sup>* can be obtained by choosing *tn*,*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> *<sup>n</sup>* for *k*≤ *n*, *tn*,*<sup>k</sup>* ¼ 0 for *k*>*n* and I*<sup>δ</sup>* by choosing *tn*,*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> *k sn* for *<sup>k</sup>*≤*n*, *tn*,*<sup>k</sup>* <sup>¼</sup> 0 for *<sup>k</sup>*>*<sup>n</sup>* where *sn* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> k*¼1 1 *<sup>k</sup>* for *n*∈ .

**4. I– and I <sup>∗</sup> –convergence**

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

notion of statistical convergence.

**Definition 1.4.**

*S*∈ F Ið Þ.

(see [47]).

same limit.

**129**

I–convergence.

(F) If a sequence ð Þ *xn* <sup>∞</sup>

converges to *<sup>L</sup>*, then ð Þ *xn* <sup>∞</sup>

sequence of real numbers (see [3]).

**Definition 1.3.** We say that a sequence ð Þ *xn* <sup>∞</sup>

*d A*ð Þ¼ ð Þ*ε* 0, where *A*ð Þ¼ *ε* f g *n*∈ :j*xn* � *L*j≥*ε* .

i. We say that a sequence ð Þ *xn* <sup>∞</sup>

the limit is in the usual sense.

belongs to the ideal I.

The notion of statistical convergence was introduced in [1, 2] and the notion of I–convergence introduced in [3] corresponds to the natural generalization of the

*<sup>n</sup>*¼<sup>1</sup> is *statistically convergent* to a

*<sup>n</sup>*¼<sup>1</sup> of real numbers is

*n*¼1

<sup>¼</sup> *<sup>L</sup>* � *<sup>K</sup>*.

<sup>¼</sup> *<sup>L</sup>* � *<sup>K</sup>*.

*<sup>n</sup>*¼<sup>1</sup> has a subsequence which

*<sup>n</sup>*¼<sup>1</sup> is <sup>I</sup>*–convergent* to a number *<sup>L</sup>*<sup>∈</sup> and we

Let us recall notions of statistical convergence, <sup>I</sup>– and <sup>I</sup> <sup>∗</sup> –convergence of

number *L*∈ and we write lim stat*xn* ¼ *L*, provided that for each *ε*>0 we have

ii. Let <sup>I</sup> be an admissible ideal on . A sequence ð Þ *xn* <sup>∞</sup>

write I � lim *xn* ¼ *L*, if for each *ε*>0 the set *A*ð Þ¼ *ε* f g *n* ∈ :j*xn* � *L*j≥*ε*

said to be <sup>I</sup> <sup>∗</sup> *–convergent* to *<sup>L</sup>*<sup>∈</sup> , if there is a set *<sup>H</sup>* <sup>∈</sup>I, such that for *<sup>M</sup>* <sup>¼</sup> n*H* ¼ f g *m*<sup>1</sup> < *m*<sup>2</sup> < ⋯ < *mk* < ⋯ ∈ F Ið Þ we have lim *<sup>k</sup>*!<sup>∞</sup> *xmk* ¼ *L*, where

In the definition of usual convergence the set *A*ð Þ*ε* is finite, it means that it is small from point of view of cardinality, *A*ð Þ*ε* ∈I *<sup>f</sup>* . Similarly in the definition of statistical convergence the set *A*ð Þ*ε* has asymptotic density zero, it is small from point of view of density, *A*ð Þ*ε* ∈I*d*. The natural generalization of these notions is the following, let I be an admissible ideal (e.g. anyone from Example 1.2) then for each *ε*>0 we ask whether the set *A*ð Þ*ε* belongs in the ideal I. In this way we obtain the notion of the I–convergence. For the following use, we note that the concept of I–convergence can be extended for such sequences that are not defined for all *n* ∈ ,

but only for "almost" all *<sup>n</sup>*<sup>∈</sup> . This means that instead of a sequence ð Þ *xn* <sup>∞</sup>

we have ð Þ *xs <sup>s</sup>*∈*<sup>S</sup>*, where *s* runs over all positive integers belonging to *S*⊆ and

the usual convergence. All notions which are used next we considered in real

i. If I � lim *xn* ¼ *L* and I � lim *yn* ¼ *K*, then I � lim *xn* � *yn*

ii. If I � lim *xn* ¼ *L* and I � lim *yn* ¼ *K*, then I � lim *xn* � *yn*

(S) Every constant sequence ð Þ *x*, *x*, … , *x*, … converges to *x*. (H) The limit of any convergent sequence is uniquely determined.

*<sup>n</sup>*¼<sup>1</sup> converges to *<sup>L</sup>*.

(U) If each subsequence of the sequence ð Þ *xn* <sup>∞</sup>

The following properties are the most familiar axioms of convergence

A natural question arises which above axioms are satisfied for the concept of

*<sup>n</sup>*¼<sup>1</sup> has the limit *<sup>L</sup>*, then each of its subsequences has the

numbers . The following theorem can be easily proved.

**Theorem 1.5** (Theorem 2.1 from [9]).

Remember that I–convergence in has many properties similar to properties of

For the matrix **<sup>T</sup>** <sup>¼</sup> ð Þ *tn*,*<sup>k</sup> <sup>n</sup>*,*k*<sup>∈</sup> , where *tn*,*<sup>k</sup>* <sup>¼</sup> *<sup>φ</sup>*ð Þ*<sup>k</sup> <sup>n</sup>* for *k*≤*n*, *k*∣*n* and *tn*,*<sup>k</sup>* ¼ 0 otherwise we obtain I *<sup>φ</sup>* ideal of Schoenberg (see [2]), where *φ* is Euler function.

Another special case of I*d***<sup>T</sup>** is the following. Take an arbitrary divergent series P<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*cn*, where *cn* <sup>&</sup>gt;0 for *<sup>n</sup>*<sup>∈</sup> and put *tn*,*<sup>k</sup>* <sup>¼</sup> *ck Sn* for *<sup>k</sup>*≤*n*, where *Sn* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*ci*, and *tn*,*<sup>k</sup>* <sup>¼</sup> 0 for *<sup>k</sup>*>*n*.


$$\mathcal{T}\_f \nsubseteq \mathcal{T}\_c^{(q\_1)} \nsubseteq \mathcal{T}\_c^{(q\_2)} \nsubseteq \mathcal{T}\_c \nsubseteq \mathcal{T}\_d \nsubseteq \mathcal{T}\_\delta. \tag{2}$$

The fact I*<sup>c</sup>* ⊊ I*<sup>d</sup>* in Eq. (2) follows from the following result. Let *A* ⊆ and P *a*∈ *A* 1 *<sup>a</sup>* < ∞ then *d A*ð Þ¼ 0 (see [46]) thus if *A* ∈I*<sup>c</sup>* then *A* ∈ I*d*. The opposite is not true, consider the set of primes , for which we have *<sup>d</sup>*ð Þ¼ 0 but <sup>P</sup> *p*∈ 1 *<sup>p</sup>* ¼ ∞ thus ∈ I*<sup>d</sup>* but ∉ I*<sup>c</sup>* ð Þ I*<sup>c</sup>* 6¼ I*<sup>d</sup>* .

The fact that for any *<sup>q</sup>*1, *<sup>q</sup>*<sup>2</sup> <sup>∈</sup>ð0, 1i, *<sup>q</sup>*<sup>1</sup> <sup>&</sup>lt;*q*<sup>2</sup> we have <sup>I</sup> *<sup>q</sup>*ð Þ<sup>1</sup> *<sup>c</sup>* <sup>⊊</sup> <sup>I</sup> *<sup>q</sup>*ð Þ<sup>2</sup> *<sup>c</sup>* in Eq. (2) is clear. For showing that <sup>I</sup> *<sup>q</sup>*ð Þ<sup>1</sup> *<sup>c</sup>* 6¼ I *<sup>q</sup>*ð Þ<sup>2</sup> *<sup>c</sup>* it suffices to find a set *H* ¼ f g *h*<sup>1</sup> <*h*<sup>2</sup> < ⋯ <*hk* < ⋯ ⊂ such that P<sup>∞</sup> *<sup>k</sup>*¼<sup>1</sup>*<sup>h</sup>* �*q*<sup>1</sup> *<sup>k</sup>* ¼ þ<sup>∞</sup> and <sup>P</sup><sup>∞</sup> *<sup>k</sup>*¼<sup>1</sup>*<sup>h</sup>* �*q*<sup>2</sup> *<sup>k</sup>* < þ ∞. Put *hk* ¼ *k* 1 *q*1 h i. Since *h*<sup>1</sup> <*h*<sup>2</sup> < ⋯ < *hk* < ⋯ and *h q*1 *<sup>k</sup>* <sup>≤</sup>*<sup>k</sup>* we have <sup>P</sup><sup>∞</sup> *<sup>k</sup>*¼<sup>1</sup>*<sup>h</sup>* �*q*<sup>1</sup> *<sup>k</sup>* <sup>≥</sup>P<sup>∞</sup> *<sup>k</sup>*¼<sup>1</sup>*k*�<sup>1</sup> ¼ þ∞. On the other side *hk* > *k* 1 *<sup>q</sup>*<sup>1</sup> � <sup>1</sup><sup>≥</sup> <sup>1</sup> 2 *k* 1 *<sup>q</sup>*<sup>1</sup> for *k*≥2, so we obtain P<sup>∞</sup> *<sup>k</sup>*¼<sup>1</sup>*<sup>h</sup>* �*q*<sup>2</sup> *<sup>k</sup>* ≤2*<sup>q</sup>*<sup>2</sup> P<sup>∞</sup> *<sup>k</sup>*¼<sup>1</sup>*k*�*q*<sup>2</sup> *<sup>q</sup>*<sup>1</sup> < þ ∞ since *<sup>q</sup>*<sup>2</sup> *q*1 > 1.

### **4. I– and I <sup>∗</sup> –convergence**

for *<sup>n</sup>* <sup>∈</sup> . If lim *<sup>n</sup>*!<sup>∞</sup> *<sup>d</sup>*ð Þ *<sup>n</sup>*

*Number Theory and Its Applications*

*tn*,*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> *k*

function.

series P<sup>∞</sup>

f. Let <sup>¼</sup> <sup>⋃</sup><sup>∞</sup>

P *a*∈ *A* 1

g. For an *<sup>q</sup>* <sup>∈</sup>ð0, 1<sup>i</sup> the set <sup>I</sup>ð Þ*<sup>q</sup>*

∈ I*<sup>d</sup>* but ∉ I*<sup>c</sup>* ð Þ I*<sup>c</sup>* 6¼ I*<sup>d</sup>* .

*<sup>k</sup>*¼<sup>1</sup>*<sup>h</sup>* �*q*<sup>1</sup>

> 2 *k* 1

*h*<sup>1</sup> <*h*<sup>2</sup> < ⋯ < *hk* < ⋯ and *h*

For showing that <sup>I</sup> *<sup>q</sup>*ð Þ<sup>1</sup>

1 *<sup>q</sup>*<sup>1</sup> � <sup>1</sup><sup>≥</sup> <sup>1</sup>

such that P<sup>∞</sup>

side *hk* > *k*

since *<sup>q</sup>*<sup>2</sup> *q*1 > 1.

**128**

ideal (see [23]). The ideal <sup>I</sup>ð Þ<sup>1</sup>

<sup>I</sup> *<sup>f</sup>* <sup>⊊</sup> <sup>I</sup> *<sup>q</sup>*ð Þ<sup>1</sup>

The fact that for any *<sup>q</sup>*1, *<sup>q</sup>*<sup>2</sup> <sup>∈</sup>ð0, 1i, *<sup>q</sup>*<sup>1</sup> <sup>&</sup>lt;*q*<sup>2</sup> we have <sup>I</sup> *<sup>q</sup>*ð Þ<sup>1</sup>

*<sup>c</sup>* 6¼ I *<sup>q</sup>*ð Þ<sup>2</sup>

*<sup>k</sup>* ¼ þ<sup>∞</sup> and <sup>P</sup><sup>∞</sup>

*q*1

obtained by choosing *tn*,*<sup>k</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>i</sup>*¼<sup>1</sup>*ci*, and *tn*,*<sup>k</sup>* <sup>¼</sup> 0 for *<sup>k</sup>*>*n*.

I*<sup>μ</sup>* is an admissible ideal and I*<sup>μ</sup>* ⊊ I*d*.

**<sup>T</sup>** ð Þ¼ *A d***T**ð Þ *A* exists, then *d***T**ð Þ *A* is called **T***–density* of

*<sup>n</sup>* for *k*≤ *n*, *tn*,*<sup>k</sup>* ¼ 0 for *k*>*n* and I*<sup>δ</sup>* by choosing

*<sup>k</sup>* for *n*∈ .

*<sup>n</sup>* for *k*≤*n*, *k*∣*n* and *tn*,*<sup>k</sup>* ¼ 0

**<sup>T</sup>** .

*p*∈ 1

*<sup>c</sup>* in Eq. (2) is clear.

*<sup>p</sup>* ¼ ∞ thus

*k*¼1 1

*A* (see [3, 44]). Put I*d***<sup>T</sup>** ¼ f g *A* ⊂ : *d***T**ð Þ¼ *A* 0 . Then I*d***<sup>T</sup>** is a non-trivial ideal and I*d***<sup>T</sup>** contains both I*<sup>d</sup>* and I*<sup>δ</sup>* ideals as a special case. Indeed I*<sup>d</sup>* can be

otherwise we obtain I *<sup>φ</sup>* ideal of Schoenberg (see [2]), where *φ* is Euler

*<sup>n</sup>*¼<sup>1</sup>*cn*, where *cn* <sup>&</sup>gt;0 for *<sup>n</sup>*<sup>∈</sup> and put *tn*,*<sup>k</sup>* <sup>¼</sup> *ck*

a non-trivial ideal. For *<sup>μ</sup><sup>n</sup>* we can take for instance *dn*, *<sup>δ</sup><sup>n</sup>* or *<sup>d</sup>*ð Þ *<sup>n</sup>*

*<sup>c</sup>* ¼ *A* ⊂ :

*<sup>c</sup>* <sup>⊊</sup> <sup>I</sup> *<sup>q</sup>*ð Þ<sup>2</sup>

true, consider the set of primes , for which we have *<sup>d</sup>*ð Þ¼ 0 but <sup>P</sup>

*<sup>k</sup>*¼<sup>1</sup>*<sup>h</sup>* �*q*<sup>2</sup>

*<sup>k</sup>* <sup>≤</sup>*<sup>k</sup>* we have <sup>P</sup><sup>∞</sup>

*<sup>q</sup>*<sup>1</sup> for *k*≥2, so we obtain P<sup>∞</sup>

*<sup>c</sup>* ¼ *A* ⊂ :

The fact I*<sup>c</sup>* ⊊ I*<sup>d</sup>* in Eq. (2) follows from the following result. Let *A* ⊆ and

*<sup>a</sup>* < ∞ then *d A*ð Þ¼ 0 (see [46]) thus if *A* ∈I*<sup>c</sup>* then *A* ∈ I*d*. The opposite is not

denoted by I*c*. It is easy to see, that for any *q*1, *q*<sup>2</sup> ∈ð Þ 0, 1 , *q*<sup>1</sup> <*q*<sup>2</sup> we have

Another special case of I*d***<sup>T</sup>** is the following. Take an arbitrary divergent

*Sn* for *<sup>k</sup>*≤*n*, where *Sn* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

d. Let *<sup>μ</sup>* be a finitely additive normed measure on a field <sup>S</sup> <sup>⊆</sup> <sup>2</sup>. Suppose that <sup>S</sup> contains all singletons f g*n* , *n* ∈ . Then the family I*<sup>μ</sup>* ¼ f g *A* ⊂ : *μ*ð Þ¼ *A* 0 is an admissible ideal. In the case if *μ* is the Buck measure density (see [13, 45]),

e. Suppose that *<sup>μ</sup><sup>n</sup>* : <sup>2</sup> ! ½ � 0, 1 is a finitely additive normed measure for *<sup>n</sup>* <sup>∈</sup> . If for *A* ⊆ there exists *μ*ð Þ¼ *A* lim *<sup>n</sup>*!<sup>∞</sup> *μn*ð Þ *A* , then the set *A* is said to be *measurable* and *μ*ð Þ *A* is called the *measure* of *A*. Obviously *μ* is a finitely additive measure on some field <sup>S</sup> <sup>⊆</sup>2. The family <sup>I</sup>*<sup>μ</sup>* <sup>¼</sup> f g *<sup>A</sup>* <sup>⊂</sup> : *<sup>μ</sup>*ð Þ¼ *<sup>A</sup>* <sup>0</sup> is

Assume that *D <sup>j</sup>* ð Þ *j* ¼ 1, 2, … are infinite sets (e.g. we can choose *Dj* ¼ <sup>2</sup> *<sup>j</sup>*�<sup>1</sup> � ð Þ <sup>2</sup>*<sup>s</sup>* � <sup>1</sup> : *<sup>s</sup>*<sup>∈</sup> � � for *<sup>j</sup>* <sup>¼</sup> 1, 2, … ). Denote <sup>I</sup> the class of all *<sup>A</sup>* <sup>⊂</sup> such that *A* intersects only a finite number of *Dj*. Then I is an admissible ideal.

*<sup>j</sup>*¼<sup>1</sup>*Dj* be a decomposition on (i.e. *Dk* <sup>∩</sup> *Dl* <sup>¼</sup> Ø for *<sup>k</sup>* 6¼ *<sup>l</sup>*).

P

P

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>a*�*<sup>q</sup>* <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup> � � is an admissible

*<sup>c</sup>* ⊊ I*<sup>c</sup>* ⊊ I*<sup>d</sup>* ⊊ I*δ:* (2)

*<sup>c</sup>* <sup>⊊</sup> <sup>I</sup> *<sup>q</sup>*ð Þ<sup>2</sup>

1 *q*1 h i

> P<sup>∞</sup> *<sup>k</sup>*¼<sup>1</sup>*k*�*q*<sup>2</sup>

. Since

*<sup>k</sup>*¼<sup>1</sup>*k*�<sup>1</sup> ¼ þ∞. On the other

*<sup>q</sup>*<sup>1</sup> < þ ∞

*<sup>c</sup>* it suffices to find a set *H* ¼ f g *h*<sup>1</sup> <*h*<sup>2</sup> < ⋯ <*hk* < ⋯ ⊂

*<sup>k</sup>* < þ ∞. Put *hk* ¼ *k*

*<sup>k</sup>*¼<sup>1</sup>*<sup>h</sup>* �*q*<sup>2</sup> *<sup>k</sup>* ≤2*<sup>q</sup>*<sup>2</sup>

*<sup>k</sup>*¼<sup>1</sup>*<sup>h</sup>* �*q*<sup>1</sup> *<sup>k</sup>* <sup>≥</sup>P<sup>∞</sup>

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>a*�<sup>1</sup> <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup> � � is usually

*sn* for *<sup>k</sup>*≤*n*, *tn*,*<sup>k</sup>* <sup>¼</sup> 0 for *<sup>k</sup>*>*<sup>n</sup>* where *sn* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

For the matrix **<sup>T</sup>** <sup>¼</sup> ð Þ *tn*,*<sup>k</sup> <sup>n</sup>*,*k*<sup>∈</sup> , where *tn*,*<sup>k</sup>* <sup>¼</sup> *<sup>φ</sup>*ð Þ*<sup>k</sup>*

The notion of statistical convergence was introduced in [1, 2] and the notion of I–convergence introduced in [3] corresponds to the natural generalization of the notion of statistical convergence.

Let us recall notions of statistical convergence, <sup>I</sup>– and <sup>I</sup> <sup>∗</sup> –convergence of sequence of real numbers (see [3]).

**Definition 1.3.** We say that a sequence ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> is *statistically convergent* to a number *L*∈ and we write lim stat*xn* ¼ *L*, provided that for each *ε*>0 we have *d A*ð Þ¼ ð Þ*ε* 0, where *A*ð Þ¼ *ε* f g *n*∈ :j*xn* � *L*j≥*ε* .

**Definition 1.4.**


In the definition of usual convergence the set *A*ð Þ*ε* is finite, it means that it is small from point of view of cardinality, *A*ð Þ*ε* ∈I *<sup>f</sup>* . Similarly in the definition of statistical convergence the set *A*ð Þ*ε* has asymptotic density zero, it is small from point of view of density, *A*ð Þ*ε* ∈I*d*. The natural generalization of these notions is the following, let I be an admissible ideal (e.g. anyone from Example 1.2) then for each *ε*>0 we ask whether the set *A*ð Þ*ε* belongs in the ideal I. In this way we obtain the notion of the I–convergence. For the following use, we note that the concept of I–convergence can be extended for such sequences that are not defined for all *n* ∈ , but only for "almost" all *<sup>n</sup>*<sup>∈</sup> . This means that instead of a sequence ð Þ *xn* <sup>∞</sup> *n*¼1 we have ð Þ *xs <sup>s</sup>*∈*<sup>S</sup>*, where *s* runs over all positive integers belonging to *S*⊆ and *S*∈ F Ið Þ.

Remember that I–convergence in has many properties similar to properties of the usual convergence. All notions which are used next we considered in real numbers . The following theorem can be easily proved.

**Theorem 1.5** (Theorem 2.1 from [9]).

i. If I � lim *xn* ¼ *L* and I � lim *yn* ¼ *K*, then I � lim *xn* � *yn* <sup>¼</sup> *<sup>L</sup>* � *<sup>K</sup>*.

ii. If I � lim *xn* ¼ *L* and I � lim *yn* ¼ *K*, then I � lim *xn* � *yn* <sup>¼</sup> *<sup>L</sup>* � *<sup>K</sup>*.

The following properties are the most familiar axioms of convergence (see [47]).

(S) Every constant sequence ð Þ *x*, *x*, … , *x*, … converges to *x*.

(H) The limit of any convergent sequence is uniquely determined.

(F) If a sequence ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> has the limit *<sup>L</sup>*, then each of its subsequences has the same limit.

(U) If each subsequence of the sequence ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> has a subsequence which converges to *<sup>L</sup>*, then ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> converges to *<sup>L</sup>*.

A natural question arises which above axioms are satisfied for the concept of I–convergence.

**Theorem 1.6** (see [14] and Proposition 3.1 from [3], where the concept of <sup>I</sup>–convergence has been investigated in a metric space) *Let* <sup>I</sup> <sup>⊂</sup>2 *be an admissible ideal.*

X *a* >*nk a*∈ *Ak*

*<sup>a</sup>*�*<sup>q</sup>* <sup>þ</sup> <sup>X</sup> *a*>*n*<sup>2</sup> *a* ∈ *A*<sup>2</sup>

*<sup>k</sup>*¼1½ð � *nk*, *nk*þ1<sup>i</sup> <sup>∩</sup> *Ak* . Then

*a*�*<sup>q</sup>* ≤ X *a*>*n*<sup>1</sup> *a*∈ *A*<sup>1</sup>

> < 1 2 þ 1 <sup>22</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup>

prove that lim *<sup>k</sup>*!<sup>∞</sup>*xmk* <sup>¼</sup> *<sup>L</sup>*. Let *<sup>ε</sup>*<sup>&</sup>gt; 0. Choose *<sup>k</sup>*<sup>0</sup> <sup>∈</sup> such that <sup>1</sup>

X *a* ∈ *H*

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

Then *mk* belongs to some interval *n <sup>j</sup>*, *n <sup>j</sup>*þ<sup>1</sup>

It is easy to prove the following lemma.

**5. I–convergence of arithmetical functions**

the normal order is as follows. The sequence ð Þ *xn* <sup>∞</sup>

<sup>I</sup>*<sup>d</sup>* � lim *<sup>ω</sup>*ð Þ *<sup>n</sup>*

<sup>I</sup>*<sup>d</sup>* � lim *h n*ð Þ

log log *n*

log *n*

*yn*

concerning this notion see [34, 38, 48].

**Definition 1.14.** The sequence ð Þ *xn* <sup>∞</sup>

Let *<sup>H</sup>* <sup>¼</sup> <sup>∪</sup> <sup>∞</sup>

Thus *<sup>H</sup>* <sup>∈</sup>Ið Þ*<sup>q</sup>*

thus lim *<sup>k</sup>*!<sup>∞</sup>*xmk* ¼ *L*.

ð Þ 1 � *ε yn* <*xn* < ð Þ 1 þ *ε yn*.

if and only if <sup>I</sup>*<sup>d</sup>* � lim *xn*

the normal order.

and

**131**

I<sup>2</sup> � lim *xn* ¼ *L.*

**Corollary 1.12** *Ideals* <sup>I</sup>ð Þ*<sup>q</sup>*

*a*�*<sup>q</sup>* <

1

*A <sup>j</sup>* ð Þ *j*≥*k*<sup>0</sup> . Hence *mk* belongs to n*A <sup>j</sup>*, and then ∣*xmk* � *L*∣<*ε* for every *mk* >*nk*<sup>0</sup> ,

**Lemma 1.13** (see [3]). *If* I<sup>1</sup> ⊆I<sup>2</sup> *then the statement* I<sup>1</sup> � lim *xn* ¼ *L implies*

We can obtain a good information about behaviour and properties of the wellknown arithmetical functions by investigating I–convergence of these functions or some sequences connected with these functions. Recall the concept of normal order.

*ε*>0 and almost all (almost all in the sense of asymptotic density) values *n* we have

formulated using the concept of statistical convergence, which coincides with I*d*– convergence. For equivalent definitions of the normal order and more examples

In the papers [21, 27, 28] and in the monograph [38] there are studied various kinds of convergence of arithmetical functions which were mentioned at the beginning. The following equalities were proved in the paper [28] by using the concept of

Schinzel and Šalát in [28] pointed out that one of equivalent definitions to have

*<sup>n</sup>*¼<sup>1</sup> has *the normal order yn*

¼ 1. The results concerning the normal order will be

¼ I*<sup>d</sup>* � lim <sup>Ω</sup>ð Þ *<sup>n</sup>*

¼ I*<sup>d</sup>* � lim *H n*ð Þ

log log *<sup>n</sup>* <sup>¼</sup> <sup>1</sup>

log *<sup>n</sup>* <sup>¼</sup> <sup>0</sup>*:*

� �<sup>∞</sup>

*<sup>n</sup>*¼<sup>1</sup> has the normal order *yn*

*<sup>n</sup>*¼<sup>1</sup> if for every

� �<sup>∞</sup> *n*¼1

*<sup>c</sup>* . Put *M* ¼ n*H* ¼ f g *m*<sup>1</sup> < *m*<sup>2</sup> < ⋯ < *mk* < ⋯ . Now it suffices to

*<sup>c</sup> for q*∈ð0, 1i *have the property (AP).*

1 2*k :*

*<sup>a</sup>*�*<sup>q</sup>* <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>X</sup>

<sup>2</sup>*<sup>k</sup>* <sup>þ</sup> <sup>⋯</sup> <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>*:*

*a* >*nk a* ∈ *Ak*

� � where *j*≥*k*<sup>0</sup> and does not belong to

*<sup>a</sup>*�*<sup>q</sup>* <sup>þ</sup> <sup>⋯</sup>

<sup>2</sup>*k*<sup>0</sup> <sup>&</sup>lt;*ε*. Let *mk* <sup>&</sup>gt;*nk*<sup>0</sup> .

i. I–convergence satisfies (S), (H) and (U).

ii. If I contains an infinite set, then I–convergence does not satisfy (F).

**Theorem 1.7** (see [3]) *Let* <sup>I</sup> *be an admissible ideal in . If* <sup>I</sup> <sup>∗</sup> � lim *xn* <sup>¼</sup> *L then* I � lim *xn* ¼ *L.*

The following example shows that the converse of Theorem 1.7 is not true. **Example 1.8.** Let I¼I be an ideal from Example 1.2 f). Define ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> as follows: For *<sup>n</sup>*<sup>∈</sup> *<sup>D</sup> <sup>j</sup>* we put *xn* <sup>¼</sup> <sup>1</sup> *<sup>j</sup>* for *j* ¼ 1, 2, … . Then obviously I � lim *xn* ¼ 0. But we show that <sup>I</sup> <sup>∗</sup> � lim *xn* <sup>¼</sup> 0 does not hold.

If *H* ∈I then directly from the definition of I there exists *p* ∈ such that *H* ⊆ *D*<sup>1</sup> ∪ *D*<sup>2</sup> ∪⋯∪ *Dp*. But then *Dp*þ<sup>1</sup> ⊆ n*H* ¼ f g *m*<sup>1</sup> < *m*<sup>2</sup> < ⋯ < *mk* < ⋯ ∈ F Ið Þ and so we have *xmk* <sup>¼</sup> <sup>1</sup> *<sup>p</sup>*þ<sup>1</sup> for infinitely many indices *<sup>k</sup>*<sup>∈</sup> . Therefore lim *<sup>k</sup>*!<sup>∞</sup>*xmk* <sup>¼</sup> <sup>0</sup> cannot be true.

In [3] was formulated a necessary and sufficient condition for an admissible ideal <sup>I</sup> under which <sup>I</sup>– and <sup>I</sup> <sup>∗</sup> –convergence to be equivalent. Recall this condition (AP) that is similar to the condition (APO) in [7, 35].

**Definition 1.9** (see also [40]) An admissible ideal <sup>I</sup> <sup>⊂</sup>2 is said to satisfy the condition (AP) if for every countable family of mutually disjoint sets f g *A*1, *A*2, … belonging to I there exists a countable family of sets f g *B*1, *B*2, … such that symmetric difference *<sup>A</sup> <sup>j</sup>ΔBj* is finite for *<sup>j</sup>*<sup>∈</sup> and *<sup>B</sup>* <sup>¼</sup> <sup>∪</sup> <sup>∞</sup> *<sup>j</sup>*¼<sup>1</sup>*B <sup>j</sup>* ∈I.

**Remark.** Observe that each *B <sup>j</sup>* from the previous Definition belong to I. **Theorem 1.10** (see [14]) *From* I � lim *xn* <sup>¼</sup> *L the statement* <sup>I</sup> <sup>∗</sup> � lim *xn* <sup>¼</sup> *<sup>L</sup> follows if and only if* I *satisfies the condition (AP).*

In [44] it is proved that <sup>I</sup>*<sup>d</sup>***<sup>T</sup>** – and <sup>I</sup> <sup>∗</sup> *<sup>d</sup>***<sup>T</sup>** –convergence are equivalent in provided that **T** ¼ ð Þ *tn*,*<sup>k</sup> <sup>n</sup>*,*<sup>k</sup>* <sup>∈</sup> from Example 1.2 c) is a non-negative triangular matrix with P*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*tn*,*<sup>k</sup>* <sup>¼</sup> 1 for *<sup>n</sup>*<sup>∈</sup> . From this we get that <sup>I</sup>*d*, <sup>I</sup>*δ*, <sup>I</sup>*φ*–convergence coinside with I ∗ *<sup>d</sup>* , <sup>I</sup> <sup>∗</sup> *<sup>δ</sup>* , <sup>I</sup> <sup>∗</sup> *<sup>φ</sup>* –convergence, respectively. On the other hand for further ideals from Example 1.2 e.g. I*u*, I and I*μ*, respectively, we have that they do not fulfill the assertion that their <sup>I</sup>–convergence coincides with <sup>I</sup> <sup>∗</sup> –convergence. Since these ideals do not fulfill condition (AP) (see [13, 38, 40]).

The following Theorem shows that also for all ideals <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* for *q*∈ð0, 1i the concepts <sup>I</sup>– and <sup>I</sup> <sup>∗</sup> –convergence coincide.

**Theorem 1.11** (see, [20, 23]) *For any q*<sup>∈</sup> <sup>ð</sup>0, 1<sup>i</sup> *the notions* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup> – and* <sup>I</sup>ð Þ*<sup>q</sup>* <sup>∗</sup> *<sup>c</sup> – convergence are equivalent.*

**Proof.** It suffices to prove that for any <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* , *<sup>q</sup>*<sup>∈</sup> <sup>ð</sup>0, 1<sup>i</sup> and any sequence ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> of real numbers such that <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim *xn* ¼ *L* for *q*∈ð0, 1i there exists a set *M* ¼ f g *<sup>m</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>m</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>⋯</sup> <sup>&</sup>lt; *mk* <sup>&</sup>lt; <sup>⋯</sup> <sup>⊆</sup> such that n*<sup>M</sup>* <sup>∈</sup>Ið Þ*<sup>q</sup> <sup>c</sup>* and lim *<sup>k</sup>*!<sup>∞</sup>*xmk* ¼ *L*.

For any positive integer *<sup>k</sup>* let *<sup>ε</sup><sup>k</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup>*<sup>k</sup>* and *Ak* <sup>¼</sup> *<sup>n</sup>* <sup>∈</sup> :j*xn* � *<sup>L</sup>*j<sup>≥</sup> <sup>1</sup> 2*k* n o. As <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim *xn* <sup>¼</sup> *<sup>L</sup>*, we have *Ak* <sup>∈</sup> <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* , i.e.

$$\sum\_{a \in A\_k} a^{-q} < \infty.$$

Therefore there exists an infinite sequence *n*<sup>1</sup> <*n*<sup>2</sup> < ⋯ <*nk* < ⋯ of integers such that for every *k* ¼ 1, 2, …

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

**Theorem 1.6** (see [14] and Proposition 3.1 from [3], where the concept of

ii. If I contains an infinite set, then I–convergence does not satisfy (F).

**Theorem 1.7** (see [3]) *Let* <sup>I</sup> *be an admissible ideal in . If* <sup>I</sup> <sup>∗</sup> � lim *xn* <sup>¼</sup> *L then*

*<sup>j</sup>* for *j* ¼ 1, 2, … . Then obviously I � lim *xn* ¼ 0.

*<sup>j</sup>*¼<sup>1</sup>*B <sup>j</sup>* ∈I.

*<sup>d</sup>***<sup>T</sup>** –convergence are equivalent in provided

*<sup>c</sup>* , *<sup>q</sup>*<sup>∈</sup> <sup>ð</sup>0, 1<sup>i</sup> and any sequence ð Þ *xn* <sup>∞</sup>

*<sup>c</sup>* and lim *<sup>k</sup>*!<sup>∞</sup>*xmk* ¼ *L*.

n o

*<sup>c</sup>* for *q*∈ð0, 1i the

*<sup>c</sup> – and* <sup>I</sup>ð Þ*<sup>q</sup>* <sup>∗</sup> *<sup>c</sup> –*

2*k*

.

*<sup>n</sup>*¼<sup>1</sup> of

*<sup>p</sup>*þ<sup>1</sup> for infinitely many indices *<sup>k</sup>*<sup>∈</sup> . Therefore lim *<sup>k</sup>*!<sup>∞</sup>*xmk* <sup>¼</sup> <sup>0</sup>

*<sup>n</sup>*¼<sup>1</sup> as

The following example shows that the converse of Theorem 1.7 is not true. **Example 1.8.** Let I¼I be an ideal from Example 1.2 f). Define ð Þ *xn* <sup>∞</sup>

If *H* ∈I then directly from the definition of I there exists *p* ∈ such that *H* ⊆ *D*<sup>1</sup> ∪ *D*<sup>2</sup> ∪⋯∪ *Dp*. But then *Dp*þ<sup>1</sup> ⊆ n*H* ¼ f g *m*<sup>1</sup> < *m*<sup>2</sup> < ⋯ < *mk* < ⋯ ∈ F Ið Þ and

In [3] was formulated a necessary and sufficient condition for an admissible ideal <sup>I</sup> under which <sup>I</sup>– and <sup>I</sup> <sup>∗</sup> –convergence to be equivalent. Recall this condition

**Definition 1.9** (see also [40]) An admissible ideal <sup>I</sup> <sup>⊂</sup>2 is said to satisfy the condition (AP) if for every countable family of mutually disjoint sets f g *A*1, *A*2, … belonging to I there exists a countable family of sets f g *B*1, *B*2, … such that sym-

**Remark.** Observe that each *B <sup>j</sup>* from the previous Definition belong to I. **Theorem 1.10** (see [14]) *From* I � lim *xn* <sup>¼</sup> *L the statement* <sup>I</sup> <sup>∗</sup> � lim *xn* <sup>¼</sup> *<sup>L</sup>*

that **T** ¼ ð Þ *tn*,*<sup>k</sup> <sup>n</sup>*,*<sup>k</sup>* <sup>∈</sup> from Example 1.2 c) is a non-negative triangular matrix with

Example 1.2 e.g. I*u*, I and I*μ*, respectively, we have that they do not fulfill the assertion that their <sup>I</sup>–convergence coincides with <sup>I</sup> <sup>∗</sup> –convergence. Since these

*<sup>c</sup>* , i.e.

*a*�*<sup>q</sup>* < ∞*:*

Therefore there exists an infinite sequence *n*<sup>1</sup> <*n*<sup>2</sup> < ⋯ <*nk* < ⋯ of integers such

X *a*∈ *Ak*

*<sup>k</sup>*¼<sup>1</sup>*tn*,*<sup>k</sup>* <sup>¼</sup> 1 for *<sup>n</sup>*<sup>∈</sup> . From this we get that <sup>I</sup>*d*, <sup>I</sup>*δ*, <sup>I</sup>*φ*–convergence coinside with

*<sup>φ</sup>* –convergence, respectively. On the other hand for further ideals from

*<sup>c</sup>* � lim *xn* ¼ *L* for *q*∈ð0, 1i there exists a set *M* ¼

<sup>2</sup>*<sup>k</sup>* and *Ak* <sup>¼</sup> *<sup>n</sup>* <sup>∈</sup> :j*xn* � *<sup>L</sup>*j<sup>≥</sup> <sup>1</sup>

<sup>I</sup>–convergence has been investigated in a metric space) *Let* <sup>I</sup> <sup>⊂</sup>2 *be an*

i. I–convergence satisfies (S), (H) and (U).

But we show that <sup>I</sup> <sup>∗</sup> � lim *xn* <sup>¼</sup> 0 does not hold.

(AP) that is similar to the condition (APO) in [7, 35].

metric difference *<sup>A</sup> <sup>j</sup>ΔBj* is finite for *<sup>j</sup>*<sup>∈</sup> and *<sup>B</sup>* <sup>¼</sup> <sup>∪</sup> <sup>∞</sup>

ideals do not fulfill condition (AP) (see [13, 38, 40]).

concepts <sup>I</sup>– and <sup>I</sup> <sup>∗</sup> –convergence coincide.

**Proof.** It suffices to prove that for any <sup>I</sup>ð Þ*<sup>q</sup>*

f g *<sup>m</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>m</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>⋯</sup> <sup>&</sup>lt; *mk* <sup>&</sup>lt; <sup>⋯</sup> <sup>⊆</sup> such that n*<sup>M</sup>* <sup>∈</sup>Ið Þ*<sup>q</sup>*

For any positive integer *<sup>k</sup>* let *<sup>ε</sup><sup>k</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>c</sup>* � lim *xn* <sup>¼</sup> *<sup>L</sup>*, we have *Ak* <sup>∈</sup> <sup>I</sup>ð Þ*<sup>q</sup>*

*convergence are equivalent.*

real numbers such that <sup>I</sup>ð Þ*<sup>q</sup>*

that for every *k* ¼ 1, 2, …

The following Theorem shows that also for all ideals <sup>I</sup>ð Þ*<sup>q</sup>*

**Theorem 1.11** (see, [20, 23]) *For any q*<sup>∈</sup> <sup>ð</sup>0, 1<sup>i</sup> *the notions* <sup>I</sup>ð Þ*<sup>q</sup>*

*follows if and only if* I *satisfies the condition (AP).* In [44] it is proved that <sup>I</sup>*<sup>d</sup>***<sup>T</sup>** – and <sup>I</sup> <sup>∗</sup>

*admissible ideal.*

I � lim *xn* ¼ *L.*

so we have *xmk* <sup>¼</sup> <sup>1</sup>

cannot be true.

P*<sup>n</sup>*

I ∗ *<sup>d</sup>* , <sup>I</sup> <sup>∗</sup> *<sup>δ</sup>* , <sup>I</sup> <sup>∗</sup>

As <sup>I</sup>ð Þ*<sup>q</sup>*

**130**

follows: For *<sup>n</sup>*<sup>∈</sup> *<sup>D</sup> <sup>j</sup>* we put *xn* <sup>¼</sup> <sup>1</sup>

*Number Theory and Its Applications*

$$\sum\_{a>u\_k} a^{-q} < \frac{1}{2^k}.$$

Let *<sup>H</sup>* <sup>¼</sup> <sup>∪</sup> <sup>∞</sup> *<sup>k</sup>*¼1½ð � *nk*, *nk*þ1<sup>i</sup> <sup>∩</sup> *Ak* . Then

$$\sum\_{a \in H} a^{-q} \le \sum\_{a > u\_1 \atop a \ll A\_1} a^{-q} + \sum\_{a > u\_2 \atop a \ll A\_2} a^{-q} + \dots + \sum\_{a > u\_k} a^{-q} + \dotsb$$

$$ < \frac{1}{2} + \frac{1}{2^2} + \dotsb + \frac{1}{2^k} + \dotsb < + \infty.$$

Thus *<sup>H</sup>* <sup>∈</sup>Ið Þ*<sup>q</sup> <sup>c</sup>* . Put *M* ¼ n*H* ¼ f g *m*<sup>1</sup> < *m*<sup>2</sup> < ⋯ < *mk* < ⋯ . Now it suffices to prove that lim *<sup>k</sup>*!<sup>∞</sup>*xmk* <sup>¼</sup> *<sup>L</sup>*. Let *<sup>ε</sup>*<sup>&</sup>gt; 0. Choose *<sup>k</sup>*<sup>0</sup> <sup>∈</sup> such that <sup>1</sup> <sup>2</sup>*k*<sup>0</sup> <sup>&</sup>lt;*ε*. Let *mk* <sup>&</sup>gt;*nk*<sup>0</sup> . Then *mk* belongs to some interval *n <sup>j</sup>*, *n <sup>j</sup>*þ<sup>1</sup> � � where *j*≥*k*<sup>0</sup> and does not belong to *A <sup>j</sup>* ð Þ *j*≥*k*<sup>0</sup> . Hence *mk* belongs to n*A <sup>j</sup>*, and then ∣*xmk* � *L*∣<*ε* for every *mk* >*nk*<sup>0</sup> , thus lim *<sup>k</sup>*!<sup>∞</sup>*xmk* ¼ *L*.

**Corollary 1.12** *Ideals* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup> for q*∈ð0, 1i *have the property (AP).* It is easy to prove the following lemma.

**Lemma 1.13** (see [3]). *If* I<sup>1</sup> ⊆I<sup>2</sup> *then the statement* I<sup>1</sup> � lim *xn* ¼ *L implies* I<sup>2</sup> � lim *xn* ¼ *L.*

#### **5. I–convergence of arithmetical functions**

We can obtain a good information about behaviour and properties of the wellknown arithmetical functions by investigating I–convergence of these functions or some sequences connected with these functions. Recall the concept of normal order.

**Definition 1.14.** The sequence ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> has *the normal order yn* � �<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> if for every *ε*>0 and almost all (almost all in the sense of asymptotic density) values *n* we have ð Þ 1 � *ε yn* <*xn* < ð Þ 1 þ *ε yn*.

Schinzel and Šalát in [28] pointed out that one of equivalent definitions to have the normal order is as follows. The sequence ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> has the normal order *yn* � �<sup>∞</sup> *n*¼1 if and only if <sup>I</sup>*<sup>d</sup>* � lim *xn yn* ¼ 1. The results concerning the normal order will be formulated using the concept of statistical convergence, which coincides with I*d*– convergence. For equivalent definitions of the normal order and more examples concerning this notion see [34, 38, 48].

In the papers [21, 27, 28] and in the monograph [38] there are studied various kinds of convergence of arithmetical functions which were mentioned at the beginning. The following equalities were proved in the paper [28] by using the concept of the normal order.

$$\mathcal{I}\_d - \lim \frac{\alpha(n)}{\log \log n} = \mathcal{I}\_d - \lim \frac{\Omega(n)}{\log \log n} = 1$$

and

$$\mathcal{T}\_d - \lim \frac{h(n)}{\log n} = \mathcal{T}\_d - \lim \frac{H(n)}{\log n} = 0.$$

Similarly for the functions *f n*ð Þ and *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>* . In [27] it is proved the following equality:

We are going to estimate the first factor of Eq. (3)

1 *N* X*α*0 *α*¼0 *α*2 *k* X þ*N*

� *<sup>k</sup> pα* � �

*<sup>p</sup><sup>α</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>O</sup>*ð Þ<sup>1</sup> � �

> 1 *N*

*k* X þ*N*

*n*¼*k*þ1

*n*¼2

≤*c*

*ap*ð Þ *n* log *n*

Under the Lemma 1.13 it is clear that if there exists the <sup>I</sup>ð Þ*<sup>q</sup>*

log *n* � �<sup>∞</sup>

for all *q*∈ð0, 1i.

*<sup>c</sup>* � lim *h n*ð Þ

*n*¼2

1 ð Þ log *n*

> 1 *N*

**Remark.** It is known that I*<sup>u</sup>* ⊊ I*<sup>d</sup>* (see e.g. [5, 6]) but the ideals I*<sup>c</sup>* and I*<sup>u</sup>* are not disjoint, and moreover I*<sup>u</sup>* ⊈I*<sup>c</sup>* and I*<sup>c</sup>* ⊈I*u*. For example the set of all prime numbers belongs to I*<sup>u</sup>* but not belongs to I*c*. On the other hand there exists the set

for any *q*∈ð0, 1i, then it is equal to the I*d*–limit of the same sequence. There are no

and *H n*ð Þ log *n* � �<sup>∞</sup>

*<sup>k</sup>*¼<sup>1</sup>*Bk*, where *Bk* <sup>¼</sup> *<sup>k</sup>*<sup>3</sup> <sup>þ</sup> 1, *<sup>k</sup>*<sup>3</sup> <sup>þ</sup> 2, … , *<sup>k</sup>*<sup>3</sup> <sup>þ</sup> *<sup>k</sup>* � � which not belongs to <sup>I</sup>*<sup>u</sup>* but it

*<sup>c</sup>* –convergence of special sequences described in the introduction.

s

*<sup>n</sup>*¼*k*þ<sup>1</sup> *ap*ð Þ¼ *<sup>n</sup> <sup>α</sup>*

� � � � � �

� *<sup>k</sup>* <sup>þ</sup> *<sup>N</sup> p<sup>α</sup>*þ<sup>1</sup> � �

¼ 1

*<sup>α</sup>*¼0*α*<sup>2</sup> <sup>¼</sup> *<sup>P</sup>*ð Þ *<sup>α</sup>*<sup>0</sup> , where *P x*ð Þ¼ *x x*ð Þ <sup>þ</sup><sup>1</sup> ð Þ <sup>2</sup>*x*þ<sup>1</sup>

1 ¼

*<sup>N</sup> <sup>N</sup>* <sup>1</sup> � <sup>1</sup>

� *<sup>k</sup> p<sup>α</sup>*þ<sup>1</sup>

*p* � �X*<sup>α</sup>*<sup>0</sup>

*α*¼0

<sup>2</sup> ! 0, since <sup>1</sup>

1 ð Þ log *n* 2

*<sup>c</sup>* ⊊ I*<sup>d</sup>* for all *q*∈ð0, 1i and Lemma 1.13 it is useful to

*n*¼2

log *<sup>n</sup>* <sup>¼</sup> 0, *for all q*∈ð0, 1i*:*

log 2 � � and moreover they both are statistically conver-

*<sup>c</sup>* –convergence, but only for the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X*<sup>N</sup>*þ<sup>1</sup> *n*¼2

¼

*O*ð Þ1 *N*

<sup>6</sup> and simple estimations give

X*α*0 *α*¼0 *α*2 *:*

<sup>2</sup> ! 0*:* (5)

*<sup>c</sup>* –limit of some sequence

. In [28] it was proved that these

*<sup>α</sup>*<sup>2</sup> <sup>1</sup> *p<sup>α</sup>* þ

*ap*ð Þ *<sup>n</sup>* � �<sup>2</sup> <sup>¼</sup> *<sup>O</sup>*ð Þ<sup>1</sup> *:* (4)

ð Þ log *n*

! 0,

1 *N*

1 *N* X*α*0 *α*¼0

1 *N* X*α*0 *α*¼0

Formula P*<sup>α</sup>*<sup>0</sup>

So we get

*<sup>p</sup><sup>α</sup>* <sup>≤</sup>P<sup>∞</sup> *α*¼0 *α*2 *<sup>p</sup><sup>α</sup>* < ∞.

> 1 *N*

uniformly in *k*.

Under the fact that <sup>I</sup>ð Þ*<sup>q</sup>*

Consider the sequences *h n*ð Þ

*n*¼2

**Theorem 1.18** (see [20]). *We have*

gent to zero. The same result we have for <sup>I</sup>ð Þ*<sup>q</sup>*

<sup>I</sup>ð Þ*<sup>q</sup>*

sequences are dense on 0, <sup>1</sup>

*h n*ð Þ log *n* � �<sup>∞</sup>

*<sup>B</sup>* <sup>¼</sup> <sup>∪</sup> <sup>∞</sup>

belongs to I*c*.

investigate <sup>I</sup>ð Þ*<sup>q</sup>*

other options.

sequence

**133**

*k* X þ*N*

*n*¼*k*þ1

P*<sup>α</sup>*<sup>0</sup> *<sup>α</sup>*¼0*α*<sup>2</sup> <sup>1</sup> *k* X þ*N*

*n*¼*k*þ1

*ap*ð Þ *<sup>n</sup>* � �<sup>2</sup> ffi

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

> *<sup>α</sup>*<sup>2</sup> *<sup>k</sup>* <sup>þ</sup> *<sup>N</sup> pα* � �

> > *<sup>p</sup><sup>α</sup>* � *<sup>N</sup>*

Estimate the second factor Eq. (3)

1 ð Þ log *n*

> 1 *N*

<sup>2</sup> ≤ 1 *N N* X þ1

Let *N* ! ∞, from Eqs. (4) and (5) we obtain

*n*¼*k*þ1

*k* X þ*N*

*<sup>α</sup>*<sup>2</sup> *<sup>N</sup>*

$$\mathcal{I}\_d - \lim \, \frac{\log \log f(n)}{\log \log n} = \mathcal{I}\_d - \lim \, \frac{\log \log f^\*(n)}{\log \log n} = 1 + \log 2.1$$

Let us recall one more result from [26], let *p* be a prime number, there was proved that the sequence log *p* � *ap*ð Þ *n* log *n* � �<sup>∞</sup> *n*¼2 is I*d*–convergent to 0. Moreover the sequence log *p* � *ap*ð Þ *n* log *n* � �<sup>∞</sup> *n*¼2 is <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergent to 0 for *<sup>q</sup>* <sup>¼</sup> 1 and it is not <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* – convergent for all *q*∈ð Þ 0, 1 , as it was shown in [21]. In [19] it was proved that this sequence is also I*u*–convergent.

The following theorem shows that the assertions using the notion I*<sup>u</sup>* instead of <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* , *q*∈ð0, 1i need to use a different technique for their proofs. First of all we recall a new kind of convergence so called the uniformly strong *ℓ*–Cesàro convergence. This convergence is an analog of the notion of strong almost convergence (see [6]).

**Definition 1.15.** A sequence ð Þ *xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> is said to be *uniformly strong <sup>ℓ</sup>*–*Cesàro convergent* ð Þ <sup>0</sup><sup>&</sup>lt; *<sup>ℓ</sup>*<sup>&</sup>lt; <sup>∞</sup> to a number *<sup>L</sup>* if lim *<sup>N</sup>*!<sup>∞</sup> <sup>1</sup> *N* P*<sup>k</sup>*þ*<sup>N</sup> <sup>n</sup>*¼*k*þ<sup>1</sup>j j *xi* � *<sup>L</sup> <sup>ℓ</sup>* <sup>¼</sup> 0 uniformly in *<sup>k</sup>*.

The following Theorem shows a connection between uniformly strong *ℓ*–Cesàro convergence and I*u*–convergence.

**Theorem 1.16** (see [6]). *If x*ð Þ*<sup>n</sup>* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> *is a bounded sequence, then x*ð Þ*<sup>n</sup>* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> *is* <sup>I</sup>*u– convergent to L if and only if x*ð Þ*<sup>n</sup>* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> *is uniformly strong <sup>ℓ</sup>–Cesàro convergent to L for some ℓ,* 0< *ℓ*< ∞*.*

The sequence log *p* � *ap*ð Þ *n* log *n* � �<sup>∞</sup> *n*¼2 is I*u*–convergent to zero i.e. for arbitrary *ε*> 0 the set *A*ð Þ¼ *ε n* ∈ : log *p* � *ap*ð Þ *n* log *<sup>n</sup>* ≥ *ε*>0 n o has uniform density equal to zero.

**Theorem 1.17** (see [19]). *We have* I*<sup>u</sup>* � lim log *p* � *ap*ð Þ *n* log *<sup>n</sup>* ¼ 0*.*

**Proof.** The sequence log *p* � *ap*ð Þ *n* log *n* � �<sup>∞</sup> *n*¼2 is bounded. Using Theorem 1.16, it is sufficient to show that the sequence log *p* � *ap*ð Þ *n* log *n* � �<sup>∞</sup> *n*¼2 is uniformly strong *ℓ*–Cesàro convergent to 0 for *ℓ* ¼ 1. For the reason that all members of log *p* � *ap*ð Þ *n* log *n* � �<sup>∞</sup> *n*¼2 are positive, we shall prove that lim *<sup>N</sup>*!<sup>∞</sup> <sup>1</sup> *N* P*<sup>k</sup>*þ*<sup>N</sup> n*¼*k*þ1 *ap*ð Þ *n* log *<sup>n</sup>* ¼ 0, uniformly in *k*. *ap*ð Þ¼ *n α* if *<sup>p</sup><sup>α</sup>*∥*n*. Let *<sup>α</sup>*<sup>0</sup> <sup>¼</sup> log *<sup>N</sup>* log *p* h i. This immediately implies that *<sup>p</sup><sup>α</sup>*<sup>0</sup> <sup>≤</sup> *<sup>N</sup>* <sup>&</sup>lt;*p<sup>α</sup>*0<sup>þ</sup>1. Then for all *n* ∈ð � *k*, *k* þ *N* we have *ap*ð Þ¼ *n α*<*α*<sup>0</sup> with the possible exception of one *n*<sup>1</sup> ∈ð � *k*, *k* þ *N* for which we could have *ap*ð Þ¼ *n*<sup>1</sup> *α*<sup>1</sup> >*α*0. Assume that there exist two such numbers *n*1, *n*<sup>2</sup> ∈ð � *k*, *k* þ *N* for which *ap*ð Þ¼ *n*<sup>1</sup> *α*<sup>1</sup> > *α*<sup>0</sup> and *ap*ð Þ¼ *n*<sup>2</sup> *<sup>α</sup>*<sup>2</sup> <sup>&</sup>gt;*α*0, then *<sup>n</sup>*<sup>1</sup> <sup>¼</sup> *<sup>m</sup>*1*p<sup>α</sup>*<sup>1</sup> , *<sup>n</sup>*<sup>2</sup> <sup>¼</sup> *<sup>m</sup>*2*p<sup>α</sup>*<sup>2</sup> hence *<sup>p</sup><sup>α</sup>*0þ<sup>1</sup>∣*n*<sup>1</sup> � *<sup>n</sup>*2. We have *<sup>p</sup><sup>α</sup>*0þ<sup>1</sup> <sup>&</sup>lt;∣*n*<sup>1</sup> � *<sup>n</sup>*2∣ ≤ *<sup>N</sup>*, what is a contradiction with *<sup>p</sup><sup>α</sup>*0þ<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*. When we omit such an *<sup>n</sup>*<sup>1</sup> from the sum, the error is less than <sup>1</sup> *N ap*ð Þ *n*<sup>1</sup> log *<sup>n</sup>*<sup>1</sup> <sup>≤</sup> <sup>1</sup> *N α*1 *<sup>α</sup>*<sup>1</sup> log *p*. Using the Hölder's inequality we get

$$\frac{1}{N} \sum\_{n=k+1}^{k+N} \frac{a\_p(n)}{\log n} \le \sqrt{\frac{1}{N} \sum\_{n=k+1}^{k+N} \left(a\_p(n)\right)^2} \sqrt{\frac{1}{N} \sum\_{n=k+1}^{k+N} \frac{1}{\left(\log n\right)^2}}.\tag{3}$$

We are going to estimate the first factor of Eq. (3)

$$\begin{split} &\frac{1}{N} \sum\_{n=k+1}^{k+N} \left( a\_p(n) \right)^2 \cong \frac{1}{N} \sum\_{a=0}^{a\_0} \alpha^2 \sum\_{\substack{n=k+1\\a\_p(n)=a}}^{k+N} 1 = \\ &\frac{1}{N} \sum\_{a=0}^{a\_0} \alpha^2 \left( \left[ \frac{k+N}{p^a} \right] - \left[ \frac{k}{p^a} \right] - \left( \left[ \frac{k+N}{p^{a+1}} \right] - \left[ \frac{k}{p^{a+1}} \right] \right) \right) = \\ &\frac{1}{N} \sum\_{a=0}^{a\_0} \alpha^2 \left( \frac{N}{p^a} - \frac{N}{p^{a+1}} + O(1) \right) = \frac{1}{N} N \left( 1 - \frac{1}{p} \right) \sum\_{a=0}^{a\_0} \alpha^2 \frac{1}{p^a} + \frac{O(1)}{N} \sum\_{a=0}^{a0} \alpha^2. \end{split}$$

Formula P*<sup>α</sup>*<sup>0</sup> *<sup>α</sup>*¼0*α*<sup>2</sup> <sup>¼</sup> *<sup>P</sup>*ð Þ *<sup>α</sup>*<sup>0</sup> , where *P x*ð Þ¼ *x x*ð Þ <sup>þ</sup><sup>1</sup> ð Þ <sup>2</sup>*x*þ<sup>1</sup> <sup>6</sup> and simple estimations give P*<sup>α</sup>*<sup>0</sup> *<sup>α</sup>*¼0*α*<sup>2</sup> <sup>1</sup> *<sup>p</sup><sup>α</sup>* <sup>≤</sup>P<sup>∞</sup> *α*¼0 *α*2 *<sup>p</sup><sup>α</sup>* < ∞.

So we get

Similarly for the functions *f n*ð Þ and *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>* . In [27] it is proved the following

Let us recall one more result from [26], let *p* be a prime number, there was

*n*¼2

convergent for all *q*∈ð Þ 0, 1 , as it was shown in [21]. In [19] it was proved that this

The following theorem shows that the assertions using the notion I*<sup>u</sup>* instead of

*N* P*<sup>k</sup>*þ*<sup>N</sup>*

The following Theorem shows a connection between uniformly strong *ℓ*–Cesàro

*<sup>c</sup>* , *q*∈ð0, 1i need to use a different technique for their proofs. First of all we recall a new kind of convergence so called the uniformly strong *ℓ*–Cesàro convergence. This convergence is an analog of the notion of strong almost convergence

*ap*ð Þ *n* log *n* � �<sup>∞</sup>

¼ I*<sup>d</sup>* � lim log log *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>*

log log *<sup>n</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> log 2*:*

is I*d*–convergent to 0. Moreover the

*<sup>n</sup>*¼<sup>1</sup> is said to be *uniformly strong <sup>ℓ</sup>*–*Cesàro conver-*

*<sup>n</sup>*¼*k*þ<sup>1</sup>j j *xi* � *<sup>L</sup> <sup>ℓ</sup>* <sup>¼</sup> 0 uniformly in *<sup>k</sup>*.

*<sup>c</sup>* –

*<sup>n</sup>*¼<sup>1</sup> *is* <sup>I</sup>*u–*

*<sup>c</sup>* –convergent to 0 for *<sup>q</sup>* <sup>¼</sup> 1 and it is not <sup>I</sup>ð Þ*<sup>q</sup>*

*<sup>n</sup>*¼<sup>1</sup> *is a bounded sequence, then x*ð Þ*<sup>n</sup>* <sup>∞</sup>

*<sup>n</sup>*¼<sup>1</sup> *is uniformly strong <sup>ℓ</sup>–Cesàro convergent to L for*

*ap*ð Þ *n* log *<sup>n</sup>* ¼ 0*.*

is I*u*–convergent to zero i.e. for arbitrary *ε*> 0

has uniform density equal to zero.

is bounded. Using Theorem 1.16, it is

is uniformly strong *ℓ*–Cesàro

log *<sup>n</sup>* ¼ 0, uniformly in *k*. *ap*ð Þ¼ *n α*

*<sup>α</sup>*<sup>1</sup> log *p*. Using the Hölder's inequality

1 ð Þ log *n* 2

*ap*ð Þ *n* log *n* � �<sup>∞</sup>

*n*¼2 are

*:* (3)

equality:

<sup>I</sup>*<sup>d</sup>* � lim log log *f n*ð Þ

proved that the sequence log *p* �

*Number Theory and Its Applications*

*ap*ð Þ *n* log *n* � �<sup>∞</sup>

**Definition 1.15.** A sequence ð Þ *xn* <sup>∞</sup>

convergence and I*u*–convergence. **Theorem 1.16** (see [6]). *If x*ð Þ*<sup>n</sup>* <sup>∞</sup>

*convergent to L if and only if x*ð Þ*<sup>n</sup>* <sup>∞</sup>

The sequence log *p* �

the set *A*ð Þ¼ *ε n* ∈ : log *p* �

**Proof.** The sequence log *p* �

positive, we shall prove that lim *<sup>N</sup>*!<sup>∞</sup> <sup>1</sup>

from the sum, the error is less than <sup>1</sup>

1 *N* *k* X þ*N*

*n*¼*k*þ1

*ap*ð Þ *n* log *n* ≤

log *p* h i

if *<sup>p</sup><sup>α</sup>*∥*n*. Let *<sup>α</sup>*<sup>0</sup> <sup>¼</sup> log *<sup>N</sup>*

we get

**132**

sufficient to show that the sequence log *p* �

*gent* ð Þ <sup>0</sup><sup>&</sup>lt; *<sup>ℓ</sup>*<sup>&</sup>lt; <sup>∞</sup> to a number *<sup>L</sup>* if lim *<sup>N</sup>*!<sup>∞</sup> <sup>1</sup>

*ap*ð Þ *n* log *n* � �<sup>∞</sup>

n o

**Theorem 1.17** (see [19]). *We have* I*<sup>u</sup>* � lim log *p* �

*n*¼2

*ap*ð Þ *n* log *<sup>n</sup>* ≥ *ε*>0

*ap*ð Þ *n* log *n* � �<sup>∞</sup>

convergent to 0 for *ℓ* ¼ 1. For the reason that all members of log *p* �

*n*¼2

*N* P*<sup>k</sup>*þ*<sup>N</sup> n*¼*k*þ1

all *n* ∈ð � *k*, *k* þ *N* we have *ap*ð Þ¼ *n α*<*α*<sup>0</sup> with the possible exception of one *n*<sup>1</sup> ∈ð � *k*, *k* þ *N* for which we could have *ap*ð Þ¼ *n*<sup>1</sup> *α*<sup>1</sup> >*α*0. Assume that there exist two such numbers *n*1, *n*<sup>2</sup> ∈ð � *k*, *k* þ *N* for which *ap*ð Þ¼ *n*<sup>1</sup> *α*<sup>1</sup> > *α*<sup>0</sup> and *ap*ð Þ¼ *n*<sup>2</sup>

*<sup>α</sup>*<sup>2</sup> <sup>&</sup>gt;*α*0, then *<sup>n</sup>*<sup>1</sup> <sup>¼</sup> *<sup>m</sup>*1*p<sup>α</sup>*<sup>1</sup> , *<sup>n</sup>*<sup>2</sup> <sup>¼</sup> *<sup>m</sup>*2*p<sup>α</sup>*<sup>2</sup> hence *<sup>p</sup><sup>α</sup>*0þ<sup>1</sup>∣*n*<sup>1</sup> � *<sup>n</sup>*2. We have *<sup>p</sup><sup>α</sup>*0þ<sup>1</sup>

*N ap*ð Þ *n*<sup>1</sup> log *<sup>n</sup>*<sup>1</sup> <sup>≤</sup> <sup>1</sup> *N α*1

X*<sup>k</sup>*þ*<sup>N</sup> n*¼*k*þ1

1 *N*

<sup>&</sup>lt;∣*n*<sup>1</sup> � *<sup>n</sup>*2∣ ≤ *<sup>N</sup>*, what is a contradiction with *<sup>p</sup><sup>α</sup>*0þ<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*. When we omit such an *<sup>n</sup>*<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*ap*ð Þ *<sup>n</sup>* � �<sup>2</sup>

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 *N*

s

X*<sup>k</sup>*þ*<sup>N</sup> n*¼*k*þ1

*ap*ð Þ *n* log *n* � �<sup>∞</sup>

*n*¼2

. This immediately implies that *p<sup>α</sup>*<sup>0</sup> ≤ *N* <*p<sup>α</sup>*0<sup>þ</sup>1. Then for

*ap*ð Þ *n*

*n*¼2

sequence log *p* �

sequence is also I*u*–convergent.

<sup>I</sup>ð Þ*<sup>q</sup>*

(see [6]).

*some ℓ,* 0< *ℓ*< ∞*.*

log log *n*

is <sup>I</sup>ð Þ*<sup>q</sup>*

$$\frac{1}{N} \sum\_{n=k+1}^{k+N} \left( a\_p(n) \right)^2 = O(1). \tag{4}$$

Estimate the second factor Eq. (3)

$$\frac{1}{N} \sum\_{n=k+1}^{k+N} \frac{1}{\left(\log n\right)^2} \le \frac{1}{N} \sum\_{n=2}^{N+1} \frac{1}{\left(\log n\right)^2} \to 0, \quad \text{since} \quad \frac{1}{\left(\log n\right)^2} \to 0. \tag{5}$$

Let *N* ! ∞, from Eqs. (4) and (5) we obtain

$$\frac{1}{N} \sum\_{n=k+1}^{k+N} \frac{a\_p(n)}{\log n} \le c \sqrt{\frac{1}{N} \sum\_{n=2}^{N+1} \frac{1}{\left(\log n\right)^2}} \to 0,$$

uniformly in *k*.

**Remark.** It is known that I*<sup>u</sup>* ⊊ I*<sup>d</sup>* (see e.g. [5, 6]) but the ideals I*<sup>c</sup>* and I*<sup>u</sup>* are not disjoint, and moreover I*<sup>u</sup>* ⊈I*<sup>c</sup>* and I*<sup>c</sup>* ⊈I*u*. For example the set of all prime numbers belongs to I*<sup>u</sup>* but not belongs to I*c*. On the other hand there exists the set *<sup>B</sup>* <sup>¼</sup> <sup>∪</sup> <sup>∞</sup> *<sup>k</sup>*¼<sup>1</sup>*Bk*, where *Bk* <sup>¼</sup> *<sup>k</sup>*<sup>3</sup> <sup>þ</sup> 1, *<sup>k</sup>*<sup>3</sup> <sup>þ</sup> 2, … , *<sup>k</sup>*<sup>3</sup> <sup>þ</sup> *<sup>k</sup>* � � which not belongs to <sup>I</sup>*<sup>u</sup>* but it belongs to I*c*.

Under the fact that <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* ⊊ I*<sup>d</sup>* for all *q*∈ð0, 1i and Lemma 1.13 it is useful to investigate <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence of special sequences described in the introduction. Under the Lemma 1.13 it is clear that if there exists the <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –limit of some sequence for any *q*∈ð0, 1i, then it is equal to the I*d*–limit of the same sequence. There are no other options.

Consider the sequences *h n*ð Þ log *n* � �<sup>∞</sup> *n*¼2 and *H n*ð Þ log *n* � �<sup>∞</sup> *n*¼2 . In [28] it was proved that these sequences are dense on 0, <sup>1</sup> log 2 � � and moreover they both are statistically convergent to zero. The same result we have for <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence, but only for the sequence *h n*ð Þ log *n* � �<sup>∞</sup> *n*¼2 for all *q*∈ð0, 1i.

**Theorem 1.18** (see [20]). *We have*

$$\mathcal{T}\_c^{(q)} - \lim\_{n \to \infty} \frac{h(n)}{\log n} = 0, \text{ for all } \ q \in (0, 1).$$

**Proof.** Let *k*∈ and *k*≥2. It is easy to see that the following equality holds

$$\mathbf{1} + \sum\_{n \in \mathbb{N} \atop h(n) \ge k} n^{-q} = \prod\_{p \in \mathbb{P}} \left( \mathbf{1} + \frac{\mathbf{1}}{p^{kq}} + \frac{\mathbf{1}}{p^{(k+1)q}} + \dotsb \right),\tag{6}$$

Every non-negative integer *<sup>n</sup>* can be represented as *<sup>n</sup>* <sup>¼</sup> *ab*<sup>2</sup>

*H b*<sup>2</sup> � � <sup>þ</sup> <sup>1</sup>

If *n* ∈ *A*ð Þ*ε* then from *H n*ð Þ≥*ε* � log *n* we have

*<sup>A</sup>*ð Þ*<sup>ε</sup>* <sup>⊆</sup>*<sup>B</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>∈</sup> : *<sup>n</sup>* <sup>¼</sup> *ab*<sup>2</sup>

X *n* ∈*B*

X *n*∈*B*

X∞ *b*¼1

**Theorem 1.22** (see [20]). *The sequences <sup>ω</sup>*ð Þ *<sup>n</sup>*

**Proof.** We prove this assertion only for *<sup>ω</sup>*ð Þ *<sup>n</sup>*

Lemma 1.13 we can assume that <sup>I</sup>*<sup>c</sup>* � lim *<sup>ω</sup>*ð Þ *<sup>n</sup>*

*H b*<sup>2</sup> � � *<sup>b</sup>*<sup>2</sup> <sup>≤</sup>

Moreover *B* ∈I*<sup>c</sup>* and because *A*ð Þ*ε* ⊆ *B* we have *A*ð Þ*ε* ∈ I*c*.

log log *n* � �<sup>∞</sup>

1 *n* 1 *<sup>n</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *b*¼1

> *j*¼1 1

<sup>≤</sup> <sup>X</sup><sup>∞</sup> *b*¼1

> X∞ *b*¼1

<sup>1</sup> <sup>⋯</sup>*pak*

1 *b*2

*H b*<sup>2</sup> � �

2 log 2 X∞ *b*¼1

We have shown that the sum in Eq. (8) is finite and therefore the sum in Eq. (7)

*n*¼2

log *b*

, <sup>Ω</sup>ð Þ *<sup>n</sup>* log log *n* � �<sup>∞</sup>

log log *n* � �<sup>∞</sup>

log log *n* � �<sup>∞</sup>

is analogous. Let *q* ¼ 1. On the basis of the Theorem 2.2 of [28] and

log *ab*<sup>2</sup> � �≤

*H n*ð Þ<sup>∈</sup> *H b*<sup>2</sup> � �, *H b*<sup>2</sup> � � <sup>þ</sup> <sup>1</sup> � �*:*

*<sup>ε</sup>* and so log *<sup>a</sup>*<sup>≤</sup>

, log *a*≤

*<sup>n</sup>*∈*<sup>B</sup>n*�<sup>1</sup> <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>. We have

1 *b*2

( )

X log *<sup>a</sup>*<sup>≤</sup> *H b*<sup>2</sup> ð Þþ<sup>1</sup> *ε*

*H b*<sup>2</sup> � � <sup>þ</sup> <sup>1</sup>

<sup>6</sup> < þ ∞, it is enough to prove that the

*<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> !

*H b*<sup>2</sup> � � <sup>þ</sup> <sup>1</sup> *<sup>ε</sup> :*

*<sup>ε</sup>* , *<sup>b</sup>*<sup>∈</sup>

*<sup>j</sup>* ≤ 1 þ log *k* for the harmonic series. Then we

*<sup>b</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>*:* (8)

*<sup>k</sup>* <sup>≥</sup> <sup>2</sup>*H n*ð Þ and from this *H n*ð Þ<sup>≤</sup> log *<sup>n</sup>*

*<sup>b</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>*:*

*n*¼2

*n*¼2

*n*¼2

log log *<sup>n</sup>* <sup>¼</sup> 1. Take *<sup>ε</sup>* <sup>∈</sup> 0, <sup>1</sup>

is following.

*n*¼2

. The proof for the sequence

� � and consider

2

*are not*

*and* <sup>Ω</sup>ð Þ *<sup>n</sup>* log log *n* � �<sup>∞</sup>

*:*

*:* (7)

log 2. Therefore

*H b*<sup>2</sup> � � <sup>þ</sup> <sup>1</sup>

1 *a :*

free number. Hence *H a*ð Þ¼ 1 and

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

It is enough to prove that P

We use the inequality *Sk* <sup>¼</sup> <sup>P</sup>*<sup>k</sup>*

*<sup>b</sup>*<sup>2</sup> <sup>¼</sup> *<sup>π</sup>*<sup>2</sup>

For any *<sup>n</sup>*<sup>∈</sup> we have *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup><sup>a</sup>*<sup>1</sup>

The situation for sequences *<sup>ω</sup>*ð Þ *<sup>n</sup>*

*<sup>c</sup> –convergent for all q*∈ð i 0, 1 *.*

have the following inequality

Because the P <sup>1</sup>

is also finite.

<sup>I</sup>ð Þ*<sup>q</sup>*

Ωð Þ *n* log log *n* � �<sup>∞</sup>

the set

**135**

*n*¼2

Therefore

, where *a* is a square-

where denotes the set of all primes.

The right-hand side of the equality Eq. (6) equals

$$\prod\_{p \in \mathbb{P}} \left( 1 + \frac{1}{p^{kq}} \cdot \frac{1}{1 - \frac{1}{p^q}} \right) = \prod\_{p \in \mathbb{P}} \left( 1 + \frac{1}{p^{(k-1)q} \cdot (p^q - 1)} \right).$$

Then for *q*> <sup>1</sup> *<sup>k</sup>*, the product on the right-hand side of the previous equality converges. Thus, the series on the left-hand side of Eq. (6) converges.

Let *<sup>ε</sup>*>0. Put *<sup>A</sup>*ð Þ¼ *<sup>ε</sup> <sup>n</sup>*<sup>∈</sup> : *h n*ð Þ log *<sup>n</sup>* ≥ *ε*>0 n o. There exists an *<sup>n</sup>*ð Þ*<sup>k</sup>* <sup>0</sup> ∈ for all *k*≥2 such that for all *n* >*n*ð Þ*<sup>k</sup>* <sup>0</sup> and *n*∈ *A*ð Þ*ε* we have *h n*ð Þ≥ *ε* � log *n* >*k* (it is sufficient to put *n*ð Þ*<sup>k</sup>* <sup>0</sup> ¼ *e k ε* h i).

From this *<sup>A</sup>*ð Þ*<sup>ε</sup>* <sup>∩</sup> *<sup>n</sup>*ð Þ*<sup>k</sup>* <sup>0</sup> <sup>þ</sup> 1, *<sup>n</sup>*ð Þ*<sup>k</sup>* <sup>0</sup> þ 2, … n o <sup>⊆</sup>f g *<sup>n</sup>*<sup>∈</sup> : *h n*ð Þ<sup>≥</sup> *<sup>k</sup>* for all *<sup>k</sup>*≥2, *<sup>k</sup>*<sup>∈</sup> . Therefore P *<sup>n</sup>*<sup>∈</sup> *<sup>A</sup>*ð Þ*<sup>ε</sup> <sup>n</sup>*�*<sup>q</sup>* <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup> for all *<sup>k</sup>*<sup>≥</sup> 2 and <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim *h n*ð Þ log *<sup>n</sup>* ¼ 0 since the series Eq. (6) converges for all *q*> <sup>1</sup> *<sup>k</sup>*. If *<sup>k</sup>* ! <sup>∞</sup> for sufficient large then <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim *h n*ð Þ log *<sup>n</sup>* ¼ 0 for all *q*∈ ð0, 1i.

**Corollary 1.19.** We have

$$\mathcal{T}\_{\epsilon}^{(q)\*} - \lim\_{} \frac{h(n)}{\log n} = \mathbf{0} \text{ } for \text{ } all \text{ } q \in (0, 1).$$

For the sequence *H n*ð Þ log *n* � �<sup>∞</sup> *n*¼2 we get the result of different character. � �<sup>∞</sup>

**Theorem 1.20** (see [20]). *The sequence H n*ð Þ log *n n*¼2 *is not* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup> –convergent for every q*∈ð Þ 0, 1 *.*

**Proof.** In the paper [21] is proved, that the sequence log *p* � *ap*ð Þ *n* log *n* � �<sup>∞</sup> *n*¼2 is not <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* – convergent for any *<sup>q</sup>*∈ð Þ 0, 1 . The sequence *ap*ð Þ *<sup>n</sup>* log *n* � �<sup>∞</sup> *n*¼2 is also not <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergent to zero. The inequality *H n*ð Þ≥*ap*ð Þ *n* holds for all *n* ¼ 1, 2, … and for any prime number *p*. Then we have *H n*ð Þ log *<sup>n</sup>* <sup>≥</sup> *ap*ð Þ *<sup>n</sup>* log *<sup>n</sup>* for all *n* ¼ 2, 3, … . This implies that the sequence *H n*ð Þ log *n* � �<sup>∞</sup> *n*¼2 is also not <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergent to zero for every *q*∈ð Þ 0, 1 .

**Theorem 1.21** (see [20]). *For q* ¼ 1*, we obtain*

$$\mathcal{T}\_c - \lim \frac{H(n)}{\log n} = \mathbf{0}.$$

**Proof.** We will show that

$$A(\varepsilon) = \left\{ n \in \mathbb{N} : \frac{H(n)}{\log n} \ge \varepsilon \right\} \in \mathcal{I}\_{\varepsilon}$$

for any *ε*>0.

Every non-negative integer *<sup>n</sup>* can be represented as *<sup>n</sup>* <sup>¼</sup> *ab*<sup>2</sup> , where *a* is a squarefree number. Hence *H a*ð Þ¼ 1 and

$$H(n) \in \left\{ H(b^2), H(b^2) + 1 \right\}.$$

If *n* ∈ *A*ð Þ*ε* then from *H n*ð Þ≥*ε* � log *n* we have

$$
\log\left(ab^2\right) \le \frac{H(b^2) + 1}{\varepsilon} \text{ and so } \log a \le \frac{H(b^2) + 1}{\varepsilon}.
$$

Therefore

**Proof.** Let *k*∈ and *k*≥2. It is easy to see that the following equality holds

1 þ 1 *pkq* þ

<sup>¼</sup> <sup>Y</sup> *p*∈

log *<sup>n</sup>* ≥ *ε*>0 n o

1 þ

*<sup>k</sup>*, the product on the right-hand side of the previous equality

1 *p*ð Þ *<sup>k</sup>*þ<sup>1</sup> *<sup>q</sup>*

� �

þ ⋯

1 *p*ð Þ *<sup>k</sup>*�<sup>1</sup> *<sup>q</sup>* � *p*ð Þ *<sup>q</sup>* � 1 � �

. There exists an *n*ð Þ*<sup>k</sup>*

*<sup>c</sup>* � lim *h n*ð Þ

*is not* <sup>I</sup>ð Þ*<sup>q</sup>*

⊆f g *n*∈ : *h n*ð Þ≥ *k* for all *k*≥2, *k*∈ .

<sup>0</sup> and *n*∈ *A*ð Þ*ε* we have *h n*ð Þ≥ *ε* � log *n* >*k* (it is sufficient to

*<sup>k</sup>*. If *<sup>k</sup>* ! <sup>∞</sup> for sufficient large then <sup>I</sup>ð Þ*<sup>q</sup>*

log *<sup>n</sup>* <sup>¼</sup> <sup>0</sup> *for all q*<sup>∈</sup> <sup>ð</sup>0, 1i*:*

log *n* � �<sup>∞</sup>

> log *n* � �<sup>∞</sup>

log *<sup>n</sup>* <sup>¼</sup> <sup>0</sup>*:*

*H n*ð Þ log *n*

� �

≥*ε*

∈I*<sup>c</sup>*

zero. The inequality *H n*ð Þ≥*ap*ð Þ *n* holds for all *n* ¼ 1, 2, … and for any prime num-

<sup>I</sup>*<sup>c</sup>* � lim *H n*ð Þ

*A*ð Þ¼ *ε n* ∈ :

*<sup>c</sup>* –convergent to zero for every *q*∈ð Þ 0, 1 .

we get the result of different character.

*n*¼2

*n*¼2

log *<sup>n</sup>* for all *n* ¼ 2, 3, … . This implies that the sequence

, (6)

<sup>0</sup> ∈ for all *k*≥2

log *<sup>n</sup>* ¼ 0 since the series

*<sup>c</sup>* � lim *h n*ð Þ

*<sup>c</sup> –convergent for every*

*n*¼2

is not <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –

*<sup>c</sup>* –convergent to

*ap*ð Þ *n* log *n* � �<sup>∞</sup>

is also not <sup>I</sup>ð Þ*<sup>q</sup>*

log *<sup>n</sup>* ¼ 0

*:*

*<sup>n</sup>*�*<sup>q</sup>* <sup>¼</sup> <sup>Y</sup> *p*∈

converges. Thus, the series on the left-hand side of Eq. (6) converges.

<sup>0</sup> þ 2, …

*<sup>n</sup>*<sup>∈</sup> *<sup>A</sup>*ð Þ*<sup>ε</sup> <sup>n</sup>*�*<sup>q</sup>* <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup> for all *<sup>k</sup>*<sup>≥</sup> 2 and <sup>I</sup>ð Þ*<sup>q</sup>*

<sup>1</sup> <sup>þ</sup> <sup>X</sup> *n* ∈ *h n*ð Þ≥*k*

where denotes the set of all primes.

1 þ 1 *pkq* � <sup>1</sup> <sup>1</sup> � <sup>1</sup> *pq*

Y *p*∈

*Number Theory and Its Applications*

Let *<sup>ε</sup>*>0. Put *<sup>A</sup>*ð Þ¼ *<sup>ε</sup> <sup>n</sup>*<sup>∈</sup> : *h n*ð Þ

Then for *q*> <sup>1</sup>

such that for all *n* >*n*ð Þ*<sup>k</sup>*

From this *<sup>A</sup>*ð Þ*<sup>ε</sup>* <sup>∩</sup> *<sup>n</sup>*ð Þ*<sup>k</sup>*

Eq. (6) converges for all *q*> <sup>1</sup>

**Corollary 1.19.** We have

For the sequence *H n*ð Þ

ber *p*. Then we have *H n*ð Þ

is also not <sup>I</sup>ð Þ*<sup>q</sup>*

**Proof.** We will show that

<sup>0</sup> ¼ *e k ε* h i ).

Therefore P

for all *q*∈ ð0, 1i.

*q*∈ð Þ 0, 1 *.*

*H n*ð Þ log *n* � �<sup>∞</sup>

**134**

*n*¼2

for any *ε*>0.

put *n*ð Þ*<sup>k</sup>*

The right-hand side of the equality Eq. (6) equals

!

<sup>0</sup> <sup>þ</sup> 1, *<sup>n</sup>*ð Þ*<sup>k</sup>*

<sup>I</sup>ð Þ*<sup>q</sup>* <sup>∗</sup> *<sup>c</sup>* � lim *h n*ð Þ

*n*¼2

**Proof.** In the paper [21] is proved, that the sequence log *p* �

log *n* � �<sup>∞</sup>

**Theorem 1.20** (see [20]). *The sequence H n*ð Þ

convergent for any *<sup>q</sup>*∈ð Þ 0, 1 . The sequence *ap*ð Þ *<sup>n</sup>*

log *<sup>n</sup>* <sup>≥</sup> *ap*ð Þ *<sup>n</sup>*

**Theorem 1.21** (see [20]). *For q* ¼ 1*, we obtain*

n o

$$A(\varepsilon) \subseteq B = \left\{ n \in \mathbb{N} : n = ab^2, \quad \log a \le \frac{H(b^2) + 1}{\varepsilon}, \ b \in \mathbb{N} \right\}.$$

It is enough to prove that P *<sup>n</sup>*∈*<sup>B</sup>n*�<sup>1</sup> <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>. We have

$$\sum\_{n \in B} \frac{1}{n} = \sum\_{b=1}^{\infty} \frac{1}{b^2} \sum\_{\log a \le \frac{H(b^2) + 1}{r}} \frac{1}{a} \cdot 1$$

We use the inequality *Sk* <sup>¼</sup> <sup>P</sup>*<sup>k</sup> j*¼1 1 *<sup>j</sup>* ≤ 1 þ log *k* for the harmonic series. Then we have the following inequality

$$\sum\_{n\in B} \frac{1}{n} \le \sum\_{b=1}^{\infty} \frac{1}{b^2} \left( \frac{H(b^2) + 1}{\varepsilon} + 1 \right). \tag{7}$$

Because the P <sup>1</sup> *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> *<sup>π</sup>*<sup>2</sup> <sup>6</sup> < þ ∞, it is enough to prove that the

$$\sum\_{b=1}^{\infty} \frac{H(b^2)}{b^2} < +\infty. \tag{8}$$

For any *<sup>n</sup>*<sup>∈</sup> we have *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup><sup>a</sup>*<sup>1</sup> <sup>1</sup> <sup>⋯</sup>*pak <sup>k</sup>* <sup>≥</sup> <sup>2</sup>*H n*ð Þ and from this *H n*ð Þ<sup>≤</sup> log *<sup>n</sup>* log 2. Therefore

$$\sum\_{b=1}^{\infty} \frac{H(b^2)}{b^2} \le \frac{2}{\log 2} \sum\_{b=1}^{\infty} \frac{\log b}{b^2} < +\infty.$$

We have shown that the sum in Eq. (8) is finite and therefore the sum in Eq. (7) is also finite.

Moreover *B* ∈I*<sup>c</sup>* and because *A*ð Þ*ε* ⊆ *B* we have *A*ð Þ*ε* ∈ I*c*. The situation for sequences *<sup>ω</sup>*ð Þ *<sup>n</sup>* log log *n* � �<sup>∞</sup> *n*¼2 , <sup>Ω</sup>ð Þ *<sup>n</sup>* log log *n* � �<sup>∞</sup> *n*¼2 is following. **Theorem 1.22** (see [20]). *The sequences <sup>ω</sup>*ð Þ *<sup>n</sup>* log log *n* � �<sup>∞</sup> *n*¼2 *and* <sup>Ω</sup>ð Þ *<sup>n</sup>* log log *n* � �<sup>∞</sup> *n*¼2 *are not* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup> –convergent for all q*∈ð i 0, 1 *.*

**Proof.** We prove this assertion only for *<sup>ω</sup>*ð Þ *<sup>n</sup>* log log *n* � �<sup>∞</sup> *n*¼2 . The proof for the sequence Ωð Þ *n* log log *n* � �<sup>∞</sup> *n*¼2 is analogous. Let *q* ¼ 1. On the basis of the Theorem 2.2 of [28] and Lemma 1.13 we can assume that <sup>I</sup>*<sup>c</sup>* � lim *<sup>ω</sup>*ð Þ *<sup>n</sup>* log log *<sup>n</sup>* <sup>¼</sup> 1. Take *<sup>ε</sup>* <sup>∈</sup> 0, <sup>1</sup> 2 � � and consider the set

$$A(\varepsilon) = \left\{ n \in \mathbb{N} : \left| \frac{o(n)}{\log \log n} - 1 \right| \ge \varepsilon \right\}.$$

According these inequalities by comparison test of the convergence of the series

2 *,* <sup>I</sup>ð Þ*<sup>q</sup>*

**Proof.** Let 0 <*q*<1 and let *M* have the same meaning as in the proof of Theorem

*n*∈ *M* 1

<sup>2</sup>*<sup>q</sup>* <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup>

<sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *M k*ð Þ <sup>1</sup>

*M*ð Þ� 2 *M*ð Þ1

*kq* � <sup>1</sup> ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup>* � �*:*

In virtue of Lagrange's mean value theorem we have

Therefore the series Eq. (10) can be written in the form

*kq*

X *n*∈ *M*

> 2 � �, *<sup>n</sup>*

not exceeding *x*. But *V x*ð Þ≥*s* � 3, where *s* is the integer satisfying *s* 2 � � <sup>≤</sup>*x*<sup>&</sup>lt;

*x*<

is greater then or equal to the number *V x*ð Þ of all numbers of the form *<sup>n</sup>*

*s s*ð Þ þ 1

1 *nq* <sup>≤</sup>*<sup>q</sup>*

*M k*ð Þ*qz<sup>q</sup>*�<sup>1</sup> *k*

ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

X∞ *k*¼1

ffiffiffi *k*

*n* � 2 � �, *<sup>t</sup>*

<sup>2</sup> , *<sup>s</sup>*<sup>&</sup>gt; ffiffiffiffiffi

*s* þ 1 2 � �*:*

<sup>2</sup>*<sup>x</sup>* <sup>p</sup> � <sup>1</sup>*:*

number of this form belongs to *M*. Consequently for *x*>4, *x*∈ the number *M x*ð Þ

*k*¼1

*M k*ð Þ *<sup>k</sup>*<sup>1</sup>þ*<sup>q</sup> :*

*<sup>c</sup>* –convergence.

2 *.*

*nq*. We write it in the form

*<sup>k</sup><sup>q</sup>* � <sup>1</sup> ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup>* � �

*<sup>k</sup>* , *k*<*zk* <*k* þ 1 ð Þ *k* ¼ 1, 2, … *:*

*kq*

*qM k*ð Þ

<sup>p</sup> , ð Þ *<sup>k</sup>* <sup>¼</sup> 1, 2, … (in the proof of Theorem

*t* � 1

� �, *<sup>n</sup>*≥4 occurs in Pascal'<sup>s</sup>

ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup> z* 1�*q k*

2

� � � � . Therefore every

1 � �, *<sup>t</sup>*

*M k*ð Þ� *M k*ð Þ � 1

*<sup>c</sup>* � lim ΓðÞ¼ *t* 2 *holds and*

*<sup>k</sup><sup>q</sup>* <sup>þ</sup> <sup>⋯</sup>

þ ⋯

(10)

*:* (11)

2 � �, *<sup>n</sup>* <sup>≥</sup><sup>4</sup>

in Eq. (9) follows.

X *n* ∈ *M*

But *z* 1�*q <sup>k</sup>* <sup>&</sup>gt; *<sup>k</sup>*<sup>1</sup>�*<sup>q</sup>*

1 *nq* <sup>¼</sup> *<sup>M</sup>*ð Þ<sup>1</sup>

<sup>I</sup>ð Þ*<sup>q</sup>*

Now we shall use the concept of <sup>I</sup>ð Þ*<sup>q</sup>*

1.23. Let us examine the series P

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

¼ *M*ð Þ1

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼1

**Theorem 1.24** (see [24]). *For every q*> <sup>1</sup>

1*<sup>q</sup>* þ

1 <sup>1</sup>*<sup>q</sup>* � <sup>1</sup> 2*q* � �

*M k*ð Þ <sup>1</sup>

ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup>* � *<sup>k</sup><sup>q</sup>* <sup>¼</sup> *qzq*�<sup>1</sup>

1 *<sup>k</sup><sup>q</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼1

, ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup>* <sup>&</sup>gt;*k<sup>q</sup>* and so

1.23). Consider that every binomial coefficient *<sup>t</sup>* <sup>¼</sup> *<sup>n</sup>*

X *k*∈ *M*

We have already seen, that *M k*ð Þ≤ *c*<sup>1</sup>

triangle at least four times as *<sup>n</sup>*

From this we get

So we obtain

**137**

*<sup>c</sup>* � lim <sup>Γ</sup>ðÞ¼ *<sup>t</sup>* <sup>2</sup> *does not hold for any q,* <sup>0</sup> <sup>&</sup>lt;*q*<sup>≤</sup> <sup>1</sup>

Put *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup>*, where *<sup>p</sup>* is a prime number, then *<sup>ω</sup>*ð Þ¼ *<sup>p</sup>* 1 and <sup>1</sup> log log *<sup>p</sup>* � 1 � � � � � �≥*<sup>ε</sup>* holds for all prime numbers *p*>*p*0. Therefore the set *A<sup>ε</sup>* contains all prime numbers greater than *<sup>p</sup>*0. For these *<sup>p</sup>* we have: <sup>P</sup> *p*>*p*<sup>0</sup> 1 *<sup>p</sup>* ¼ þ∞ and so *A*ð Þ*ε* ∉ I*c*. From this <sup>I</sup>*<sup>c</sup>* � lim *<sup>ω</sup>*ð Þ *<sup>n</sup>* log log *<sup>n</sup>* 6¼ 1. Under the inclusion <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* <sup>⊊</sup> <sup>I</sup>ð Þ<sup>1</sup> *<sup>c</sup>* � I*<sup>c</sup>* and according to Lemma 1.13 we have <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim *<sup>ω</sup>*ð Þ *<sup>n</sup>* log log *<sup>n</sup>* 6¼ 1 for *q* ∈ð0, 1i. This complete the proof.

Further possibility where the results can be strengthened by the way that the statistical convergence in them is replaced by <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence is the concept of the famous *Pascal's triangle*. The *n*-th row of the Pascal's triangle consists of the numbers *n* 0 � �, *<sup>n</sup>* 1 � �, … , *<sup>n</sup> n* � 1 � �, *<sup>n</sup> n* � �. Their sum equals to 2*<sup>n</sup>* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>1</sup> *<sup>n</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> k*¼0 *n k* � �. Let Γð Þ*t* denote the number of times the positive integer *t*, *t*>2 occurs in the Pascal's triangle. That is, <sup>Γ</sup>ð Þ*<sup>t</sup>* is the number of binomial coefficients *<sup>n</sup> k* � � satisfying *n* � �

*k* ¼ *t*. From this point of view Γ is the function which maps the set in the set ∪ f g *ℵ*<sup>0</sup> ð Þ Γð Þ¼ 1 *ℵ*<sup>0</sup> . Let us observe that for every *t*∈ , Γð Þ*t* ≥1.

In [32] it is proved that the average and normal order of the function Γ is 2. Since the normal order is 2, we have

$$\mathcal{T}\_d - \lim \Gamma(t) = 2$$

(see [28]). We are going to show two results which strengthen the result of [32] and their proofs are outlined in [24].

**Theorem 1.23** (see [24]). I*<sup>c</sup>* � lim ΓðÞ¼ *t* 2*:*

**Proof.** The values of the function Γ are positive integers for *t* 6¼ 1. Thus for *ε* >0 the set *A<sup>ε</sup>* ¼ f g *t*∈ :jΓðÞ�*t* 2j≥*ε* is a subset of the set *H* ¼ f g1 ∪ f g2 ∪ *M*, where *M* ¼ f g *t*∈ : Γð Þ*t* >2 . Note that Γð Þ¼ 2 1. Therefore is suffices to show that P *n*∈ *H* 1 *<sup>n</sup>* < þ ∞. Evidently this is equivalent with

$$\sum\_{n \in M} \frac{1}{n} < +\infty. \tag{9}$$

We shall prove Eq. (9). Firstly, we write the left-hand site of Eq. (9) in the form

$$\sum\_{n \in M} \frac{1}{n} = \frac{M(1)}{1} + \frac{M(2) - M(1)}{2} + \dots + \frac{M(k) - M(k-1)}{k} + \dots$$

$$= \frac{M(1)}{1 \cdot 2} + \frac{M(2)}{2 \cdot 3} + \dots + \frac{M(k)}{k \cdot (k+1)} + \dots$$

In [32] it is shown that *M x*ð Þ¼ *<sup>O</sup>* ffiffiffi *x* p ð Þ. Therefore there exists such *c*<sup>1</sup> >0 that for every *k*∈ , *M k*ð Þ≤ *c*<sup>1</sup> ffiffiffi *k* <sup>p</sup> holds. But then

$$\frac{M(k)}{k \cdot (k+1)} \le \frac{c\_1}{k^{\frac{3}{2}}} \quad (k=1,2,\dots).$$

*<sup>A</sup>*ð Þ¼ *<sup>ε</sup> <sup>n</sup>* <sup>∈</sup> : *<sup>ω</sup>*ð Þ *<sup>n</sup>*

Put *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup>*, where *<sup>p</sup>* is a prime number, then *<sup>ω</sup>*ð Þ¼ *<sup>p</sup>* 1 and <sup>1</sup>

greater than *<sup>p</sup>*0. For these *<sup>p</sup>* we have: <sup>P</sup>

*Number Theory and Its Applications*

, … , *<sup>n</sup>*

the normal order is 2, we have

their proofs are outlined in [24].

X *n*∈ *M*

every *k*∈ , *M k*ð Þ≤ *c*<sup>1</sup>

1 *<sup>n</sup>* <sup>¼</sup> *<sup>M</sup>*ð Þ<sup>1</sup> 1 þ

> <sup>¼</sup> *<sup>M</sup>*ð Þ<sup>1</sup> <sup>1</sup> � <sup>2</sup> <sup>þ</sup>

In [32] it is shown that *M x*ð Þ¼ *<sup>O</sup>* ffiffiffi

ffiffiffi *k*

**Theorem 1.23** (see [24]). I*<sup>c</sup>* � lim ΓðÞ¼ *t* 2*:*

*<sup>n</sup>* < þ ∞. Evidently this is equivalent with

*<sup>c</sup>* � lim *<sup>ω</sup>*ð Þ *<sup>n</sup>*

*n* � 1 � �

statistical convergence in them is replaced by <sup>I</sup>ð Þ*<sup>q</sup>*

log log *<sup>n</sup>* 6¼ 1. Under the inclusion <sup>I</sup>ð Þ*<sup>q</sup>*

, *<sup>n</sup> n* � �

triangle. That is, <sup>Γ</sup>ð Þ*<sup>t</sup>* is the number of binomial coefficients *<sup>n</sup>*

∪ f g *ℵ*<sup>0</sup> ð Þ Γð Þ¼ 1 *ℵ*<sup>0</sup> . Let us observe that for every *t*∈ , Γð Þ*t* ≥1.

<sup>I</sup>*<sup>c</sup>* � lim *<sup>ω</sup>*ð Þ *<sup>n</sup>*

*n* 0 � �

*n k* � �

P *n*∈ *H* 1

**136**

1.13 we have <sup>I</sup>ð Þ*<sup>q</sup>*

, *<sup>n</sup>* 1 � � � � � �

for all prime numbers *p*>*p*0. Therefore the set *A<sup>ε</sup>* contains all prime numbers

*p*>*p*<sup>0</sup> 1

Further possibility where the results can be strengthened by the way that the

famous *Pascal's triangle*. The *n*-th row of the Pascal's triangle consists of the numbers

Let Γð Þ*t* denote the number of times the positive integer *t*, *t*>2 occurs in the Pascal's

¼ *t*. From this point of view Γ is the function which maps the set in the set

In [32] it is proved that the average and normal order of the function Γ is 2. Since

I*<sup>d</sup>* � lim ΓðÞ¼ *t* 2

(see [28]). We are going to show two results which strengthen the result of [32] and

X *n*∈ *M*

*M*ð Þ� 2 *M*ð Þ1

*M*ð Þ2 <sup>2</sup> � <sup>3</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup>

<sup>p</sup> holds. But then

*M k*ð Þ *<sup>k</sup>* � ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>≤</sup> 1 *n*

**Proof.** The values of the function Γ are positive integers for *t* 6¼ 1. Thus for *ε* >0 the set *A<sup>ε</sup>* ¼ f g *t*∈ :jΓðÞ�*t* 2j≥*ε* is a subset of the set *H* ¼ f g1 ∪ f g2 ∪ *M*, where *M* ¼ f g *t*∈ : Γð Þ*t* >2 . Note that Γð Þ¼ 2 1. Therefore is suffices to show that

We shall prove Eq. (9). Firstly, we write the left-hand site of Eq. (9) in the form

*M k*ð Þ *<sup>k</sup>* � ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>⋯</sup>*:*

ð Þ *k* ¼ 1, 2, … *:*

<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup>

*x* p

*c*1 *k* 3 2 log log *<sup>n</sup>* � <sup>1</sup>

*<sup>c</sup>* <sup>⊊</sup> <sup>I</sup>ð Þ<sup>1</sup>

log log *<sup>n</sup>* 6¼ 1 for *q* ∈ð0, 1i. This complete the proof.

. Their sum equals to 2*<sup>n</sup>* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>1</sup> *<sup>n</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

� �

� � � � ≥*ε*

*:*

� � �

*<sup>p</sup>* ¼ þ∞ and so *A*ð Þ*ε* ∉ I*c*. From this

log log *<sup>p</sup>* � 1

*<sup>c</sup>* � I*<sup>c</sup>* and according to Lemma

*<sup>c</sup>* –convergence is the concept of the

*k* � �

< þ ∞*:* (9)

*<sup>k</sup>* <sup>þ</sup> <sup>⋯</sup>

*M k*ð Þ� *M k*ð Þ � 1

ð Þ. Therefore there exists such *c*<sup>1</sup> >0 that for

� �

*k*¼0

satisfying

*n k* � � .

�≥*<sup>ε</sup>* holds

According these inequalities by comparison test of the convergence of the series in Eq. (9) follows.

Now we shall use the concept of <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence.

**Theorem 1.24** (see [24]). *For every q*> <sup>1</sup> 2 *,* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim ΓðÞ¼ *t* 2 *holds and* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim <sup>Γ</sup>ðÞ¼ *<sup>t</sup>* <sup>2</sup> *does not hold for any q,* <sup>0</sup><*q*<sup>≤</sup> <sup>1</sup> 2 *.*

**Proof.** Let 0 <*q*<1 and let *M* have the same meaning as in the proof of Theorem 1.23. Let us examine the series P *n*∈ *M* 1 *nq*. We write it in the form

$$\sum\_{n \in M} \frac{1}{n^q} = \frac{M(1)}{1^q} + \frac{M(2) - M(1)}{2^q} + \dots + \frac{M(k) - M(k-1)}{k^q} + \dotsb$$

$$= M(1) \left(\frac{1}{1^q} - \frac{1}{2^q}\right) + \dots + M(k) \left(\frac{1}{k^q} - \frac{1}{(k+1)^q}\right) + \dotsb \tag{10}$$

$$= \sum\_{k=1}^{\infty} M(k) \left(\frac{1}{k^q} - \frac{1}{(k+1)^q}\right).$$

In virtue of Lagrange's mean value theorem we have

$$k(k+1)^q - k^q = qz\_k^{q-1}, \quad k < z\_k < k+1 \quad (k = 1, 2, \dots).$$

Therefore the series Eq. (10) can be written in the form

$$\sum\_{k \in M} \frac{1}{k^q} = \sum\_{k=1}^{\infty} \frac{M(k) q z\_k^{q-1}}{k^q (k+1)^q} = \sum\_{k=1}^{\infty} \frac{qM(k)}{k^q (k+1)^q z\_k^{1-q}}.\tag{11}$$

But *z* 1�*q <sup>k</sup>* <sup>&</sup>gt; *<sup>k</sup>*<sup>1</sup>�*<sup>q</sup>* , ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup>* <sup>&</sup>gt;*k<sup>q</sup>* and so

$$\sum\_{n \in M} \frac{1}{n^q} \le q \sum\_{k=1}^\infty \frac{M(k)}{k^{1+q}}.$$

We have already seen, that *M k*ð Þ≤ *c*<sup>1</sup> ffiffiffi *k* <sup>p</sup> , ð Þ *<sup>k</sup>* <sup>¼</sup> 1, 2, … (in the proof of Theorem 1.23). Consider that every binomial coefficient *<sup>t</sup>* <sup>¼</sup> *<sup>n</sup>* 2 � �, *<sup>n</sup>*≥4 occurs in Pascal'<sup>s</sup> triangle at least four times as *<sup>n</sup>* 2 � �, *<sup>n</sup> n* � 2 � �, *<sup>t</sup>* 1 � �, *<sup>t</sup> t* � 1 � � � � . Therefore every number of this form belongs to *M*. Consequently for *x*>4, *x*∈ the number *M x*ð Þ is greater then or equal to the number *V x*ð Þ of all numbers of the form *<sup>n</sup>* 2 � �, *<sup>n</sup>* <sup>≥</sup><sup>4</sup> not exceeding *x*. But *V x*ð Þ≥*s* � 3, where *s* is the integer satisfying

$$
\binom{s}{2} \le x < \binom{s+1}{2} \cdot 2
$$

From this we get

$$x < \frac{s(s+1)}{2}, \quad s > \sqrt{2\kappa} - 1.$$

So we obtain

$$M(\mathfrak{x}) \ge V(\mathfrak{x}) \ge \mathfrak{s} - 3 > \sqrt{2\mathfrak{x}} - 4, \qquad \frac{M(\mathfrak{x})}{\sqrt{\mathfrak{x}}} > \sqrt{2} - \frac{4}{\mathfrak{x}} \ge \underbrace{\sqrt{2} - 1}\_{c\_2} > 0. \cdot \mathfrak{x}$$

Now it is clear that if *x*≥ 4 2 � � then

$$c\_2\sqrt{\mathcal{X}} \le \mathcal{M}(\mathfrak{x}) \le c\_1\sqrt{\mathcal{X}}, \quad c\_2 = \sqrt{2} - 1 > 0. \tag{12}$$

**Proof.** According to Theorem 2.2 of [27] again suppose that the

*<sup>c</sup>* � lim log log *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>*

where *q*∈ ð0, 1i. The proof is going similar as in the previous Theorem. Put *n* ¼ *pi*

, *<sup>p</sup> <sup>j</sup>* are distinct prime numbers. Then *<sup>f</sup>* <sup>∗</sup> ð Þ¼ *<sup>n</sup> <sup>f</sup>* <sup>∗</sup> *pi*

*<sup>p</sup>* ð Þ*<sup>j</sup>*

*<sup>n</sup>*�*<sup>q</sup>* <sup>≥</sup> <sup>X</sup><sup>∞</sup>

number of divisors of *<sup>n</sup>*). The following equality holds: log *f n*ð Þ¼ *d n*ð Þ

log log *<sup>n</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> log *d n*ð Þ

From Theorem 1.25 we have the following statement.

*γ*ð Þ� *n* 1

X∞ *n*¼2

*j*¼1 *p j* 6¼2

log log *pi p j*

> � � � �

*<sup>A</sup>*ð Þ¼ *<sup>ε</sup> <sup>n</sup>*<sup>∈</sup> : log log *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>*

log log *<sup>n</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> log 2,

log log *<sup>n</sup>* � ð Þ <sup>1</sup> <sup>þ</sup> log 2

*p j*

, *pi* <sup>¼</sup> <sup>2</sup> � �*:*

<sup>2</sup> <sup>þ</sup> log *d n*ð Þþ log log *<sup>n</sup>*, *<sup>n</sup>* <sup>&</sup>gt;*<sup>e</sup>*

<sup>þ</sup> log <sup>1</sup> 2 log log *<sup>n</sup>* , *<sup>n</sup>* <sup>&</sup>gt;*<sup>e</sup>*

*is not* <sup>I</sup>ð Þ*<sup>q</sup>*

*τ*ð Þ� *n* 1

*τ*ð Þ� *n* 1 *<sup>n</sup><sup>α</sup>* ,

*<sup>c</sup>* –convergence of functions *γ*ð Þ *n* and *τ*ð Þ *n* . The following

*<sup>n</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>π</sup>*<sup>2</sup>

6 *:*

� �*:*

1 2*pj*

diverges, we have *<sup>A</sup>*ð Þ*<sup>ε</sup>* <sup>∉</sup> <sup>I</sup>ð Þ*<sup>q</sup>*

log log *<sup>n</sup>* 6¼ 1 þ log 2 and the proof is complete. There exists a relationship between functions *f n*ð Þ and *d n*ð Þ (where *d n*ð Þ is the

log log *n*

*n*¼2

*n*¼2

In connection with these results we have investigated the convergence of series

*n*¼2

*<sup>n</sup><sup>α</sup>* , <sup>X</sup><sup>∞</sup>

log log *n* � �<sup>∞</sup>

*<sup>n</sup>* <sup>¼</sup> 1, <sup>X</sup><sup>∞</sup>

The following results concerning the functions *γ*ð Þ *n* and *τ*ð Þ *n* . In [33, Theorem 3, 5] there are proofs of the following results:

*γ*ð Þ� *n* 1

*p j* ,

*<sup>p</sup>* ð Þ*<sup>j</sup> pi p j* ¼

*p j* � � <sup>¼</sup> *f pi*

¼ 1. Let *ε*∈ð Þ 0, log 2 and define the set

� � � � ≥*ε*

, *i* 6¼ *j*. For *q*∈ð0, 1i we have:

*<sup>c</sup>* for all *q*∈ ð0, 1i. There-

*e :*

*e :*

*<sup>c</sup> –convergent for all q* ∈ð0, 1i*.*

<sup>2</sup> � log *n*, ð Þ *n* >*e*

<sup>I</sup>ð Þ*<sup>q</sup>*

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

, *<sup>i</sup>* ¼6 *<sup>j</sup>*. Hence log log *<sup>f</sup>* <sup>∗</sup> *pi*

This set contains all numbers of the type *pi*

*j*¼1 1 2*p <sup>j</sup>*

log log *f n*ð Þ¼ log <sup>1</sup>

log log *f n*ð Þ

**Corollary 1.27.** *The sequence* log *d n*ð Þ

X∞ *n*¼2

X *n*∈ *A*ð Þ*ε*

*i* ¼6 *j*, where *pi*

¼ *pi p j*

Since the series P<sup>∞</sup>

*<sup>c</sup>* � lim log log *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>*

(see [34]). From this we have

fore <sup>I</sup>ð Þ*<sup>q</sup>*

Therefore

for any *α*∈ ð Þ 0, 1 ,

**139**

that we need for <sup>I</sup>ð Þ*<sup>q</sup>*

results are outlined in [21].

*pi p <sup>j</sup> pi <sup>p</sup>* ð Þ*<sup>j</sup> pi p j*

Therefore by Eq. (11) we get

$$\sum\_{n \in M} \frac{1}{n^q} = q \sum\_{k=1}^{\infty} \frac{M(k)}{k^q (k+1)^q z\_k^{1-q}}.$$

But *zk* <*k* þ 1, hence *z* 1�*q <sup>k</sup>* <sup>&</sup>lt;ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup>�*<sup>q</sup>* and so

$$\sum\_{n \in M} \frac{1}{n^q} \ge q \sum\_{k=1}^\infty \frac{M(k)}{\left(k+1\right)^{1+q}} \dots$$

From this owing to Eq. (12) we obtain

$$\sum\_{n \in M} \frac{1}{n^q} \ge q \sum\_{k=1}^\infty \frac{c\_2 \sqrt{k}}{(k+1)^{1+q}} \ge q \frac{c\_2}{2} \sum\_{k=1}^\infty \frac{1}{(k+1)^{\frac{1}{2}+q}} = +\infty \quad \text{if} \quad 0 < q \le \frac{1}{2}.$$

Thus P *n* ∈ *M* 1 *nq* ¼ þ∞, and so <sup>P</sup> *n*∈ *A<sup>ε</sup>* 1 *nq* ¼ þ∞ for every *ε*>0. Similar results we can prove for functions *f n*ð Þ and *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>* .

**Theorem 1.25** (see [20, 27]). *The sequence* log log *f n*ð Þ log log *n* � �<sup>∞</sup> *n*¼2 *is not* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup> –convergent for all q*∈ð0, 1i*.*

**Proof.** According to Theorem 2.1 of [27] suppose that the

$$\mathcal{T}\_c^{(q)} - \lim \frac{\log \log f(n)}{\log \log n} = \mathbf{1} + \log \mathbf{2},$$

where *q*∈ ð0, 1i. Let *ε*∈ð Þ 0, log 2 and define the set

$$A(\varepsilon) = \left\{ n \in \mathbb{N} : \left| \frac{\log \log f(n)}{\log \log n} - (1 + \log 2) \right| \ge \varepsilon \right\}.$$

Put *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup>*, where *<sup>p</sup>* is a prime number, then *f p*ð Þ¼ *<sup>p</sup>* and log log *<sup>p</sup>* log log *<sup>p</sup>* ¼ 1. Therefore the set *A*ð Þ*ε* contains all prime numbers. Next we have:

$$\sum\_{n \in A(\mathfrak{e})} n^{-q} \ge \sum\_{j=1}^{\infty} p\_j^{-q} \ge \sum\_{j=1}^{\infty} p\_j^{-1} = +\infty, \quad q \in (0, 1).$$

Hence *<sup>A</sup>*ð Þ*<sup>ε</sup>* <sup>∉</sup> <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* and <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim log log *f n*ð Þ log log *<sup>n</sup>* 6¼ 1 þ log 2 for all *q*∈ð0, 1i.

**Theorem 1.26** (see [20, 27]). *The sequence* log log *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>* log log *n* � �<sup>∞</sup> *n*¼2 *is not* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup> –convergent for all q*∈ð i 0, 1 *.*

*M x*ð Þ≥*V x*ð Þ≥*<sup>s</sup>* � <sup>3</sup><sup>&</sup>gt; ffiffiffiffiffi

*c*2 ffiffiffi

4 2 � �

*<sup>x</sup>* <sup>p</sup> <sup>≤</sup> *M x*ð Þ≤*c*<sup>1</sup>

1 *nq* <sup>¼</sup> *<sup>q</sup>*

X *n*∈ *M*

X *n*∈ *M*

1�*q*

From this owing to Eq. (12) we obtain

*c*2 ffiffiffi *k* p ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup>þ*<sup>q</sup>* <sup>≥</sup>*<sup>q</sup>*

*nq* ¼ þ∞, and so <sup>P</sup>

<sup>I</sup>ð Þ*<sup>q</sup>*

where *q*∈ ð0, 1i. Let *ε*∈ð Þ 0, log 2 and define the set

the set *A*ð Þ*ε* contains all prime numbers. Next we have:

*<sup>n</sup>*�*<sup>q</sup>* <sup>≥</sup> <sup>X</sup><sup>∞</sup>

*<sup>c</sup>* and <sup>I</sup>ð Þ*<sup>q</sup>*

*j*¼1 *p j*

**Theorem 1.26** (see [20, 27]). *The sequence* log log *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>*

X *n*∈ *A*ð Þ*ε*

Hence *<sup>A</sup>*ð Þ*<sup>ε</sup>* <sup>∉</sup> <sup>I</sup>ð Þ*<sup>q</sup>*

*for all q*∈ð i 0, 1 *.*

**138**

*<sup>A</sup>*ð Þ¼ *<sup>ε</sup> <sup>n</sup>*<sup>∈</sup> : log log *f n*ð Þ

� � � �

Put *<sup>n</sup>* <sup>¼</sup> *<sup>p</sup>*, where *<sup>p</sup>* is a prime number, then *f p*ð Þ¼ *<sup>p</sup>* and log log *<sup>p</sup>*

�*<sup>q</sup>* <sup>≥</sup> <sup>X</sup><sup>∞</sup>

*<sup>c</sup>* � lim log log *f n*ð Þ

*j*¼1 *p j*

X∞ *k*¼1

Now it is clear that if *x*≥

*Number Theory and Its Applications*

Therefore by Eq. (11) we get

But *zk* <*k* þ 1, hence *z*

X *n*∈ *M*

Thus P

*all q*∈ð0, 1i*.*

1 *nq* <sup>≥</sup>*<sup>q</sup>*

*n* ∈ *M* 1 <sup>2</sup>*<sup>x</sup>* <sup>p</sup> � 4, *M x*ð Þ

ffiffiffi *<sup>x</sup>* <sup>p</sup> , *<sup>c</sup>*<sup>2</sup> <sup>¼</sup> ffiffi

X∞ *k*¼1

*<sup>k</sup>* <sup>&</sup>lt;ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup>�*<sup>q</sup>* and so

*c*2 2 X∞ *k*¼1

*n*∈ *A<sup>ε</sup>* 1

*<sup>c</sup>* � lim log log *f n*ð Þ

Similar results we can prove for functions *f n*ð Þ and *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>* . **Theorem 1.25** (see [20, 27]). *The sequence* log log *f n*ð Þ

**Proof.** According to Theorem 2.1 of [27] suppose that the

1 *nq* <sup>≥</sup>*<sup>q</sup>* *kq*

X∞ *k*¼1

then

ffiffiffi *<sup>x</sup>* <sup>p</sup> <sup>&</sup>gt; ffiffi 2 <sup>p</sup> � <sup>4</sup> *x* ≥ ffiffi 2 <sup>p</sup> � <sup>1</sup> |fflfflffl{zfflfflffl} *<sup>c</sup>*<sup>2</sup>

*M k*ð Þ

ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup> z* 1�*q k :*

*M k*ð Þ ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup>þ*<sup>q</sup> :*

1 ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup> 2 þ*q*

*nq* ¼ þ∞ for every *ε*>0.

log log *n* � �<sup>∞</sup>

log log *<sup>n</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> log 2,

log log *<sup>n</sup>* � ð Þ <sup>1</sup> <sup>þ</sup> log 2

� �

*n*¼2

� � � � ≥*ε*

�<sup>1</sup> ¼ þ∞, *<sup>q</sup>*∈ð0, 1i*:*

log log *<sup>n</sup>* 6¼ 1 þ log 2 for all *q*∈ð0, 1i.

*n*¼2

log log *n* � �<sup>∞</sup> *:*

*is not* <sup>I</sup>ð Þ*<sup>q</sup>*

log log *<sup>p</sup>* ¼ 1. Therefore

*<sup>c</sup> –convergent*

2

>0*:*

<sup>p</sup> � <sup>1</sup>>0*:* (12)

¼ þ∞ if 0< *q*≤

*is not* <sup>I</sup>ð Þ*<sup>q</sup>*

1 2 *:*

*<sup>c</sup> –convergent for*

**Proof.** According to Theorem 2.2 of [27] again suppose that the

$$\mathcal{T}\_{\varepsilon}^{(q)} - \lim \frac{\log \log f^\*(n)}{\log \log n} = 1 + \log 2,$$

where *q*∈ ð0, 1i. The proof is going similar as in the previous Theorem. Put *n* ¼ *pi p j* , *i* ¼6 *j*, where *pi* , *<sup>p</sup> <sup>j</sup>* are distinct prime numbers. Then *<sup>f</sup>* <sup>∗</sup> ð Þ¼ *<sup>n</sup> <sup>f</sup>* <sup>∗</sup> *pi p j* � � <sup>¼</sup> *f pi <sup>p</sup>* ð Þ*<sup>j</sup> pi p j* ¼ *pi p <sup>j</sup> pi <sup>p</sup>* ð Þ*<sup>j</sup> pi p j* ¼ *pi p j* , *<sup>i</sup>* ¼6 *<sup>j</sup>*. Hence log log *<sup>f</sup>* <sup>∗</sup> *pi <sup>p</sup>* ð Þ*<sup>j</sup>* log log *pi p j* ¼ 1. Let *ε*∈ð Þ 0, log 2 and define the set

$$A(\varepsilon) = \left\{ n \in \mathbb{N} : \left| \frac{\log \log f^\*(n)}{\log \log n} - (1 + \log 2) \right| \ge \varepsilon \right\}.$$

This set contains all numbers of the type *pi p j* , *i* 6¼ *j*. For *q*∈ð0, 1i we have:

$$\sum\_{n \in A(\iota)} n^{-q} \ge \sum\_{j=1}^{\infty} \frac{1}{2p\_j}, \quad \left(p\_i = 2\right).$$

Since the series P<sup>∞</sup> *j*¼1 1 2*p <sup>j</sup>* diverges, we have *<sup>A</sup>*ð Þ*<sup>ε</sup>* <sup>∉</sup> <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* for all *q*∈ ð0, 1i. Therefore <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* � lim log log *<sup>f</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>* log log *<sup>n</sup>* 6¼ 1 þ log 2 and the proof is complete.

There exists a relationship between functions *f n*ð Þ and *d n*ð Þ (where *d n*ð Þ is the number of divisors of *<sup>n</sup>*). The following equality holds: log *f n*ð Þ¼ *d n*ð Þ <sup>2</sup> � log *n*, ð Þ *n* >*e* (see [34]). From this we have

$$
\log \log f(n) = \log \frac{1}{2} + \log d(n) + \log \log n, \quad n > e^{\varepsilon}.
$$

Therefore

$$\frac{\log \log f(n)}{\log \log n} = 1 + \frac{\log d(n)}{\log \log n} + \frac{\log \frac{1}{2}}{\log \log n}, \quad n > e^{\epsilon}.$$

From Theorem 1.25 we have the following statement.

**Corollary 1.27.** *The sequence* log *d n*ð Þ log log *n* � �<sup>∞</sup> *n*¼2 *is not* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup> –convergent for all q* ∈ð0, 1i*.* The following results concerning the functions *γ*ð Þ *n* and *τ*ð Þ *n* . In [33, Theorem 3, 5] there are proofs of the following results:

$$\sum\_{n=2}^{\infty} \frac{\chi(n) - 1}{n} = 1, \qquad \sum\_{n=2}^{\infty} \frac{\pi(n) - 1}{n} = 1 + \frac{\pi^2}{6}.$$

In connection with these results we have investigated the convergence of series for any *α*∈ ð Þ 0, 1 ,

$$\sum\_{n=2}^{\infty} \frac{\wp(n) - 1}{n^a}, \qquad \sum\_{n=2}^{\infty} \frac{\mathfrak{r}(n) - 1}{n^a},$$

that we need for <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence of functions *γ*ð Þ *n* and *τ*ð Þ *n* . The following results are outlined in [21].

**Theorem 1.28.** *The series*

$$\sum\_{n=2}^{\infty} \frac{\gamma(n) - 1}{n^a}$$

diverges for 0<*α* ≤ <sup>1</sup> <sup>2</sup> and converges for *α*> <sup>1</sup> 2 . **Proof.**

a. Let 0 <*α* ≤ <sup>1</sup> 2 . Put *<sup>K</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> : *<sup>k</sup>*<sup>∈</sup> , *<sup>k</sup>*><sup>1</sup> � �. A simple estimation gives

$$\sum\_{n=2}^{\infty} \frac{\gamma(n) - 1}{n^a} \ge \sum\_{n \in K} \frac{\gamma(n) - 1}{n^a}.$$

Clearly *γ*ð Þ *n* ≥2 for *n* ∈*K*. Therefore

$$\sum\_{n=2}^{\infty} \frac{\gamma(n) - 1}{n^a} \ge \sum\_{n \in K} \frac{1}{n^a} = \sum\_{k=2}^{\infty} \frac{1}{k^{2a}} \ge \sum\_{k=2}^{\infty} \frac{1}{k} = +\infty. \tag{13}$$

X∞ *n*¼2

Theorem 1.28. Therefore *<sup>γ</sup>*ð Þ *<sup>n</sup>* is <sup>I</sup>ð Þ*<sup>q</sup>*

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

**Theorem 1.30** The series

diverges for 0<*α* ≤ <sup>1</sup>

P<sup>∞</sup> *n*¼2

1

*τ*ð Þ� *n* 1

Denote by *bk* <sup>¼</sup> <sup>1</sup>

therefore the series P*<sup>τ</sup>*ð Þ�*<sup>n</sup>* <sup>1</sup>

ii. <sup>I</sup>ð Þ*<sup>q</sup>*

**6. Conclusions**

**141**

i. I*c*–*convergent to 1*,

**Remark.** We have lim stat γð Þ¼ *n* 1.

1 *<sup>n</sup><sup>α</sup>* <sup>≤</sup> <sup>X</sup><sup>∞</sup> *k*¼2

X∞ *n*¼2

<sup>2</sup> and converges for *α*> <sup>1</sup>

*τ*ð Þ� *n* 1

*<sup>n</sup><sup>α</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

<sup>2</sup>*k<sup>α</sup>* � <sup>1</sup> *<sup>k</sup><sup>α</sup> <sup>k</sup><sup>α</sup>* ð Þ � <sup>1</sup> <sup>2</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*ak* <sup>¼</sup> <sup>2</sup> � <sup>1</sup>

*<sup>k</sup>*2*<sup>α</sup>* and consider that lim *<sup>k</sup>*!<sup>∞</sup> *ak*

2 � � *and* <sup>I</sup>ð Þ*<sup>q</sup>*

*kα* <sup>1</sup> � <sup>1</sup> *kα* � �<sup>2</sup> � <sup>1</sup>

It turns out that the study of I–convergence of arithmetical functions or some sequences related to these arithmetical functions for different kinds of ideals I

*k*¼2

We shall try to use a similar method to Mycielski's proof of the convergence of

**Proof.** Let 0 <*α* <1. We write the given series in the form

X∞ *n*¼2

*<sup>n</sup><sup>α</sup>* to explain the equality Eq. (15). Since *<sup>s</sup>*

X∞ *s*¼2

*<sup>t</sup><sup>α</sup> <sup>t</sup>*ð Þ *<sup>α</sup>*�<sup>1</sup> the right-hand side of Eq. (15) is equal to

For the *k*-th term of P*ak* we have

converges (diverges) if and only if the series P<sup>∞</sup>

*<sup>n</sup><sup>α</sup>* .

**Proof.** Similar to the proof of Corollary 1.29.

P*bk* is convergent (divergent) for any *α* > <sup>1</sup>

**Corollary 1.31.** *The sequence τ*ð Þ *n is*

*<sup>c</sup>* –*divergent for q*∈ 0, <sup>1</sup>

**Remark.** We have lim stat *τ*ð Þ¼ *n* 1*.*

X∞ *s*¼2

*<sup>c</sup>* –convergent to 1 if *q* ∈ <sup>1</sup>

2 .

X∞ *s*¼2

> *s*¼2 *ak*

*k*2*α :*

<sup>2</sup> 0< *α*≤ <sup>1</sup>

2

*<sup>c</sup> –convergent to* 1 *for q*∈ <sup>1</sup>

*s*

*<sup>k</sup>α<sup>s</sup>* ¼ � *<sup>k</sup> α d dt* 1 *tαs* � �

<sup>2</sup> , 1 � �.

*<sup>k</sup>α<sup>s</sup>* , (15)

*bk* <sup>¼</sup> 2. Hence the series <sup>P</sup><sup>∞</sup>

*<sup>s</sup>*¼<sup>2</sup>*bk* converges (diverges). Since

� � so does the series P*ak* and

<sup>2</sup> , 1 � �.

*<sup>t</sup>*¼*<sup>k</sup>* and <sup>P</sup><sup>∞</sup>

*s*¼2 1 *<sup>t</sup>α<sup>s</sup>* ¼

*<sup>s</sup>*¼<sup>2</sup>*ak*

The convergence of the series on the right-hand side we proved previously in

*τ*ð Þ� *n* 1 *nα*

1 *kαs :*

b. Let *α* > <sup>1</sup> 2 . We will use the formula

$$\sum\_{n=2}^{\infty} \frac{\gamma(n) - 1}{n^a} = \sum\_{k=2}^{\infty} \sum\_{s=2}^{\infty} \frac{1}{k^a} = \sum\_{k=2}^{\infty} \frac{1}{k^a (k^a - 1)}.\tag{14}$$

For a sufficiently large number *k k*ð Þ <sup>&</sup>gt;*k*<sup>0</sup> we have *<sup>k</sup><sup>α</sup> <sup>k</sup>α*�<sup>1</sup> <sup>&</sup>lt;2. We can estimate the series on the right-hand side of Eq. (14) with

$$\sum\_{k=2}^{\infty} \frac{1}{k^a (k^a - 1)} < \sum\_{k=2}^{k\_0} \frac{1}{k^a (k^a - 1)} + 2 \sum\_{k > k\_0} \frac{1}{k^{2a}}.$$

Since 2*α*>1 we get

$$\sum\_{n=2}^{\infty} \frac{\gamma(n) - 1}{n^a} < + \infty.$$

**Corollary 1.29.** *The sequence γ*ð Þ *n is*

i. I*c*–convergent to 1,

ii. <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –divergent for *q*∈ 0, <sup>1</sup> 2 � � and <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergent to 1 for *q*∈ <sup>1</sup> <sup>2</sup> , 1 � �.

#### **Proof.**


$$\sum\_{n \in A\_{\varepsilon}} \frac{1}{n^a} \ge \sum\_{n \in K} \frac{1}{n^a} \ge +\infty.$$

Therefore *<sup>γ</sup>*ð Þ *<sup>n</sup>* is <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –divergent. If <sup>1</sup> <sup>2</sup> <*q*< 1, *q* ¼ *α* then *A<sup>ε</sup>* ⊂ *H* and I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

$$\sum\_{n=2}^{\infty} \frac{1}{n^a} \le \sum\_{k=2}^{\infty} \sum\_{s=2}^{\infty} \frac{1}{k^{as}} \dots$$

The convergence of the series on the right-hand side we proved previously in Theorem 1.28. Therefore *<sup>γ</sup>*ð Þ *<sup>n</sup>* is <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergent to 1 if *q* ∈ <sup>1</sup> <sup>2</sup> , 1 � �.

**Remark.** We have lim stat γð Þ¼ *n* 1.

**Theorem 1.30** The series

**Theorem 1.28.** *The series*

*Number Theory and Its Applications*

diverges for 0<*α* ≤ <sup>1</sup>

2

Clearly *γ*ð Þ *n* ≥2 for *n* ∈*K*. Therefore

X∞ *n*¼2

> X∞ *n*¼2

series on the right-hand side of Eq. (14) with

X∞ *k*¼2

**Corollary 1.29.** *The sequence γ*ð Þ *n is*

*<sup>c</sup>* –divergent for *q*∈ 0, <sup>1</sup>

*γ*ð Þ� *n* 1 *<sup>n</sup><sup>α</sup>* <sup>≥</sup>

. We will use the formula

*γ*ð Þ� *n* 1

For a sufficiently large number *k k*ð Þ <sup>&</sup>gt;*k*<sup>0</sup> we have *<sup>k</sup><sup>α</sup>*

1 *<sup>k</sup><sup>α</sup> <sup>k</sup><sup>α</sup>* ð Þ � <sup>1</sup> <sup>&</sup>lt; <sup>X</sup>

*<sup>n</sup><sup>α</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

X∞ *n*¼2

2 � � and <sup>I</sup>ð Þ*<sup>q</sup>*

*<sup>s</sup>* f g : *<sup>n</sup>* <sup>∈</sup> , *<sup>t</sup>*>1, *<sup>s</sup>* <sup>&</sup>gt;<sup>1</sup> and <sup>P</sup>

X *n*∈ *A<sup>ε</sup>*

*<sup>c</sup>* –divergent. If <sup>1</sup>

1 *<sup>n</sup><sup>α</sup>* <sup>≥</sup>

*k*¼2

*k*0

*k*¼2

*γ*ð Þ� *n* 1

X∞ *s*¼2

1 *<sup>k</sup>α<sup>s</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼2

1 *<sup>k</sup><sup>α</sup> <sup>k</sup><sup>α</sup>* ð Þ � <sup>1</sup> <sup>þ</sup> <sup>2</sup>

*<sup>n</sup><sup>α</sup>* <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>*:*

i. Let *ε*>0. The set of numbers f g *n* ∈ , *n*> 1 :j*γ*ð Þ� *n* 1j≥ *ε* is a subset of

ii. Let *<sup>ε</sup>*>0 and denote *<sup>A</sup><sup>ε</sup>* <sup>¼</sup> f g *<sup>n</sup>* <sup>∈</sup> :j*γ*ð Þ� *<sup>n</sup>* <sup>1</sup>j≥*<sup>ε</sup>* . When 0 <sup>&</sup>lt;*q*<sup>≤</sup> <sup>1</sup>

X *n* ∈*K* *a* ∈ *H* 1

*<sup>n</sup><sup>α</sup>* <sup>≥</sup> <sup>þ</sup> <sup>∞</sup>*:*

<sup>2</sup> <*q*< 1, *q* ¼ *α* then *A<sup>ε</sup>* ⊂ *H* and

I*c*–convergence Cor. 1.29 i. (Cor. is the abbreviation for Corollary) follows.

numbers *<sup>n</sup>*<sup>∈</sup> *<sup>K</sup>*, *<sup>K</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> : *<sup>k</sup>*<sup>∈</sup> *<sup>N</sup>*, *<sup>k</sup>*><sup>1</sup> � � considering Eq. (13) for *<sup>q</sup>* <sup>¼</sup> *<sup>α</sup>* holds

1

a. Let 0 <*α* ≤ <sup>1</sup>

b. Let *α* > <sup>1</sup>

2

Since 2*α*>1 we get

i. I*c*–convergent to 1,

*H* ¼ *n* ¼ *t*

Therefore *<sup>γ</sup>*ð Þ *<sup>n</sup>* is <sup>I</sup>ð Þ*<sup>q</sup>*

ii. <sup>I</sup>ð Þ*<sup>q</sup>*

**Proof.**

**140**

**Proof.**

X∞ *n*¼2

<sup>2</sup> and converges for *α*> <sup>1</sup>

*γ*ð Þ� *n* 1 *<sup>n</sup><sup>α</sup>* <sup>≥</sup>

> X *n* ∈*K*

1 *<sup>n</sup><sup>α</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼2

X∞ *n*¼2

*γ*ð Þ� *n* 1 *nα*

> 2 .

*γ*ð Þ� *n* 1 *<sup>n</sup><sup>α</sup> :*

1 *<sup>k</sup>*<sup>2</sup>*<sup>α</sup>* <sup>≥</sup> <sup>X</sup><sup>∞</sup> *k*¼2

1

1

X *k* >*k*<sup>0</sup>

*<sup>c</sup>* –convergent to 1 for *q*∈ <sup>1</sup>

1 *k*2*α :*

*<sup>k</sup>* ¼ þ∞*:* (13)

*<sup>k</sup><sup>α</sup> <sup>k</sup><sup>α</sup>* ð Þ � <sup>1</sup> *:* (14)

*<sup>k</sup>α*�<sup>1</sup> <sup>&</sup>lt;2. We can estimate the

<sup>2</sup> , 1 � �.

<sup>2</sup> then for the

*<sup>a</sup>* < þ ∞. From the definition of

. Put *<sup>K</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> : *<sup>k</sup>*<sup>∈</sup> , *<sup>k</sup>*><sup>1</sup> � �. A simple estimation gives

X *n*∈*K*

$$\sum\_{n=2}^{\infty} \frac{\tau(n) - 1}{n^a}$$

diverges for 0<*α* ≤ <sup>1</sup> <sup>2</sup> and converges for *α*> <sup>1</sup> 2 . **Proof.** Let 0 <*α* <1. We write the given series in the form

$$\sum\_{n=2}^{\infty} \frac{\tau(n) - 1}{n^a} = \sum\_{k=2}^{\infty} \sum\_{s=2}^{\infty} \frac{s}{k^{\varpi}},\tag{15}$$

We shall try to use a similar method to Mycielski's proof of the convergence of P<sup>∞</sup> *n*¼2 *τ*ð Þ� *n* 1 *<sup>n</sup><sup>α</sup>* to explain the equality Eq. (15). Since *<sup>s</sup> <sup>k</sup>α<sup>s</sup>* ¼ � *<sup>k</sup> α d dt* 1 *tαs* � � *<sup>t</sup>*¼*<sup>k</sup>* and <sup>P</sup><sup>∞</sup> *s*¼2 1 *<sup>t</sup>α<sup>s</sup>* ¼ 1 *<sup>t</sup><sup>α</sup> <sup>t</sup>*ð Þ *<sup>α</sup>*�<sup>1</sup> the right-hand side of Eq. (15) is equal to

$$\sum\_{s=2}^{\infty} \frac{2k^a - 1}{k^a \left(k^a - 1\right)^2} = \sum\_{s=2}^{\infty} a\_k$$

For the *k*-th term of P*ak* we have

$$a\_k = \frac{2 - \frac{1}{k^a}}{\left(1 - \frac{1}{k^a}\right)^2} \cdot \frac{1}{k^{2a}} \cdot 1$$

Denote by *bk* <sup>¼</sup> <sup>1</sup> *<sup>k</sup>*2*<sup>α</sup>* and consider that lim *<sup>k</sup>*!<sup>∞</sup> *ak bk* <sup>¼</sup> 2. Hence the series <sup>P</sup><sup>∞</sup> *<sup>s</sup>*¼<sup>2</sup>*ak* converges (diverges) if and only if the series P<sup>∞</sup> *<sup>s</sup>*¼<sup>2</sup>*bk* converges (diverges). Since P*bk* is convergent (divergent) for any *α* > <sup>1</sup> <sup>2</sup> 0< *α*≤ <sup>1</sup> 2 � � so does the series P*ak* and therefore the series P*<sup>τ</sup>*ð Þ�*<sup>n</sup>* <sup>1</sup> *<sup>n</sup><sup>α</sup>* .

**Corollary 1.31.** *The sequence τ*ð Þ *n is*

i. I*c*–*convergent to 1*,

ii. <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –*divergent for q*∈ 0, <sup>1</sup> 2 � � *and* <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup> –convergent to* 1 *for q*∈ <sup>1</sup> <sup>2</sup> , 1 � �.

**Proof.** Similar to the proof of Corollary 1.29. **Remark.** We have lim stat *τ*ð Þ¼ *n* 1*.*

#### **6. Conclusions**

It turns out that the study of I–convergence of arithmetical functions or some sequences related to these arithmetical functions for different kinds of ideals I

(see [18]) gives a deeper insight into the behaviour and properties of these arithmetical functions.

**References**

361-375

669-686

3-12

**11**(1):1-7

struktury; 1968

[1] Fast H. Sur la convergence statistique. Colloquia Mathematica.

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

certain functions and related Summability methods. American Mathematical Monthly. 1959;**66**(5):

I–convergence. Real Analysis Exchange. Bratislava; 2000;**26**(2):

[4] Bourbaki N. Éléments de

[2] Schoenberg IJ. The Integrability of

methods. Journal of Mathematical Analysis and Applications. 2020;**484**(2).

[12] Giuliano R, Grekos G. On the upper

functions. Mathematica Slovaca. 2017;

[13] Paštéka M, Šalát T, Visnyai T. Remarks on Buck's measure density and a generalization of asymptotic density.

Tatra Mountains Mathematical Publications. 2005;**31**(2):87-101

[14] Kostyrko P, Mačaj M, Šalát T. Statistical convergence and

[15] Kostyrko P, Mačaj M, Šalát T, Strauch O. On statistical limit points. Proceedings of American Mathematical

Society. 2001;**129**(9):2647-2654

sequences of real numbers. Mathematica Slovaca. 1980;**30**(2):

139-150

2003;**6**:43-52

pp. 43-48

[16] Šalát T. On statistically convergent

[17] Šalát T, Visnyai T. Subadditive measures on N and the convergence of series with positive terms. Acta Math.

convergence of arithmetical functions. Periodica Mathematica Hungarica. 2020

[19] Baláž V. Remarks on uniform density *u*. In: Proceedings IAM Workshop on Institute of Information Engineering, Automation and Mathematics. Bratislava: Slovak University of Technology; 2007.

[20] Balá<sup>ž</sup> V, Gogola J, Visnyai T. <sup>I</sup>ð Þ*<sup>q</sup>*

convergence of arithmetical functions.

*<sup>c</sup>* –

[18] Tóth JT, Filip F, Bukor J, Zsilinszky L. On I<sup>&</sup>lt;*<sup>q</sup>* and I≤*<sup>q</sup>*

[Accessed: 24 January 2020]

I–convergence; 2000. Available from: http://thales.doa.fmph.uniba.sk/macaj/ ICON.pdf [Accessed: 20 January 2020]

[Accessed: 15 April 2020]

**67**(5):1105-1128

and lower exponential density

[3] Kostyrko P, Wilczyński W, Šalát T.

mathématique: Topologie générale. In: Livre III, (Russian translation) Obščaja topologija. Moskow, Nauka: Osnovnye

[5] Baláž V, Strauch O, Šalát T. Remarks on several types of convergence of bounded sequences. Acta Mathematica Universitatis Ostraviensis. 2006;**14**(1):

[6] Baláž V, Šalát T. Uniform density *u* and corresponding I*u*–convergence. Mathematical Communications. 2006;

[7] Connor J. The statistical and strong *p*-Cesaro convergence of sequences.

[8] Fridy JA. On statistical convergence.

Sleziak M. I–convergence and extremal I–limit points. Mathematica Slovaca.

convergence. Analysis. 1991;**11**(1):59-66

[11] Filipów R, Tryba J. Representation of ideal convergence as a union and intersection of matrix summability

Analysis. 1988;**8**(1–2):47-64

Analysis. 1985;**5**(4):301-314

2005;**55**(4):443-464

**143**

[9] Kostyrko P, Mačaj M, Šalát T,

[10] Fridy J, Miller H. A matrix characterization of statistical

1951;**2**(3–4):241-244

On the other hand Algebraic number theory has many deep applications in cryptology. Many basic algorithms, which are widely used, have its security due to ANT. The theory of arithmetic functions has many connections to the classical ciphers, and to the general theory as well.

### **Acknowledgements**

This part was partially supported by The Slovak Research and Development Agency under the grant VEGA No. 2/0109/18.

### **Author details**

Vladimír Baláž\*† and Tomáš Visnyai† Institute of Information Engineering, Automation, and Mathematics, Faculty of Chemical and Food Technology STU in Bratislava, Bratislava, Slovakia

\*Address all correspondence to: vladimir.balaz@stuba.sk

† These authors contributed equally.

© 2020 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.

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

### **References**

(see [18]) gives a deeper insight into the behaviour and properties of these arith-

On the other hand Algebraic number theory has many deep applications in cryptology. Many basic algorithms, which are widely used, have its security due to ANT. The theory of arithmetic functions has many connections to the classical

This part was partially supported by The Slovak Research and Development

Institute of Information Engineering, Automation, and Mathematics, Faculty of

© 2020 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,

Chemical and Food Technology STU in Bratislava, Bratislava, Slovakia

\*Address all correspondence to: vladimir.balaz@stuba.sk

metical functions.

*Number Theory and Its Applications*

**Acknowledgements**

**Author details**

**142**

Vladimír Baláž\*† and Tomáš Visnyai†

† These authors contributed equally.

provided the original work is properly cited.

ciphers, and to the general theory as well.

Agency under the grant VEGA No. 2/0109/18.

[1] Fast H. Sur la convergence statistique. Colloquia Mathematica. 1951;**2**(3–4):241-244

[2] Schoenberg IJ. The Integrability of certain functions and related Summability methods. American Mathematical Monthly. 1959;**66**(5): 361-375

[3] Kostyrko P, Wilczyński W, Šalát T. I–convergence. Real Analysis Exchange. Bratislava; 2000;**26**(2): 669-686

[4] Bourbaki N. Éléments de mathématique: Topologie générale. In: Livre III, (Russian translation) Obščaja topologija. Moskow, Nauka: Osnovnye struktury; 1968

[5] Baláž V, Strauch O, Šalát T. Remarks on several types of convergence of bounded sequences. Acta Mathematica Universitatis Ostraviensis. 2006;**14**(1): 3-12

[6] Baláž V, Šalát T. Uniform density *u* and corresponding I*u*–convergence. Mathematical Communications. 2006; **11**(1):1-7

[7] Connor J. The statistical and strong *p*-Cesaro convergence of sequences. Analysis. 1988;**8**(1–2):47-64

[8] Fridy JA. On statistical convergence. Analysis. 1985;**5**(4):301-314

[9] Kostyrko P, Mačaj M, Šalát T, Sleziak M. I–convergence and extremal I–limit points. Mathematica Slovaca. 2005;**55**(4):443-464

[10] Fridy J, Miller H. A matrix characterization of statistical convergence. Analysis. 1991;**11**(1):59-66

[11] Filipów R, Tryba J. Representation of ideal convergence as a union and intersection of matrix summability

methods. Journal of Mathematical Analysis and Applications. 2020;**484**(2). [Accessed: 15 April 2020]

[12] Giuliano R, Grekos G. On the upper and lower exponential density functions. Mathematica Slovaca. 2017; **67**(5):1105-1128

[13] Paštéka M, Šalát T, Visnyai T. Remarks on Buck's measure density and a generalization of asymptotic density. Tatra Mountains Mathematical Publications. 2005;**31**(2):87-101

[14] Kostyrko P, Mačaj M, Šalát T. Statistical convergence and I–convergence; 2000. Available from: http://thales.doa.fmph.uniba.sk/macaj/ ICON.pdf [Accessed: 20 January 2020]

[15] Kostyrko P, Mačaj M, Šalát T, Strauch O. On statistical limit points. Proceedings of American Mathematical Society. 2001;**129**(9):2647-2654

[16] Šalát T. On statistically convergent sequences of real numbers. Mathematica Slovaca. 1980;**30**(2): 139-150

[17] Šalát T, Visnyai T. Subadditive measures on N and the convergence of series with positive terms. Acta Math. 2003;**6**:43-52

[18] Tóth JT, Filip F, Bukor J, Zsilinszky L. On I<sup>&</sup>lt;*<sup>q</sup>* and I≤*<sup>q</sup>* convergence of arithmetical functions. Periodica Mathematica Hungarica. 2020 [Accessed: 24 January 2020]

[19] Baláž V. Remarks on uniform density *u*. In: Proceedings IAM Workshop on Institute of Information Engineering, Automation and Mathematics. Bratislava: Slovak University of Technology; 2007. pp. 43-48

[20] Balá<sup>ž</sup> V, Gogola J, Visnyai T. <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* – convergence of arithmetical functions. Journal of Number Theory. 2018;**183**: 74-83

[21] Fehér Z, László B, Mačaj M, Šalát T. Remarks on arithmetical functions *ap*ð Þ *n* , *γ*ð Þ *n* , *τ*ð Þ *n* . Annals of Mathematics and Informaticae. 2006;**33**:35-43

[22] Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton: Princeton University Press; 1981

[23] Gogola J, Mačaj M, Visnyai T. On <sup>I</sup>ð Þ*<sup>q</sup> <sup>c</sup>* –convergence. Annals of Mathematics and Informaticae. 2011;**38**:27-36

[24] Gubo Š, Mačaj M, Šalát T, Tomanová J. On binomial coefficients. Acta Math. 2003;**6**:33-42

[25] Renling J. Applications of nonstandard analysis in additive number theory. The Bulletin of Symbolic Logic. 2000;**6**(3):331-341

[26] Šalát T. On the function *pap*ð Þ *<sup>n</sup>* <sup>k</sup>*n n*ð Þ <sup>&</sup>gt;<sup>1</sup> . Mathematica Slovaca. 1994;**44**(2):143-151

[27] Šalát T, Tomanová J. On the product of divisors of a positive integer. Mathematica Slovaca. 2002;**52**(3): 271-287

[28] Schinzel A, Šalát T. Remarks on maximum and minimum exponents in factoring. Mathematica Slovaca. 1994; **44**(5):505-514

[29] Petri H. Asymptotic properties of welfare relations. Economic Theory. 2019;**67**(4):853-874

[30] Ramsey FP. A mathematical theory of saving. The Economic Journal. 1928; **38**(152):543-559

[31] Fey M. May's theorem with an infinite population. Social Choice and Welfare. 2004;**23**:275-293

[32] Abbott H, Erdös P, Hanson D. On the number of times an integer occurs as a binomial coefficient. American Mathematical Monthly. 1974;**81**(3): 256-261

the American Mathematical Society.

I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

[45] Buck RC. The measure theoretic approach to density. American Journal of Mathematics. 1946;**68**(4):560-580

[46] Powell BJ, Šalát T. Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers. Publications de l'Institut Mathématique Nouvelle Série. 1991;**50**:

[47] Mikusiński P. Axiomatic theory of convergence. Pr Nauk Uniw ŚI Katow.

[48] Mitrinoviċ DS, Sándor J, Crstici B.

Handbook of Number Theory (Mathematics and its Applications). Vol. 351. Dordrecht: Kluwer Academic

1995;**347**(5):1811-1819

60-70

1982;**12**:13-21

Publishers; 1995

**145**

[33] Mycielski J. Sur les représentations des nombres naturels par des puissances à base et exposant naturels. Colloquium Mathematicum. 1951;**2**(3–4):254-260

[34] Hardy GH, Wright EM. An Introduction to the Theory of Numbers. 5th ed. Oxford: Clarendon Press; 1979

[35] Ostmann HH. Additive Zahlentheorie I. Berlin: Springer; 1956

[36] Paštéka M. On Four Approaches to Density. Bratislava: Peter Lang AG; 2014

[37] Paštéka M. Density and Related Topics. Nakladatelství Academia: Praha; 2017

[38] Strauch O, Porubský Š. Distribution of Sequences: A Sampler. Bern, Switzerland: Peter Lang; 2005

[39] Brown T, Freedman A. Arithmetic progressions in lacunary sets. Rocky Mountain Journal of Mathematics. 1987; **17**(3):587-596

[40] Freedman AR, Sember JJ. Densities and summability. Pacific Journal of Mathematics. 1981;**95**(2):293-305

[41] Kuratowski C. Topologie I. Warszawa: Panstwowe Wydawnictwa Naukowe; 1958

[42] Nagata JI. Modern General Topology. 2nd ed. Amsterdam: Elsevier; 1985

[43] Petersen GM. Regular Matrix Transformations. London: McGraw-Hill; 1966

[44] Miller HI. A measure theoretical subsequence characterization of statistical convergence. Transactions of I*–Convergence of Arithmetical Functions DOI: http://dx.doi.org/10.5772/intechopen.91932*

the American Mathematical Society. 1995;**347**(5):1811-1819

Journal of Number Theory. 2018;**183**:

*Number Theory and Its Applications*

a binomial coefficient. American Mathematical Monthly. 1974;**81**(3):

[34] Hardy GH, Wright EM. An

[35] Ostmann HH. Additive

[33] Mycielski J. Sur les représentations des nombres naturels par des puissances à base et exposant naturels. Colloquium Mathematicum. 1951;**2**(3–4):254-260

Introduction to the Theory of Numbers. 5th ed. Oxford: Clarendon Press; 1979

Zahlentheorie I. Berlin: Springer; 1956

[36] Paštéka M. On Four Approaches to Density. Bratislava: Peter Lang AG;

[37] Paštéka M. Density and Related Topics. Nakladatelství Academia: Praha;

[38] Strauch O, Porubský Š. Distribution

[39] Brown T, Freedman A. Arithmetic progressions in lacunary sets. Rocky Mountain Journal of Mathematics. 1987;

[40] Freedman AR, Sember JJ. Densities and summability. Pacific Journal of Mathematics. 1981;**95**(2):293-305

[41] Kuratowski C. Topologie I. Warszawa: Panstwowe Wydawnictwa

[42] Nagata JI. Modern General

[43] Petersen GM. Regular Matrix Transformations. London: McGraw-

[44] Miller HI. A measure theoretical subsequence characterization of statistical convergence. Transactions of

Topology. 2nd ed. Amsterdam: Elsevier;

of Sequences: A Sampler. Bern, Switzerland: Peter Lang; 2005

256-261

2014

2017

**17**(3):587-596

Naukowe; 1958

1985

Hill; 1966

[21] Fehér Z, László B, Mačaj M, Šalát T. Remarks on arithmetical functions *ap*ð Þ *n* , *γ*ð Þ *n* , *τ*ð Þ *n* . Annals of Mathematics and Informaticae. 2006;**33**:35-43

[22] Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton: Princeton

[23] Gogola J, Mačaj M, Visnyai T. On

Tomanová J. On binomial coefficients.

and Informaticae. 2011;**38**:27-36

[24] Gubo Š, Mačaj M, Šalát T,

[25] Renling J. Applications of nonstandard analysis in additive number theory. The Bulletin of Symbolic Logic. 2000;**6**(3):331-341

[26] Šalát T. On the function

of divisors of a positive integer. Mathematica Slovaca. 2002;**52**(3):

[28] Schinzel A, Šalát T. Remarks on maximum and minimum exponents in factoring. Mathematica Slovaca. 1994;

[29] Petri H. Asymptotic properties of welfare relations. Economic Theory.

[30] Ramsey FP. A mathematical theory of saving. The Economic Journal. 1928;

[31] Fey M. May's theorem with an infinite population. Social Choice and

[32] Abbott H, Erdös P, Hanson D. On the number of times an integer occurs as

Welfare. 2004;**23**:275-293

1994;**44**(2):143-151

271-287

**44**(5):505-514

2019;**67**(4):853-874

**38**(152):543-559

**144**

*pap*ð Þ *<sup>n</sup>* <sup>k</sup>*n n*ð Þ <sup>&</sup>gt;<sup>1</sup> . Mathematica Slovaca.

[27] Šalát T, Tomanová J. On the product

Acta Math. 2003;**6**:33-42

*<sup>c</sup>* –convergence. Annals of Mathematics

University Press; 1981

74-83

<sup>I</sup>ð Þ*<sup>q</sup>*

[45] Buck RC. The measure theoretic approach to density. American Journal of Mathematics. 1946;**68**(4):560-580

[46] Powell BJ, Šalát T. Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers. Publications de l'Institut Mathématique Nouvelle Série. 1991;**50**: 60-70

[47] Mikusiński P. Axiomatic theory of convergence. Pr Nauk Uniw ŚI Katow. 1982;**12**:13-21

[48] Mitrinoviċ DS, Sándor J, Crstici B. Handbook of Number Theory (Mathematics and its Applications). Vol. 351. Dordrecht: Kluwer Academic Publishers; 1995

**Chapter 9**

**Abstract**

**1. Introduction**

**147**

Identification of

*Triantafyllos K. Makarios*

Eigen-Frequencies and

Conditions at Supports

is given via the above-mentioned three degrees of freedom.

modal analysis of the continuous beam, inverted pendulum

**Keywords:** equivalent masses of continuous amphi-hinge vertical pendulum, identification of mode-shapes, distributed mass and stiffness, continuous systems,

An ideal three degrees of freedom system that is equivalent with the modal behavior of an infinity number of degree of freedom of two cases of pendulums is proposed for each case. The first pendulum has hinges at the two ends, while the second pendulum is a cantilever (inverted pendulum). Both pendulums are presented analytically in the present article. This equivalent three DoF system can

Mode-Shapes of Beams with

and Elasticity and for Various

Continuous Distribution of Mass

In the present article, an equivalent three degrees of freedom (DoF) system of two different cases of inverted pendulums is presented for each separated case. The first case of inverted pendulum refers to an amphi-hinge pendulum that possesses distributed mass and stiffness along its height, while the second case of inverted pendulum refers to an inverted pendulum with distributed mass and stiffness along its height. These vertical pendulums have infinity number of degree of freedoms. Based on the free vibration of the above-mentioned pendulums according to partial differential equation, a mathematically equivalent three-degree of freedom system is given for each case, where its equivalent mass matrix is analytically formulated with reference on specific mass locations along the pendulum height. Using the three DoF model, the first three fundamental frequencies of the real pendulum can be identified with very good accuracy. Furthermore, taking account the 3 3 mass matrix, it is possible to estimate the possible pendulum damages using a known technique of identification mode-shapes via records of response accelerations. Moreover, the way of instrumentation with a local network by three accelerometers

#### **Chapter 9**

## Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution of Mass and Elasticity and for Various Conditions at Supports

*Triantafyllos K. Makarios*

### **Abstract**

In the present article, an equivalent three degrees of freedom (DoF) system of two different cases of inverted pendulums is presented for each separated case. The first case of inverted pendulum refers to an amphi-hinge pendulum that possesses distributed mass and stiffness along its height, while the second case of inverted pendulum refers to an inverted pendulum with distributed mass and stiffness along its height. These vertical pendulums have infinity number of degree of freedoms. Based on the free vibration of the above-mentioned pendulums according to partial differential equation, a mathematically equivalent three-degree of freedom system is given for each case, where its equivalent mass matrix is analytically formulated with reference on specific mass locations along the pendulum height. Using the three DoF model, the first three fundamental frequencies of the real pendulum can be identified with very good accuracy. Furthermore, taking account the 3 3 mass matrix, it is possible to estimate the possible pendulum damages using a known technique of identification mode-shapes via records of response accelerations. Moreover, the way of instrumentation with a local network by three accelerometers is given via the above-mentioned three degrees of freedom.

**Keywords:** equivalent masses of continuous amphi-hinge vertical pendulum, identification of mode-shapes, distributed mass and stiffness, continuous systems, modal analysis of the continuous beam, inverted pendulum

#### **1. Introduction**

An ideal three degrees of freedom system that is equivalent with the modal behavior of an infinity number of degree of freedom of two cases of pendulums is proposed for each case. The first pendulum has hinges at the two ends, while the second pendulum is a cantilever (inverted pendulum). Both pendulums are presented analytically in the present article. This equivalent three DoF system can

#### *Number Theory and Its Applications*

be used in instrumentation of such pendulums, where the concept of the concentrated masses is not existing, with a local network of three accelerometers. This issue is a main problem that appears very common during the instrumentation of inverted pendulums or bridge beams or steel stairs [1–3] or wind energy powers [4, 5] in order to identify the real vibration mode-shapes and the fundamental eigen-frequencies of the structure via records of response accelerograms at specific positions due to ambient excitation [6, 7].

of this beam, we consider an infinitesimal part of the vertical beam, at location *x* from the origin *o*, that has isolated by two very nearest parallel sections. The infinitesimal length of this part is the *dx*. On this infinitesimal length, we notice the flexural moment *M x*ð Þ , *t* , the shear force *Q x*ð Þ , *t* with their differential increments, while the axial force *Ν*ð Þ *x*, *t* is ignored, because it does not affect the horizontal beam vibration along z-axis. Moreover, noted the resulting force *Pz*ð Þ *x*, *t* of the

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution…*

where the resulting force *Pz*ð Þ *x*, *t* acts at the total beam infinitesimal part. Furthermore, according to D'Alembert principle, the resulting inertia force

Here, we agree that the time derivatives of the displacements are going to symbolize with full stops, while the spatial derivatives of the displacements are going to symbolize with accent. Next, the damping and the second order differential are ignored, so the

Moreover, the moment equilibrium with reference to weight center (w.c.) of the

*∂Q ∂x dx* � � �

> *<sup>Q</sup>* <sup>¼</sup> *<sup>∂</sup><sup>M</sup> ∂x*

According to Euler-Bernoulli Bending theory (where the shear deformations are

*∂*2 *uz*ð Þ *x*, *t*

*∂Q ∂x*

*uz*ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>=</sup>∂<sup>t</sup>*

force equilibrium on the infinitesimal part of the beam along z-axis gives:

<sup>X</sup>*Fz* <sup>¼</sup> <sup>0</sup> ) *<sup>Q</sup>* <sup>þ</sup> *Pz*ð Þ� *<sup>x</sup>*, *<sup>t</sup> <sup>Q</sup>* <sup>þ</sup>

*∂Q*

*dx*

<sup>2</sup> <sup>þ</sup> *<sup>Q</sup>* <sup>þ</sup>

ignored), it is well-known that the following basic equation is true:

*M x*ð Þ¼ , *t ΕΙ<sup>y</sup>* �

Eqs. (4) and (5) are inserted into Eq. (3), so the motion equation without

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> *pz*ð Þ� *<sup>x</sup>*, *<sup>t</sup> <sup>m</sup>* � *<sup>u</sup>*€*z*ð Þ) *<sup>x</sup>*, *<sup>t</sup>*

*<sup>∂</sup>*<sup>4</sup>*uz*ð Þ *<sup>x</sup>*, *<sup>t</sup>*

infinitesimal part of the beam (see **Figure 1**) gives:

damping for the examined vertical beam is given:

*∂*2 *<sup>∂</sup>x*<sup>2</sup> *ΕΙ<sup>y</sup>* �

**149**

*m ∂*2 *uz*ð Þ *x*, *t <sup>∂</sup>t*<sup>2</sup> <sup>þ</sup> *ΕΙ<sup>y</sup>*

*∂*2 *M*

*∂*2 *uz*ð Þ *x*, *t ∂x*<sup>2</sup> � �

*m* � *u*€*z*ð Þþ *x*, *t ΕΙ<sup>y</sup>* � *u*<sup>0000</sup>

*Pz*ð Þ¼ *x*, *t pz*ð Þ� *x*, *t dx* (1)

<sup>2</sup> ) *Fa*ð Þ¼ � *<sup>x</sup>*, *<sup>t</sup>* ð Þ� *<sup>m</sup>* � *dx <sup>u</sup>*€*z*ð Þ *<sup>x</sup>*, *<sup>t</sup>* (2)

*dx* � � <sup>þ</sup> *Fa*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>0</sup> )

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *pz*ð Þ� *<sup>x</sup>*, *<sup>t</sup> <sup>m</sup>* � *<sup>u</sup>*€*z*ð Þ *<sup>x</sup>*, *<sup>t</sup>* (3)

<sup>2</sup> � *<sup>M</sup>* <sup>þ</sup>

*∂M ∂x dx* � � <sup>¼</sup> <sup>0</sup> )

*<sup>∂</sup>x*<sup>2</sup> (5)

*<sup>z</sup>* ð Þ¼ *x*, *t pz*ð Þ *x*, *t* (6)

(4)

*dx*

¼ *pz*ð Þ� *x*, *t m* � *u*€*z*ð Þ) *x*, *t*

*<sup>∂</sup>x*<sup>4</sup> <sup>¼</sup> *pz*ð Þ) *<sup>x</sup>*, *<sup>t</sup>*

external dynamic loading. Therefore, we can write:

*DOI: http://dx.doi.org/10.5772/intechopen.92185*

*Fa*ð Þ *x*, *t* is noted, where:

*Fa*ð Þ¼ � *<sup>x</sup>*, *<sup>t</sup>* ð Þ� *<sup>m</sup>* � *dx <sup>∂</sup>*<sup>2</sup>

<sup>X</sup>*My* <sup>¼</sup> <sup>0</sup> ) *<sup>M</sup>* <sup>þ</sup> *<sup>Q</sup>* �

#### **2. First case: modal analysis of undamped amphi-hinge vertical pendulum with distributed mass and stiffness**

According to the theory of continuous systems [8, 9], consider a straight amphi-hinge vertical pendulum that is loaded by an external continuous dynamic loading *pz*ð Þ *x*, *t* , with reference to a Cartesian three-dimensional reference system *oxyz*, (**Figure 1**).

The vertical pendulum possesses a distributed mass *m x*ð Þ per unit height, which in the special case of uniform distribution is given as *m x*ð Þ¼ *m* in tons per meter (tn/m). Furthermore, according to Bernoulli Technical Bending Theory, the beam has section flexural stiffness *ΕΙy*ð Þ *x* , where in the special case of an uniform distribution of the stiffness it is given as *ΕΙy*ð Þ¼ *x ΕΙy*, where *E* is the material modulus of elasticity and *Ι<sup>y</sup>* is the section moment of inertia about y-axis (**Figure 1**). Next, we are examining a such amphi-hinge vertical pendulum that possesses constant value of distributed mass along its height, as well as constant value of distributed section flexural stiffness *ΕΙy*. Due to fact that the vertical pendulum mass is continuously distributed, this pendulum/beam has infinity number of degrees of freedom for vibration along the horizontal *oz*-axis. In order to formulate of the motion equation

**Figure 1.** *Amphi-hinge vertical pendulum with distributed mass and section flexural stiffness.*

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution… DOI: http://dx.doi.org/10.5772/intechopen.92185*

of this beam, we consider an infinitesimal part of the vertical beam, at location *x* from the origin *o*, that has isolated by two very nearest parallel sections. The infinitesimal length of this part is the *dx*. On this infinitesimal length, we notice the flexural moment *M x*ð Þ , *t* , the shear force *Q x*ð Þ , *t* with their differential increments, while the axial force *Ν*ð Þ *x*, *t* is ignored, because it does not affect the horizontal beam vibration along z-axis. Moreover, noted the resulting force *Pz*ð Þ *x*, *t* of the external dynamic loading. Therefore, we can write:

$$P\_x(\mathbf{x}, t) = p\_x(\mathbf{x}, t) \cdot d\mathbf{x} \tag{1}$$

where the resulting force *Pz*ð Þ *x*, *t* acts at the total beam infinitesimal part.

Furthermore, according to D'Alembert principle, the resulting inertia force *Fa*ð Þ *x*, *t* is noted, where:

$$F\_a(\mathbf{x}, t) = (-\overline{m} \cdot d\mathbf{x}) \cdot \partial^2 u\_x(\mathbf{x}, t) / \partial t^2 \Rightarrow F\_a(\mathbf{x}, t) = (-\overline{m} \cdot d\mathbf{x}) \cdot \ddot{u}\_x(\mathbf{x}, t) \tag{2}$$

Here, we agree that the time derivatives of the displacements are going to symbolize with full stops, while the spatial derivatives of the displacements are going to symbolize with accent. Next, the damping and the second order differential are ignored, so the force equilibrium on the infinitesimal part of the beam along z-axis gives:

$$
\sum F\_x = \mathbf{0} \implies Q + P\_x(\mathbf{x}, t) - \left(Q + \frac{\partial Q}{\partial \mathbf{x}} d\mathbf{x}\right) + F\_d(\mathbf{x}, t) = \mathbf{0} \Rightarrow
$$

$$
\frac{\partial Q}{\partial \mathbf{x}} = p\_x(\mathbf{x}, t) - \overline{m} \cdot \ddot{u}\_x(\mathbf{x}, t) \tag{3}
$$

Moreover, the moment equilibrium with reference to weight center (w.c.) of the infinitesimal part of the beam (see **Figure 1**) gives:

$$
\sum M\_{\mathcal{V}} = 0 \Rightarrow M + Q \cdot \frac{dx}{2} + \left(Q + \frac{\partial Q}{\partial \mathbf{x}} dx\right) \cdot \frac{dx}{2} - \left(M + \frac{\partial M}{\partial \mathbf{x}} dx\right) = 0 \Rightarrow
$$

$$
Q = \frac{\partial M}{\partial \mathbf{x}}\tag{4}
$$

According to Euler-Bernoulli Bending theory (where the shear deformations are ignored), it is well-known that the following basic equation is true:

$$M(\mathbf{x},t) = EI\_\mathcal{y} \cdot \frac{\partial^2 u\_x(\mathbf{x},t)}{\partial \mathbf{x}^2} \tag{5}$$

Eqs. (4) and (5) are inserted into Eq. (3), so the motion equation without damping for the examined vertical beam is given:

$$\frac{\partial^2 M}{\partial \mathbf{x}^2} = p\_x(\mathbf{x}, t) - \overline{m} \cdot \ddot{u}\_x(\mathbf{x}, t) \Rightarrow$$

$$\frac{\partial^2}{\partial \mathbf{x}^2} \left( EI\_\mathcal{Y} \cdot \frac{\partial^2 u\_x(\mathbf{x}, t)}{\partial \mathbf{x}^2} \right) = p\_x(\mathbf{x}, t) - \overline{m} \cdot \ddot{u}\_x(\mathbf{x}, t) \Rightarrow$$

$$\overline{m} \frac{\partial^2 u\_x(\mathbf{x}, t)}{\partial t^2} + EI\_\mathcal{Y} \frac{\partial^4 u\_x(\mathbf{x}, t)}{\partial \mathbf{x}^4} = p\_x(\mathbf{x}, t) \Rightarrow$$

$$\overline{m} \cdot \ddot{u}\_x(\mathbf{x}, t) + EI\_\mathcal{Y} \cdot u\_x^{\prime\prime\prime}(\mathbf{x}, t) = p\_x(\mathbf{x}, t) \tag{6}$$

be used in instrumentation of such pendulums, where the concept of the concentrated masses is not existing, with a local network of three accelerometers. This issue is a main problem that appears very common during the instrumentation of inverted pendulums or bridge beams or steel stairs [1–3] or wind energy powers [4, 5] in order to identify the real vibration mode-shapes and the fundamental eigen-frequencies of the structure via records of response accelerograms at specific

**2. First case: modal analysis of undamped amphi-hinge vertical**

According to the theory of continuous systems [8, 9], consider a straight amphi-hinge vertical pendulum that is loaded by an external continuous dynamic loading *pz*ð Þ *x*, *t* , with reference to a Cartesian three-dimensional reference system *oxyz*, (**Figure 1**).

The vertical pendulum possesses a distributed mass *m x*ð Þ per unit height, which in the special case of uniform distribution is given as *m x*ð Þ¼ *m* in tons per meter (tn/m). Furthermore, according to Bernoulli Technical Bending Theory, the beam has section flexural stiffness *ΕΙy*ð Þ *x* , where in the special case of an uniform distribution of the stiffness it is given as *ΕΙy*ð Þ¼ *x ΕΙy*, where *E* is the material modulus of elasticity and *Ι<sup>y</sup>* is the section moment of inertia about y-axis (**Figure 1**). Next, we are examining a such amphi-hinge vertical pendulum that possesses constant value of distributed mass along its height, as well as constant value of distributed section flexural stiffness *ΕΙy*. Due to fact that the vertical pendulum mass is continuously distributed, this pendulum/beam has infinity number of degrees of freedom for vibration along the horizontal *oz*-axis. In order to formulate of the motion equation

**pendulum with distributed mass and stiffness**

*Amphi-hinge vertical pendulum with distributed mass and section flexural stiffness.*

positions due to ambient excitation [6, 7].

*Number Theory and Its Applications*

**Figure 1.**

**148**

Eq. (6) is a partial differential equation that describes the motion *uz*ð Þ *x*, *t* of the vertical beam that is loaded with the external dynamic loading *pz*ð Þ *x*, *t* . In order to arise a unique solution from Eq. (6), the support conditions must be used at the two beam ends. It is worthy to note that the classical case of a beam with distributed mass and section flexural stiffness, under external horizontal excitation (**Figure 2**) on the two supports is mathematically equivalent with the vibration that is described by Eq. (6). Indeed, in the case of **Figure 2**, the total displacement *u*tot *<sup>z</sup>* ð Þ *x*, *t* of the beam at *x*-location is given:

$$u\_{\mathbf{z}}^{\text{tot}}(\mathbf{x},t) = u\_{\mathbf{g}}(t) + u\_{\mathbf{z}}(\mathbf{x},t) \tag{7}$$

Following, Eqs. (4) and (5) are inserting into Eq. (8), thus we are taken:

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution…*

*ug*ð Þ*t ∂t*<sup>2</sup> þ

By the comparison of Eqs. (6) and (9), we notice that the undamped beam

equivalent with the undamped vibration of the same beam where the two supports are fixed and the beam is loaded with the equivalent distributed dynamic loading

In the case of the horizontal pendulum/beam free vibration without damping,

Furthermore, we ask the unknown spatial time-function *uz*ð Þ *x*, *t* , which is the

where *φ*ð Þ *x* is an unknown spatial function and *q t*ð Þ is an unknown timefunction. Eq*.* (12) has been derived two times with reference to time-dimension *t*

and, next, divided with the number *m* � *φ*ð Þ� *x q t*ð Þ, thus we are getting:

The left part of Eq. (15) is a time-function, but the right part is a spatialfunction. In order to true Eq. (15) for all time values as well as for all spatial positions, the two parts of Eq. (15) must be equal with a constant *λ*. Thus, Eq. (15) is

*q t*ð Þ <sup>¼</sup> *ΕΙ<sup>y</sup>* � *<sup>φ</sup>*0000ð Þ *<sup>x</sup> m* � *φ*ð Þ *x*

�€*q t*ð Þ

vibration due to horizontal motion of the two supports is mathematically

*p*eqð Þ¼� *x*, *t m* �

*<sup>∂</sup>*<sup>4</sup>*uz*ð Þ *<sup>x</sup>*, *<sup>t</sup>*

*∂*2 *uz*ð Þ *x*, *t ∂t*2

*<sup>∂</sup>x*<sup>4</sup> ¼ �*<sup>m</sup>* �

*∂*2 *ug* ð Þ*t*

*<sup>∂</sup>*<sup>4</sup>*uz*ð Þ *<sup>x</sup>*, *<sup>t</sup>*

*uz*ð Þ *x*, *t*

*m* � *φ*ð Þ� *x* €*q t*ðÞþ *ΕΙ<sup>y</sup>* � *φ*0000ð Þ� *x q t*ðÞ¼ 0 (14)

*q t*ð Þ <sup>¼</sup> *<sup>λ</sup>* ) €*q t*ðÞþ *<sup>λ</sup>* � *q t*ðÞ¼ <sup>0</sup> (16)

*<sup>m</sup>* � *<sup>φ</sup>*ð Þ *<sup>x</sup>* <sup>¼</sup> *<sup>λ</sup>* ) *ΕΙ<sup>y</sup>* � *<sup>φ</sup>*0000ð Þ *<sup>x</sup>* � *<sup>λ</sup>* � *<sup>m</sup>* � *<sup>φ</sup>*ð Þ¼ *<sup>x</sup>* 0 (17)

)

*<sup>∂</sup>t*<sup>2</sup> (9)

*<sup>∂</sup>t*<sup>2</sup> (10)

*<sup>∂</sup>x*<sup>4</sup> <sup>¼</sup> <sup>0</sup> (11)

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> *<sup>φ</sup>*00ð Þ� *<sup>x</sup> q t*ð Þ (13)

(15)

*uz*ð Þ¼ *x*, *t φ*ð Þ� *x q t*ð Þ (12)

*∂*2 *ug* ð Þ*t*

*∂*2 *M*

*DOI: http://dx.doi.org/10.5772/intechopen.92185*

*m ∂*2 *uz*ð Þ *x*, *t <sup>∂</sup>t*<sup>2</sup> <sup>þ</sup> *ΕΙ<sup>y</sup>*

we consider the first part of Eq. (9) that must be null:

*m ∂*2 *uz*ð Þ *x*, *t <sup>∂</sup>t*<sup>2</sup> <sup>þ</sup> *ΕΙ<sup>y</sup>*

solution of Eq. (11), must have the form of separated variants:

and more two times with reference to spatial-dimension *x*, so:

*<sup>∂</sup>t*<sup>2</sup> <sup>¼</sup> *<sup>φ</sup>*ð Þ� *<sup>x</sup>* €*q t*ð Þ, *<sup>∂</sup>*<sup>2</sup>

*∂*2 *uz*ð Þ *x*, *t*

Eqs. (13) are inserted into Eq. (11), giving:

separated at two following differential equations:

*ΕΙ<sup>y</sup>* � *φ*0000ð Þ *x*

**151**

�€*q t*ð Þ

*p*eqð Þ *x*, *t* :

*<sup>∂</sup>x*<sup>2</sup> ¼ �*<sup>m</sup>* � *<sup>∂</sup>*<sup>2</sup>

where *ug* ð Þ*t* is the displacement at the base, same for the two supports.

But, it is known that the inertia forces of the beam are depended by the total displacement *u*tot *<sup>z</sup>* ð Þ *x*, *t* , while the distributed dynamic loading is null, *pz*ð Þ¼ *x*, *t* 0. Thus, Eq. (3) is transformed into:

$$\frac{\partial Q}{\partial \mathbf{x}} = p\_x(\mathbf{x}, t) - \overline{m} \cdot \frac{\partial^2 u\_x^{\text{tot}}(\mathbf{x}, t)}{\partial t^2} \Rightarrow$$

$$\frac{\partial Q}{\partial \mathbf{x}} = \mathbf{0} - \overline{m} \cdot \frac{\partial^2 u\_\mathbf{g}(t)}{\partial t^2} - \overline{m} \cdot \frac{\partial^2 u\_x(\mathbf{x}, t)}{\partial t^2} \tag{8}$$

**Figure 2.** *Amphi-hinge vertical pendulum subjected with the same horizontal ground motion u*gð Þ*t on the two end-supports.*

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution… DOI: http://dx.doi.org/10.5772/intechopen.92185*

Following, Eqs. (4) and (5) are inserting into Eq. (8), thus we are taken:

$$\frac{\partial^2 M}{\partial \mathbf{x}^2} = -\overline{m} \cdot \left( \frac{\partial^2 u\_\mathcal{g}(t)}{\partial t^2} + \frac{\partial^2 u\_x(\mathbf{x}, t)}{\partial t^2} \right) \Rightarrow$$

$$\overline{m} \frac{\partial^2 u\_x(\mathbf{x}, t)}{\partial t^2} + EI\_\mathcal{g} \frac{\partial^4 u\_x(\mathbf{x}, t)}{\partial \mathbf{x}^4} = -\overline{m} \cdot \frac{\partial^2 u\_\mathcal{g}(t)}{\partial t^2} \tag{9}$$

By the comparison of Eqs. (6) and (9), we notice that the undamped beam vibration due to horizontal motion of the two supports is mathematically equivalent with the undamped vibration of the same beam where the two supports are fixed and the beam is loaded with the equivalent distributed dynamic loading *p*eqð Þ *x*, *t* :

$$p\_{\rm eq}(\varkappa, t) = -\overline{m} \cdot \frac{\partial^2 u\_{\rm g}(t)}{\partial t^2} \tag{10}$$

In the case of the horizontal pendulum/beam free vibration without damping, we consider the first part of Eq. (9) that must be null:

$$
\overline{m}\frac{\partial^2 u\_x(\varkappa, t)}{\partial t^2} + EI\_\jmath \frac{\partial^4 u\_x(\varkappa, t)}{\partial \varkappa^4} = 0 \tag{11}
$$

Furthermore, we ask the unknown spatial time-function *uz*ð Þ *x*, *t* , which is the solution of Eq. (11), must have the form of separated variants:

$$
\mu\_x(\mathbf{x}, t) = q(\mathbf{x}) \cdot q(\mathbf{t}) \tag{12}
$$

where *φ*ð Þ *x* is an unknown spatial function and *q t*ð Þ is an unknown timefunction. Eq*.* (12) has been derived two times with reference to time-dimension *t* and more two times with reference to spatial-dimension *x*, so:

$$\frac{\partial^2 u\_x(\mathbf{x}, t)}{\partial t^2} = \rho(\mathbf{x}) \cdot \ddot{q}(t), \quad \frac{\partial^2 u\_x(\mathbf{x}, t)}{\partial \mathbf{x}^2} = \rho^{\prime\prime}(\mathbf{x}) \cdot q(t) \tag{13}$$

Eqs. (13) are inserted into Eq. (11), giving:

$$
\overline{m} \cdot \boldsymbol{\varrho}(\mathbf{x}) \cdot \ddot{\boldsymbol{q}}(\mathbf{t}) + EI\_{\mathcal{Y}} \cdot \boldsymbol{\varrho}^{\prime\prime\prime\prime}(\mathbf{x}) \cdot \boldsymbol{q}(\mathbf{t}) = \mathbf{0} \tag{14}
$$

and, next, divided with the number *m* � *φ*ð Þ� *x q t*ð Þ, thus we are getting:

$$\frac{-\ddot{q}(t)}{q(t)} = \frac{EI\_\circ \cdot \rho^{\prime\prime\prime}(\infty)}{\overline{m} \cdot \rho(\infty)}\tag{15}$$

The left part of Eq. (15) is a time-function, but the right part is a spatialfunction. In order to true Eq. (15) for all time values as well as for all spatial positions, the two parts of Eq. (15) must be equal with a constant *λ*. Thus, Eq. (15) is separated at two following differential equations:

$$\frac{-\ddot{q}(t)}{q(t)} = \lambda \Rightarrow \ddot{q}(t) + \lambda \cdot q(t) = 0\tag{16}$$

$$\frac{EI\_\chi \cdot \phi^{\prime\prime\prime\prime}(\mathbf{x})}{\overline{m} \cdot \phi(\mathbf{x})} = \lambda \Rightarrow EI\_\chi \cdot \phi^{\prime\prime\prime(\mathbf{x})} - \lambda \cdot \overline{m} \cdot \phi(\mathbf{x}) = \mathbf{0} \tag{17}$$

Eq. (6) is a partial differential equation that describes the motion *uz*ð Þ *x*, *t* of the vertical beam that is loaded with the external dynamic loading *pz*ð Þ *x*, *t* . In order to arise a unique solution from Eq. (6), the support conditions must be used at the two beam ends. It is worthy to note that the classical case of a beam with distributed mass and section flexural stiffness, under external horizontal excitation (**Figure 2**)

*<sup>z</sup>* ð Þ¼ *x*, *t ug*ðÞþ*t uz*ð Þ *x*, *t* (7)

*<sup>z</sup>* ð Þ *x*, *t* , while the distributed dynamic loading is null, *pz*ð Þ¼ *x*, *t* 0.

*∂*2 *uz*ð Þ *x*, *t*

*<sup>∂</sup>t*<sup>2</sup> (8)

*∂*2 *u*tot *<sup>z</sup>* ð Þ *x*, *t ∂t*<sup>2</sup> ) *<sup>z</sup>* ð Þ *x*, *t*

on the two supports is mathematically equivalent with the vibration that is described by Eq. (6). Indeed, in the case of **Figure 2**, the total displacement *u*tot

where *ug* ð Þ*t* is the displacement at the base, same for the two supports. But, it is known that the inertia forces of the beam are depended by the total

> *∂*2 *ug*ð Þ*t <sup>∂</sup>t*<sup>2</sup> � *<sup>m</sup>* �

*Amphi-hinge vertical pendulum subjected with the same horizontal ground motion u*gð Þ*t on the two*

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *pz*ð Þ� *<sup>x</sup>*, *<sup>t</sup> <sup>m</sup>* �

*u*tot

*∂Q*

*∂Q*

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup> � *<sup>m</sup>* �

of the beam at *x*-location is given:

*Number Theory and Its Applications*

Thus, Eq. (3) is transformed into:

displacement *u*tot

**Figure 2.**

**150**

*end-supports.*

However, the time equation (16) indicates a free vibration of an ideal single degree of freedom system that has eigen-frequency <sup>¼</sup> ffiffi *λ* <sup>p</sup> . Inserting the eigenfrequency *ω* into Eq. (17) gives:

$$EI\_{\mathcal{Y}} \cdot \boldsymbol{\varrho}^{\prime\prime\prime}(\mathbf{x}) - \boldsymbol{\alpha}^2 \cdot \overline{\boldsymbol{m}} \cdot \boldsymbol{\varrho}(\mathbf{x}) = \mathbf{0} \Rightarrow \boldsymbol{\varrho}^{\prime\prime\prime}(\mathbf{x}) - \frac{\boldsymbol{\omega}^2 \cdot \overline{\boldsymbol{m}}}{EI\_{\mathcal{Y}}} \cdot \boldsymbol{\varrho}(\mathbf{x}) = \mathbf{0} \tag{18}$$

Next, we set the positive parameter *β* such as to be equal:

$$
\beta^4 = \frac{\alpha^2 \cdot \overline{m}}{EI\_\circ} \tag{19}
$$

By Eqs. (22) and (26) directly arise *C*<sup>2</sup> ¼ 0 and *C*<sup>4</sup> ¼ 0, thus the general solution

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution…*

In addition, the parameters *C*1,*C*<sup>3</sup> are calculated by the support conditions of the

second support of the beam. Therefore, for *x* ¼ *L* the vertical displacement *u L*ð Þ¼ , *t* 0 be true. Thus, from Eq. (12) arises that *φ*ð Þ¼ *L* 0 and Eq. (27) gives:

> *∂*2 *φ*ð Þ *L*

However, when Eqs. (28) and (30) are re-written again, we get:

And added part to part these two above-mentioned equations arise:

existing. Therefore, *C*<sup>3</sup> has to equal with zero, so Eq. (28) is formed:

However, Eq. (35) is transformed to Eq. (36):

*<sup>β</sup>*<sup>4</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> � *<sup>m</sup> ΕΙ<sup>y</sup>*

*<sup>ω</sup><sup>n</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup> � *<sup>π</sup>*<sup>2</sup>

*<sup>L</sup>*<sup>2</sup> �

each *n*-value.

**153**

*<sup>β</sup><sup>L</sup>* <sup>¼</sup> *<sup>n</sup>* � *<sup>π</sup>* ) *<sup>β</sup>*<sup>2</sup>

But, the term sinh *βL* is not equal with zero, because then vibration is not

by Eq. (27), either sin *βL* ¼ 0 that means the following equation must be true:

By the definition of parameter *β*, we can calculate the eigen-frequency *ω*:

) *<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> *ΕΙ<sup>y</sup>* � *<sup>β</sup>*<sup>4</sup>

ffiffiffiffiffiffiffi *ΕΙ<sup>y</sup> m*

r

*m*

Thus, inserting Eq. (36) into Eq. (37), the eigen-frequency *ω<sup>n</sup>* directly arises for

Moreover, by Eq. (34) arise that either *C*<sup>1</sup> ¼ 0 that is mpossibility because *φ*ð Þ *x* ¼6 0

where *φ*00ð Þ *L* is directly getting from Eq*.* (25) that is equivalent with zero:

*φ*ð Þ¼ *x C*<sup>1</sup> sin *βx* þ *C*<sup>3</sup> sinh *βx* (27)

*φ*ð Þ¼ *L C*<sup>1</sup> sin *βL* þ *C*<sup>3</sup> sinh *βL* ¼ 0 (28)

*<sup>φ</sup>*00ð Þ¼ *<sup>L</sup> <sup>C</sup>*<sup>1</sup> �*β*<sup>2</sup> � � � sin *<sup>β</sup><sup>L</sup>* <sup>þ</sup> *<sup>C</sup>*3*β*<sup>2</sup> � sinh *<sup>β</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup> (30)

*C*<sup>1</sup> � sin *βL* þ *C*<sup>3</sup> � sinh *βL* ¼ 0 (31) �*C*<sup>1</sup> � sin *βL* þ *C*<sup>3</sup> � sinh *βL* ¼ 0 (32)

2 � *C*<sup>3</sup> � sinh *βL* ¼ 0 (33)

*φ*ð Þ¼ *L C*<sup>1</sup> � sin *βL* ¼ 0 (34)

*βL* ¼ *n* � *π n* ¼ 1, 2, 3, … (35)

*<sup>L</sup>*<sup>2</sup> (36)

(37)

ffiffiffiffiffiffiffi *ΕΙ<sup>y</sup> m*

*n* ¼ 1, 2, 3, … (38)

r

*<sup>L</sup>*<sup>2</sup> <sup>¼</sup> *<sup>n</sup>*<sup>2</sup> � *<sup>π</sup>*<sup>2</sup> ) *<sup>β</sup>*<sup>2</sup> <sup>¼</sup> *<sup>n</sup>*<sup>2</sup> � *<sup>π</sup>*<sup>2</sup>

) *<sup>ω</sup>* <sup>¼</sup> *<sup>β</sup>*<sup>2</sup> �

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> <sup>0</sup> ) *ΕΙ<sup>y</sup>* � *<sup>φ</sup>*00ð Þ¼ *<sup>L</sup>* <sup>0</sup> (29)

of Eq. (20) is the following:

*DOI: http://dx.doi.org/10.5772/intechopen.92185*

In continuous, Eq. (23) gives:

*M L*ð Þ¼ , *t ΕΙ<sup>y</sup>* �

because the parameters *ω*2, *m*, *ΕΙ<sup>y</sup>* are always positive. By the mathematic theory it is known that the general solution of Eq*.* (18) has the following form:

$$\rho(\mathbf{x}) = \mathbf{C}\_1 \sin \beta \mathbf{x} + \mathbf{C}\_2 \cos \beta \mathbf{x} + \mathbf{C}\_3 \sinh \beta \mathbf{x} + \mathbf{C}\_4 \cosh \beta \mathbf{x} \tag{20}$$

where the four unknown parameters *C*1,*C*2,*C*3,*C*<sup>4</sup> must be calculated. In order to achieve this, four support conditions of the beam have to used. Indeed, for *x* ¼ 0 and *x* ¼ *L* the displacement *uz*ð Þ 0, *t* of the amphi-hinge vertical pendulum as well as the flexural moment *M*ð Þ 0, *t* , both are equal zero. The spatial function *φ*ð Þ *x* , which is the solution of Eq. (20), gives the modal elastic line of the beam. Having as known data that the following equation is true:

$$\sinh\beta\mathbb{x} = \frac{e^{\beta\mathbb{x}} - e^{-\beta\mathbb{x}}}{2}, \cosh\beta\mathbb{x} = \frac{e^{\beta\mathbb{x}} + e^{-\beta\mathbb{x}}}{2} \tag{21}$$

The spatial function of the modal elastic line for *x* ¼ 0 is:

$$\rho(\mathbf{0}) = \mathbf{C}\_1 \sin \mathbf{0} + \mathbf{C}\_2 \cos \mathbf{0} + \mathbf{C}\_3 \sinh \mathbf{0} + \mathbf{C}\_4 \cosh \mathbf{0} = \mathbf{0} \Rightarrow$$

$$\mathbf{C}\_2 + \mathbf{C}\_4 = \mathbf{0} \tag{22}$$

and also for *x* ¼ 0, the function of the flexural moment due to examined modal elastic line of the beam is given by Eq*.* (5):

$$M(\mathbf{0}, t) = EI\_\circ \cdot \frac{\partial^2 \rho(\mathbf{0})}{\partial \mathbf{x}^2} = \mathbf{0} \Rightarrow EI\_\circ \cdot \rho''(\mathbf{0}) = \mathbf{0} \tag{23}$$

Eq. (20) has been derived two times with reference to spatial-dimension *x*, thus arise:

$$\boldsymbol{\rho}'\left(\mathbf{x}\right) = \mathbf{C}\_1 \cdot \boldsymbol{\beta} \cdot \cos\beta \mathbf{x} + \mathbf{C}\_2 \cdot (-\boldsymbol{\beta}) \cdot \sin\beta \mathbf{x} + \mathbf{C}\_3 \cdot \boldsymbol{\beta} \cdot \cosh\beta \mathbf{x} + \mathbf{C}\_4 \cdot \boldsymbol{\beta} \cdot \sinh\beta \mathbf{x} \tag{24}$$

and

$$\rho''(\mathbf{x}) = \mathbf{C}\_1 \left(-\boldsymbol{\beta}^2\right) \cdot \sin\beta \mathbf{x} + \mathbf{C}\_2 \left(-\boldsymbol{\beta}^2\right) \cdot \cos\beta \mathbf{x} + \mathbf{C}\_3 \boldsymbol{\beta}^2 \cdot \sinh\beta \mathbf{x} + \mathbf{C}\_4 \boldsymbol{\beta}^2 \cdot \cosh\beta \mathbf{x} \tag{25}$$

Therefore, Eq. (23) is transformed:

$$EI\_{\mathcal{Y}} \cdot \beta^2 \cdot (\mathbf{C\_4} - \mathbf{C\_2}) = \mathbf{0} \tag{26}$$

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution… DOI: http://dx.doi.org/10.5772/intechopen.92185*

By Eqs. (22) and (26) directly arise *C*<sup>2</sup> ¼ 0 and *C*<sup>4</sup> ¼ 0, thus the general solution of Eq. (20) is the following:

$$
\rho(\mathbf{x}) = \mathbf{C}\_1 \sin \beta \mathbf{x} + \mathbf{C}\_3 \sinh \beta \mathbf{x} \tag{27}
$$

In addition, the parameters *C*1,*C*<sup>3</sup> are calculated by the support conditions of the second support of the beam. Therefore, for *x* ¼ *L* the vertical displacement *u L*ð Þ¼ , *t* 0 be true. Thus, from Eq. (12) arises that *φ*ð Þ¼ *L* 0 and Eq. (27) gives:

$$
\rho(L) = C\_1 \sin \beta L + C\_3 \sinh \beta L = 0 \tag{28}
$$

In continuous, Eq. (23) gives:

However, the time equation (16) indicates a free vibration of an ideal single

*<sup>β</sup>*<sup>4</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> � *<sup>m</sup> ΕΙ<sup>y</sup>*

because the parameters *ω*2, *m*, *ΕΙ<sup>y</sup>* are always positive. By the mathematic theory

where the four unknown parameters *C*1,*C*2,*C*3,*C*<sup>4</sup> must be calculated. In order to achieve this, four support conditions of the beam have to used. Indeed, for *x* ¼ 0 and *x* ¼ *L* the displacement *uz*ð Þ 0, *t* of the amphi-hinge vertical pendulum as well as the flexural moment *M*ð Þ 0, *t* , both are equal zero. The spatial function *φ*ð Þ *x* , which is the solution of Eq. (20), gives the modal elastic line of the beam. Having as

*φ*ð Þ¼ 0 *C*<sup>1</sup> sin 0 þ *C*<sup>2</sup> cos 0 þ *C*<sup>3</sup> sinh 0 þ *C*<sup>4</sup> cosh 0 ¼ 0 )

and also for *x* ¼ 0, the function of the flexural moment due to examined modal

Eq. (20) has been derived two times with reference to spatial-dimension *x*, thus

*φ*<sup>0</sup> ð Þ¼ *x C*<sup>1</sup> � *β* � cos *βx* þ *C*<sup>2</sup> � �ð Þ� *β* sin *βx* þ *C*<sup>3</sup> � *β* � cosh *βx* þ *C*<sup>4</sup> � *β* � sinh *βx*

*<sup>φ</sup>*00ð Þ¼ *<sup>x</sup> <sup>C</sup>*<sup>1</sup> �*β*<sup>2</sup> � � � sin *<sup>β</sup><sup>x</sup>* <sup>þ</sup> *<sup>C</sup>*<sup>2</sup> �*β*<sup>2</sup> � � � cos *<sup>β</sup><sup>x</sup>* <sup>þ</sup> *<sup>C</sup>*3*β*<sup>2</sup> � sinh *<sup>β</sup><sup>x</sup>* <sup>þ</sup> *<sup>C</sup>*4*β*<sup>2</sup> � cosh *<sup>β</sup><sup>x</sup>*

*φ*ð Þ¼ *x C*<sup>1</sup> sin *βx* þ *C*<sup>2</sup> cos *βx* þ *C*<sup>3</sup> sinh *βx* þ *C*<sup>4</sup> cosh *βx* (20)

<sup>2</sup> , cosh *<sup>β</sup><sup>x</sup>* <sup>¼</sup> *<sup>e</sup>β<sup>x</sup>* <sup>þ</sup> *<sup>e</sup>*�*β<sup>x</sup>*

*C*<sup>2</sup> þ *C*<sup>4</sup> ¼ 0 (22)

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> <sup>0</sup> ) *ΕΙ<sup>y</sup>* � *<sup>φ</sup>*00ð Þ¼ <sup>0</sup> <sup>0</sup> (23)

*ΕΙ<sup>y</sup>* � *<sup>β</sup>*<sup>2</sup> � ð Þ¼ *<sup>C</sup>*<sup>4</sup> � *<sup>C</sup>*<sup>2</sup> <sup>0</sup> (26)

*λ*

*ΕΙ<sup>y</sup>*

<sup>p</sup> . Inserting the eigen-

� *φ*ð Þ¼ *x* 0 (18)

<sup>2</sup> (21)

(19)

(24)

(25)

degree of freedom system that has eigen-frequency <sup>¼</sup> ffiffi

*ΕΙ<sup>y</sup>* � *<sup>φ</sup>*<sup>0000</sup> ð Þ� *<sup>x</sup> <sup>ω</sup>*<sup>2</sup> � *<sup>m</sup>* � *<sup>φ</sup>*ð Þ¼ *<sup>x</sup>* <sup>0</sup> ) *<sup>φ</sup>*0000ð Þ� *<sup>x</sup> <sup>ω</sup>*<sup>2</sup> � *<sup>m</sup>*

Next, we set the positive parameter *β* such as to be equal:

known data that the following equation is true:

elastic line of the beam is given by Eq*.* (5):

Therefore, Eq. (23) is transformed:

arise:

and

**152**

*M*ð Þ¼ 0, *t ΕΙ<sup>y</sup>* �

sinh *<sup>β</sup><sup>x</sup>* <sup>¼</sup> *<sup>e</sup>β<sup>x</sup>* � *<sup>e</sup>*�*β<sup>x</sup>*

The spatial function of the modal elastic line for *x* ¼ 0 is:

*∂*2 *φ*ð Þ 0

it is known that the general solution of Eq*.* (18) has the following form:

frequency *ω* into Eq. (17) gives:

*Number Theory and Its Applications*

$$\mathbf{M}(L,\mathbf{t}) = EI\_{\mathcal{Y}} \cdot \frac{\partial^2 \rho(L)}{\partial \mathbf{x}^2} = \mathbf{0} \Rightarrow EI\_{\mathcal{Y}} \cdot \rho''(L) = \mathbf{0} \tag{29}$$

where *φ*00ð Þ *L* is directly getting from Eq*.* (25) that is equivalent with zero:

$$\boldsymbol{\uprho}''(L) = \mathbf{C}\_1 \left(-\boldsymbol{\uprho}^2\right) \cdot \sin\boldsymbol{\uprho}L + \mathbf{C}\_3 \boldsymbol{\uprho}^2 \cdot \sinh\boldsymbol{\uprho}L = \mathbf{0} \tag{30}$$

However, when Eqs. (28) and (30) are re-written again, we get:

$$\mathbf{C}\_1 \cdot \sin \beta \mathbf{L} + \mathbf{C}\_3 \cdot \sinh \beta \mathbf{L} = \mathbf{0} \tag{31}$$

$$-C\_1 \cdot \sin \beta L + C\_3 \cdot \sinh \beta L = 0\tag{32}$$

And added part to part these two above-mentioned equations arise:

$$\mathcal{Q} \cdot \mathcal{C}\_{\mathcal{S}} \cdot \sinh \beta L = \mathbf{0} \tag{33}$$

But, the term sinh *βL* is not equal with zero, because then vibration is not existing. Therefore, *C*<sup>3</sup> has to equal with zero, so Eq. (28) is formed:

$$
\rho(L) = \mathbf{C}\_1 \cdot \sin \beta L = \mathbf{0} \tag{34}
$$

Moreover, by Eq. (34) arise that either *C*<sup>1</sup> ¼ 0 that is mpossibility because *φ*ð Þ *x* ¼6 0 by Eq. (27), either sin *βL* ¼ 0 that means the following equation must be true:

$$
\beta \mathbf{L} = n \cdot \boldsymbol{\pi} \qquad n = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots \tag{35}
$$

However, Eq. (35) is transformed to Eq. (36):

$$
\beta L = n \cdot \pi \Rightarrow \beta^2 L^2 = n^2 \cdot \pi^2 \Rightarrow \beta^2 = \frac{n^2 \cdot \pi^2}{L^2} \tag{36}
$$

By the definition of parameter *β*, we can calculate the eigen-frequency *ω*:

$$\beta^4 = \frac{\alpha^2 \cdot \overline{m}}{EI\_\circ} \Rightarrow \alpha^2 = \frac{EI\_\circ \cdot \beta^4}{\overline{m}} \Rightarrow \alpha = \beta^2 \cdot \sqrt{\frac{EI\_\circ}{\overline{m}}} \tag{37}$$

Thus, inserting Eq. (36) into Eq. (37), the eigen-frequency *ω<sup>n</sup>* directly arises for each *n*-value.

$$
\rho\_n = \frac{n^2 \cdot \pi^2}{L^2} \cdot \sqrt{\frac{EI\_y}{\overline{m}}} \qquad \qquad n = 1, 2, 3, \dots \tag{38}
$$

**Figure 3.** *Modal analysis of continuous amphi-hinge vertical pendulum. The first four mode-shapes.*

Therefore, the vibration mode-shape of the examined vertical pendulum arises by Eq. (27)—since previous inserting Eq. (35)—thus:

$$\rho\_n(\mathbf{x}) = \mathbf{C}\_1 \sin \beta \mathbf{x} = \mathbf{C}\_1 \sin \frac{n \cdot \pi \cdot \mathbf{x}}{L} \qquad n = 1, 2, 3, \dots \tag{39}$$

The pendulum displacement vector **u** of the three degrees of freedom, as well as

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution…*

Furthermore, the pendulum flexibility matrix *f* can be calculated using a suitable method (**Figure 4**), and the inverse matrix gives the stiffness matrix *k* of the

The equations of motion for the case of the free undamped vibration of the ideal

where the eigen-frequencies *ω<sup>n</sup>* and the three mode-shapes *φ<sup>n</sup>* are known by Eq. (38) and (39) and **Figure 4**. Therefore, the unique unknown parameter is the

*m*eq 0 0 0 *m*eq 0 0 0 *m*eq

1 0*:*6875 0*:*6875 0*:*6875 0*:*5625 0*:*4375 0*:*6875 0*:*4375 0*:*5625

18*:*285714 �12*:*571429 �12*:*571429 �12*:*571429 13*:*142857 5*:*142857 �12*:*571429 5*:*142857 13*:*142857

**m u**€ð Þþ*t* **k u**ðÞ¼ *t* **0** (43)

*<sup>n</sup>***<sup>m</sup>** � �**φ***<sup>n</sup>* <sup>¼</sup> **<sup>0</sup>** *<sup>n</sup>* <sup>¼</sup> 1, 2, 3*:* (44)

(40)

(41)

(42)

the diagonal beam mass matrix *m* is written:

*DOI: http://dx.doi.org/10.5772/intechopen.92185*

three degrees of freedom beam.

*k*1,1 *k*1,2 *k*1,3 *k*2,1 *k*2,2 *k*2,3 *k*3,1 *k*3,2 *k*3,3

The eigen-problem is written as:

*D*1,1 *D*1,2 *D*1,3 *D*2,1 *D*2,2 *D*2,3 *D*3,1 *D*3,2 *D*3,3

> 3 7 <sup>5</sup> <sup>¼</sup> <sup>48</sup>*EIy <sup>L</sup>*<sup>3</sup> �

> > **<sup>k</sup>** � *<sup>ω</sup>*<sup>2</sup>

*f* ¼

**Figure 4.**

2 6 4

beam is given as:

mass *m*eq. Thus,

**155**

*k* ¼

**u** ¼

*u*1 *u*2 *u*3

*The equivalent three-degree of freedom system on amphi-hinge vertical pendulum.*

9 >>=

>>;

, **m** ¼

8 >><

>>:

3 7 7 <sup>5</sup> <sup>¼</sup> *<sup>L</sup>*<sup>3</sup> 48*EIy* �

The value of *C*<sup>1</sup> is arbitrary, and we usually get it equal to unit. Thus, for each value of parameter *n*, a mode-shape with its eigen-frequency is resulted. The fundamental (first) mode-shape is resulted for *n* ¼ 1, which shows a half sinusoidal wave, the second mode-shape shows a full sinusoidal wave, etc. (**Figure 3**). The order of the eigen-frequencies is *ω*1, *ω*<sup>2</sup> ¼ 4*ω*1, *ω*<sup>3</sup> ¼ 9*ω*1, *ω*<sup>4</sup> ¼ 16*ω*1, etc.

#### **3. The equivalent three degrees of freedom system for amphi-hinges pendulum**

At vertical pendulums or beams where the fundamental horizontal mode-shape does not activate the 90% of the total beam mass, we ask to consider the three first mode-shapes. Thus, for this purpose, we must define an ideal equivalent three degrees of freedom beam, which is going to give the first three frequencies and the first three mode-shapes of the examined beam. Therefore, which is the ideal three degrees of freedom system, where its three mode-shapes coincide with the real first three frequencies of the vertical pendulum with distributed mass and flexural stiffness?

In order to answer the above-mentioned question, consider a weightless vertical pendulum with height *L* and constant section in elevation, where carry three concentrated masses that each one has the same mass-value *m*eq, located per distance 0.25*L*, between one to one, and each one mass possesses an horizontal degree of freedom (**Figure 4**).

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution… DOI: http://dx.doi.org/10.5772/intechopen.92185*

**Figure 4.** *The equivalent three-degree of freedom system on amphi-hinge vertical pendulum.*

The pendulum displacement vector **u** of the three degrees of freedom, as well as the diagonal beam mass matrix *m* is written:

$$\mathbf{u} = \begin{Bmatrix} u\_1 \\ u\_2 \\ u\_3 \end{Bmatrix}, \mathbf{m} = \begin{bmatrix} m\_{\text{eq}} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & m\_{\text{eq}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & m\_{\text{eq}} \end{bmatrix} \tag{40}$$

Furthermore, the pendulum flexibility matrix *f* can be calculated using a suitable method (**Figure 4**), and the inverse matrix gives the stiffness matrix *k* of the three degrees of freedom beam.

$$\begin{aligned} \mathbf{f} &= \begin{bmatrix} D\_{1,1} & D\_{1,2} & D\_{1,3} \\ D\_{2,1} & D\_{2,2} & D\_{2,3} \\ D\_{3,1} & D\_{3,2} & D\_{3,3} \end{bmatrix} = \frac{L^3}{48EI\_{\mathcal{Y}}} \cdot \begin{bmatrix} 1 & 0.6875 & 0.6875 \\ 0.6875 & 0.5625 & 0.4375 \\ 0.6875 & 0.4375 & 0.5625 \end{bmatrix} \tag{41} \\\\ \mathbf{k} &= \begin{bmatrix} k\_{1,1} & k\_{1,2} & k\_{1,3} \\ k\_{2,1} & k\_{2,2} & k\_{2,3} \\ k\_{3,1} & k\_{3,2} & k\_{3,3} \end{bmatrix} = \frac{48EI\_{\mathcal{Y}}}{L^3} \cdot \begin{bmatrix} 18.285714 & -12.571429 & -12.571429 \\ -12.571429 & 13.142857 & 5.142857 \\ -12.571429 & 5.142857 & 13.142857 \end{bmatrix} \tag{42} \end{aligned}$$

The equations of motion for the case of the free undamped vibration of the ideal beam is given as:

$$\mathbf{m}\,\ddot{\mathbf{u}}(t) + \mathbf{k}\,\mathbf{u}(t) = \mathbf{0} \tag{43}$$

The eigen-problem is written as:

$$(\mathbf{k} - \alpha\_n^2 \mathbf{m})\boldsymbol{\upvarphi}\_n = \mathbf{0} \qquad n = 1, 2, 3. \tag{44}$$

where the eigen-frequencies *ω<sup>n</sup>* and the three mode-shapes *φ<sup>n</sup>* are known by Eq. (38) and (39) and **Figure 4**. Therefore, the unique unknown parameter is the mass *m*eq. Thus,

Therefore, the vibration mode-shape of the examined vertical pendulum arises

The value of *C*<sup>1</sup> is arbitrary, and we usually get it equal to unit. Thus, for each value of parameter *n*, a mode-shape with its eigen-frequency is resulted. The fundamental (first) mode-shape is resulted for *n* ¼ 1, which shows a half sinusoidal wave, the second mode-shape shows a full sinusoidal wave, etc. (**Figure 3**). The

*<sup>L</sup> <sup>n</sup>* <sup>¼</sup> 1, 2, 3, … (39)

by Eq. (27)—since previous inserting Eq. (35)—thus:

**pendulum**

**Figure 3.**

*Number Theory and Its Applications*

stiffness?

**154**

freedom (**Figure 4**).

*<sup>φ</sup>n*ð Þ¼ *<sup>x</sup> <sup>C</sup>*<sup>1</sup> sin *<sup>β</sup><sup>x</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>1</sup> sin *<sup>n</sup>* � *<sup>π</sup>* � *<sup>x</sup>*

*Modal analysis of continuous amphi-hinge vertical pendulum. The first four mode-shapes.*

order of the eigen-frequencies is *ω*1, *ω*<sup>2</sup> ¼ 4*ω*1, *ω*<sup>3</sup> ¼ 9*ω*1, *ω*<sup>4</sup> ¼ 16*ω*1, etc.

**3. The equivalent three degrees of freedom system for amphi-hinges**

At vertical pendulums or beams where the fundamental horizontal mode-shape does not activate the 90% of the total beam mass, we ask to consider the three first mode-shapes. Thus, for this purpose, we must define an ideal equivalent three degrees of freedom beam, which is going to give the first three frequencies and the first three mode-shapes of the examined beam. Therefore, which is the ideal three degrees of freedom system, where its three mode-shapes coincide with the real first three frequencies of the vertical pendulum with distributed mass and flexural

In order to answer the above-mentioned question, consider a weightless vertical pendulum with height *L* and constant section in elevation, where carry three concentrated masses that each one has the same mass-value *m*eq, located per distance 0.25*L*, between one to one, and each one mass possesses an horizontal degree of

$$\det\left(\mathbf{k} - \alpha\_1^2 \mathbf{m}\right) = \mathbf{0} \Rightarrow \tag{45}$$

$$m\_{\rm eq}^3 + A \cdot m\_{\rm eq}^2 + B \cdot m\_{\rm eq} + C = 0 \tag{46}$$

where

$$A = -\frac{k\_{11} + k\_{22} + k\_{33}}{\alpha\_1^2}, B = \frac{k\_{11}k\_{33} + k\_{11}k\_{22} + k\_{22}k\_{33} - k\_{12}^2 - k\_{13}^2 - k\_{23}^2}{\alpha\_1^4}$$

$$C = -\frac{k\_{11}k\_{22}k\_{33} + 2k\_{12}k\_{13}k\_{23} - k\_{11}k\_{23}^2 - k\_{22}k\_{13}^2 - k\_{33}k\_{12}^2}{\alpha\_1^6}$$

The numerical solution of Eq. (46) gives three roots for parameter *m*eq, where only the first root is acceptable, because the other two values rejected since do not have natural meaning (appear values greater from the total pendulum mass *mL*). Thus, the only one acceptable root is given as:

$$m\_{\text{eq}} = 0.24984748 \cdot (\overline{m}L) \tag{47}$$

material modulus of elasticity and *Ι<sup>y</sup>* is the section moment of inertia about y-axis (**Figure 1**). Next, we are examining such an inverted pendulum that possesses constant value of distributed mass along its height, as well as constant value of distributed section flexural stiffness *ΕΙy*. Due to fact that the cantilever mass is continuously distributed, this inverted pendulum/beam has infinity number of degrees of freedom for vibration along the horizontal *oz*-axis. In order to formulate of the motion equation of this beam, we consider an infinitesimal part of the vertical beam, at location *x*

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution…*

*DOI: http://dx.doi.org/10.5772/intechopen.92185*

The infinitesimal length of this part is the *dx*. On this infinitesimal length, we notice the flexural moment *M x*ð Þ , *t* , the shear force *Q x*ð Þ , *t* with their differential increments, while the axial force *Ν*ð Þ *x*, *t* is ignored, because it does not affect the horizontal cantilever vibration along z-axis. Moreover, noted the resulting force *Pz*ð Þ *x*, *t* of the external dynamic loading. Eqs. (1)–(20) describe mathematically the modal analysis of the inverted pendulum. Afterward, the four unknown parameters *C*1,*C*2,*C*3,*C*<sup>4</sup> of Eq. (20) must be calculated taking account the conditions of the cantilever. In order to achieve this, four support conditions of the cantilever have to be used. Indeed, for the end at *x* ¼ 0 the displacement *uz*ð Þ 0, *t* and the slope *u*<sup>0</sup>

*φ*ð Þ¼ 0 *C*<sup>1</sup> sin 0 þ *C*<sup>2</sup> cos 0 þ *C*<sup>3</sup> sinh 0 þ *C*<sup>4</sup> cosh 0 ¼ 0 ) *C*<sup>2</sup> þ *C*<sup>4</sup> ¼ 0 (48)

Moreover, for the free end at *x* ¼ *L* of the cantilever, the flexural moment *M L*ð Þ , *t* as well as the shear force *Q L*ð Þ , *t* must be both zero. So, from Eqs. (20) and

ð Þ¼ 0 *β*ð Þ¼ *C*<sup>1</sup> þ *C*<sup>3</sup> 0 ) *C*<sup>1</sup> þ *C*<sup>3</sup> ¼ 0 (49)

*<sup>z</sup>*ð Þ 0, *t*

from the origin *o*, that has isolated by two very nearest parallel sections.

*Vertical cantilever (inverted pendulum) with distributed mass and section flexural stiffness.*

of displacement profile must be zero. So, Eq. (20) gives:

*φ*0

(5) and afterward from Eq. (48) we take:

and

**157**

**Figure 5.**

Therefore, inserting the ideal equivalent mass *m*eq by Eq. (47) at three degrees of freedom system of **Figure 4**, the three eigen-frequencies coincide with the real values of the initial vertical pendulum that has distributed mass and flexural stiffness.

#### **4. Discussion about the amphi-hinge vertical pendulums**

By the previous mathematic analysis, an ideal three degrees of freedom system that is equivalent with the modal behavior of the amphi-hinge vertical pendulum with distributed mass and flexural stiffness along its height has been presented. This ideal three degrees of freedom system can be used in instrumentation of a such vertical tower, which does not possess concentrated masses. In the framework of the identification of mode-shapes of an amphi-hinge pendulum, the equivalent mass by Eq. (47) permits to locate accelerometers per 0.25L (as shown at **Figure 4**) and there measure the response horizontal acceleration histories, in order to calculate the real first three mode-shapes of the vertical pendulum in order to avoid the instability and resonance-vibrations between the examined vertical pendulum and the supported rockets or space bus before the last launched.

#### **5. Second case: modal analysis of undamped inverted pendulum with distributed mass and stiffness**

According to the theory of continuous systems [8, 9], consider a straight vertical cantilever (inverted pendulum) that is loaded by an external continuous dynamic loading *pz*ð Þ *x*, *t* , with reference to a Cartesian three-dimensional reference system *oxyz*, (**Figure 5**).

The vertical inverted pendulum possesses a distributed mass *m x*ð Þ per unit height, which in the special case of uniform distribution is given as *m x*ð Þ¼ *m* in tons per meter (tn/m). Furthermore, according to Bernoulli Technical Bending Theory, this cantilever has section flexural stiffness *ΕΙy*ð Þ *x* , where in the special case of an uniform distribution of the stiffness it is given as *ΕΙy*ð Þ¼ *x ΕΙy*, where *E* is the

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution… DOI: http://dx.doi.org/10.5772/intechopen.92185*

**Figure 5.** *Vertical cantilever (inverted pendulum) with distributed mass and section flexural stiffness.*

material modulus of elasticity and *Ι<sup>y</sup>* is the section moment of inertia about y-axis (**Figure 1**). Next, we are examining such an inverted pendulum that possesses constant value of distributed mass along its height, as well as constant value of distributed section flexural stiffness *ΕΙy*. Due to fact that the cantilever mass is continuously distributed, this inverted pendulum/beam has infinity number of degrees of freedom for vibration along the horizontal *oz*-axis. In order to formulate of the motion equation of this beam, we consider an infinitesimal part of the vertical beam, at location *x* from the origin *o*, that has isolated by two very nearest parallel sections.

The infinitesimal length of this part is the *dx*. On this infinitesimal length, we notice the flexural moment *M x*ð Þ , *t* , the shear force *Q x*ð Þ , *t* with their differential increments, while the axial force *Ν*ð Þ *x*, *t* is ignored, because it does not affect the horizontal cantilever vibration along z-axis. Moreover, noted the resulting force *Pz*ð Þ *x*, *t* of the external dynamic loading. Eqs. (1)–(20) describe mathematically the modal analysis of the inverted pendulum. Afterward, the four unknown parameters *C*1,*C*2,*C*3,*C*<sup>4</sup> of Eq. (20) must be calculated taking account the conditions of the cantilever. In order to achieve this, four support conditions of the cantilever have to be used. Indeed, for the end at *x* ¼ 0 the displacement *uz*ð Þ 0, *t* and the slope *u*<sup>0</sup> *<sup>z</sup>*ð Þ 0, *t* of displacement profile must be zero. So, Eq. (20) gives:

$$\rho(\mathbf{0}) = \mathbf{C}\_1 \sin \mathbf{0} + \mathbf{C}\_2 \cos \mathbf{0} + \mathbf{C}\_3 \sinh \mathbf{0} + \mathbf{C}\_4 \cosh \mathbf{0} = \mathbf{0} \Rightarrow \mathbf{C}\_2 + \mathbf{C}\_4 = \mathbf{0} \tag{48}$$

and

det **<sup>k</sup>** � *<sup>ω</sup>*<sup>2</sup>

, *<sup>B</sup>* <sup>¼</sup> *<sup>k</sup>*11*k*<sup>33</sup> <sup>þ</sup> *<sup>k</sup>*11*k*<sup>22</sup> <sup>þ</sup> *<sup>k</sup>*22*k*<sup>33</sup> � *<sup>k</sup>*<sup>2</sup>

*ω*6 1

The numerical solution of Eq. (46) gives three roots for parameter *m*eq, where only the first root is acceptable, because the other two values rejected since do not have natural meaning (appear values greater from the total pendulum mass *mL*).

Therefore, inserting the ideal equivalent mass *m*eq by Eq. (47) at three degrees of

By the previous mathematic analysis, an ideal three degrees of freedom system that is equivalent with the modal behavior of the amphi-hinge vertical pendulum with distributed mass and flexural stiffness along its height has been presented. This ideal three degrees of freedom system can be used in instrumentation of a such vertical tower, which does not possess concentrated masses. In the framework of the identification of mode-shapes of an amphi-hinge pendulum, the equivalent mass by Eq. (47) permits to locate accelerometers per 0.25L (as shown at **Figure 4**) and there measure the response horizontal acceleration histories, in order to calculate the real first three mode-shapes of the vertical pendulum in order to avoid the instability and resonance-vibrations between the examined vertical pendulum and

**5. Second case: modal analysis of undamped inverted pendulum with**

The vertical inverted pendulum possesses a distributed mass *m x*ð Þ per unit height, which in the special case of uniform distribution is given as *m x*ð Þ¼ *m* in tons per meter (tn/m). Furthermore, according to Bernoulli Technical Bending Theory, this cantilever has section flexural stiffness *ΕΙy*ð Þ *x* , where in the special case of an uniform distribution of the stiffness it is given as *ΕΙy*ð Þ¼ *x ΕΙy*, where *E* is the

According to the theory of continuous systems [8, 9], consider a straight vertical cantilever (inverted pendulum) that is loaded by an external continuous dynamic loading *pz*ð Þ *x*, *t* , with reference to a Cartesian three-dimensional reference system

freedom system of **Figure 4**, the three eigen-frequencies coincide with the real values of the initial vertical pendulum that has distributed mass and flexural

eq <sup>þ</sup> *<sup>Α</sup>* � *<sup>m</sup>*<sup>2</sup>

*<sup>C</sup>* ¼ � *<sup>k</sup>*11*k*22*k*<sup>33</sup> <sup>þ</sup> <sup>2</sup>*k*12*k*13*k*<sup>23</sup> � *<sup>k</sup>*11*k*<sup>2</sup>

**4. Discussion about the amphi-hinge vertical pendulums**

the supported rockets or space bus before the last launched.

**distributed mass and stiffness**

*oxyz*, (**Figure 5**).

**156**

*m*3

*<sup>A</sup>* ¼ � *<sup>k</sup>*<sup>11</sup> <sup>þ</sup> *<sup>k</sup>*<sup>22</sup> <sup>þ</sup> *<sup>k</sup>*<sup>33</sup> *ω*2 1

*Number Theory and Its Applications*

Thus, the only one acceptable root is given as:

where

stiffness.

<sup>1</sup>**<sup>m</sup>** <sup>¼</sup> <sup>0</sup> ) (45)

*ω*4 1

<sup>23</sup> � *<sup>k</sup>*22*k*<sup>2</sup>

*m*eq ¼ 0*:*24984748 � ð Þ *mL* (47)

eq þ *Β* � *m*eq þ *C* ¼ 0 (46)

<sup>12</sup> � *<sup>k</sup>*<sup>2</sup>

12

<sup>13</sup> � *<sup>k</sup>*33*k*<sup>2</sup>

<sup>13</sup> � *<sup>k</sup>*<sup>2</sup> 23

$$
\rho'(\mathbf{0}) = \beta(\mathbf{C}\_1 + \mathbf{C}\_3) = \mathbf{0} \Rightarrow \mathbf{C}\_1 + \mathbf{C}\_3 = \mathbf{0} \tag{49}
$$

Moreover, for the free end at *x* ¼ *L* of the cantilever, the flexural moment *M L*ð Þ , *t* as well as the shear force *Q L*ð Þ , *t* must be both zero. So, from Eqs. (20) and (5) and afterward from Eq. (48) we take:

$$M(L,t) = EI\_\mathcal{Y} \cdot \frac{\partial^2 \rho(L)}{\partial \mathbf{x}^2} = \mathbf{0} \Rightarrow EI\_\mathcal{Y} \cdot \rho''(L) = \mathbf{0} \Rightarrow$$

$$\mathbf{C}\_1(\sin \beta L + \sinh \beta L) + \mathbf{C}\_2(\cos \beta L + \cosh \beta L) = \mathbf{0} \tag{50}$$

and

$$\mathbf{Q}(L,t) = EI\_{\mathcal{Y}} \cdot \frac{\partial^3 \rho(L)}{\partial \mathbf{x}^3} = \mathbf{0} \Rightarrow EI\_{\mathcal{Y}} \cdot \rho'''(L) = \mathbf{0} \Rightarrow$$

$$\mathbf{C}\_1(\cos \beta L + \cosh \beta L) + \mathbf{C}\_2(-\sin \beta L + \sinh \beta L) = \mathbf{0} \tag{51}$$

However, re-writing Eqs. (50) and (51) again, we get the matrix form:

$$
\begin{bmatrix}
(\sin\beta L + \sinh\beta L) & (\cos\beta L + \cosh\beta L) \\
(\cos\beta L + \cosh\beta L) & (-\sin\beta L + \sinh\beta L)
\end{bmatrix}
\begin{Bmatrix} C\_1 \\ C\_2 \end{Bmatrix} = \begin{Bmatrix} 0 \\ 0 \end{Bmatrix} \tag{52}
$$

Eq. (52) is a real eigenvalue problem. In order to calculate the eigenvalues, parameters *C*<sup>1</sup> and *C*<sup>2</sup> must not both equal zero. Thus, the determinant of the matrix by Eq. (52) must be zero, where it drives to the following frequency equation:

$$\mathbf{1} + (\cos \beta L) \cdot (\cosh \beta L) = \mathbf{0} \tag{53}$$

**6. The equivalent three degrees of freedom system of inverted**

*Modal analysis of continuous inverted pendulum. The first four mode-shapes.*

the inverted pendulum with distributed mass and flexural stiffness?

as well as the diagonal beam mass matrix *m* are written:

8 ><

>:

*u*1 *u*2 *u*3

3 7 <sup>5</sup> <sup>¼</sup> *<sup>L</sup>*<sup>3</sup> 3*EIy* �

9 >=

>;

, **m** ¼

**u** ¼

At inverted pendulums cantilevers, where the fundamental horizontal modeshape does not activate the 90% of the total cantilever mass, we ask to consider the three first mode-shapes. Thus, for this purpose, we must define an ideal equivalent three degrees of freedom beam, which is going to give the three mode-shapes of the examined beam. Therefore, which is the ideal three degrees of freedom system, where its three eigen-frequencies coincide with the real first three frequencies of

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution…*

*DOI: http://dx.doi.org/10.5772/intechopen.92185*

In order to answer the above-mentioned question, consider a weightless vertical inverted pendulum with height *L* and constant section in elevation, where carry three concentrated masses that each one has the same mass-value *m*eq, located per distance 0.333*L*, between one to one, and each one mass possesses an horizontal

The inverted pendulum displacement vector **u** of the three degrees of freedom,

2 6 4

Furthermore, the inverted pendulum flexibility matrix *f* can be calculated using a suitable method (**Figure 7**), and the inverse matrix gives the stiffness matrix *k* of

> 2 6 4

*m*eq 0 0 0 *m*eq 0 0 0 *m*eq 3 7

1 0*:*5185 0*:*1481 0*:*5185 0*:*2963 0*:*0926 0*:*1481 0*:*0926 0*:*0370

<sup>5</sup> (58)

3 7

<sup>5</sup> (59)

**pendulum**

**Figure 6.**

degree of freedom (**Figure 7**).

the three degrees of freedom beam.

*D*1,1 *D*1,2 *D*1,3 *D*2,1 *D*2,2 *D*2,3 *D*3,1 *D*3,2 *D*3,3

*f* ¼

**159**

2 6 4

However, Eq. (54) can be solved numerically only, where we obtain the first four roots (*n* ¼ 1, 2, 3, 4Þ:

$$
\beta\_1 L = 1.8751, \quad \beta\_2 L = 4.6941, \quad \beta\_3 L = 7.8548 \text{ and } \quad \beta\_4 L = 10.996
\tag{54}
$$

By the definition of parameter *β*, we can calculate the first four circular eigenfrequency *ω* using Eq. (36):

$$\beta^4 = \frac{\alpha^2 \cdot \overline{m}}{EI\_\circ} \Rightarrow \alpha^2 = \frac{EI\_\circ \cdot \beta^4}{\overline{m}} \Rightarrow \alpha = \beta^2 \cdot \sqrt{\frac{EI\_\circ}{\overline{m}}} \tag{55}$$

Thus, inserting Eq. (54) into Eq. (55), the first four eigen-frequency *ω<sup>n</sup>* is directly arise for each *n*-value.

$$\alpha\_1 = \frac{3.516}{L^2} \cdot \sqrt{\frac{EI\_\circ}{\overline{m}}}, \ \alpha\_2 = \frac{22.03}{L^2} \cdot \sqrt{\frac{EI\_\circ}{\overline{m}}},$$

$$\alpha\_3 = \frac{61.70}{L^2} \cdot \sqrt{\frac{EI\_\circ}{\overline{m}}}, \ \alpha\_4 = \frac{120.9}{L^2} \cdot \sqrt{\frac{EI\_\circ}{\overline{m}}}\tag{56}$$

Therefore, the vibration mode-shape of the examined vertical inverted pendulum arises by Eq. (20)—since previous inserting Eq. (56)—thus:

$$\rho\_n(\mathbf{x}) = C\_1 \left[ \cosh \beta\_n \mathbf{x} - \cos \beta\_n \mathbf{x} - \frac{\cosh \beta\_n L + \cos \beta\_n L}{\sinh \beta\_n L + \sin \beta\_n L} (\sinh \beta\_n \mathbf{x} - \sin \beta\_n \mathbf{x}) \right] \tag{57}$$

The value of *C*<sup>1</sup> is an arbitrary constant, and we usually get it equal to unit. Thus, for each value of parameter *n*, a mode-shape with its eigen-frequency is resulted. The fundamental (first) mode-shape is resulted for *n* ¼ 1, etc. (**Figure 6**).

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution… DOI: http://dx.doi.org/10.5772/intechopen.92185*

**Figure 6.** *Modal analysis of continuous inverted pendulum. The first four mode-shapes.*

#### **6. The equivalent three degrees of freedom system of inverted pendulum**

At inverted pendulums cantilevers, where the fundamental horizontal modeshape does not activate the 90% of the total cantilever mass, we ask to consider the three first mode-shapes. Thus, for this purpose, we must define an ideal equivalent three degrees of freedom beam, which is going to give the three mode-shapes of the examined beam. Therefore, which is the ideal three degrees of freedom system, where its three eigen-frequencies coincide with the real first three frequencies of the inverted pendulum with distributed mass and flexural stiffness?

In order to answer the above-mentioned question, consider a weightless vertical inverted pendulum with height *L* and constant section in elevation, where carry three concentrated masses that each one has the same mass-value *m*eq, located per distance 0.333*L*, between one to one, and each one mass possesses an horizontal degree of freedom (**Figure 7**).

The inverted pendulum displacement vector **u** of the three degrees of freedom, as well as the diagonal beam mass matrix *m* are written:

$$\mathbf{u} = \begin{Bmatrix} u\_1 \\ u\_2 \\ u\_3 \end{Bmatrix}, \mathbf{m} = \begin{bmatrix} m\_{\text{eq}} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & m\_{\text{eq}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & m\_{\text{eq}} \end{bmatrix} \tag{58}$$

Furthermore, the inverted pendulum flexibility matrix *f* can be calculated using a suitable method (**Figure 7**), and the inverse matrix gives the stiffness matrix *k* of the three degrees of freedom beam.

$$f = \begin{bmatrix} D\_{1,1} & D\_{1,2} & D\_{1,3} \\ D\_{2,1} & D\_{2,2} & D\_{2,3} \\ D\_{3,1} & D\_{3,2} & D\_{3,3} \end{bmatrix} = \frac{L^3}{3EI\_\circ} \cdot \begin{bmatrix} 1 & 0.5185 & 0.1481 \\ 0.5185 & 0.2963 & 0.0926 \\ 0.1481 & 0.0926 & 0.0370 \end{bmatrix} \tag{59}$$

*M L*ð Þ¼ , *t ΕΙ<sup>y</sup>* �

*Number Theory and Its Applications*

*Q L*ð Þ¼ , *t ΕΙ<sup>y</sup>* �

and

four roots (*n* ¼ 1, 2, 3, 4Þ:

frequency *ω* using Eq. (36):

directly arise for each *n*-value.

**158**

*<sup>β</sup>*<sup>4</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> � *<sup>m</sup> ΕΙ<sup>y</sup>*

> *<sup>ω</sup>*<sup>1</sup> <sup>¼</sup> <sup>3</sup>*:*<sup>516</sup> *<sup>L</sup>*<sup>2</sup> �

*<sup>ω</sup>*<sup>3</sup> <sup>¼</sup> <sup>61</sup>*:*<sup>70</sup> *<sup>L</sup>*<sup>2</sup> �

*<sup>φ</sup>n*ð Þ¼ *<sup>x</sup> <sup>C</sup>*<sup>1</sup> cosh *<sup>β</sup>nx* � cos *<sup>β</sup>nx* � cosh *<sup>β</sup>nL* <sup>þ</sup> cos *<sup>β</sup>nL*

*∂*2 *φ*ð Þ *L*

*∂*3 *φ*ð Þ *L*

ð Þ sin *βL* þ sinh *βL* ð Þ cos *βL* þ cosh *βL* ð cos *βL* þ cosh *βL*Þ �ð Þ sin *βL* þ sinh *βL*

" # *C*<sup>1</sup>

However, re-writing Eqs. (50) and (51) again, we get the matrix form:

Eq. (52) is a real eigenvalue problem. In order to calculate the eigenvalues, parameters *C*<sup>1</sup> and *C*<sup>2</sup> must not both equal zero. Thus, the determinant of the matrix by Eq. (52) must be zero, where it drives to the following frequency equation:

However, Eq. (54) can be solved numerically only, where we obtain the first

*β*1*L* ¼ 1*:*8751, *β*2*L* ¼ 4*:*6941, *β*3*L* ¼ 7*:*8548 and *β*4*L* ¼ 10*:*996 (54)

By the definition of parameter *β*, we can calculate the first four circular eigen-

*m*

, *<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> <sup>22</sup>*:*<sup>03</sup>

, *<sup>ω</sup>*<sup>4</sup> <sup>¼</sup> <sup>120</sup>*:*<sup>9</sup>

� �

The value of *C*<sup>1</sup> is an arbitrary constant, and we usually get it equal to unit. Thus, for each value of parameter *n*, a mode-shape with its eigen-frequency is resulted.

) *<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> *ΕΙ<sup>y</sup>* � *<sup>β</sup>*<sup>4</sup>

Thus, inserting Eq. (54) into Eq. (55), the first four eigen-frequency *ω<sup>n</sup>* is

ffiffiffiffiffiffiffi *ΕΙ<sup>y</sup> m*

> ffiffiffiffiffiffiffi *ΕΙ<sup>y</sup> m*

Therefore, the vibration mode-shape of the examined vertical inverted pendulum arises by Eq. (20)—since previous inserting Eq. (56)—thus:

The fundamental (first) mode-shape is resulted for *n* ¼ 1, etc. (**Figure 6**).

r

r

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> <sup>0</sup> ) *ΕΙ<sup>y</sup>* � *<sup>φ</sup>*00ð Þ¼ *<sup>L</sup>* <sup>0</sup> )

*C*1ðsin *βL* þ sinh *βL*Þ þ *C*2ð cos *βL* þ cosh *βL*Þ ¼ 0 (50)

*<sup>∂</sup>x*<sup>3</sup> <sup>¼</sup> <sup>0</sup> ) *ΕΙ<sup>y</sup>* � *<sup>φ</sup>*000ð Þ¼ *<sup>L</sup>* <sup>0</sup> )

*C*2

1 þ ð Þ� cos *βL* ð Þ¼ cosh *βL* 0 (53)

) *<sup>ω</sup>* <sup>¼</sup> *<sup>β</sup>*<sup>2</sup> �

*<sup>L</sup>*<sup>2</sup> �

*<sup>L</sup>*<sup>2</sup> �

( )

<sup>¼</sup> <sup>0</sup> 0

ffiffiffiffiffiffiffi *ΕΙ<sup>y</sup> m*

r

ffiffiffiffiffiffiffi *ΕΙ<sup>y</sup> m*

,

ffiffiffiffiffiffiffi *ΕΙ<sup>y</sup> m*

sinh *<sup>β</sup>nL* <sup>þ</sup> sin *<sup>β</sup>nL* ð Þ sinh *<sup>β</sup>nx* � sin *<sup>β</sup>nx*

r

r

( )

(52)

(55)

(56)

(57)

*C*1ð cos *βL* þ cosh *βL*Þ þ *C*2ð� sin *βL* þ sinh *βL*Þ ¼ 0 (51)

**Figure 7.** *The equivalent three-degree of freedom system on inverted pendulum.*

$$\mathbf{k} = \begin{bmatrix} k\_{1,1} & k\_{1,2} & k\_{1,3} \\ k\_{2,1} & k\_{2,2} & k\_{2,3} \\ k\_{3,1} & k\_{3,2} & k\_{3,3} \end{bmatrix} = \frac{3EI\_y}{L^3} \cdot \begin{bmatrix} 14.53846 & -33.23077 & 24.92308 \\ -33.23077 & 91.38462 & -95.53846 \\ 24.92308 & -95.53846 & 166.15385 \end{bmatrix} \tag{60}$$

The equations of motion for the case of the free undamped vibration of the ideal beam is given as:

$$\mathbf{m}\,\ddot{\mathbf{u}}(t) + \mathbf{k}\,\mathbf{u}(t) = \mathbf{0} \tag{61}$$

*m*eq ¼ 0*:*1868388 � ð Þ *mL* (65)

Therefore, inserting the ideal equivalent mass *m*eq by Eq. (65) at three degrees of

On the contrary to above-mentioned about the examined cantilever, it is worth noting that in the case where ask an ideal single degree of freedom system that has eigen-frequency equal to fundamental eigen-frequency of the real cantilever we

• Eigen-frequency of single degree of freedom system with k the cantilever lateral stiffness and *m*eq, sdof the concentrated mass at the top of the cantilever:

> <sup>¼</sup> <sup>3</sup>*EIy=L*<sup>3</sup> *m*eq,sdof

> > ffiffiffiffiffiffiffi *ΕΙ<sup>y</sup> m*

*m*eq,sdof ¼ 0*:*2426742 � ð Þ *mL* (68)

r

(66)

(67)

*m*eq,sdof

*<sup>ω</sup>* <sup>¼</sup> <sup>3</sup>*:*<sup>516</sup> *<sup>L</sup>*<sup>2</sup> �

• Fundamental (first) eigen-frequency (see Eq. (56)) of the real cantilever with

• Thus, inserting Eq. (67) into Eq. (66) the equivalent concentrated mass meq,sdof

The present article has presented a mathematic ideal three degrees of freedom system that is equivalent with the modal behavior of two cases of pendulums. First, an amphi-hinge vertical pendulum with distributed mass and flexural stiffness along its height has been examined. Second, an inverted pendulum that can be simulates a tower (or cantilever) has been examined too. For each case, an ideal three degrees of freedom system has been proposed that can be used in instrumentation of such a vertical tower, which does not possess concentrated masses. In the framework of the identification of mode-shapes of the above-mentioned pendulums, the equivalent mass by Eqs. (47) and (65) permits to locate accelerometers (as shown at **Figures 4** and **7**) and measures the response acceleration histories, in

The author wishes to acknowledge the German Research Foundation (DFG) program on Initiation of International Collaboration entitled "Data-driven analysis models for slender structures using explainable artificial intelligence", Project No. 417973400 for the period 2019–2021, Prof. Dr.-Ing. Kay Smarsly, Bauhaus Univer-

*<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*

continuous distribution of mass and flexural stiffness:

at the top of the cantilever is given [9]:

order to calculate the real first three frequencies.

freedom system of **Figure 7**, the three eigen-frequencies coincide with the real values of the initial vertical inverted pendulum that has distributed mass and flex-

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution…*

ural stiffness.

**7. Conclusions**

**Acknowledgements**

**161**

sity Weimar, project coordinator.

write the following equations:

*DOI: http://dx.doi.org/10.5772/intechopen.92185*

The eigen-problem is written as:

$$(\mathbf{k} - \alpha\_n^2 \mathbf{m})\boldsymbol{\upvarphi}\_n = \mathbf{0} \quad n = 1, 2, 3. \tag{62}$$

where the eigen-frequencies *ω<sup>n</sup>* and the three mode-shapes *φ<sup>n</sup>* are known by Eq. (56) and **Figure 6**. Therefore, the unique unknown parameter is the mass *m*eq. Thus,

$$\det\left(\mathbf{k} - \omega\_1^2 \mathbf{m}\right) = \mathbf{0} \Rightarrow \tag{63}$$

$$m\_{\rm eq}^3 + A \cdot m\_{\rm eq}^2 + B \cdot m\_{\rm eq} + C = 0 \tag{64}$$

where

$$A = -\frac{k\_{11} + k\_{22} + k\_{33}}{\alpha\_1^2}, B = \frac{k\_{11}k\_{33} + k\_{11}k\_{22} + k\_{22}k\_{33} - k\_{12}^2 - k\_{13}^2 - k\_{23}^2}{\alpha\_1^4}$$

$$C = -\frac{k\_{11}k\_{22}k\_{33} + 2k\_{12}k\_{13}k\_{23} - k\_{11}k\_{23}^2 - k\_{22}k\_{13}^2 - k\_{33}k\_{12}^2}{\alpha\_1^6}$$

The numerical solution of Eq. (64) gives three roots for parameter *m*eq, where only the first root is acceptable, because the other two values rejected since do not have natural meaning (appear values greater from the total pendulum mass *mL*). Thus, the only one acceptable root is given:

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution… DOI: http://dx.doi.org/10.5772/intechopen.92185*

$$m\_{\rm eq} = 0.1868388 \cdot (\overline{m}L) \tag{65}$$

Therefore, inserting the ideal equivalent mass *m*eq by Eq. (65) at three degrees of freedom system of **Figure 7**, the three eigen-frequencies coincide with the real values of the initial vertical inverted pendulum that has distributed mass and flexural stiffness.

On the contrary to above-mentioned about the examined cantilever, it is worth noting that in the case where ask an ideal single degree of freedom system that has eigen-frequency equal to fundamental eigen-frequency of the real cantilever we write the following equations:

• Eigen-frequency of single degree of freedom system with k the cantilever lateral stiffness and *m*eq, sdof the concentrated mass at the top of the cantilever:

$$\alpha^2 = \frac{k}{m\_{\text{eq,sdof}}} = \frac{\text{3EI}\_{\text{y}}/L^3}{m\_{\text{eq,sdof}}} \tag{66}$$

• Fundamental (first) eigen-frequency (see Eq. (56)) of the real cantilever with continuous distribution of mass and flexural stiffness:

$$
\rho = \frac{3.516}{L^2} \cdot \sqrt{\frac{EI\_y}{\overline{m}}} \tag{67}
$$

• Thus, inserting Eq. (67) into Eq. (66) the equivalent concentrated mass meq,sdof at the top of the cantilever is given [9]:

$$m\_{\text{eq,sdof}} = 0.2426742 \cdot (\overline{m}L) \tag{68}$$

#### **7. Conclusions**

*k* ¼

**Figure 7.**

2 6 4

beam is given as:

where

**160**

*k*1,1 *k*1,2 *k*1,3 *k*2,1 *k*2,2 *k*2,3 *k*3,1 *k*3,2 *k*3,3

*Number Theory and Its Applications*

The eigen-problem is written as:

*<sup>A</sup>* ¼ � *<sup>k</sup>*<sup>11</sup> <sup>þ</sup> *<sup>k</sup>*<sup>22</sup> <sup>þ</sup> *<sup>k</sup>*<sup>33</sup> *ω*2 1

Thus, the only one acceptable root is given:

3 7 <sup>5</sup> <sup>¼</sup> <sup>3</sup>*EIy <sup>L</sup>*<sup>3</sup> �

*The equivalent three-degree of freedom system on inverted pendulum.*

2 6 4

**<sup>k</sup>** � *<sup>ω</sup>*<sup>2</sup>

*m*3

The equations of motion for the case of the free undamped vibration of the ideal

where the eigen-frequencies *ω<sup>n</sup>* and the three mode-shapes *φ<sup>n</sup>* are known by Eq. (56) and **Figure 6**. Therefore, the unique unknown parameter is the mass *m*eq. Thus,

, *<sup>B</sup>* <sup>¼</sup> *<sup>k</sup>*11*k*<sup>33</sup> <sup>þ</sup> *<sup>k</sup>*11*k*<sup>22</sup> <sup>þ</sup> *<sup>k</sup>*22*k*<sup>33</sup> � *<sup>k</sup>*<sup>2</sup>

*ω*6 1

The numerical solution of Eq. (64) gives three roots for parameter *m*eq, where only the first root is acceptable, because the other two values rejected since do not have natural meaning (appear values greater from the total pendulum mass *mL*).

det **<sup>k</sup>** � *<sup>ω</sup>*<sup>2</sup>

eq <sup>þ</sup> *<sup>Α</sup>* � *<sup>m</sup>*<sup>2</sup>

*<sup>C</sup>* ¼ � *<sup>k</sup>*11*k*22*k*<sup>33</sup> <sup>þ</sup> <sup>2</sup>*k*12*k*13*k*<sup>23</sup> � *<sup>k</sup>*11*k*<sup>2</sup>

14*:*53846 �33*:*23077 24*:*92308 �33*:*23077 91*:*38462 �95*:*53846 24*:*92308 �95*:*53846 166*:*15385

**m u**€ð Þþ*t* **k u**ðÞ¼ *t* **0** (61)

<sup>1</sup>**<sup>m</sup>** � � <sup>¼</sup> <sup>0</sup> ) (63)

*ω*4 1

<sup>23</sup> � *<sup>k</sup>*22*k*<sup>2</sup>

eq þ *Β* � *m*eq þ *C* ¼ 0 (64)

<sup>12</sup> � *<sup>k</sup>*<sup>2</sup>

12

<sup>13</sup> � *<sup>k</sup>*33*k*<sup>2</sup>

<sup>13</sup> � *<sup>k</sup>*<sup>2</sup> 23

*<sup>n</sup>***<sup>m</sup>** � �**φ***<sup>n</sup>* <sup>¼</sup> **<sup>0</sup>** *<sup>n</sup>* <sup>¼</sup> 1, 2, 3*:* (62)

3 7 <sup>5</sup> (60)

> The present article has presented a mathematic ideal three degrees of freedom system that is equivalent with the modal behavior of two cases of pendulums. First, an amphi-hinge vertical pendulum with distributed mass and flexural stiffness along its height has been examined. Second, an inverted pendulum that can be simulates a tower (or cantilever) has been examined too. For each case, an ideal three degrees of freedom system has been proposed that can be used in instrumentation of such a vertical tower, which does not possess concentrated masses. In the framework of the identification of mode-shapes of the above-mentioned pendulums, the equivalent mass by Eqs. (47) and (65) permits to locate accelerometers (as shown at **Figures 4** and **7**) and measures the response acceleration histories, in order to calculate the real first three frequencies.

#### **Acknowledgements**

The author wishes to acknowledge the German Research Foundation (DFG) program on Initiation of International Collaboration entitled "Data-driven analysis models for slender structures using explainable artificial intelligence", Project No. 417973400 for the period 2019–2021, Prof. Dr.-Ing. Kay Smarsly, Bauhaus University Weimar, project coordinator.

*Number Theory and Its Applications*

### **Author details**

Triantafyllos K. Makarios Institute of Structural Analysis and Dynamics of Structures, School of Civil Engineering, Faculty of Engineer, Aristotle University of Thessaloniki, Thessaloniki Town, Greece

**References**

[1] Manolis GD, Makarios TK, Terzi V, Karetsou I. Mode shape identification of an existing three-story flexible steel stairway as a continuous dynamic system. Theoretical and Applied Mechanics. 2015;**42**(3):151-166

*DOI: http://dx.doi.org/10.5772/intechopen.92185*

and Applications. New York: Nova Science Publisher; 2013. pp. 77-113.

[8] Chopra A. Dynamics of Structures. Theory and Applications to Earthquake Engineering. International Edition. Englewood Cliffs, New Jersey: Prentice-

[9] Clough R, Penzien J. Dynamics of Structures. Third edition. Berkeley,

ISBN: 978-1-62808-128-2

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution…*

Hall, Inc.; 1995. p. 07632

USA: McGraw-Hill; 1995

[2] Makarios TK, Manolis G, Karetsou I, Papanikolaou M, Terzi V. Modelling and identification of the dynamic response of an existing three-story steel stairway. In: COMPDYN 2015, 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering; 25–27 May.

Crete Island, Greece; 2015

[3] Makarios T, Manolis G, Terzi V, Karetsou I. Identification of dynamic characteristics of a continuous system: Case study for a flexible steel stairway.

[4] Makarios T, Baniotopoulos C. Wind energy structures: Modal analysis by the continuous model approach. Journal of Vibration and Control. 2014;**20**(3):

[5] Makarios T, Baniotopoulos C. Modal analysis of wind turbine tower via its continuous model with partially fixed foundation. International Journal of Innovative Research in Advanced Engineering. 2015;**2**(1):14-25

[6] Makarios T. Identification of the mode shapes of spatial tall multi-storey buildings due to earthquakes. The new "modal time-histories" method. Journal of the Structural Design of Tall & Special Buildings. 2012;**21**(9):621-641

[7] Makarios T. Chapter 4: Identification of building dynamic characteristics by using the modal response acceleration time-histories in the seismic excitation and the wind dynamic loading cases. In: Accelerometers; Principles, Structure

In: 16th World Conference on Earthquake; 9–13 January; 16WCEE 2017 (paper 1011). Santiago Chile; 2017

395-405

**163**

\*Address all correspondence to: makariostr@civil.auth.gr

© 2020 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.

*Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution… DOI: http://dx.doi.org/10.5772/intechopen.92185*

### **References**

[1] Manolis GD, Makarios TK, Terzi V, Karetsou I. Mode shape identification of an existing three-story flexible steel stairway as a continuous dynamic system. Theoretical and Applied Mechanics. 2015;**42**(3):151-166

[2] Makarios TK, Manolis G, Karetsou I, Papanikolaou M, Terzi V. Modelling and identification of the dynamic response of an existing three-story steel stairway. In: COMPDYN 2015, 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering; 25–27 May. Crete Island, Greece; 2015

[3] Makarios T, Manolis G, Terzi V, Karetsou I. Identification of dynamic characteristics of a continuous system: Case study for a flexible steel stairway. In: 16th World Conference on Earthquake; 9–13 January; 16WCEE 2017 (paper 1011). Santiago Chile; 2017

[4] Makarios T, Baniotopoulos C. Wind energy structures: Modal analysis by the continuous model approach. Journal of Vibration and Control. 2014;**20**(3): 395-405

[5] Makarios T, Baniotopoulos C. Modal analysis of wind turbine tower via its continuous model with partially fixed foundation. International Journal of Innovative Research in Advanced Engineering. 2015;**2**(1):14-25

[6] Makarios T. Identification of the mode shapes of spatial tall multi-storey buildings due to earthquakes. The new "modal time-histories" method. Journal of the Structural Design of Tall & Special Buildings. 2012;**21**(9):621-641

[7] Makarios T. Chapter 4: Identification of building dynamic characteristics by using the modal response acceleration time-histories in the seismic excitation and the wind dynamic loading cases. In: Accelerometers; Principles, Structure

and Applications. New York: Nova Science Publisher; 2013. pp. 77-113. ISBN: 978-1-62808-128-2

[8] Chopra A. Dynamics of Structures. Theory and Applications to Earthquake Engineering. International Edition. Englewood Cliffs, New Jersey: Prentice-Hall, Inc.; 1995. p. 07632

[9] Clough R, Penzien J. Dynamics of Structures. Third edition. Berkeley, USA: McGraw-Hill; 1995

**Author details**

**162**

Triantafyllos K. Makarios

*Number Theory and Its Applications*

Thessaloniki Town, Greece

provided the original work is properly cited.

Institute of Structural Analysis and Dynamics of Structures, School of Civil Engineering, Faculty of Engineer, Aristotle University of Thessaloniki,

© 2020 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: makariostr@civil.auth.gr

**Chapter 10**

*Cheon Seoung Ryoo*

**Abstract**

Some Identities Involving

2-Variable Modified Degenerate

and Distribution of Their Zeros

In this chapter, we introduce the 2-variable modified degenerate Hermite polynomials and obtain some new symmetric identities for 2-variable modified degenerate Hermite polynomials. In order to give explicit identities for 2-variable modified degenerate Hermite polynomials, differential equations arising from the generating functions of 2-variable modified degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the

**Keywords:** differential equations, symmetric identities, modified degenerate

where *ε* is unrestricted. Hermite equation is encountered in the study of quantum mechanical harmonic oscillator, where *ε* represent the energy of the oscillator. The ordinary Hermite numbers *Hn* and Hermite polynomials *Hn*ð Þ *x* are

*<sup>t</sup>*ð Þ <sup>2</sup>*x*�*<sup>t</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*n*¼0

*Hn*ð Þ *x t n*

*e*

*e* �*t* 2 <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0 *Hn t n n*!

ð Þþ *x* ð Þ *ε* � 1 *u x*ð Þ¼ 0, *x*∈ ½ � �∞, ∞ , (1)

*<sup>n</sup>*! (2)

*:* (3)

Hermite Polynomials Arising

from Differential Equations

2-variable modified degenerate Hermite equations.

Hermite polynomials, complex zeros

The Hermite equation is defined as

usually defined by the generating functions

*u*00ð Þ� *x* 2*xu*<sup>0</sup>

**1. Introduction**

and

**165**

Clearly, *Hn* ¼ *Hn*ð Þ 0 .

#### **Chapter 10**

## Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros

*Cheon Seoung Ryoo*

#### **Abstract**

In this chapter, we introduce the 2-variable modified degenerate Hermite polynomials and obtain some new symmetric identities for 2-variable modified degenerate Hermite polynomials. In order to give explicit identities for 2-variable modified degenerate Hermite polynomials, differential equations arising from the generating functions of 2-variable modified degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the 2-variable modified degenerate Hermite equations.

**Keywords:** differential equations, symmetric identities, modified degenerate Hermite polynomials, complex zeros

#### **1. Introduction**

The Hermite equation is defined as

$$
\mu''(\mathbf{x}) - 2\mathbf{x}u'(\mathbf{x}) + (\varepsilon - \mathbf{1})u(\mathbf{x}) = \mathbf{0}, \mathbf{x} \in [-\infty, \infty], \tag{1}
$$

where *ε* is unrestricted. Hermite equation is encountered in the study of quantum mechanical harmonic oscillator, where *ε* represent the energy of the oscillator. The ordinary Hermite numbers *Hn* and Hermite polynomials *Hn*ð Þ *x* are usually defined by the generating functions

$$e^{t(2\mathbf{x}-t)} = \sum\_{n=0}^{\infty} H\_n(\mathbf{x}) \frac{t^n}{n!} \tag{2}$$

and

$$e^{-t^2} = \sum\_{n=0}^{\infty} H\_n \frac{t^n}{n!}.\tag{3}$$

Clearly, *Hn* ¼ *Hn*ð Þ 0 .

#### *Number Theory and Its Applications*

It is known that these numbers and polynomials play an important role in various fields of mathematics and physics, including number theory, combinations, special functions, and differential equations. Many interested properties about that have been studied (see [1–5]). The ordinary Hermite polynomials *Hn*ð Þ *x* satisfy the Hermite differential equation

$$\frac{d^2H(\mathbf{x})}{d\mathbf{x}^2} - 2\mathbf{x}\frac{dH(\mathbf{x})}{d\mathbf{x}} + 2nH(\mathbf{x}) = \mathbf{0}, n = \mathbf{0}, \mathbf{1}, \mathbf{2}, \dots \tag{4}$$

Hence ordinary Hermite polynomials *Hn*ð Þ *x* satisfy the second-order ordinary differential equation

$$
u'' - 2\kappa 
u' + 2\kappa 
u = 0.\tag{5}$$

Since log 1ð Þ <sup>þ</sup>*<sup>λ</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92687*

**polynomials**

Since 1ð Þ <sup>þ</sup> *<sup>μ</sup> xt*

**167**

by means of the generating function

X∞ *n*¼0

Hermite polynomials H*n*ð Þ *x*, *y*j*μ* are totally different.

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> yt=<sup>μ</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*m*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *m*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *l*¼0

need the binomial theorem: for a variable *y*,

H*n*ð Þ *x*, *y*j*μ*

also obtained.

*<sup>λ</sup>* ! 1 as *λ* approaches to 0, it is apparent that (10) descends to (7).

Mathematicians have studied the differential equations arising from the generating functions of special numbers and polynomials (see [10–14]). Now, a new class of 2-variable modified degenerate Hermite polynomials are constructed based on the results so far. We can induce the differential equations generated from the generating function of 2-variable modified degenerate Hermite polynomials. By using the coefficients of this differential equation, we obtain explicit identities for the 2-variable modified degenerate Hermite polynomials. The rest of the paper is organized as follows. In Section 2, we construct the 2-variable modified degenerate Hermite polynomials and obtain basic properties of these polynomials. In Section 3, we give some symmetric identities for 2-variable modified degenerate Hermite polynomials. In Section 4, we derive the differential equations generated from the generating function of 2-variable modified degenerate Hermite polynomials. Using the coefficients of this differential equation, we have explicit identities for the 2 variable modified degenerate Hermite polynomials. In Section 5, we investigate the zeros of the 2-variable modified degenerate Hermite equations by using computer. Further, we observe the pattern of scattering phenomenon for the zeros of 2 variable modified degenerate Hermite equations. Our paper will finish with Section 6, where the conclusions and future directions of this work are showed.

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising…*

**2. Basic properties for the 2-variable modified degenerate Hermite**

In this section, a new class of the 2-variable modified degenerate Hermite polynomials are considered. Furthermore, some properties of these polynomials are

We define the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ*

*t n n*!

degenerate Hermite polynomials H*n*ð Þ *x*, *y*, *μ* and 2-variable modified degenerate

*ty μ* � �

> X*m l*¼0

X∞ *m*¼*l*

Now, we recall that the *μ*-analogue of the falling factorial sequences as follows:

ð Þ *x*j*μ* <sup>0</sup> ¼ 1,ð Þ *x*j*μ <sup>n</sup>* ¼ *x x*ð Þ � *μ* ð Þ *x* � 2*μ* ⋯ð Þ *x* � ð Þ *n* � 1 *μ* ,ð Þ *n*≥1 *:* (12)

*μm m*!

*<sup>S</sup>*1ð Þ *<sup>m</sup>*, *<sup>l</sup> ty*

*<sup>S</sup>*1ð Þ *<sup>m</sup>*, *<sup>l</sup> yl*

!

! *<sup>t</sup>*

*μ* � �*<sup>l</sup> μm m*!

> *<sup>μ</sup><sup>m</sup>*�*<sup>l</sup> <sup>l</sup>*! *m*!

*l l*! *:* (13)

Note that lim *<sup>μ</sup>*!<sup>1</sup>ð Þ *x*j*μ <sup>n</sup>* ¼ *x x*ð Þ � 1 ð Þ *x* � 2 ⋯ð Þ¼ *x* � ð Þ *n* � 1 ð Þ *x <sup>n</sup>*,ð Þ *n*≥ 1 *:* We also

*m*

<sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> xt*

*<sup>μ</sup>* ! *ext* as *<sup>μ</sup>* ! 0, it is clear that (11) reduces to (6). Observe that

*μ e yt*2

*:* (11)

We remind that the 2-variable Hermite polynomials *Hn*ð Þ *x*, *y* defined by the generating function (see [2])

$$\sum\_{n=0}^{\infty} H\_n(\mathbf{x}, \boldsymbol{y}) \frac{t^n}{n!} = e^{t(\mathbf{x} + \boldsymbol{y}t)} \tag{6}$$

are the solution of heat equation

$$\frac{\partial}{\partial \boldsymbol{\eta}} H\_{\boldsymbol{n}}(\boldsymbol{\chi}, \boldsymbol{\chi}) = \frac{\partial^2}{\partial \boldsymbol{\chi}^2} H\_{\boldsymbol{n}}(\boldsymbol{\chi}, \boldsymbol{\chi}), \quad H\_{\boldsymbol{n}}(\boldsymbol{\chi}, \mathbf{0}) = \boldsymbol{\mathfrak{x}}^{\boldsymbol{n}}.\tag{7}$$

Observe that

$$H\_n(2\mathfrak{x}, -\mathfrak{1}) = H\_n(\mathfrak{x}).\tag{8}$$

Motivated by their importance and potential applications in certain problems of probability, combinatorics, number theory, differential equations, numerical analysis and other areas of mathematics and physics, several kinds of some special numbers and polynomials were recently studied by many authors (see [1–8]). Many mathematicians have studied in the area of the degenerate Stiling, degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Genocchi polynomials, and degenerate tangent polynomials (see [6, 7, 9]).

Recently, Hwang and Ryoo [10] proposed the 2-variable degenerate Hermite polynomials H*n*ð Þ *x*, *y*, *λ* by means of the generating function

$$\sum\_{n=0}^{\infty} \mathcal{H}\_n(\infty, y, \lambda) \frac{t^n}{n!} = (1 + \lambda)^{\frac{t(x+yt)}{\lambda}}.\tag{9}$$

Since 1ð Þ <sup>þ</sup> *<sup>μ</sup> <sup>t</sup> <sup>μ</sup>* ! *<sup>e</sup><sup>t</sup>* as *<sup>μ</sup>* ! 0, it is evident that (9) reduces to (6). The 2-variable degenerate Hermite polynomials H*n*ð Þ *x*, *y*, *λ* in generating function (9) are the solution of equation

$$\begin{split} \frac{\partial}{\partial \boldsymbol{\eta}} \mathcal{H}\_{\boldsymbol{n}}(\boldsymbol{\varkappa}, \boldsymbol{\jmath}, \boldsymbol{\lambda}) &= \frac{\boldsymbol{\lambda}}{\log \left( \mathbf{1} + \boldsymbol{\lambda} \right)} \frac{\partial^2}{\partial \boldsymbol{\varkappa}^2} \mathcal{H}\_{\boldsymbol{n}}(\boldsymbol{\varkappa}, \boldsymbol{\jmath}, \boldsymbol{\lambda}), \\\\ \mathcal{H}\_{\boldsymbol{n}}(\boldsymbol{\varkappa}, \mathbf{0}, \boldsymbol{\lambda}) &= \left( \frac{\log \left( \mathbf{1} + \boldsymbol{\lambda} \right)}{\boldsymbol{\lambda}} \right)^{\boldsymbol{n}} \boldsymbol{\varkappa}^{\boldsymbol{n}}. \end{split} \tag{10}$$

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising… DOI: http://dx.doi.org/10.5772/intechopen.92687*

Since log 1ð Þ <sup>þ</sup>*<sup>λ</sup> <sup>λ</sup>* ! 1 as *λ* approaches to 0, it is apparent that (10) descends to (7).

Mathematicians have studied the differential equations arising from the generating functions of special numbers and polynomials (see [10–14]). Now, a new class of 2-variable modified degenerate Hermite polynomials are constructed based on the results so far. We can induce the differential equations generated from the generating function of 2-variable modified degenerate Hermite polynomials. By using the coefficients of this differential equation, we obtain explicit identities for the 2-variable modified degenerate Hermite polynomials. The rest of the paper is organized as follows. In Section 2, we construct the 2-variable modified degenerate Hermite polynomials and obtain basic properties of these polynomials. In Section 3, we give some symmetric identities for 2-variable modified degenerate Hermite polynomials. In Section 4, we derive the differential equations generated from the generating function of 2-variable modified degenerate Hermite polynomials. Using the coefficients of this differential equation, we have explicit identities for the 2 variable modified degenerate Hermite polynomials. In Section 5, we investigate the zeros of the 2-variable modified degenerate Hermite equations by using computer. Further, we observe the pattern of scattering phenomenon for the zeros of 2 variable modified degenerate Hermite equations. Our paper will finish with Section 6, where the conclusions and future directions of this work are showed.

#### **2. Basic properties for the 2-variable modified degenerate Hermite polynomials**

In this section, a new class of the 2-variable modified degenerate Hermite polynomials are considered. Furthermore, some properties of these polynomials are also obtained.

We define the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* by means of the generating function

$$\sum\_{n=0}^{\infty} \mathbf{H}\_{\pi}(\infty, \mathfrak{y}|\mu) \frac{t^{n}}{n!} = (\mathbf{1} + \mu)^{\frac{\omega}{\mu}} e^{\mathfrak{y}t^{2}}.\tag{11}$$

Since 1ð Þ <sup>þ</sup> *<sup>μ</sup> xt <sup>μ</sup>* ! *ext* as *<sup>μ</sup>* ! 0, it is clear that (11) reduces to (6). Observe that degenerate Hermite polynomials H*n*ð Þ *x*, *y*, *μ* and 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* are totally different.

Now, we recall that the *μ*-analogue of the falling factorial sequences as follows:

$$(\mathfrak{x}|\mu)\_0 = \mathfrak{1}, (\mathfrak{x}|\mu)\_n = \mathfrak{x}(\mathfrak{x} - \mu)(\mathfrak{x} - 2\mu) \cdots (\mathfrak{x} - (n - 1)\mu), (n \ge 1). \tag{12}$$

Note that lim *<sup>μ</sup>*!<sup>1</sup>ð Þ *x*j*μ <sup>n</sup>* ¼ *x x*ð Þ � 1 ð Þ *x* � 2 ⋯ð Þ¼ *x* � ð Þ *n* � 1 ð Þ *x <sup>n</sup>*,ð Þ *n*≥ 1 *:* We also need the binomial theorem: for a variable *y*,

$$\begin{split} (1+\mu)^{\mathsf{y}t/\mu} &= \sum\_{m=0}^{\infty} \left(\frac{t\mathsf{y}}{\mu}\right)\_{m} \frac{\mu^{m}}{m!} \\ &= \sum\_{m=0}^{\infty} \left(\sum\_{l=0}^{m} S\_{1}(m,l) \left(\frac{t\mathsf{y}}{\mu}\right)^{l} \frac{\mu^{m}}{m!} \right) \\ &= \sum\_{l=0}^{\infty} \left(\sum\_{m=l}^{\infty} S\_{1}(m,l) \mathsf{y}^{l} \mu^{m-l} \frac{l!}{m!} \right) \frac{t^{l}}{l!} .\end{split} \tag{13}$$

It is known that these numbers and polynomials play an important role in various fields of mathematics and physics, including number theory, combinations, special functions, and differential equations. Many interested properties about that have been studied (see [1–5]). The ordinary Hermite polynomials *Hn*ð Þ *x* satisfy the

Hence ordinary Hermite polynomials *Hn*ð Þ *x* satisfy the second-order ordinary

We remind that the 2-variable Hermite polynomials *Hn*ð Þ *x*, *y* defined by the

*t n n*! ¼ *e*

Motivated by their importance and potential applications in certain problems of probability, combinatorics, number theory, differential equations, numerical analysis and other areas of mathematics and physics, several kinds of some special numbers and polynomials were recently studied by many authors (see [1–8]). Many mathematicians have studied in the area of the degenerate Stiling, degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Genocchi poly-

Recently, Hwang and Ryoo [10] proposed the 2-variable degenerate Hermite

*n n*!

log 1ð Þ þ *λ*

*λ* � �*<sup>n</sup>*

¼ ð Þ 1 þ *λ*

*<sup>μ</sup>* ! *<sup>e</sup><sup>t</sup>* as *<sup>μ</sup>* ! 0, it is evident that (9) reduces to (6). The 2-variable

*∂*2

*xn:*

*t x*ð Þ þ*yt*

*<sup>∂</sup>x*<sup>2</sup> <sup>H</sup>*n*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>λ</sup>* ,

*<sup>λ</sup> :* (9)

(10)

*Hn*ð Þ *x*, *y*

*∂*2

*dx* <sup>þ</sup> <sup>2</sup>*nH x*ð Þ¼ 0, *<sup>n</sup>* <sup>¼</sup> 0, 1, 2, … *:* (4)

*u*<sup>00</sup> � 2*xu*<sup>0</sup> þ 2*nu* ¼ 0*:* (5)

*<sup>∂</sup>x*<sup>2</sup> *Hn*ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *Hn*ð Þ¼ *<sup>x</sup>*, 0 *xn:* (7)

*Hn*ð Þ¼ 2*x*, �1 *Hn*ð Þ *x :* (8)

*t x*ð Þ <sup>þ</sup>*yt* (6)

*dH x*ð Þ

X∞ *n*¼0

*Hn*ð Þ¼ *x*, *y*

nomials, and degenerate tangent polynomials (see [6, 7, 9]).

polynomials H*n*ð Þ *x*, *y*, *λ* by means of the generating function

<sup>H</sup>*n*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>λ</sup> <sup>t</sup>*

<sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>λ</sup> <sup>λ</sup>*

<sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, 0, *<sup>λ</sup>* log 1ð Þ <sup>þ</sup> *<sup>λ</sup>*

degenerate Hermite polynomials H*n*ð Þ *x*, *y*, *λ* in generating function (9) are the

X∞ *n*¼0

*∂ ∂y*

Hermite differential equation

*Number Theory and Its Applications*

generating function (see [2])

are the solution of heat equation

*∂ ∂y*

differential equation

Observe that

Since 1ð Þ <sup>þ</sup> *<sup>μ</sup> <sup>t</sup>*

solution of equation

**166**

*d*2 *H x*ð Þ *dx*<sup>2</sup> � <sup>2</sup>*<sup>x</sup>*

We remember that the classical Stirling numbers of the first kind *S*1ð Þ *n*, *k* and the second kind *S*2ð Þ *n*, *k* are defined by the relations (see [6–13])

$$\mathbf{S}(\boldsymbol{\mathfrak{x}})\_n = \sum\_{k=0}^n \mathbf{S}\_1(n,k)\boldsymbol{\mathfrak{x}}^k \text{ and } \boldsymbol{\mathfrak{x}}^n = \sum\_{k=0}^n \mathbf{S}\_2(n,k)(\boldsymbol{\mathfrak{x}})\_k,\tag{14}$$

By comparing the coefficients of *<sup>t</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92687*

*<sup>μ</sup>* ¼ 1, we get

**Theorem 2.** For any positive integer *n*, we have

X∞ *m*¼*n*�2*k*

*l*¼0

*l*¼0

*n* 2 h i

*k*¼0

for 2-variable modified degenerate Hermite polynomials.

*bm*H*<sup>m</sup> ax*, *a*<sup>2</sup>

Then the expression for Gð Þ *t*, *μ* is symmetric in *a* and *b*

*<sup>y</sup>*j*<sup>μ</sup>* � � ð Þ *bt <sup>m</sup>*

*<sup>y</sup>*j*<sup>μ</sup>* � � ð Þ *at <sup>m</sup>*

H*<sup>m</sup> ax*, *a*<sup>2</sup>

H*<sup>m</sup> bx*, *b*<sup>2</sup>

**Proof.** Let *a*, *b*> 0 (*a* 6¼ *b*). We start with

Gð Þ¼ *<sup>t</sup>*, *<sup>μ</sup>* <sup>X</sup><sup>∞</sup>

Gð Þ¼ *<sup>t</sup>*, *<sup>μ</sup>* <sup>X</sup><sup>∞</sup>

**169**

*m*¼0

By the similar way, we get that

*m*¼0

*yk*

*n l*

*n l*

*l*¼0

!

*n* 2 h i

*k*¼0

Since lim *<sup>μ</sup>*!<sup>0</sup> log 1ð Þ <sup>þ</sup>*<sup>μ</sup>*

<sup>1</sup>*:* <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup>

<sup>2</sup>*:* <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*2, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup>*<sup>n</sup>*

<sup>3</sup>*:* <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*2, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup>*<sup>n</sup>*

<sup>4</sup>*:* <sup>H</sup>*<sup>n</sup> <sup>x</sup>*, *<sup>y</sup>*<sup>1</sup> <sup>þ</sup> *<sup>y</sup>*2j*<sup>μ</sup>* � � <sup>¼</sup> <sup>X</sup>

**polynomials**

<sup>5</sup>*:* <sup>H</sup>*<sup>n</sup> <sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*2, *<sup>y</sup>*<sup>1</sup> <sup>þ</sup> *<sup>y</sup>*2j*<sup>μ</sup>* � � <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

explanation.

*n n*!

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising…*

X*n* 2½ �

*ykxn*�2*<sup>k</sup> k*!ð Þ *n* � 2*k* !

*k*¼0

The following basic properties of the 2-variable degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* are induced form (11). Therefore, it is enough to delete involved detail

*<sup>S</sup>*1ð Þ *<sup>m</sup>*, *<sup>n</sup>* � <sup>2</sup>*<sup>k</sup> xn*�2*<sup>k</sup>*

*μ* � �*<sup>l</sup>*

<sup>H</sup>*<sup>n</sup>*�*<sup>l</sup>*ð Þ *<sup>x</sup>*1, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup><sup>∞</sup>

<sup>H</sup>*<sup>n</sup>*�2*<sup>k</sup> <sup>x</sup>*, *<sup>y</sup>*1j*<sup>μ</sup>* � � *<sup>y</sup> <sup>k</sup>*

*n l*

**3. Symmetric identities for 2-variable modified degenerate Hermite**

**Theorem 3.** Let *a*, *b*>0 (*a* 6¼ *b*). The following identity holds true:

Gð Þ¼ *<sup>t</sup>*, *<sup>μ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abxt*

In this section, we give some new symmetric identities for 2-variable modified degenerate Hermite polynomials. We also get some explicit formulas and properties

*<sup>y</sup>*j*<sup>μ</sup>* � � <sup>¼</sup> *am*H*<sup>m</sup> bx*, *<sup>b</sup>*<sup>2</sup>

*<sup>m</sup>*! <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*<sup>m</sup>*! <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*m*¼0

*m*¼0

*bm*H*<sup>m</sup> ax*, *a*<sup>2</sup>

*a<sup>m</sup>*H*<sup>m</sup> bx*, *b*<sup>2</sup>

*<sup>y</sup>*j*<sup>μ</sup>* � � *<sup>t</sup>*

*<sup>y</sup>*j*<sup>μ</sup>* � � *<sup>t</sup>*

*<sup>μ</sup> e a*2*b*<sup>2</sup> *yt*2

!

! log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*Hn*ð Þ¼ *x*, *y n*!

, the expected result of Theorem 1 is achieved.

*<sup>μ</sup><sup>m</sup>*�ð Þ *<sup>n</sup>*�2*<sup>k</sup> <sup>n</sup>*!

2H*<sup>n</sup>*�*<sup>l</sup>*ð Þ *x*1, *y*j*μ :*

*<sup>S</sup>*1ð Þ *<sup>m</sup>*, *<sup>l</sup> <sup>x</sup><sup>l</sup>*

*:*

<sup>H</sup>*<sup>l</sup> <sup>x</sup>*1, *<sup>y</sup>*1j*<sup>μ</sup>* � �H*<sup>n</sup>*�*<sup>l</sup> <sup>x</sup>*2, *<sup>y</sup>*2j*<sup>μ</sup>* � �*:*

*xl*

*m*¼*l*

<sup>2</sup> *n*! *k*!ð Þ *n* � 2*k* ! *m*!*k*! *:*

2*μ<sup>m</sup>*�*<sup>l</sup> <sup>l</sup>*! *m*! *:*

*<sup>y</sup>*j*<sup>μ</sup>* � �*:* (23)

*:* (24)

*m m*!

*m m*! *:* (25)

*:* (26)

*:* (21)

□

(22)

respectively. We also have

$$\sum\_{n=m}^{\infty} \mathcal{S}\_2(n,m) \frac{t^n}{n!} = \frac{\left(e^t - 1\right)^m}{m!} \text{and} \sum\_{n=m}^{\infty} \mathcal{S}\_1(n,m) \frac{t^n}{n!} = \frac{\left(\log\left(1+t\right)\right)^m}{m!}.\tag{15}$$

As another application of the differential equation for H*n*ð Þ *x*, *y*j*μ* is as follows: Note that

$$G(t, \mathfrak{x}, \mathfrak{y}, \mathfrak{\mu}) = (\mathfrak{1} + \mathfrak{\mu})^{\frac{\mathfrak{x}}{\mathfrak{\mu}}} e^{\mathfrak{y}^{\mathfrak{x}^2}} \tag{16}$$

satisfies

$$\frac{\partial G(t, \varkappa, y, \mu)}{\partial y} - \left(\frac{\log\left(1 + \mu\right)}{\mu}\right)^2 \frac{\partial^2 G(t, \varkappa, y, \mu)}{\partial \varkappa^2} = 0. \tag{17}$$

Substitute the series in (11) for *G t*ð Þ , *x*, *y*, *μ* to get

$$\frac{\partial}{\partial \boldsymbol{\eta}} \mathcal{H}\_{\boldsymbol{n}}(\boldsymbol{\kappa}, \boldsymbol{\chi} | \boldsymbol{\mu}) = \left( \frac{\boldsymbol{\mu}}{\log \left( 1 + \boldsymbol{\mu} \right)} \right)^{2} \frac{\partial^{2}}{\partial \boldsymbol{\kappa}^{2}} \mathcal{H}\_{\boldsymbol{n}}(\boldsymbol{\kappa}, \boldsymbol{\chi} | \boldsymbol{\mu}) .$$

Thus the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* in generating function (11) are the solution of equation

$$\begin{aligned} \left(\frac{\log\left(1+\mu\right)}{\mu}\right)^2 \frac{\partial}{\partial \boldsymbol{\uprho}} \mathcal{H}\_{\boldsymbol{n}}(\boldsymbol{x}, \boldsymbol{y}|\mu) - \frac{\partial^2}{\partial \boldsymbol{x}^2} \mathcal{H}\_{\boldsymbol{n}}(\boldsymbol{x}, \boldsymbol{y}|\mu) &= \mathbf{0}, \\ \mathcal{H}\_{\boldsymbol{n}}(\boldsymbol{x}, \boldsymbol{0}|\mu) = \left(\frac{\log\left(1+\mu\right)}{\mu}\right)^n \boldsymbol{x}^n. \end{aligned} \tag{18}$$

The generating function (11) is useful for deriving several properties of the 2 variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* . For example, we have the following expression for these polynomials:

**Theorem 1**. For any positive integer *n*, we have

$$\mathbf{H}\_{\mathfrak{n}}(\mathbf{x}, \boldsymbol{\chi} | \boldsymbol{\mu}) = \sum\_{k=0}^{\left[\frac{n}{2}\right]} \left( \frac{\log \left( 1 + \boldsymbol{\mu} \right)}{\boldsymbol{\mu}} \right)^{n-2k} \boldsymbol{\varkappa}^{n-2k} \boldsymbol{\jmath}^{k} \frac{n!}{k!(n-2k)!},\tag{19}$$

where ½ � denotes taking the integer part. **Proof.** By (11) and (13), we have

$$\begin{split} \sum\_{n=0}^{\infty} \mathsf{H}\_{n}(\mathbf{x}, \boldsymbol{y} | \boldsymbol{\mu}) \frac{t^{n}}{n!} &= (1 + \boldsymbol{\mu})^{\frac{\boldsymbol{\mu}}{2}} \boldsymbol{e}^{\boldsymbol{\mu}^{\boldsymbol{\mu}}} \\ &= \sum\_{k=0}^{\infty} \mathsf{y}^{k} \frac{t^{2k}}{k!} \sum\_{l=0}^{\infty} \left( \frac{\log \left( 1 + \boldsymbol{\mu} \right)}{\boldsymbol{\mu}} \right)^{l} \boldsymbol{x}^{l} \frac{t^{l}}{l!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{\left[ \frac{n\pi}{2} \right]} \mathsf{y}^{k} \left( \frac{\log \left( 1 + \boldsymbol{\mu} \right)}{\boldsymbol{\mu}} \right)^{n-2k} \boldsymbol{x}^{n-2k} \frac{n!}{k!(n-2k)!} \right) \frac{t^{n}}{n!} . \end{split} \tag{20}$$

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising… DOI: http://dx.doi.org/10.5772/intechopen.92687*

By comparing the coefficients of *<sup>t</sup> n n*! , the expected result of Theorem 1 is achieved.

Since lim *<sup>μ</sup>*!<sup>0</sup> log 1ð Þ <sup>þ</sup>*<sup>μ</sup> <sup>μ</sup>* ¼ 1, we get

We remember that the classical Stirling numbers of the first kind *S*1ð Þ *n*, *k* and the

*k*¼0

*t n n*!

*μ e yt*2

*G t*ð Þ , *x*, *y*, *μ*

*<sup>∂</sup>x*<sup>2</sup> <sup>H</sup>*n*ð Þ *<sup>x</sup>*, *<sup>y</sup>*j*<sup>μ</sup> :*

*<sup>∂</sup>x*<sup>2</sup> <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*j*<sup>μ</sup>* 0,

*S*2ð Þ *n*, *k* ð Þ *x <sup>k</sup>*, (14)

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* (17)

*<sup>m</sup>*! *:* (15)

(16)

(18)

, (19)

(20)

<sup>¼</sup> ð Þ log 1ð Þ <sup>þ</sup> *<sup>t</sup> <sup>m</sup>*

*<sup>S</sup>*1ð Þ *<sup>n</sup>*, *<sup>k</sup> xk* and*xn* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*n*¼*m*

*G t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> xt*

*μ* � �<sup>2</sup> *∂*<sup>2</sup>

> *μ* log 1ð Þ þ *μ* � �<sup>2</sup> *∂*<sup>2</sup>

Thus the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* in

H*n*ð Þ� *x*, *y*j*μ*

The generating function (11) is useful for deriving several properties of the 2 variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* . For example, we

> log 1ð Þ þ *μ μ* � �*<sup>l</sup>*

*yk* log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ* � �*n*�2*<sup>k</sup>*

*∂*2

*x<sup>n</sup>*�2*<sup>k</sup>*

*xl t l l*!

*xn*�2*<sup>k</sup> <sup>n</sup>*!

*k*!ð Þ *n* � 2*k* !

1

CCCA *t n n*! *:*

*<sup>y</sup><sup>k</sup> <sup>n</sup>*! *k*!ð Þ *n* � 2*k* !

*xn:*

As another application of the differential equation for H*n*ð Þ *x*, *y*j*μ* is as follows:

*S*1ð Þ *n*, *m*

second kind *S*2ð Þ *n*, *k* are defined by the relations (see [6–13])

ð Þ *<sup>x</sup> <sup>n</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*<sup>∂</sup>G t*ð Þ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*

*∂ ∂y*

Substitute the series in (11) for *G t*ð Þ , *x*, *y*, *μ* to get

H*n*ð Þ¼ *x*, *y*j*μ*

generating function (11) are the solution of equation

have the following expression for these polynomials: **Theorem 1**. For any positive integer *n*, we have

2½ �

*k*¼0

<sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup>*<sup>n</sup>*

where ½ � denotes taking the integer part.

<sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> xt μ e yt*2

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼0 *yk t* 2*k k*! X∞ *l*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0 X *n* 2 h i

0

BBB@

*k*¼0

**Proof.** By (11) and (13), we have

*t n n*!

H*n*ð Þ *x*, *y*j*μ*

X∞ *n*¼0

**168**

log 1ð Þ þ *μ μ* � �<sup>2</sup> *∂*

respectively. We also have

*Number Theory and Its Applications*

*t n n*!

*S*2ð Þ *n*, *m*

X∞ *n*¼*m*

Note that

satisfies

*k*¼0

<sup>¼</sup> *<sup>e</sup>*ð Þ *<sup>t</sup>* � <sup>1</sup> *<sup>m</sup>*

*<sup>m</sup>*! and <sup>X</sup><sup>∞</sup>

*<sup>∂</sup><sup>y</sup>* � log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*∂y*

*μ* � �*<sup>n</sup>*

log 1ð Þ þ *μ μ* � �*<sup>n</sup>*�2*<sup>k</sup>*

<sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, 0j*<sup>μ</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

$$H\_n(\mathbf{x}, \boldsymbol{y}) = n! \sum\_{k=0}^{\left[\frac{n}{2}\right]} \frac{\boldsymbol{y}^k \boldsymbol{x}^{n-2k}}{k!(n-2k)!} \,. \tag{21}$$

□

The following basic properties of the 2-variable degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* are induced form (11). Therefore, it is enough to delete involved detail explanation.

**Theorem 2.** For any positive integer *n*, we have

<sup>1</sup>*:* <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup> *n* 2 h i *k*¼0 X∞ *m*¼*n*�2*k yk <sup>S</sup>*1ð Þ *<sup>m</sup>*, *<sup>n</sup>* � <sup>2</sup>*<sup>k</sup> xn*�2*<sup>k</sup> <sup>μ</sup><sup>m</sup>*�ð Þ *<sup>n</sup>*�2*<sup>k</sup> <sup>n</sup>*! *m*!*k*! *:* <sup>2</sup>*:* <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*2, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup>*<sup>n</sup> l*¼0 *n l* ! log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ* � �*<sup>l</sup> xl* 2H*<sup>n</sup>*�*<sup>l</sup>*ð Þ *x*1, *y*j*μ :* <sup>3</sup>*:* <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*2, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup>*<sup>n</sup> l*¼0 *n l* !<sup>H</sup>*<sup>n</sup>*�*<sup>l</sup>*ð Þ *<sup>x</sup>*1, *<sup>y</sup>*j*<sup>μ</sup>* <sup>X</sup><sup>∞</sup> *m*¼*l <sup>S</sup>*1ð Þ *<sup>m</sup>*, *<sup>l</sup> <sup>x</sup><sup>l</sup>* 2*μ<sup>m</sup>*�*<sup>l</sup> <sup>l</sup>*! *m*! *:* <sup>4</sup>*:* <sup>H</sup>*<sup>n</sup> <sup>x</sup>*, *<sup>y</sup>*<sup>1</sup> <sup>þ</sup> *<sup>y</sup>*2j*<sup>μ</sup>* � � <sup>¼</sup> <sup>X</sup> *n* 2 h i *k*¼0 <sup>H</sup>*<sup>n</sup>*�2*<sup>k</sup> <sup>x</sup>*, *<sup>y</sup>*1j*<sup>μ</sup>* � � *<sup>y</sup> <sup>k</sup>* <sup>2</sup> *n*! *k*!ð Þ *n* � 2*k* ! *:* <sup>5</sup>*:* <sup>H</sup>*<sup>n</sup> <sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*2, *<sup>y</sup>*<sup>1</sup> <sup>þ</sup> *<sup>y</sup>*2j*<sup>μ</sup>* � � <sup>¼</sup> <sup>X</sup>*<sup>n</sup> l*¼0 *n l* !<sup>H</sup>*<sup>l</sup> <sup>x</sup>*1, *<sup>y</sup>*1j*<sup>μ</sup>* � �H*<sup>n</sup>*�*<sup>l</sup> <sup>x</sup>*2, *<sup>y</sup>*2j*<sup>μ</sup>* � �*:* (22)

#### **3. Symmetric identities for 2-variable modified degenerate Hermite polynomials**

In this section, we give some new symmetric identities for 2-variable modified degenerate Hermite polynomials. We also get some explicit formulas and properties for 2-variable modified degenerate Hermite polynomials.

**Theorem 3.** Let *a*, *b*>0 (*a* 6¼ *b*). The following identity holds true:

$$a^m \mathbf{H}\_m(a\boldsymbol{\kappa}, a^2 \boldsymbol{\jmath}|\boldsymbol{\mu}) = a^m \mathbf{H}\_m\left(b\boldsymbol{\kappa}, b^2 \boldsymbol{\jmath}|\boldsymbol{\mu}\right). \tag{23}$$

**Proof.** Let *a*, *b*> 0 (*a* 6¼ *b*). We start with

$$\mathcal{G}(\mathfrak{t},\mathfrak{\mu}) = (\mathfrak{1} + \mathfrak{\mu})^{\frac{\text{det}\mathfrak{t}}{\mathfrak{\mu}}} \mathfrak{e}^{a^2 b^2 \mathfrak{\mu}^2} . \tag{24}$$

Then the expression for Gð Þ *t*, *μ* is symmetric in *a* and *b*

$$\mathcal{G}(t,\mu) = \sum\_{m=0}^{\infty} \mathbf{H}\_m(a\mathbf{x}, a^2\mathbf{y}|\mu) \frac{(bt)^m}{m!} = \sum\_{m=0}^{\infty} b^m \mathbf{H}\_m(a\mathbf{x}, a^2\mathbf{y}|\mu) \frac{t^m}{m!}.\tag{25}$$

By the similar way, we get that

$$\mathcal{G}(t,\mu) = \sum\_{m=0}^{\infty} \mathbf{H}\_m(b\mathbf{x}, b^2\mathbf{y}|\mu) \frac{(at)^m}{m!} = \sum\_{m=0}^{\infty} a^m \mathbf{H}\_m(b\mathbf{x}, b^2\mathbf{y}|\mu) \frac{t^m}{m!}.\tag{26}$$

By comparing the coefficients of *<sup>t</sup> m <sup>m</sup>*! in last two equations, the expected result of Theorem 3 is achieved. □

Again, we now use

$$\mathcal{F}(t,\mu) = \frac{abt(1+\mu)^{\frac{abt}{\mu}}e^{a^2b^2yt^2}\left((1+\mu)^{\frac{abt}{\mu}}-1\right)}{\left((1+\mu)^{\frac{at}{\mu}}-1\right)\left((1+\mu)^{\frac{bt}{\mu}}-1\right)}.\tag{27}$$

For *μ*∈ , we introduce the modified degenerate Bernoulli polynomials given by the generating function

$$\sum\_{n=0}^{\infty} \beta\_n(\varkappa|\mu) \frac{t^n}{n!} = \frac{t}{(1+\mu)^{\frac{t}{\mu}} - 1} (1+\mu)^{\frac{xt}{\mu}}, (\text{see}[6, 7]). \tag{28}$$

When *x* ¼ 0 and *βn*ð Þ¼ *μ βn*ð Þ 0j*μ* are called the modified degenerate Bernoulli numbers. Note that

$$\lim\_{\mu \to 0} \beta\_n(\mu) = B\_n,\tag{29}$$

In a similar fashion we have

*DOI: http://dx.doi.org/10.5772/intechopen.92687*

*abt* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> bt*

*βn*ð Þ *μ*

X*n i*¼0

*<sup>μ</sup>* � 1 � � ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abxt*

> X *i*

*m*¼0

*m*

*ai b<sup>n</sup>*þ1�*<sup>i</sup>*

*m* !*bi*

> *m* !*ai*

> > *m*

 !*bi a<sup>n</sup>*þ1�*<sup>i</sup>*

!

By comparing the coefficients of *<sup>t</sup>*

equations, we have the below theorem.

*n i*

! *i*

*n i*

! *i*

*n i*

! *i*

*n i*

**degenerate Hermite polynomials**

! *i*

ð Þ *bt <sup>n</sup> n*!

> *n i*

! *i*

*m* !*bi*

*an*þ1�*<sup>i</sup>*

By taking the limit as *μ* ! 0, we have the following corollary.

*bn*þ1�*<sup>i</sup>*

**4. Differential equations associated with 2-variable modified**

In this section, we construct the differential equations with coefficients *ai*ð Þ *N*, *x*, *y*, *μ* arising from the generating functions of the 2-variable modified

*G t*ð Þ� , *x*, *y*, *μ a*0ð Þ *N*, *x*, *y*, *μ G t*ð Þ� , *x*, *y*, *μ* ⋯

*NG t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>0</sup>*:*

*G* ¼ *G t*ð Þ , *x*, *y*, *μ* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> xt*

> *t n n*!

H*n*ð Þ *x*, *y*j*μ*

*μ e yt*2

, *μ*, *x*, *t*∈ *:*

By using the coefficients of this differential equation, we can get explicit identities for the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*, *μ* .

**Corollary 5.** Let *a*, *b*> 0 (*a* 6¼ *b*). The the following identity holds true:

X∞ *n*¼0

*<sup>μ</sup> e a*2*b*<sup>2</sup>

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising…*

H*<sup>n</sup> ax*, *a*<sup>2</sup>

*m*

**Theorem 4.** Let *a*, *b*>0(*a* 6¼ *b*). The the following identity holds true:

*<sup>β</sup>m*ð Þ *<sup>μ</sup>* <sup>H</sup>*<sup>i</sup>*�*<sup>m</sup> bx*, *<sup>b</sup>*<sup>2</sup>

*<sup>β</sup>m*ð Þ *<sup>μ</sup>* <sup>H</sup>*<sup>i</sup>*�*<sup>m</sup> ax*, *<sup>a</sup>*<sup>2</sup>

<sup>S</sup>*<sup>n</sup>*�*<sup>i</sup>*ð Þ *<sup>a</sup>* � <sup>1</sup> *BmHi*�*<sup>m</sup> bx*, *<sup>b</sup>*<sup>2</sup>

<sup>S</sup>*<sup>n</sup>*�*<sup>i</sup>*ð Þ *<sup>b</sup>* � <sup>1</sup> *BmHi*�*<sup>m</sup> ax*, *<sup>a</sup>*<sup>2</sup>

*yt*<sup>2</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abt*

*<sup>y</sup>*j*<sup>μ</sup>* � � ð Þ *bt <sup>n</sup>*

*an*þ1�*<sup>i</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> at*

*n*!

! *<sup>t</sup>*

*<sup>μ</sup>* � 1 � �

*<sup>μ</sup>* � 1 � �

> X∞ *n*¼0

*<sup>β</sup>m*ð Þ *<sup>μ</sup>* <sup>H</sup>*<sup>i</sup>*�*<sup>m</sup> ax*, *<sup>a</sup>*<sup>2</sup>

*<sup>m</sup>*! on the right hand sides of the last two

*<sup>y</sup>*j*<sup>μ</sup>* � �*σ<sup>n</sup>*�*<sup>i</sup>*ð Þ *<sup>a</sup>* � <sup>1</sup>j*<sup>μ</sup>*

*<sup>y</sup>*j*<sup>μ</sup>* � �*σ<sup>n</sup>*�*<sup>i</sup>*ð Þ *<sup>b</sup>* � <sup>1</sup>j*<sup>μ</sup> :*

*y* � �

*y* � �*:*

*σk*ð Þ *b* � 1j*μ*

ð Þ *at <sup>n</sup> n*!

> *n n*! *:*

(33)

(34)

(35)

(36)

(37)

*<sup>y</sup>*j*<sup>μ</sup>* � �*σ<sup>n</sup>*�*<sup>i</sup>*ð Þ *<sup>b</sup>* � <sup>1</sup>j*<sup>μ</sup>*

Fð Þ¼ *t*, *μ*

¼ *a* X∞ *n*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

X*n i*¼0

<sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼0

X *i*

*m*¼0

X*n i*¼0

<sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼0

X *i*

*m*¼0

X *i*

*m*¼0

X *i*

*m*¼0

degenerate Hermite polynomials:

Recall that

**171**

*∂ ∂t* � �*<sup>N</sup>*

�*a*2*<sup>N</sup>*ð Þ *N*, *x*, *y*, *μ t*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

where *Bn* are called the Bernoulli numbers. The first few of them are

$$\begin{aligned} \beta\_{0}(\mathbf{x}|\mu) &= \frac{\mu}{\log(1+\mu)}, \\ \beta\_{1}(\mathbf{x}|\mu) &= -\frac{1}{2} + \mathbf{x}, \\ \beta\_{2}(\mathbf{x}|\mu) &= \frac{\log\left(1+\mu\right)}{6\mu} - \frac{\mathbf{x}\log\left(1+\mu\right)}{\mu} + \frac{\mathbf{x}^{2}\log\left(1+\mu\right)}{\mu}, \\ \beta\_{3}(\mathbf{x}|\mu) &= \frac{\mathbf{x}\log\left(1+\mu\right)^{2}}{2\mu^{2}} - \frac{3\mathbf{x}^{2}\log\left(1+\mu\right)^{2}}{2\mu^{2}} + \frac{\mathbf{x}^{3}\log\left(1+\mu\right)^{2}}{\mu^{2}}, \\ \beta\_{4}(\mathbf{x}|\mu) &= -\frac{\log\left(1+\mu\right)^{3}}{30\mu^{3}} + \frac{\mathbf{x}^{2}\log\left(1+\mu\right)^{3}}{\mu^{3}} - \frac{2\mathbf{x}^{3}\log\left(1+\mu\right)^{3}}{\mu^{3}} + \frac{\mathbf{x}^{4}\log\left(1+\mu\right)^{3}}{\mu^{3}}. \end{aligned} \tag{30}$$

For each integer *<sup>k</sup>*<sup>≥</sup> 0, S*k*ð Þ¼ *<sup>n</sup>* <sup>0</sup>*<sup>k</sup>* <sup>þ</sup> <sup>1</sup>*<sup>k</sup>* <sup>þ</sup> <sup>2</sup>*<sup>k</sup>* <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>k</sup>* is called sum of integers. A modified generalized falling factorial sum *σk*ð Þ *n*, *μ* can be defined by the generating function

$$\sum\_{k=0}^{\infty} \sigma\_k(n|\mu) \frac{t^k}{k!} = \frac{(\mathbf{1} + \mu)^{\frac{(n+1)t}{\mu}} - \mathbf{1}}{(\mathbf{1} + \mu)^{\frac{t}{\mu}} - \mathbf{1}}.\tag{31}$$

Note that lim *<sup>μ</sup>*!<sup>0</sup>*σk*ð Þ¼ *n*j*μ* S*k*ð Þ *n :* From Fð Þ *t*, *μ* , we get the following result:

$$\begin{split} \mathcal{F}(t,\mu) &= \frac{abt(1+\mu)^{\frac{ab}{\mu}}e^{a^{b}b^{\frac{ab}{\mu}}} \left( (1+\mu)^{\frac{ab}{\mu}} - 1 \right)}{\left( (1+\mu)^{\frac{ab}{\mu}} - 1 \right) \left( (1+\mu)^{\frac{ab}{\mu}} - 1 \right)} \\ &= \frac{abt}{\left( (1+\mu)^{\frac{ab}{\mu}} - 1 \right)} (1+\mu)^{\frac{ab}{\mu}} e^{a^{2}b^{\frac{ab}{\mu}}} \frac{\left( (1+\mu)^{\frac{ab}{\mu}} - 1 \right)}{\left( (1+\mu)^{\frac{ab}{\mu}} - 1 \right)} \\ &= b \sum\_{n=0}^{\infty} \beta\_{n}(\mu) \frac{(at)^{n}}{n!} \sum\_{n=0}^{\infty} \mathrm{H}\_{n}(bx, b^{2}y|\mu) \frac{(at)^{n}}{n!} \sum\_{n=0}^{\infty} \sigma\_{k}(a-1|\mu) \frac{(bt)^{n}}{n!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{i=0}^{n} \sum\_{m=0}^{i} \binom{n}{i} \binom{i}{m} a^{i} b^{n+1-i} \rho\_{m}(\mu) \mathrm{H}\_{i-m}(bx, b^{2}y|\mu) \sigma\_{n-i}(a-1|\mu) \right) \frac{t^{n}}{n!}. \end{split} \tag{32}$$

**170**

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising… DOI: http://dx.doi.org/10.5772/intechopen.92687*

In a similar fashion we have

By comparing the coefficients of *<sup>t</sup>*

X∞ *n*¼0

*μ* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>* ,

*x* log 1ð Þ þ *μ*

1 <sup>2</sup> <sup>þ</sup> *<sup>x</sup>*,

*<sup>β</sup>*2ð Þ¼ *<sup>x</sup>*j*<sup>μ</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*<sup>β</sup>*4ð Þ¼� *<sup>x</sup>*j*<sup>μ</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*abt*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abxt*

<sup>¼</sup> *abt* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> at*

¼ *b* X∞ *n*¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> at*

*βn*ð Þ *μ*

X*n i*¼0

*<sup>μ</sup> e<sup>a</sup>*2*b*<sup>2</sup> *yt*2

*<sup>μ</sup>* � 1 � �

> ð Þ *at <sup>n</sup> n*!

> > *n i* ! *i*

X∞ *n*¼0

*<sup>μ</sup>* � 1 � � ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abxt*

> X*i m*¼0

Fð Þ¼ *t*, *μ*

*βn*ð Þ *x*j*μ*

*t n n*!

<sup>6</sup>*<sup>μ</sup>* � *<sup>x</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

2

3 30*μ*<sup>3</sup> þ

> X∞ *k*¼0

*σk*ð Þ *n*j*μ*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abt*

*<sup>μ</sup> e a*2*b*<sup>2</sup>

H*<sup>n</sup> bx*, *b*<sup>2</sup>

*m* ! *ai b<sup>n</sup>*þ1�*<sup>i</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> bt*

*t k k*!

*<sup>μ</sup>* � 1 � �

*<sup>y</sup>*j*<sup>μ</sup>* � � ð Þ *at <sup>n</sup>*

*<sup>μ</sup>* � 1 � �

*μ*

<sup>2</sup>*μ*<sup>2</sup> � <sup>3</sup>*x*<sup>2</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

Again, we now use

*Number Theory and Its Applications*

the generating function

numbers. Note that

*β*0ð Þ¼ *x*j*μ*

*β*1ð Þ¼� *x*j*μ*

*β*3ð Þ¼ *x*j*μ*

generating function

Fð Þ¼ *t*, *μ*

**170**

*m*

*abt*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abxt*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> at*

<sup>¼</sup> *<sup>t</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> <sup>t</sup>*

lim*μ*!0

where *Bn* are called the Bernoulli numbers. The first few of them are

þ

2*μ*<sup>2</sup> þ

For each integer *<sup>k</sup>*<sup>≥</sup> 0, S*k*ð Þ¼ *<sup>n</sup>* <sup>0</sup>*<sup>k</sup>* <sup>þ</sup> <sup>1</sup>*<sup>k</sup>* <sup>þ</sup> <sup>2</sup>*<sup>k</sup>* <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>k</sup>* is called sum of integers. A modified generalized falling factorial sum *σk*ð Þ *n*, *μ* can be defined by the

Note that lim *<sup>μ</sup>*!<sup>0</sup>*σk*ð Þ¼ *n*j*μ* S*k*ð Þ *n :* From Fð Þ *t*, *μ* , we get the following result:

*yt*<sup>2</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abt*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> bt*

*n*!

!

*<sup>x</sup>*<sup>2</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

2

3

<sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> <sup>t</sup>*

*<sup>μ</sup>* � 1 � �

*<sup>μ</sup>* � 1 � �

*σk*ð Þ *a* � 1j*μ*

ð Þ *bt <sup>n</sup> n*!

> *t n n*! *:*

(32)

*<sup>y</sup>*j*<sup>μ</sup>* � �*σ<sup>n</sup>*�*<sup>i</sup>*ð Þ *<sup>a</sup>* � <sup>1</sup>j*<sup>μ</sup>*

X∞ *n*¼0

*<sup>β</sup>m*ð Þ *<sup>μ</sup>* <sup>H</sup>*<sup>i</sup>*�*<sup>m</sup> bx*, *<sup>b</sup>*<sup>2</sup>

*<sup>x</sup>*<sup>2</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup> <sup>μ</sup>* ,

*<sup>μ</sup>*<sup>3</sup> � <sup>2</sup>*x*<sup>3</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

ð Þ *n*þ1 *t <sup>μ</sup>* � 1

*<sup>x</sup>*<sup>3</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

Theorem 3 is achieved. □

*<sup>μ</sup> e<sup>a</sup>*2*b*<sup>2</sup> *yt*2

*<sup>μ</sup>* � 1 � �

For *μ*∈ , we introduce the modified degenerate Bernoulli polynomials given by

*<sup>μ</sup>* � 1

When *x* ¼ 0 and *βn*ð Þ¼ *μ βn*ð Þ 0j*μ* are called the modified degenerate Bernoulli

*<sup>m</sup>*! in last two equations, the expected result of

*<sup>μ</sup>* � 1 � �

*βn*ð Þ¼ *μ Bn*, (29)

2 *<sup>μ</sup>*<sup>2</sup> ,

3 *μ*<sup>3</sup> þ

*<sup>μ</sup>* � <sup>1</sup> *:* (31)

*<sup>x</sup>*<sup>4</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

3 *<sup>μ</sup>*<sup>3</sup> *:*

(30)

� � *:* (27)

*<sup>μ</sup>* , see 6, 7 ð Þ ½ � *:* (28)

*<sup>μ</sup>* � 1

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> abt*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> bt*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> xt*

$$\begin{split} \mathcal{F}(t,\mu) &= \frac{abt}{\left((1+\mu)^{\frac{bt}{r}}-1\right)} \left(1+\mu\right)^{\frac{abt}{r}} e^{a^2b^2\mu^2} \frac{\left((1+\mu)^{\frac{bt}{r}}-1\right)}{\left((1+\mu)^{\frac{at}{r}}-1\right)} \\ &= a \sum\_{n=0}^{\infty} \beta\_n(\mu) \frac{(bt)^n}{n!} \sum\_{n=0}^{\infty} \mathcal{H}\_n(ax, a^2y|\mu) \frac{\left(bt\right)^n}{n!} \sum\_{n=0}^{\infty} \sigma\_k(b-1|\mu) \frac{\left(at\right)^n}{n!} \\ &= \sum\_{n=0}^{\infty} \left(\sum\_{i=0}^n \sum\_{m=0}^i \binom{n}{i} \binom{i}{m} b^i a^{n+1-i} \beta\_m(\mu) \mathcal{H}\_{i-m}(ax, a^2y|\mu) \sigma\_{n-i}(b-1|\mu) \right) \frac{t^n}{n!}, \end{split} \tag{33}$$

By comparing the coefficients of *<sup>t</sup> m <sup>m</sup>*! on the right hand sides of the last two equations, we have the below theorem.

**Theorem 4.** Let *a*, *b*>0(*a* 6¼ *b*). The the following identity holds true:

$$\begin{split} &\sum\_{i=0}^{n} \sum\_{m=0}^{i} \binom{n}{i} \binom{i}{m} a^i b^{n+1-i} \beta\_m(\mu) \mathcal{H}\_{i-m} \left( b \boldsymbol{\omega}, b^2 \boldsymbol{\uprho} \middle| \boldsymbol{\uprho} \right) \sigma\_{n-i}(\boldsymbol{a} - \boldsymbol{1} | \boldsymbol{\mu}) \\ &= \sum\_{i=0}^{n} \sum\_{m=0}^{i} \binom{n}{i} \binom{i}{m} b^i a^{n+1-i} \beta\_m(\mu) \mathcal{H}\_{i-m} \left( a \boldsymbol{\upomega}, a^2 \boldsymbol{\uprho} \middle| \boldsymbol{\uprho} \right) \sigma\_{n-i}(\boldsymbol{b} - \boldsymbol{1} | \boldsymbol{\uprho}). \end{split} \tag{34}$$

By taking the limit as *μ* ! 0, we have the following corollary. **Corollary 5.** Let *a*, *b*> 0 (*a* 6¼ *b*). The the following identity holds true:

$$\begin{split} &\sum\_{i=0}^{n} \sum\_{m=0}^{i} \binom{n}{i} \binom{i}{m} a^i b^{n+1-i} \mathbf{S}\_{n-i}(a-1) B\_m H\_{i-m} \left( b \mathbf{x}, b^2 \mathbf{y} \right) \\ &= \sum\_{i=0}^{n} \sum\_{m=0}^{i} \binom{n}{i} \binom{i}{m} b^i a^{n+1-i} \mathbf{S}\_{n-i}(b-1) B\_m H\_{i-m} \left( a \mathbf{x}, a^2 \mathbf{y} \right). \end{split} \tag{35}$$

#### **4. Differential equations associated with 2-variable modified degenerate Hermite polynomials**

In this section, we construct the differential equations with coefficients *ai*ð Þ *N*, *x*, *y*, *μ* arising from the generating functions of the 2-variable modified degenerate Hermite polynomials:

$$\begin{split} & \left( \frac{\partial}{\partial t} \right)^{N} G(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) - a\_{0}(N, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) G(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) - \cdots \\ & - a\_{2N}(N, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) t^{N} G(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) = \mathbf{0}. \end{split} \tag{36}$$

By using the coefficients of this differential equation, we can get explicit identities for the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*, *μ* . Recall that

$$\begin{aligned} \mathcal{G} &= \mathcal{G}(t, \boldsymbol{\varkappa}, \boldsymbol{\jmath}, \mu) \\ &= (\mathbb{1} + \mu)^{\frac{\boldsymbol{\varkappa}}{\boldsymbol{\mu}}} e^{\boldsymbol{\eta}^{2}} \\ &= \sum\_{n=0}^{\infty} \mathcal{H}\_{n}(\boldsymbol{\varkappa}, \boldsymbol{\jmath}|\mu) \frac{t^{n}}{n!}, \quad \mu, \boldsymbol{\varkappa}, t \in \mathbb{C}. \end{aligned} \tag{37}$$

Then, by (37), we have

$$\begin{split} G^{(1)} &= \frac{\partial}{\partial t} G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &= \frac{\partial}{\partial t} \left( (1 + \mu)^{\frac{\mu}{\nu}} e^{\mathbf{y}^{\mu}} \right) \\ &= \left( \frac{\log \left( 1 + \mu \right)}{\mu} \mathbf{x} + 2 \mathbf{y} t \right) \left( (1 + \mu)^{\frac{\mu}{\nu}} e^{\mathbf{y}^{\mu}} \right) \\ &= \left( \frac{\log \left( 1 + \mu \right)}{\mu} \mathbf{x} + 2 \mathbf{y} t \right) \mathbf{G}(t, \mathbf{x}, \mathbf{y}, \mu), \end{split} \tag{38}$$

Now we replace *N* by *N* þ 1 in (40). We find

*DOI: http://dx.doi.org/10.5772/intechopen.92687*

*N* X þ1

*i*¼0

*μ*

*<sup>a</sup>*0ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ *<sup>a</sup>*1ð Þþ *<sup>N</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*ai*ð Þ *N* þ 1, *x*, *y*, *μ t*

By comparing the coefficients on both sides of (41) and (42), we get

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising…*

*x* � �*aN*ð Þ *<sup>N</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*

<sup>þ</sup>2*yaN*�<sup>1</sup>ð Þ *<sup>N</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* ,

*ai*ð*N* þ 1, *x*, *y*, *μ*Þ ¼ ð Þ *i* þ 1 *ai*þ<sup>1</sup>ð Þ *N*, *x*, *y*, *μ*

*x* � �*ai*ð Þ *<sup>N</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*

*G t*ð Þþ , *x*, *y*, *μ* 2*ytG t*ð Þ , *x*, *y*, *μ*

*G t*ð Þ , *x*, *y*, *μ*

*x* log 1ð Þ þ *μ*

*x* log 1ð Þ þ *μ μ* � �*<sup>i</sup>*

¼ *a*0ð Þ 1, *x*, *y*, *μ G t*ð Þþ , *x*, *y*, *μ a*1ð Þ 1, *x*, *y*, *μ tG t*ð Þ , *x*, *y*, *μ :*

*μ*

*μ*

*xN*þ<sup>1</sup> ,

*a*1ð Þ *N* � *i*, *x*, *y*, *μ*

þð Þ 2*y ai*�<sup>1</sup>ð Þ *N*, *x*, *y*, *μ* , 1ð Þ ≤*i*≤ *N* � 1 *:*

*G t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup> <sup>G</sup>*ð Þ <sup>0</sup> ð Þ¼ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup> <sup>a</sup>*0ð Þ 0, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup> G t*ð Þ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup> :* (45)

*a*0ð Þ¼ 0, *x*, *y*, *μ* 1*:* (46)

*<sup>μ</sup>* , *<sup>a</sup>*1ð Þ¼ 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>2</sup>*y:* (48)

*a*0ð Þ *N*, *x*, *y*, *μ* ,

*a*0ð Þ *N* � 1, *x*, *y*, *μ* ,

<sup>þ</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ*

*i*

*x* � �*a*0ð Þ *<sup>N</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* ,

*μ*

*G t*ð Þ , *x*, *y*, *μ :* (42)

(43)

(44)

(47)

(49)

*<sup>G</sup>*ð Þ *<sup>N</sup>*þ<sup>1</sup> <sup>¼</sup>

*aN*ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

In addition, by (37), we have

It is not difficult to show that

<sup>¼</sup> <sup>X</sup> 1

*i*¼0

Thus, by (38) and (47), we also get

From (43) and (44), we note that

*<sup>a</sup>*0ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ <sup>X</sup>

…

**173**

*a*0ð Þ¼ 1, *x*, *y*, *μ*

*<sup>a</sup>*0ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ *<sup>a</sup>*1ð Þþ *<sup>N</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup> <sup>x</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*<sup>a</sup>*0ð Þ¼ *<sup>N</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup> <sup>a</sup>*1ð*<sup>N</sup>* � 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ þ *<sup>x</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*N*

*i*¼0

<sup>þ</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ* � �*N*þ<sup>1</sup>

*x* log 1ð Þ þ *μ μ*

<sup>¼</sup> *<sup>G</sup>*ð Þ<sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*

*ai*ð Þ 1, *x*, *y*, *μ t*

*i*

By (45), we get

and

*aN*þ<sup>1</sup>ð*N* þ 1, *x*, *y*, *μ*Þ ¼ ð Þ 2*y aN*ð Þ *N*, *x*, *y*, *μ* ,

$$\begin{split} G^{(2)} &= \frac{\partial}{\partial t} G^{(1)}(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) \\ &= 2\eta G(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) + \left( \frac{\log\left(\mathbf{1} + \boldsymbol{\mu}\right)}{\mu} \mathbf{x} + 2\boldsymbol{\mathfrak{y}}t \right) G^{(1)}(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) \\ &= \left( 2\mathbf{y} + \left( \frac{\log\left(\mathbf{1} + \boldsymbol{\mu}\right)}{\mu} \right)^2 \mathbf{x}^2 \right) G(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) \\ &+ \left( \frac{\log\left(\mathbf{1} + \boldsymbol{\mu}\right)}{\mu} 4\mathbf{x}\mathbf{y} \right) t G(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) \\ &+ (2\mathbf{y})^2 t^2 G(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}). \end{split} \tag{39}$$

By continuing this process as shown in (39), we can get easily that

$$\begin{split} \mathbf{G}^{(N)} &= \left( \frac{\partial}{\partial t} \right)^{N} \mathbf{G}(t, \mathbf{x}, \boldsymbol{\mathcal{y}}, \boldsymbol{\mu}) \\ &= \sum\_{i=0}^{N} a\_{i}(N, \mathbf{x}, \boldsymbol{\mathcal{y}}, \boldsymbol{\mu}) t^{i} \mathbf{G}(t, \mathbf{x}, \boldsymbol{\mathcal{y}}, \boldsymbol{\mu}), (N = \mathbf{0}, \mathbf{1}, \mathbf{2}, \dots). \end{split} \tag{40}$$

By differentiating (40) with respect to *t*, we have

$$\begin{split} G^{(N+1)} &= \frac{\partial G^{(N)}}{\partial t} = \sum\_{i=0}^{N} a\_i(N, \mathbf{x}, \mathbf{y}, \mu)(i) t^{i-1} G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &+ \sum\_{i=0}^{N} a\_i(N, \mathbf{x}, \mathbf{y}, \mu) t^i G^{(1)}(t, \mathbf{x}, \mathbf{y}, \mu) \\ &= \sum\_{i=0}^{N} (i) a\_i(N, \mathbf{x}, \mathbf{y}, \mu) t^{i-1} G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &+ \sum\_{i=0}^{N} \left( \frac{\log(1+\mu)}{\mu} \mathbf{x} \right) a\_i(N, \mathbf{x}, \mathbf{y}, \mu) t^i G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &+ \sum\_{i=0}^{N} (2y) a\_i(N, \mathbf{x}, \mathbf{y}, \mu) t^{i+1} G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &= \sum\_{i=0}^{N-1} (i+1) a\_{i+1}(N, \mathbf{x}, \mathbf{y}, \mu) t^i G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &+ \sum\_{i=0}^{N} \left( \frac{\log(1+\mu)}{\mu} \mathbf{x} \right) a\_i(N, \mathbf{x}, \mathbf{y}, \mu) t^i G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &+ \sum\_{i=1}^{N+1} (2y) a\_{i-1}(N, \mathbf{x}, \mathbf{y}, \mu) t^i G(t, \mathbf{x}, \mathbf{y}, \mu). \end{split} \tag{41.3.1}$$

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising… DOI: http://dx.doi.org/10.5772/intechopen.92687*

Now we replace *N* by *N* þ 1 in (40). We find

$$G^{(N+1)} = \sum\_{i=0}^{N+1} a\_i (N+1, \ge, \underline{\nu}, \mu) t^i G(t, \ge, \underline{\nu}, \mu). \tag{42}$$

By comparing the coefficients on both sides of (41) and (42), we get

$$\begin{aligned} a\_0(N+1, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) &= a\_1(N, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) + \left(\frac{\log\left(1+\boldsymbol{\mu}\right)}{\mu}\mathbf{x}\right) a\_0(N, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}), \\ a\_N(N+1, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) &= \left(\frac{\log\left(1+\boldsymbol{\mu}\right)}{\mu}\mathbf{x}\right) a\_N(N, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) \\ &+ 2ya\_{N-1}(N, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}), \\ a\_{N+1}(N+1, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) &= (2\mathbf{y})a\_N(N, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}), \end{aligned} \tag{43}$$

and

Then, by (37), we have

*Number Theory and Its Applications*

*<sup>G</sup>*ð Þ<sup>2</sup> <sup>¼</sup> *<sup>∂</sup> ∂t* *<sup>G</sup>*ð Þ<sup>1</sup> <sup>¼</sup> *<sup>∂</sup> ∂t*

¼ *∂*

*<sup>G</sup>*ð Þ<sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*

<sup>¼</sup> <sup>2</sup>*<sup>y</sup>* <sup>þ</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

� �

<sup>þ</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ*

þð Þ 2*y* 2 *t* 2

*<sup>G</sup>*ð Þ *<sup>N</sup>* <sup>¼</sup> *<sup>∂</sup>*

<sup>¼</sup> <sup>X</sup> *N*

*i*¼0

<sup>¼</sup> <sup>X</sup> *N*

*i*¼0

þ X *N*

þ X *N*

þ X *N*

þ X *N*þ1

¼ *N* X�1 *i*¼0

**172**

*i*¼0

*i*¼0

*i*¼0

*i*¼1

*∂t* � �*<sup>N</sup>*

<sup>¼</sup> <sup>2</sup>*yG t*ð Þþ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

!

*μ* � �<sup>2</sup>

4*xy*

*G t*ð Þ , *x*, *y*, *μ*

*i*

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>X</sup> *N*

*i*¼0

*ai*ð Þ *N*, *x*, *y*, *μ t*

*i*�1

*x*

*x*

*ai*ð Þ *N*, *x*, *y*, *μ t*

By differentiating (40) with respect to *t*, we have

*<sup>G</sup>*ð Þ *<sup>N</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>∂</sup>G*ð Þ *<sup>N</sup>*

þ X *N*

*i*¼0

ð Þ*i ai*ð Þ *N*, *x*, *y*, *μ t*

log 1ð Þ þ *μ μ*

� �

ð Þ 2*y ai*ð Þ *N*, *x*, *y*, *μ t*

ð Þ *i* þ 1 *ai*þ<sup>1</sup>ð Þ *N*, *x*, *y*, *μ t*

log 1ð Þ þ *μ μ*

� �

ð Þ 2*y ai*�<sup>1</sup>ð Þ *N*, *x*, *y*, *μ t*

*G t*ð Þ , *x*, *y*, *μ :*

*G t*ð Þ , *x*, *y*, *μ*

*μ e yt*<sup>2</sup> � �

� �

� �

*μ*

*x*2

*tG t*ð Þ , *x*, *y*, *μ*

By continuing this process as shown in (39), we can get easily that

*x* þ 2*yt*

*x* þ 2*yt*

� �

*x* þ 2*yt*

*G t*ð Þ , *x*, *y*, *μ*

*G t*ð Þ , *x*, *y*, *μ* ,ð Þ *N* ¼ 0, 1, 2, … *:*

*ai*ð Þ *N*, *x*, *y*, *μ* ð Þ*i t*

*i*

*G t*ð Þ , *x*, *y*, *μ*

*i*þ1

*i*

*ai*ð Þ *N*, *x*, *y*, *μ t*

*G t*ð Þ , *x*, *y*, *μ*

*G t*ð Þ , *x*, *y*, *μ*

*ai*ð Þ *N*, *x*, *y*, *μ t*

*G t*ð Þ , *x*, *y*, *μ :*

*i*

*i*�1

*i*

*i*

*<sup>G</sup>*ð Þ<sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*

*G t*ð Þ , *x*, *y*, *μ*

*G t*ð Þ , *x*, *y*, *μ*

*G t*ð Þ , *x*, *y*, *μ*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> xt μ e yt*<sup>2</sup> � � (38)

(39)

(40)

(41)

*G t*ð Þ , *x*, *y*, *μ* ,

*<sup>G</sup>*ð Þ<sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*

*<sup>∂</sup><sup>t</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> xt*

<sup>¼</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ*

<sup>¼</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ*

$$\begin{split} a\_i(N+1, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{\mu}) &= (i+1)a\_{i+1}(N, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{\mu}) \\ &+ \left(\frac{\log\left(1+\mu\right)}{\mu}\boldsymbol{x}\right)a\_i(N, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{\mu}) \\ &+ (2\boldsymbol{y})a\_{i-1}(N, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{\mu}), (1 \le i \le N-1). \end{split} \tag{44}$$

In addition, by (37), we have

$$G(t, \mathfrak{x}, \mathfrak{y}, \mu) = G^{(0)}(t, \mathfrak{x}, \mathfrak{y}, \mu) = a\_0(0, \mathfrak{x}, \mathfrak{y}, \mu) G(t, \mathfrak{x}, \mathfrak{y}, \mu). \tag{45}$$

By (45), we get

$$
\mathfrak{a}\_0(\mathbf{0}, \mathfrak{x}, \mathfrak{y}, \mathfrak{\mu}) = \mathbf{1}.\tag{46}
$$

It is not difficult to show that

$$\begin{aligned} &\frac{\mu \log(1+\mu)}{\mu} G(t, \mathbf{x}, \mathbf{y}, \mu) + 2ytG(t, \mathbf{x}, \mathbf{y}, \mu) \\ &= G^{(1)}(t, \mathbf{x}, \mathbf{y}, \mu) \\ &= \sum\_{i=0}^{1} a\_i(\mathbf{1}, \mathbf{x}, \mathbf{y}, \mu) t^i G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &= a\_0(\mathbf{1}, \mathbf{x}, \mathbf{y}, \mu) G(t, \mathbf{x}, \mathbf{y}, \mu) + a\_1(\mathbf{1}, \mathbf{x}, \mathbf{y}, \mu) t G(t, \mathbf{x}, \mathbf{y}, \mu). \end{aligned} \tag{47}$$

Thus, by (38) and (47), we also get

$$a\_0(\mathbf{1}, \mathbf{x}, \boldsymbol{\jmath}, \boldsymbol{\mu}) = \frac{\boldsymbol{\varkappa} \log \left( \mathbf{1} + \boldsymbol{\mu} \right)}{\boldsymbol{\mu}}, \quad a\_1(\mathbf{1}, \boldsymbol{\varkappa}, \boldsymbol{\jmath}, \boldsymbol{\mu}) = \mathbf{2} \boldsymbol{\jmath}. \tag{48}$$

From (43) and (44), we note that

$$\begin{split} a\_0(N+1, \mathbf{x}, \boldsymbol{y}, \boldsymbol{\mu}) &= a\_1(N, \mathbf{x}, \boldsymbol{y}, \boldsymbol{\mu}) + \frac{\mathbf{x} \log(1+\boldsymbol{\mu})}{\mu} a\_0(N, \mathbf{x}, \boldsymbol{y}, \boldsymbol{\mu}), \\ a\_0(N, \mathbf{x}, \boldsymbol{y}, \boldsymbol{\mu}) &= a\_1(N-1, \mathbf{x}, \boldsymbol{y}, \boldsymbol{\mu}) + \frac{\mathbf{x} \log(1+\boldsymbol{\mu})}{\mu} a\_0(N-1, \mathbf{x}, \boldsymbol{y}, \boldsymbol{\mu}), \\ \dots \\ a\_0(N+1, \mathbf{x}, \boldsymbol{y}, \boldsymbol{\mu}) &= \sum\_{i=0}^{N} \left( \frac{\mathbf{x} \log(1+\boldsymbol{\mu})}{\mu} \right)^i a\_1(N-i, \mathbf{x}, \boldsymbol{y}, \boldsymbol{\mu}) \\ &\quad + \left( \frac{\log(1+\boldsymbol{\mu})}{\mu} \right)^{N+1} \mathbf{x}^{N+1}, \end{split} \tag{49}$$

$$\begin{split} a\_{N}(N+1,\mathbf{x},\mathbf{y},\mu) &= \frac{\mathbf{x}\log\left(1+\mu\right)}{\mu} a\_{N}(N,\mathbf{x},\mathbf{y},\mu) \\ &+ (2\mathfrak{y}) a\_{N-1}(N,\mathbf{x},\mathbf{y},\mu), \\ a\_{N-1}(N,\mathbf{x},\mathbf{y},\mu) &= \frac{\mathbf{x}\log\left(1+\mu\right)}{\mu} a\_{N-1}(N-1,\mathbf{x},\mathbf{y},\mu) \\ &+ (2\mathfrak{y}) a\_{N-2}(N-1,\mathbf{x},\mathbf{y},\mu), \dots \\ a\_{N}(N+1,\mathbf{x},\mathbf{y},\mu) &= (N+1)\mathfrak{x}(2\mathfrak{y})^{N} \left(\frac{\log\left(1+\mu\right)}{\mu}\right), \end{split} \tag{50}$$

has a solution

*<sup>a</sup>*0ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ <sup>X</sup>

*aN*ð*N* þ 1, *x*, *y*, *μ*Þ ¼ ð Þ *N* þ 1 ð Þ 2*y*

*aN*þ1ð*N* þ 1, *x*, *y*, *μ*Þ ¼ ð Þ 2*y*

*ai*ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ ð Þ *<sup>i</sup>* <sup>þ</sup> <sup>1</sup> <sup>X</sup>

*N*

*DOI: http://dx.doi.org/10.5772/intechopen.92687*

*x* log 1ð Þ þ *μ μ* � �*<sup>i</sup>*

*i*¼0

þ2*y* X *N*

<sup>þ</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ* � �*N*þ<sup>1</sup>

> *N*þ1 ,

> > *N*

*k*¼0

*k*¼0

*∂ ∂t* � �*<sup>N</sup>*

Hence we have the following theorem. **Theorem 6.** For *N* ¼ 0, 1, 2, … , we get

**Corollary 7.** For *N* ¼ 0, 1, 2, … , we have

**175**

By (58) and Theorem 5, we have

where

*<sup>G</sup>* <sup>¼</sup> *G t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> xt*

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising…*

*xN*þ<sup>1</sup> ,

*<sup>N</sup> x* log 1ð Þ þ *μ μ* � �,

*x* log 1ð Þ þ *μ μ* � �*<sup>k</sup>*

choose �2≤*x*≤2, � 1≤ *t*≤1, *μ* ¼ 1*=*10, and *y* ¼ 0*:*1. In the right picture of

*G t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>X</sup><sup>∞</sup>

þ*aN*ð Þ *N*, *x*, *y*, *μ t*

Here is a plot of the surface for this solution. In the left picture of **Figure 1**, we

*m*¼0

*a*0ð Þ *N*, *x*, *y*, *μ G t*ð Þ , *x*, *y*, *μ* þ*a*1ð Þ *N*, *x*, *y*, *μ tG t*ð Þ , *x*, *y*, *μ*

H*<sup>m</sup>*þ*<sup>N</sup>*ð Þ *x*, *y*, *μ*

*i*¼0

*x* log 1ð Þ þ *μ μ* � �*<sup>k</sup>*

**Figure 1**, we choose �2 ≤*y*≤2, � 1≤ *t*≤1, *μ* ¼ 1*=*10, and *x* ¼ 0*:*1. Making *N*-times derivative for (10) with respect to *t*, we have

þ⋯

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *m*¼0

<sup>H</sup>*<sup>m</sup>*þ*<sup>N</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>X</sup>*<sup>m</sup>*

If we take *m* ¼ 0 in (60), then we have the below corollary.

*μ e yt*2

*ai*þ1ð Þ *N* � *k*, *x*, *y*, *μ*

H*<sup>m</sup>*þ*<sup>N</sup>*ð Þ *x*, *y*, *μ*

*NG t*ð Þ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*

H*<sup>m</sup>*�*<sup>i</sup>*ð Þ *x ai*ð Þ *N*, *x*, *y*, *μ m*!

H*N*ð Þ¼ *x*, *y*, *μ a*0ð Þ *N*, *x*, *y*, *μ* , (61)

ð Þ *<sup>m</sup>* � *<sup>i</sup>* ! *:* (60)

*t m m*! *:*

*ai*�1ð Þ *N* � *k*, *x*, *y*, *μ* , 1ð Þ ≤*i* ≤ *N* � 1 *:*

*t m m*!

*:* (58)

(59)

*a*1ð Þ *N* � *i*, *x*, *y*, *μ*

, (56)

(57)

and

$$\begin{aligned} a\_{N+1}(N+1, \boldsymbol{\omega}, \boldsymbol{y}, \boldsymbol{\mu}) &= (2\boldsymbol{\eta}) a\_N(N, \boldsymbol{\omega}, \boldsymbol{y}, \boldsymbol{\mu}), \\ a\_N(N, \boldsymbol{\omega}, \boldsymbol{y}, \boldsymbol{\mu}) &= (2\boldsymbol{\eta}) a\_{N-1}(N-1, \boldsymbol{\omega}, \boldsymbol{y}, \boldsymbol{\mu}), \dots \\ a\_{N+1}(N+1, \boldsymbol{\omega}, \boldsymbol{y}, \boldsymbol{\mu}) &= \left(2\boldsymbol{\eta}\right)^{N+1}. \end{aligned} \tag{51}$$

For *i* ¼ 1 in (44), we have

$$\begin{split} a\_1(N+1, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{\mu}) &= 2 \sum\_{k=0}^{N} \left( \frac{\boldsymbol{\varkappa} \log(1+\mu)}{\mu} \right)^k a\_2(N-k, \boldsymbol{\varkappa}, \boldsymbol{\jmath}, \boldsymbol{\mu}) \\ &+ 2 \boldsymbol{\jmath} \sum\_{k=0}^{N} \left( \frac{\boldsymbol{\varkappa} \log(1+\mu)}{\mu} \right)^k a\_0(N-k, \boldsymbol{\varkappa}, \boldsymbol{\jmath}, \boldsymbol{\mu}), \end{split} \tag{52}$$

Continuing this process, we can deduce that, for 1≤*i* ≤ *N* � 1,

$$\begin{split} a\_i(N+1, \mathbf{x}, \boldsymbol{\jmath}, \mu) &= (i+1) \sum\_{k=0}^{N} \left( \frac{\mathbf{x} \log(1+\mu)}{\mu} \right)^k a\_{i+1}(N-k, \mathbf{x}, \boldsymbol{\jmath}, \mu) \\ &+ 2 \mathfrak{y} \sum\_{k=0}^{N} \left( \frac{\mathbf{x} \log(1+\mu)}{\mu} \right)^k a\_{i-1}(N-k, \mathbf{x}, \boldsymbol{\jmath}, \mu) .\end{split} \tag{53}$$

Note that, from (37)–(53), here the matrix *ai*ð Þ *<sup>j</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>0</sup> <sup>≤</sup>*i*,*<sup>j</sup>* <sup>≤</sup> *<sup>N</sup>*þ<sup>1</sup> is given by

$$\begin{pmatrix} 1 & \frac{\mathbf{x}\log\left(1+\mu\right)}{\mu} & 2\mathbf{y} + \left(\frac{\log\left(1+\mu\right)}{\mu}\right)^2 \mathbf{x}^2 & \cdots & \cdots \\\\ \mathbf{0} & 2\mathbf{y} & \left(\frac{\log\left(1+\mu\right)}{\mu}\right)\mathbf{4}\mathbf{x}\mathbf{y} & \cdots & \cdots \\\\ \mathbf{0} & \mathbf{0} & \left(2\mathbf{y}\right)^2 & \cdots & \cdots \\\\ \vdots & \vdots & \vdots & \ddots & \cdots \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \cdots & \left(2\mathbf{y}\right)^{N+1} \end{pmatrix} \tag{54}$$

Therefore, from (37)–(53), we obtain the following theorem. **Theorem 5.** For *N* ¼ 0, 1, 2, … , the differential equation

$$\left(\frac{\partial}{\partial t}\right)^{N}\mathbf{G}(t,\mathbf{x},\boldsymbol{\mathcal{y}},\boldsymbol{\mu}) - \left(\sum\_{i=0}^{N} a\_{i}(\mathbf{N},\boldsymbol{\mathcal{x}},\boldsymbol{\mathcal{y}},\boldsymbol{\mu})t^{i}\right)\mathbf{G}(t,\mathbf{x},\boldsymbol{\mathcal{y}},\boldsymbol{\mu}) = \mathbf{0} \tag{55}$$

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising… DOI: http://dx.doi.org/10.5772/intechopen.92687*

has a solution

$$G = G(t, \varkappa, y, \mu) = (1 + \mu)^{\frac{\omega}{\nu}} e^{\imath t^2},\tag{56}$$

where

*aN*ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ *<sup>x</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*aN*ð*N* þ 1, *x*, *y*, *μ*Þ ¼ ð Þ *N* þ 1 *x*ð Þ 2*y*

*aN*þ<sup>1</sup>ð*N* þ 1, *x*, *y*, *μ*Þ ¼ ð Þ 2*y*

X *N*

*k*¼0

Continuing this process, we can deduce that, for 1≤*i* ≤ *N* � 1,

*N*

*k*¼0

*x* log 1ð Þ þ *μ μ* � �*<sup>k</sup>*

0 2*<sup>y</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

00 2ð Þ*y*

Therefore, from (37)–(53), we obtain the following theorem. **Theorem 5.** For *N* ¼ 0, 1, 2, … , the differential equation

*N*

*i*¼0

*G t*ð Þ� , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>X</sup>

*x* log 1ð Þ þ *μ μ* � �*<sup>k</sup>*

*aN*�<sup>1</sup>ð Þ¼ *N*, *x*, *y*, *μ*

and

For *i* ¼ 1 in (44), we have

*Number Theory and Its Applications*

*a*1ð*N* þ 1, *x*, *y*, *μ*Þ ¼ 2

*ai*ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ ð Þ *<sup>i</sup>* <sup>þ</sup> <sup>1</sup> <sup>X</sup>

þ2*y* X *N*

0

BBBBBBBBBBBBBBBBBB@

*∂ ∂t* � �*<sup>N</sup>*

**174**

*k*¼0

<sup>1</sup> *<sup>x</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ*

þ2*y* X *N*

*k*¼0

*μ*

*x* log 1ð Þ þ *μ μ*

*aN*þ<sup>1</sup>ð*N* þ 1, *x*, *y*, *μ*Þ ¼ ð Þ 2*y aN*ð Þ *N*, *x*, *y*, *μ* , *aN*ð Þ¼ *N*, *x*, *y*, *μ* ð Þ 2*y aN*�<sup>1</sup>ð Þ *N* � 1, *x*, *y*, *μ* , …

> *x* log 1ð Þ þ *μ μ* � �*<sup>k</sup>*

*x* log 1ð Þ þ *μ μ* � �*<sup>k</sup>*

Note that, from (37)–(53), here the matrix *ai*ð Þ *<sup>j</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>0</sup> <sup>≤</sup>*i*,*<sup>j</sup>* <sup>≤</sup> *<sup>N</sup>*þ<sup>1</sup> is given by

<sup>2</sup>*<sup>y</sup>* <sup>þ</sup> log 1ð Þ <sup>þ</sup>*<sup>μ</sup> μ* � �<sup>2</sup>

*μ* � �

⋮⋮ ⋮ ⋱ �

00 0 ⋯ ð Þ 2*y*

*ai*ð Þ *N*, *x*, *y*, *μ t*

!

*i*

*aN*ð Þ *N*, *x*, *y*, *μ*

*aN*�<sup>1</sup>ð Þ *N* � 1, *x*, *y*, *μ*

*<sup>N</sup>* log 1ð Þ þ *μ μ* � �

*a*2ð Þ *N* � *k*, *x*, *y*, *μ*

*ai*þ<sup>1</sup>ð Þ *N* � *k*, *x*, *y*, *μ*

*<sup>x</sup>*<sup>2</sup> <sup>⋯</sup> �

4*xy* ⋯ �

*N*þ1

1

CCCCCCCCCCCCCCCCCCA

*G t*ð Þ¼ , *x*, *y*, *μ* 0 (55)

<sup>2</sup> <sup>⋯</sup> �

*a*0ð Þ *N* � *k*, *x*, *y*, *μ* ,

*ai*�<sup>1</sup>ð Þ *N* � *k*, *x*, *y*, *μ :*

,

(50)

(51)

(52)

(53)

(54)

þð Þ 2*y aN*�<sup>2</sup>ð Þ *N* � 1, *x*, *y*, *μ* , …

þð Þ 2*y aN*�<sup>1</sup>ð Þ *N*, *x*, *y*, *μ* ,

*N*þ1 *:*

$$\begin{split} a\_{0}(N+1,\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= \sum\_{i=0}^{N} \left( \frac{\mathbf{x}\log\left(1+\boldsymbol{\mu}\right)}{\boldsymbol{\mu}} \right)^{i} a\_{1}(N-i,\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) \\ &+ \left(\frac{\log\left(1+\boldsymbol{\mu}\right)}{\boldsymbol{\mu}}\right)^{N+1} \boldsymbol{x}^{N+1}, \\ a\_{N}(N+1,\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= \left(N+1\right)\left(2\boldsymbol{\upgamma}\right)^{N} \left(\frac{\mathbf{x}\log\left(1+\boldsymbol{\mu}\right)}{\boldsymbol{\mu}}\right), \\ a\_{N+1}(N+1,\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= \left(2\boldsymbol{\upgamma}\right)^{N+1}, \\ a\_{i}(N+1,\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= \left(i+1\right)\sum\_{k=0}^{N} \left(\frac{\mathbf{x}\log\left(1+\boldsymbol{\mu}\right)}{\boldsymbol{\mu}}\right)^{k} a\_{i+1}(N-k,\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) \\ &+ 2\boldsymbol{y}\sum\_{k=0}^{N} \left(\frac{\mathbf{x}\log\left(1+\boldsymbol{\mu}\right)}{\boldsymbol{\mu}}\right)^{k} a\_{i-1}(N-k,\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}), \left(1\leq i\leq N-1\right). \end{split} \tag{57}$$

Here is a plot of the surface for this solution. In the left picture of **Figure 1**, we choose �2≤*x*≤2, � 1≤ *t*≤1, *μ* ¼ 1*=*10, and *y* ¼ 0*:*1. In the right picture of **Figure 1**, we choose �2 ≤*y*≤2, � 1≤ *t*≤1, *μ* ¼ 1*=*10, and *x* ¼ 0*:*1.

Making *N*-times derivative for (10) with respect to *t*, we have

$$\mathcal{G}\left(\frac{\partial}{\partial t}\right)^{N}G(t,\mathfrak{x},\mathfrak{y},\mathfrak{\mu})=\sum\_{m=0}^{\infty}\mathcal{H}\_{m+N}(\mathfrak{x},\mathfrak{y},\mathfrak{\mu})\frac{t^{m}}{m!}.\tag{58}$$

By (58) and Theorem 5, we have

$$\begin{aligned} &a\_0(N, \mathbf{x}, \mathbf{y}, \mu) G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &+ a\_1(N, \mathbf{x}, \mathbf{y}, \mu) t G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &+ \cdots \\ &+ a\_N(N, \mathbf{x}, \mathbf{y}, \mu) t^N G(t, \mathbf{x}, \mathbf{y}, \mu) \\ &= \sum\_{m=0}^{\infty} \mathcal{H}\_{m+N}(\mathbf{x}, \mathbf{y}, \mu) \frac{t^m}{m!} .\end{aligned} \tag{59}$$

Hence we have the following theorem. **Theorem 6.** For *N* ¼ 0, 1, 2, … , we get

$$\mathbf{H}\_{m+N}(\mathbf{x}, \mathbf{y}, \mu) = \sum\_{i=0}^{m} \frac{\mathbf{H}\_{m-i}(\mathbf{x}) a\_i(N, \mathbf{x}, \mathbf{y}, \mu) m!}{(m-i)!}.\tag{60}$$

If we take *m* ¼ 0 in (60), then we have the below corollary. **Corollary 7.** For *N* ¼ 0, 1, 2, … , we have

$$\mathcal{H}\_N(\mathfrak{x}, \mathfrak{y}, \mu) = a\_0(N, \mathfrak{x}, \mathfrak{y}, \mu), \tag{61}$$

**Figure 1.** *The surface for the solution G t*ð Þ , *x*, *y*, *μ .*

where

$$\begin{split} a\_0(0, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) &= \mathbf{1}, \\ a\_0(N+1, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) &= \sum\_{i=0}^{N} \left( \frac{\mathbf{x} \log(1+\boldsymbol{\mu})}{\boldsymbol{\mu}} \right)^i a\_1(N-i, \mathbf{x}, \mathbf{y}, \boldsymbol{\mu}) \\ &+ \left( \frac{\log(1+\boldsymbol{\mu})}{\boldsymbol{\mu}} \right)^{N+1} \mathbf{x}^{N+1}. \end{split} \tag{62}$$

Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0. The zeros of the H*n*ð Þ¼ *x*, *y*j*μ* 0 for *n* ¼ 30, *y* ¼ 3, � 3, 3 þ *i*, � 3 � *i*, *μ* ¼ 1*=*2, and *x*∈ are displayed in **Figure 2**. In the top-left picture of **Figure 2**, we choose *n* ¼ 30 and *y* ¼ 3. In the top-right picture of **Figure 2**, we choose *n* ¼ 30 and *y* ¼ �3. In the bottom-left picture of **Figure 2**, we choose *n* ¼ 30 and *y* ¼ �3 þ *i* . In the bottom-right picture of **Figure 2**, we choose

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising…*

*DOI: http://dx.doi.org/10.5772/intechopen.92687*

Stacks of zeros of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 for 1≤ *n*≤50, *μ* ¼ 1*=*2 from a 3-D structure are presented in **Figure 3**. In the top-left picture of **Figure 3**, we choose *y* ¼ 3. In the top-right picture of **Figure 3**, we choose *y* ¼ �3. In the bottom-left picture of **Figure 3**, we choose *y* ¼

Our numerical results for approximate solutions of real zeros of the 2-variable

We observed a remarkable regular structure of the complex roots of the 2 variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 and also hope to verify same kind of regular structure of the complex roots of the 2-variable modi-

Plot of real zeros of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 for 1≤*n*≤ 50, *μ* ¼ 1*=*2 structure are presented in **Figure 4**. In the topleft picture of **Figure 4**, we choose *y* ¼ 3. In the top-right picture of **Figure 4**, we

�3 þ *i*. In the bottom-right picture of **Figure 3**, we choose *y* ¼ �3 � *i*.

modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 are displayed

fied degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 (**Table 1**).

*n* ¼ 30 and *y* ¼ �3 � *i*.

**Figure 2.**

*Zeros of* H*n*ð Þ¼ *x*, *y*j*μ* 0*.*

(**Tables 1** and **2**).

**177**

The first few of them are

$$\begin{aligned} \mathcal{H}\_{\mathbf{H}}(\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= \mathbf{1}, \\ \mathcal{H}\_{\mathbf{H}}(\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= \frac{\log\left(1+\boldsymbol{\mu}\right)}{\mu}\mathbf{x}, \\ \mathcal{H}\_{\mathbf{H}}(\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= 2\boldsymbol{y} + \frac{\left(\log\left(1+\boldsymbol{\mu}\right)\right)^{2}}{\mu^{2}}\mathbf{x}^{2}, \\ \mathcal{H}\_{\mathbf{3}}(\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= 6\mathbf{x}\mathbf{y}\frac{\log\left(1+\boldsymbol{\mu}\right)}{\mu} + \frac{\left(\log\left(1+\boldsymbol{\mu}\right)\right)^{3}}{\mu^{3}}\mathbf{x}^{3}, \\ \mathcal{H}\_{\mathbf{4}}(\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= 12\boldsymbol{y}^{2} + 12\mathbf{x}^{2}\boldsymbol{y}\frac{\left(\log\left(1+\boldsymbol{\mu}\right)\right)^{2}}{\mu^{2}} + \frac{\left(\log\left(1+\boldsymbol{\mu}\right)\right)^{4}}{\mu^{4}}\mathbf{x}^{4}, \\ \mathcal{H}\_{\mathbf{5}}(\mathbf{x},\boldsymbol{y},\boldsymbol{\mu}) &= 60\mathbf{x}\mathbf{y}^{2}\frac{\log\left(1+\boldsymbol{\mu}\right)}{\mu} + 20\mathbf{x}^{3}\boldsymbol{y}\frac{\left(\log\left(1+\boldsymbol{\mu}\right)\right)^{3}}{\mu^{3}} + \frac{\left(\log\left(1+\boldsymbol{\mu}\right)\right)^{5}}{\mu^{5}}\mathbf{x}^{5}. \end{aligned} \tag{63}$$

#### **5. Zeros of the 2-variable modified degenerate Hermite polynomials**

This section shows the benefits of supporting theoretical prediction through numerical experiments and finding new interesting pattern of the zeros of the 2 variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0. By using computer, the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* can be determined explicitly. We investigate the zeros of the 2-variable modified degenerate

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising… DOI: http://dx.doi.org/10.5772/intechopen.92687*

**Figure 2.** *Zeros of* H*n*ð Þ¼ *x*, *y*j*μ* 0*.*

where

**Figure 1.**

*a*0ð Þ¼ 0, *x*, *y*, *μ* 1,

The first few of them are

*The surface for the solution G t*ð Þ , *x*, *y*, *μ .*

*Number Theory and Its Applications*

H1ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

*μ*

H2ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>2</sup>*<sup>y</sup>* <sup>þ</sup> ð Þ log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

H3ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>6</sup>*xy* log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

H5ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>60</sup>*xy*<sup>2</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

H4ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>* <sup>12</sup>*y*<sup>2</sup> <sup>þ</sup> <sup>12</sup>*x*<sup>2</sup>

**176**

*x*,

*μ*

*y*

*μ*

2 *<sup>μ</sup>*<sup>2</sup> *<sup>x</sup>*<sup>2</sup>

ð Þ log 1ð Þ þ *μ*

<sup>þ</sup> <sup>20</sup>*x*<sup>3</sup> *y*

**5. Zeros of the 2-variable modified degenerate Hermite polynomials**

This section shows the benefits of supporting theoretical prediction through numerical experiments and finding new interesting pattern of the zeros of the 2 variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0. By using computer, the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* can be determined explicitly. We investigate the zeros of the 2-variable modified degenerate

,

<sup>þ</sup> ð Þ log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

2

3 *<sup>μ</sup>*<sup>3</sup> *<sup>x</sup>*<sup>3</sup>

*<sup>μ</sup>*<sup>2</sup> <sup>þ</sup> ð Þ log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

ð Þ log 1ð Þ þ *μ*

,

4 *<sup>μ</sup>*<sup>4</sup> *<sup>x</sup>*4,

*<sup>μ</sup>*<sup>3</sup> <sup>þ</sup> ð Þ log 1ð Þ <sup>þ</sup> *<sup>μ</sup>*

5 *<sup>μ</sup>*<sup>5</sup> *<sup>x</sup>*<sup>5</sup>

*:*

3

H0ð Þ¼ *x*, *y*, *μ* 1,

*<sup>a</sup>*0ð*<sup>N</sup>* <sup>þ</sup> 1, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>μ</sup>*Þ ¼ <sup>X</sup>

*N*

*x* log 1ð Þ þ *μ μ* � �*<sup>i</sup>*

<sup>þ</sup> log 1ð Þ <sup>þ</sup> *<sup>μ</sup> μ* � �*<sup>N</sup>*þ<sup>1</sup> *a*1ð Þ *N* � *i*, *x*, *y*, *μ*

(62)

(63)

*x<sup>N</sup>*þ<sup>1</sup> *:*

*i*¼0

Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0. The zeros of the H*n*ð Þ¼ *x*, *y*j*μ* 0 for *n* ¼ 30, *y* ¼ 3, � 3, 3 þ *i*, � 3 � *i*, *μ* ¼ 1*=*2, and *x*∈ are displayed in **Figure 2**. In the top-left picture of **Figure 2**, we choose *n* ¼ 30 and *y* ¼ 3. In the top-right picture of **Figure 2**, we choose *n* ¼ 30 and *y* ¼ �3. In the bottom-left picture of **Figure 2**, we choose *n* ¼ 30 and *y* ¼ �3 þ *i* . In the bottom-right picture of **Figure 2**, we choose *n* ¼ 30 and *y* ¼ �3 � *i*.

Stacks of zeros of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 for 1≤ *n*≤50, *μ* ¼ 1*=*2 from a 3-D structure are presented in **Figure 3**. In the top-left picture of **Figure 3**, we choose *y* ¼ 3. In the top-right picture of **Figure 3**, we choose *y* ¼ �3. In the bottom-left picture of **Figure 3**, we choose *y* ¼ �3 þ *i*. In the bottom-right picture of **Figure 3**, we choose *y* ¼ �3 � *i*.

Our numerical results for approximate solutions of real zeros of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 are displayed (**Tables 1** and **2**).

We observed a remarkable regular structure of the complex roots of the 2 variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 and also hope to verify same kind of regular structure of the complex roots of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 (**Table 1**).

Plot of real zeros of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 for 1≤*n*≤ 50, *μ* ¼ 1*=*2 structure are presented in **Figure 4**. In the topleft picture of **Figure 4**, we choose *y* ¼ 3. In the top-right picture of **Figure 4**, we

**Figure 3.** *Stacks of zeros of* H*n*ð Þ¼ *x*, *y*j*μ* 0, 1 ≤*n*≤50*.*


choose *y* ¼ �3. In the bottom-left picture of **Figure 4**, we choose *y* ¼ �3 þ *i*. In the

2*.*

Next, we calculated an approximate solution satisfying H*n*ð Þ¼ *x*, *y*j*μ* 0, *x*∈ . The results are given in **Table 2**. In **Table 2**, we choose *y* ¼ �3 and *μ* ¼ 1*=*2.

bottom-right picture of **Figure 4**, we choose *y* ¼ �3 � *i*.

*Real zeros of* <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*j*<sup>μ</sup>* 0 for 1≤*<sup>n</sup>* <sup>≤</sup>50*, <sup>μ</sup>* <sup>¼</sup> <sup>1</sup>

**Degree** *n x* 1 0

*DOI: http://dx.doi.org/10.5772/intechopen.92687*

*Approximate solutions of* H*n*ð Þ¼ *x*, *y*j*μ* 0, *x* ∈ *.*

**Table 2.**

**Figure 4.**

**179**

2 �3*:*0206, 3*:*0206 3 �5*:*2318, 0, 5*:*2318

4 �7*:*0513, � 2*:*2412, 2*:*2412, 7*:*0513 5 �8*:*6297, � 4*:*0948, 0, 4*:*0948, 8*:*6297

6 �10*:*041, � 5*:*7064, � 1*:*8628, 1*:*8628, 5*:*7064, 10*:*041 7 �11*:*329, � 7*:*1490, � 3*:*4870, 0, 3*:*4870, 7*:*1490, 11*:*329

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising…*

8 �12*:*519, � 8*:*4652, � 4*:*9433, � 1*:*6283, 1*:*6283, 4*:*9433, 8*:*4652, 12*:*519

#### **Table 1.**

*Numbers of real and complex zeros of* H*n*ð Þ¼ *x*, *y*j*μ* 0*.*

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising… DOI: http://dx.doi.org/10.5772/intechopen.92687*


#### **Table 2.**

**Figure 3.**

**Table 1.**

**178**

*Stacks of zeros of* H*n*ð Þ¼ *x*, *y*j*μ* 0, 1 ≤*n*≤50*.*

*Number Theory and Its Applications*

*Numbers of real and complex zeros of* H*n*ð Þ¼ *x*, *y*j*μ* 0*.*

*y* ¼ **3,** *μ* ¼ **1***=***2** *y* ¼ �**3,** *μ* ¼ **1***=***2**

**Degree** *n* **Real zeros Complex zeros Real zeros Complex zeros** 1 1010 2 0220 3 1230 4 0440 5 1450 6 0660 7 1670 8 0880 9 1890 10 0 10 10 0

*Approximate solutions of* H*n*ð Þ¼ *x*, *y*j*μ* 0, *x* ∈ *.*

**Figure 4.** *Real zeros of* <sup>H</sup>*n*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*j*<sup>μ</sup>* 0 for 1≤*<sup>n</sup>* <sup>≤</sup>50*, <sup>μ</sup>* <sup>¼</sup> <sup>1</sup> 2*.*

choose *y* ¼ �3. In the bottom-left picture of **Figure 4**, we choose *y* ¼ �3 þ *i*. In the bottom-right picture of **Figure 4**, we choose *y* ¼ �3 � *i*.

Next, we calculated an approximate solution satisfying H*n*ð Þ¼ *x*, *y*j*μ* 0, *x*∈ . The results are given in **Table 2**. In **Table 2**, we choose *y* ¼ �3 and *μ* ¼ 1*=*2.

#### **6. Conclusions**

In this chapter, we constructed the 2-variable modified degenerate Hermite polynomials and got some new symmetric identities for 2-variable modified degenerate Hermite polynomials. We constructed differential equations arising from the generating function of the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* . We also investigated the symmetry of the zeros of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 for various variables *x* and *y*. As a result, we found that the distribution of the zeros of 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 is very regular pattern. So, we make the following series of conjectures with numerical experiments:

Let us use the following notations. *R*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*j*<sup>μ</sup>* denotes the number of real zeros of H*n*ð Þ¼ *x*, *y*j*μ* 0 lying on the real plane *Im x*ð Þ¼ 0 and *C*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*j*<sup>μ</sup>* denotes the number of complex zeros of H*n*ð Þ¼ *x*, *y*j*μ* 0. Since *n* is the degree of the polynomial H*n*ð Þ *x*, *y*j*μ* , we have *R*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*j*<sup>μ</sup>* ¼ *n* � *C*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*j*<sup>μ</sup>* .

We can see a good regular pattern of the complex roots of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*, *μ* 0 for *y* and *μ*. Therefore, the following conjecture is possible.

**Conjecture 1.** Let *n* be odd positive integer. For *a*> 0 or *a*∈ nf g *a* j *a*< 0 , prove or disprove that

$$R\_{\mathcal{H}\_n(\mathbf{x}, a, \mu)} = \mathbf{1}, \quad \mathcal{C}\_{\mathcal{H}\_n(\mathbf{x}, a, \mu)} = 2 \begin{bmatrix} n \\ \mathbf{2} \end{bmatrix}, \tag{64}$$

where is the set of complex numbers.

**Conjecture 2.** Let *n* be odd positive integer and *a*∈ . Prove or disprove that

$$\mathbf{H}\_n(\mathbf{0}, \mathfrak{a}, \mathfrak{a}, \mathfrak{\mu}) = \mathbf{0}.\tag{65}$$

**Author details**

**181**

Cheon Seoung Ryoo

\*Address all correspondence to: ryoocs@hnu.kr

provided the original work is properly cited.

Department of Mathematics, Hannam University, Daejeon, Republic of Korea

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising…*

*DOI: http://dx.doi.org/10.5772/intechopen.92687*

© 2020 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,

As a result of investigating more *y* and *μ* variables, it is still unknown whether the conjecture 1 and conjecture 2 is true or false for all variables *y* and *μ*.

We observe that solutions of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*, *μ* 0 has not *Re x*ð Þ¼ *b* reflection symmetry for *b*∈ . It is expected that solutions of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*, *μ* 0, has not *Re x*ð Þ¼ *b* reflection symmetry (see **Figures 2**–**4**).

**Conjecture 3.** Prove that the zeros of H*n*ð Þ¼ *x*, *a*, *μ* 0, *a*∈ , has *Im x*ð Þ¼ 0 reflection symmetry analytic complex functions. Prove that the zeros of H*n*ð Þ¼ *x*, *a*, *μ*

0, *a*<0, *a*∈ n, has not *Im x*ð Þ¼ 0 reflection symmetry analytic complex functions. Finally, we consider the more general problems. How many zeros does H*n*ð Þ *x*, *y*, *μ* have? We are not able to decide if H*n*ð Þ¼ *x*, *y*, *μ* 0 has *n* distinct solutions.

We would like to know the number of complex zeros *C*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*,*<sup>μ</sup>* of H*n*ð Þ¼ *x*, *y*, *μ* 0*:* **Conjecture 4.** For *a*∈ , prove or disprove that H*n*ð Þ¼ *x*, *a*, *μ* 0 has *n* distinct solutions.

As a result of investigating more *n* variables, it is still unknown whether the conjecture is true or false for all variables *n* (see **Tables 1** and **2**).

We expect that research in these directions will make a new approach using the numerical method related to the research of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*, *μ* 0 which appear in applied mathematics and mathematical physics.

#### **Additional information**

Mathematics Subject Classification: 05A19, 11B83, 34A30, 65L99

*Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising… DOI: http://dx.doi.org/10.5772/intechopen.92687*

#### **Author details**

**6. Conclusions**

*Number Theory and Its Applications*

In this chapter, we constructed the 2-variable modified degenerate Hermite polynomials and got some new symmetric identities for 2-variable modified degenerate Hermite polynomials. We constructed differential equations arising from the generating function of the 2-variable modified degenerate Hermite polynomials H*n*ð Þ *x*, *y*j*μ* . We also investigated the symmetry of the zeros of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 for various variables *x* and *y*. As a result, we found that the distribution of the zeros of 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*j*μ* 0 is very regular pattern. So, we make the following

Let us use the following notations. *R*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*j*<sup>μ</sup>* denotes the number of real zeros of H*n*ð Þ¼ *x*, *y*j*μ* 0 lying on the real plane *Im x*ð Þ¼ 0 and *C*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*j*<sup>μ</sup>* denotes the number of complex zeros of H*n*ð Þ¼ *x*, *y*j*μ* 0. Since *n* is the degree of the polynomial H*n*ð Þ *x*, *y*j*μ* ,

We can see a good regular pattern of the complex roots of the 2-variable modi-

**Conjecture 1.** Let *n* be odd positive integer. For *a*> 0 or *a*∈ nf g *a* j *a*< 0 , prove

2 h i

H*n*ð Þ¼ 0, *a*, *μ* 0*:* (65)

, (64)

*<sup>R</sup>*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*a*,*<sup>μ</sup>* <sup>¼</sup> 1, *<sup>C</sup>*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*a*,*<sup>μ</sup>* <sup>¼</sup> <sup>2</sup> *<sup>n</sup>*

**Conjecture 2.** Let *n* be odd positive integer and *a*∈ . Prove or disprove that

As a result of investigating more *y* and *μ* variables, it is still unknown whether

We observe that solutions of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*, *μ* 0 has not *Re x*ð Þ¼ *b* reflection symmetry for *b*∈ . It is expected

**Conjecture 3.** Prove that the zeros of H*n*ð Þ¼ *x*, *a*, *μ* 0, *a*∈ , has *Im x*ð Þ¼ 0 reflec-

tion symmetry analytic complex functions. Prove that the zeros of H*n*ð Þ¼ *x*, *a*, *μ* 0, *a*<0, *a*∈ n, has not *Im x*ð Þ¼ 0 reflection symmetry analytic complex functions. Finally, we consider the more general problems. How many zeros does H*n*ð Þ *x*, *y*, *μ* have? We are not able to decide if H*n*ð Þ¼ *x*, *y*, *μ* 0 has *n* distinct solutions. We would like to know the number of complex zeros *C*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*,*<sup>μ</sup>* of H*n*ð Þ¼ *x*, *y*, *μ* 0*:* **Conjecture 4.** For *a*∈ , prove or disprove that H*n*ð Þ¼ *x*, *a*, *μ* 0 has *n* distinct

As a result of investigating more *n* variables, it is still unknown whether the

We expect that research in these directions will make a new approach using the numerical method related to the research of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*, *μ* 0 which appear in applied mathematics and math-

the conjecture 1 and conjecture 2 is true or false for all variables *y* and *μ*.

that solutions of the 2-variable modified degenerate Hermite equations H*n*ð Þ¼ *x*, *y*, *μ* 0, has not *Re x*ð Þ¼ *b* reflection symmetry (see **Figures 2**–**4**).

conjecture is true or false for all variables *n* (see **Tables 1** and **2**).

Mathematics Subject Classification: 05A19, 11B83, 34A30, 65L99

fied degenerate Hermite equations H*n*ð Þ¼ *x*, *y*, *μ* 0 for *y* and *μ*. Therefore, the

series of conjectures with numerical experiments:

where is the set of complex numbers.

we have *R*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*j*<sup>μ</sup>* ¼ *n* � *C*<sup>H</sup>*n*ð Þ *<sup>x</sup>*,*y*j*<sup>μ</sup>* .

following conjecture is possible.

or disprove that

solutions.

**180**

ematical physics.

**Additional information**

Cheon Seoung Ryoo Department of Mathematics, Hannam University, Daejeon, Republic of Korea

\*Address all correspondence to: ryoocs@hnu.kr

© 2020 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] Andrews LC. Special Functions for Engineers and Mathematicians. New York, NY, USA: Macmillan. Co.; 1985

[2] Appell P, Hermitt Kampéde Fériet J. Fonctions Hypergéométriques et Hypersphériques: Polynomes d Hermite. Paris, France: Gauthier-Villars; 1926

[3] Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG. Higher Transcendental Functions. Vol. 3. New York, NY, USA: Krieger; 1981

[4] Andrews GE, Askey R, Roy R. Special Functions. Cambridge, England: Cambridge University Press; 1999

[5] Arfken G. Mathematical Methods for Physicists. 3rd ed. Orlando, FL: Academic Press; 1985

[6] Carlitz L. Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Mathematica. 1979;**15**:51-88

[7] Young PT. Degenerate Bernoulli polynomials, generalized factorial sums, and their applications. Journal of Number Theory. 2008;**128**:738-758

[8] Ryoo CS. A numerical investigation on the structure of the zeros of the degenerate Euler-tangent mixed-type polynomials. Journal of Nonlinear Sciences and Applications. 2017;**10**: 4474-4484

[9] Ryoo CS. Notes on degenerate tangent polynomials. The Global Journal of Pure and Applied Mathematics. 2015; **11**:3631-3637

[10] Hwang KW, Ryoo CS. Differential equations associated with two variable degenerate Hermite polynomials. Mathematics. 2020;**8**:1-18. DOI: 10.3390/math8020228

[11] Hwang KW, Ryoo CS, Jung NS. Differential equations arising from the generating function of the (r,β)-Bell polynomials and distribution of zeros of equations. Mathematics. 2019;**7**:1-11. DOI: 10.3390/math7080736

**Chapter 11**

**Abstract**

types of curves.

**1. Introduction**

**183**

Ring *Rn*

Elliptic Curve over a Local Finite

The goal of this chapter is to study some arithmetic proprieties of an elliptic curve defined by a Weierstrass equation on the local ring *Rn* <sup>¼</sup> *q*½ � *<sup>X</sup> <sup>=</sup> <sup>X</sup><sup>n</sup>* ð Þ, where *n*≥ 1 is an integer. It consists of, an introduction, four sections, and a conclusion. In the first section, we review some fundamental arithmetic proprieties of finite local rings *Rn*, which will be used in the remainder of the chapter. The second section is devoted to a study the above mentioned elliptic curve on these finite local rings for arbitrary characteristics. A restriction to some specific characteristic cases will then be considered in the third section. Using these studies, we give in the fourth section some cryptography applications, and we give in the conclusion some

current research perspectives concerning the use of this kind of curves in cryptography. We can see in the conclusion of research in perspectives on these

Elliptic curves are especially important in number theory and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC), integer factorization, classical mechanics in the description of the movement of spinning tops, to produce efficient codes … For these reasons, the subject is

The purpose of cryptography is to ensure the security of communications and data stored in the presence of adversaries [1–3]. It offers a set of techniques for providing confidentiality, authenticity, and integrity services. Cryptology, also known as the science of secrecy, combines cryptography and cryptanalysis. While the role of cryptographers is to design, build, and prove cryptosystems, among other things, the goal of cryptanalysis is to "break" these systems. The history of cryptography has long been the history of secret codes and along all previous times, this has affected the fate of men and nations [4]. In fact, until 1970, the main goal of cryptography was to build a signature encryption systems [5, 6], but thanks to cryptanalysis, the army and the black cabinets of diplomats were able to wage their wars in the shadows controlling the communication networks, especially of their enemies [7, 8]. The internet revolution and the increasingly massive use of information in digital form facilitated communications but in counterparty it weakened the security level of information. Indeed, "open" networks create security holes,

**Keywords:** elliptic curve, finite ring, cryptography

well known, presented, and worth exploring.

*Abdelhakim Chillali and Lhoussain El Fadil*

[12] Ryoo CS. Differential equations associated with tangent numbers. Journal of Applied Mathematics and Informatics. 2016;**34**:487-494

[13] Ryoo CS. Some identities involving Hermitt Kampé de Fériet polynomials arising from differential equations and location of their zeros. Mathematics. 2019;**7**:1-11. DOI: 10.3390/math7010023

[14] Ryoo CS, Agarwal RP, Kang JY. Differential equations associated with Bell-Carlitz polynomials and their zeros. Neural, Parallel & Scientific Computations. 2016;**24**:453-462

#### **Chapter 11**

**References**

[1] Andrews LC. Special Functions for Engineers and Mathematicians. New York, NY, USA: Macmillan. Co.; 1985

*Number Theory and Its Applications*

generating function of the (r,β)-Bell polynomials and distribution of zeros of equations. Mathematics. 2019;**7**:1-11.

[12] Ryoo CS. Differential equations associated with tangent numbers. Journal of Applied Mathematics and Informatics. 2016;**34**:487-494

[13] Ryoo CS. Some identities involving Hermitt Kampé de Fériet polynomials arising from differential equations and location of their zeros. Mathematics. 2019;**7**:1-11. DOI: 10.3390/math7010023

[14] Ryoo CS, Agarwal RP, Kang JY. Differential equations associated with Bell-Carlitz polynomials and their zeros.

Neural, Parallel & Scientific Computations. 2016;**24**:453-462

DOI: 10.3390/math7080736

[2] Appell P, Hermitt Kampéde Fériet J. Fonctions Hypergéométriques et

Hypersphériques: Polynomes d Hermite. Paris, France: Gauthier-Villars; 1926

Oberhettinger F, Tricomi FG. Higher Transcendental Functions. Vol. 3. New

[4] Andrews GE, Askey R, Roy R. Special

[5] Arfken G. Mathematical Methods for

[3] Erdelyi A, Magnus W,

York, NY, USA: Krieger; 1981

Functions. Cambridge, England: Cambridge University Press; 1999

Physicists. 3rd ed. Orlando, FL:

[6] Carlitz L. Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas

[7] Young PT. Degenerate Bernoulli polynomials, generalized factorial sums, and their applications. Journal of Number Theory. 2008;**128**:738-758

[8] Ryoo CS. A numerical investigation on the structure of the zeros of the degenerate Euler-tangent mixed-type polynomials. Journal of Nonlinear Sciences and Applications. 2017;**10**:

[9] Ryoo CS. Notes on degenerate tangent polynomials. The Global Journal of Pure and Applied Mathematics. 2015;

[10] Hwang KW, Ryoo CS. Differential equations associated with two variable degenerate Hermite polynomials. Mathematics. 2020;**8**:1-18. DOI:

[11] Hwang KW, Ryoo CS, Jung NS. Differential equations arising from the

Mathematica. 1979;**15**:51-88

Academic Press; 1985

4474-4484

**11**:3631-3637

**182**

10.3390/math8020228

## Elliptic Curve over a Local Finite Ring *Rn*

*Abdelhakim Chillali and Lhoussain El Fadil*

#### **Abstract**

The goal of this chapter is to study some arithmetic proprieties of an elliptic curve defined by a Weierstrass equation on the local ring *Rn* <sup>¼</sup> *q*½ � *<sup>X</sup> <sup>=</sup> <sup>X</sup><sup>n</sup>* ð Þ, where *n*≥ 1 is an integer. It consists of, an introduction, four sections, and a conclusion. In the first section, we review some fundamental arithmetic proprieties of finite local rings *Rn*, which will be used in the remainder of the chapter. The second section is devoted to a study the above mentioned elliptic curve on these finite local rings for arbitrary characteristics. A restriction to some specific characteristic cases will then be considered in the third section. Using these studies, we give in the fourth section some cryptography applications, and we give in the conclusion some current research perspectives concerning the use of this kind of curves in cryptography. We can see in the conclusion of research in perspectives on these types of curves.

**Keywords:** elliptic curve, finite ring, cryptography

#### **1. Introduction**

Elliptic curves are especially important in number theory and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC), integer factorization, classical mechanics in the description of the movement of spinning tops, to produce efficient codes … For these reasons, the subject is well known, presented, and worth exploring.

The purpose of cryptography is to ensure the security of communications and data stored in the presence of adversaries [1–3]. It offers a set of techniques for providing confidentiality, authenticity, and integrity services. Cryptology, also known as the science of secrecy, combines cryptography and cryptanalysis. While the role of cryptographers is to design, build, and prove cryptosystems, among other things, the goal of cryptanalysis is to "break" these systems. The history of cryptography has long been the history of secret codes and along all previous times, this has affected the fate of men and nations [4]. In fact, until 1970, the main goal of cryptography was to build a signature encryption systems [5, 6], but thanks to cryptanalysis, the army and the black cabinets of diplomats were able to wage their wars in the shadows controlling the communication networks, especially of their enemies [7, 8]. The internet revolution and the increasingly massive use of information in digital form facilitated communications but in counterparty it weakened the security level of information. Indeed, "open" networks create security holes,

which allow access to the information. Cryptography, or the art of encrypting messages, a science that sites today in the crossroads of mathematics, computer sciences, and some applied physics, has then become a necessity for today's civilization to keep its secrets from adversaries. Confusion is often made between cryptography and cryptology, but the difference exists. Cryptology is the "science of secrecy," and combines two branches on the one hand, cryptography, which makes it possible to encrypt messages, and on the other hand, cryptanalysis, which serves to decrypt them. Our focus in this chapter is to show how some elliptic curves, mathematical objects studied particularly in algebraic geometry [9–12]. You can give several definitions depending on the person you are talking to. Cryptography indeed used elliptic curves for more than 40 years the appearance of the Diffie-Hellman key exchange protocol and the ElGamal cryptogram [13–15]. These cryptographic protocols use in particular group structures, for by applying these methods to groups defined by elliptic curves, a new speciality was born at the end of the 1980: ECC, Elliptic Curve Cryptography. Recall that Diffie-Hellman key exchange which is based on the difficulty of the discrete logarithm problem (DLP) [16–18]. The success of elliptic curves in public key cryptographic systems has then created a new interest in the study of the arithmetic of these geometric objects. The group of points on an elliptical curve is an interesting group in cryptography because there is no known sub-exponential algorithm for sound (DLP) [19–21]. In general, the DLP is difficult to be solved, but not as much as in a generic group as in the case of finite field. We know sub-exponential algorithms to solve it depending on the size of the group to use, which impose criteria for the PLD to be infeasible. The prime number p which is the characteristic of our base ring must then have at least 1024 bits, which offers a security level similar to the one given by a generic order group of 160 bits. Recall that a generic group for the DLP is a group for which there is no a specific algorithm to solve the DLP [22], so that the only available algorithms are those for all groups.

*<sup>X</sup>:<sup>Y</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*�<sup>1</sup>

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

**Corollary 2.1** *Let X* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

By formula (2), we have

have the following corollary: **Corollary 2.2** *<sup>X</sup>*<sup>3</sup> <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

∀*k*≥0,

**Lemma 2.3** *Let Y* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

*<sup>i</sup>*¼<sup>0</sup> *yi*

**Proof.**

we have

So,

**185**

Let *<sup>Y</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

**Proof.**

*i*¼0

*zi*ϵ*<sup>i</sup>*

∀*k*≥0,

, where *<sup>z</sup> <sup>j</sup>* <sup>¼</sup> <sup>X</sup>

*x*0 <sup>2</sup>*<sup>k</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

8 < :

*x*0 <sup>2</sup>*k*þ<sup>1</sup> <sup>¼</sup> <sup>2</sup>

∀*j*≥0, *x*<sup>0</sup>

For*j* ¼ 2*k*, *x*<sup>0</sup>

<sup>2</sup>*<sup>k</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

<sup>2</sup>*k*þ<sup>1</sup> ¼ 2

*, where*

<sup>2</sup>*<sup>k</sup>x*<sup>0</sup> <sup>þ</sup> <sup>P</sup>*<sup>k</sup>*�<sup>1</sup>

*<sup>l</sup>*¼<sup>0</sup> *<sup>x</sup>*<sup>0</sup> 2*l*

so, *x*<sup>0</sup>

Similarly, for*j* ¼ 2*k* þ 1, *x*<sup>0</sup>

then, *x*<sup>0</sup>

*<sup>i</sup>*¼<sup>0</sup> *<sup>x</sup>*<sup>00</sup> *i* ϵ*i*

<sup>2</sup>*k*þ<sup>1</sup> <sup>¼</sup> <sup>P</sup>*<sup>k</sup>*

*<sup>y</sup>*<sup>0</sup> <sup>¼</sup> *<sup>x</sup>*�<sup>1</sup> 0 *<sup>y</sup> <sup>j</sup>* ¼ �*x*�<sup>1</sup> 0 P *<sup>j</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *yi*

<sup>ϵ</sup>*<sup>i</sup>* be the inverse of *<sup>X</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

*zi*ϵ*<sup>i</sup>*

*<sup>i</sup>*¼<sup>0</sup> *yi*

*XY* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*�<sup>1</sup>

*i*¼0

*x*00 <sup>2</sup>*<sup>k</sup>* ¼ *x*<sup>0</sup>

*x*00

(

*j*

*i*¼0

*<sup>i</sup>*¼<sup>0</sup> *xi*ϵ*<sup>i</sup>* <sup>∈</sup>*Rn, then X*<sup>2</sup> <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

*<sup>j</sup>* <sup>¼</sup> <sup>X</sup> *j*

*i*¼0

<sup>2</sup>*<sup>k</sup>* <sup>¼</sup> <sup>X</sup> 2*k*

*<sup>k</sup>* þ 2

*i*¼0

X *k*�1

*i*¼0

X *k*

*i*¼0

Under the same hypotheses of the corollary (2.1) and by an analogous proof, we

*<sup>l</sup>*¼<sup>0</sup> *<sup>x</sup>*<sup>0</sup> 2*l*

<sup>ϵ</sup>*<sup>i</sup> the inverse of X* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

� � (

*x*2*k*þ1�2*<sup>l</sup>* þ *x*<sup>0</sup>

<sup>2</sup>*k*þ<sup>1</sup> ¼

2 X *k*þ1

*i*¼0

*x*2*k*�2*<sup>l</sup>* þ *x*<sup>0</sup>

*x <sup>j</sup>*�*<sup>i</sup>*, ∀*j*>0

*j*

*xiyj*�*<sup>i</sup>*

*z*<sup>0</sup> ¼ 1and ∀*j*>0, *z <sup>j</sup>* ¼ 0, (12)

*i*¼0

*<sup>i</sup>*¼<sup>0</sup> *xi*ϵ*<sup>i</sup>*

, where *<sup>z</sup> <sup>j</sup>* <sup>¼</sup> <sup>X</sup>

� �

<sup>2</sup>*l*þ<sup>1</sup>*x*2*k*�2*<sup>l</sup>*

*<sup>i</sup>*¼<sup>0</sup> *xi*ϵ*<sup>i</sup>*

*<sup>k</sup>* þ 2

P*<sup>k</sup>*

P*<sup>k</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *xix*2*k*�*<sup>i</sup>*

*<sup>i</sup>*¼<sup>0</sup>*xix*2*k*þ1�*<sup>i</sup>*

*xiy <sup>j</sup>*�*<sup>i</sup>* The  cauchy  product � � (2)

*xix <sup>j</sup>*�*<sup>i</sup>:* (4)

*xix*2*k*�*<sup>i</sup>*, (5)

*xix*2*k*�*<sup>i</sup>:* (6)

*xix*2*k*þ1�*<sup>i</sup>:* (8)

<sup>2</sup>*l*þ<sup>1</sup>*x*2*k*�1�2*<sup>l</sup>*

*. Then*

. Then *XY* ¼ 1, by formula (2),

*:* (11)

*xix*2*k*þ1�*<sup>i</sup>*, (7)

(3)

(9)

(10)

*<sup>i</sup>*¼<sup>0</sup> *<sup>x</sup>*<sup>0</sup> *i* ϵ*<sup>i</sup> where*

In [23], Elhassani et al. have built an encryption method based on DLP and Lattice. Boulbot et al. in [24] have studied elliptic curves on a non-local ring to compare these curves on local and non-local rings, while in [25], Sahmoudi et al. have studied these types of curves on a family of finite rings in the authors have introduced a cryptosystem on these types of curves, see [26].

In this chapter, *<sup>d</sup>* and *<sup>n</sup>* are a positive integers and *<sup>q</sup>* <sup>¼</sup> *<sup>p</sup><sup>d</sup>* is a power of a prime natural number *p*.

## **2. The ring** *Rn* <sup>¼</sup> *q*½ � *<sup>X</sup> <sup>=</sup> <sup>X</sup><sup>n</sup>* ð Þ

Let *Rn* <sup>¼</sup> *q*½ � *<sup>X</sup> <sup>=</sup> <sup>X</sup><sup>n</sup>* ð Þ be a *q*-algebra of dimension *<sup>n</sup>*, with 1, <sup>ϵ</sup>, …, <sup>ϵ</sup>*<sup>n</sup>*�<sup>1</sup> ð Þ as a *q*basis, where <sup>ϵ</sup> <sup>¼</sup> *<sup>X</sup>*, <sup>ϵ</sup>*<sup>n</sup>* <sup>¼</sup> 0, *<sup>q</sup>* is the finite field of order *<sup>q</sup>* <sup>¼</sup> *pr* , and *p* being a prime integer [27–29].

#### **2.1 Internal laws in** *Rn*

Recall that the two laws "+" and "." are naturally defined on *Rn* [30, 31]: for every two elements *<sup>X</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *xi*ϵ*<sup>i</sup>* and *<sup>Y</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *yi* ϵ*<sup>i</sup>* in *Rn*, with *x*1, …, *xn*, *y*1, …, *yn* in *q*,

$$X + Y = \sum\_{i=0}^{n-1} z\_i \mathbf{e}^i, \text{where } z\_j = x\_j + y\_j \operatorname{in} \mathbb{F}\_q \tag{1}$$

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

$$X.Y = \sum\_{i=0}^{n-1} z\_i \mathbf{e}^i, \text{where} \\ z\_j = \sum\_{i=0}^j x\_i y\_{\,-i} \text{(The \text{\textquotedblleft Cauchy product}\right)} \tag{2}$$

**Corollary 2.1** *Let X* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *xi*ϵ*<sup>i</sup>* <sup>∈</sup>*Rn, then X*<sup>2</sup> <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *<sup>x</sup>*<sup>0</sup> *i* ϵ*<sup>i</sup> where*

$$\forall k \ge 0, \begin{cases} \varkappa'\_{2k} = \varkappa\_k^2 + 2\sum\_{i=0}^{k-1} \varkappa\_i \varkappa\_{2k-i} \\ \varkappa'\_{2k+1} = 2\sum\_{i=0}^k \varkappa\_i \varkappa\_{2k+1-i} \end{cases} \tag{3}$$

#### **Proof.**

By formula (2), we have

$$\forall j \ge 0, \boldsymbol{\varkappa}'\_j = \sum\_{i=0}^{j} \boldsymbol{\varkappa}\_i \boldsymbol{\varkappa}\_{j-i}. \tag{4}$$

$$\text{For}j = 2k, \mathbf{x}'\_{2k} = \sum\_{i=0}^{2k} \mathbf{x}\_i \mathbf{x}\_{2k-i},\tag{5}$$

$$\text{iso}, \mathbf{x}'\_{2k} = \mathbf{x}\_k^2 + 2\sum\_{i=0}^{k-1} \mathbf{x}\_i \mathbf{x}\_{2k-i}.\tag{6}$$

$$\text{Similarly, for}\\j = 2k + 1, \mathbf{x}'\_{2k+1} = \sum\_{i=0}^{2k+1} \mathbf{x}\_i \mathbf{x}\_{2k+1-i},\tag{7}$$

$$\text{then}, \mathbf{x}'\_{2k+1} = 2 \sum\_{i=0}^{k} \mathbf{x}\_i \mathbf{x}\_{2k+1-i}. \tag{8}$$

Under the same hypotheses of the corollary (2.1) and by an analogous proof, we have the following corollary:

**Corollary 2.2** *<sup>X</sup>*<sup>3</sup> <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *<sup>x</sup>*<sup>00</sup> *i* ϵ*i , where*

$$\forall k \ge 0, \begin{cases} \varkappa\_{2k}^{\prime\prime} = \varkappa\_{2k}^{\prime}\infty\_{0} + \sum\_{l=0}^{k-1} \left( \varkappa\_{2l}^{\prime}\varkappa\_{2k-2l} + \varkappa\_{2l+1}^{\prime}\varkappa\_{2k-1-2l} \right) \\ \varkappa\_{2k+1}^{\prime\prime} = \sum\_{l=0}^{k} \left( \varkappa\_{2l}^{\prime}\varkappa\_{2k+1-2l} + \varkappa\_{2l+1}^{\prime}\varkappa\_{2k-2l} \right) \end{cases} \tag{9}$$

**Lemma 2.3** *Let Y* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *yi* <sup>ϵ</sup>*<sup>i</sup> the inverse of X* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *xi*ϵ*<sup>i</sup> . Then*

$$\begin{cases} \mathcal{Y}\_0 = \mathfrak{x}\_0^{-1} \\ \mathcal{Y}\_j = -\mathfrak{x}\_0^{-1} \sum\_{i=0}^{j-1} \mathcal{Y}\_i \mathfrak{x}\_{j-i}, \qquad \forall j > 0 \end{cases} \tag{10}$$

**Proof.**

Let *<sup>Y</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *yi* <sup>ϵ</sup>*<sup>i</sup>* be the inverse of *<sup>X</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *xi*ϵ*<sup>i</sup>* . Then *XY* ¼ 1, by formula (2), we have

$$XY = \sum\_{i=0}^{n-1} z\_i \mathbf{e}^i, \text{where } \\ z\_j = \sum\_{i=0}^j \boldsymbol{\kappa}\_j \boldsymbol{y}\_{j-i}. \tag{11}$$

So,

$$z\_0 = \mathbf{1} \text{ and } \forall j > 0, z\_j = 0,\tag{12}$$

which means that,

$$\begin{cases} \mathcal{Y}\_0 = \mathfrak{x}\_0^{-1} \\ \mathcal{Y}\_j = -\mathfrak{x}\_0^{-1} \sum\_{i=0}^{j-1} \mathcal{Y}\_i \mathfrak{x}\_{j-i}, \qquad \forall j > 0 \end{cases} \tag{13}$$

Note that in addition, for every *k*≥1,

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

So, *π<sup>k</sup>* and *k<sup>π</sup>* are tow rings morphisms. **Theorem 2.7** *Let n*≥2 *be an integer,*

*Y*2

*<sup>B</sup>* <sup>¼</sup> *<sup>y</sup>*<sup>2</sup>

*<sup>Z</sup>* <sup>¼</sup> *<sup>Y</sup>*<sup>~</sup> <sup>þ</sup> *yn*�<sup>1</sup>ϵ*<sup>n</sup>*�<sup>1</sup> <sup>2</sup> *<sup>Z</sup>*<sup>~</sup> <sup>þ</sup> *zn*�<sup>1</sup>ϵ*<sup>n</sup>*�<sup>1</sup>

<sup>0</sup>*xn*�<sup>1</sup>ϵ*<sup>n</sup>*�<sup>1</sup>

<sup>þ</sup> *bn*�<sup>1</sup>*z*<sup>3</sup>

*Y*2

þ ð3*x*<sup>2</sup>

þ ~ *bZ*~<sup>3</sup>

> þ ~ *bZ*~<sup>3</sup>

<sup>0</sup> � <sup>3</sup>*z*<sup>2</sup>

*<sup>C</sup>* ¼ � <sup>3</sup>*x*<sup>2</sup>

*<sup>D</sup>* <sup>¼</sup> *bn*�<sup>1</sup>*z*<sup>3</sup>

<sup>0</sup>*zn*�<sup>1</sup> <sup>þ</sup> <sup>2</sup>*y*0*z*0*yn*�<sup>1</sup> ϵ*<sup>n</sup>*�<sup>1</sup>

<sup>þ</sup> <sup>2</sup>*zn*�<sup>1</sup>*z*0*a*0*x*<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*0*xn*�<sup>1</sup>*z*<sup>2</sup>

<sup>0</sup> <sup>þ</sup> <sup>3</sup>*z*<sup>2</sup>

þ ~ *bZ*~<sup>3</sup>

*<sup>a</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>~</sup> <sup>þ</sup> *an*�<sup>1</sup>ϵ*<sup>n</sup>*�1, *<sup>b</sup>* <sup>¼</sup> <sup>~</sup>

*Then*

*Y*~2

where,

and

**Proof.** We have:

If

*y*2

**187**

then, *<sup>Y</sup>*~<sup>2</sup>

*Y*~2

*Y*2

<sup>¼</sup> *<sup>Y</sup>*~<sup>2</sup>

<sup>¼</sup> *<sup>X</sup>*~<sup>3</sup>

*bZ*<sup>3</sup> <sup>¼</sup> <sup>~</sup> *bZ*~<sup>3</sup>

<sup>þ</sup> *<sup>a</sup>*~*X*~*Z*~<sup>2</sup>

<sup>þ</sup> *<sup>a</sup>*~*X*~*Z*~<sup>2</sup>

<sup>0</sup>*zn*�<sup>1</sup> � <sup>2</sup>*y*0*z*0*yn*�<sup>1</sup>Þϵ*<sup>n</sup>*�<sup>1</sup> and therefore,

*aXZ*<sup>2</sup> <sup>¼</sup> *<sup>a</sup>*~*X*~*Z*~<sup>2</sup>

*<sup>Z</sup>*<sup>~</sup> <sup>¼</sup> *<sup>X</sup>*~<sup>3</sup>

*<sup>Z</sup>*<sup>~</sup> <sup>¼</sup> *<sup>X</sup>*~<sup>3</sup>

*<sup>Z</sup>*<sup>~</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup>

*<sup>X</sup>*<sup>3</sup> <sup>¼</sup> *<sup>X</sup>*<sup>~</sup> <sup>þ</sup> *xn*�<sup>1</sup>ϵ*<sup>n</sup>*�<sup>1</sup> <sup>3</sup>

<sup>þ</sup> <sup>3</sup>*x*<sup>2</sup>

*<sup>Z</sup>*<sup>~</sup> <sup>¼</sup> *<sup>X</sup>*~<sup>3</sup>

*<sup>Z</sup>* <sup>¼</sup> *<sup>Z</sup>*<sup>~</sup> <sup>þ</sup> *zn*�<sup>1</sup>ϵ*<sup>n</sup>*�<sup>1</sup> *be elements of Rn with:*

<sup>þ</sup> *<sup>a</sup>*~*X*~*Z*~<sup>2</sup>

*<sup>k</sup><sup>π</sup>* <sup>¼</sup> *<sup>π</sup>*2∘*π*3∘*π*4……∘*π<sup>k</sup>* (16)

ϵ*<sup>n</sup>*�<sup>1</sup> (18)

*A* ¼ 2*y*0*z*0, (19)

<sup>0</sup>*b*<sup>0</sup> � 2*z*0*a*0*x*0, (20)

(21)

<sup>0</sup> <sup>þ</sup> *an*�<sup>1</sup>*x*0*z*<sup>2</sup>

ϵ*<sup>n</sup>*�<sup>1</sup> (25)

0

, (24)

<sup>0</sup> <sup>þ</sup> *an*�1*x*0*z*<sup>2</sup>

03*z*<sup>2</sup>

<sup>0</sup>*zn*�1*b*0�

<sup>0</sup>*:* (22)

(23)

*:* (17)

*<sup>b</sup>* <sup>þ</sup> *bn*�<sup>1</sup>ϵ*<sup>n</sup>*�1, *<sup>X</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>~</sup> <sup>þ</sup> *xn*�<sup>1</sup>ϵ*<sup>n</sup>*�1, *<sup>Y</sup>* <sup>¼</sup> *<sup>Y</sup>*<sup>~</sup> <sup>þ</sup> *yn*�<sup>1</sup>ϵ*<sup>n</sup>*�<sup>1</sup> *and*

*<sup>Z</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *aXZ*<sup>2</sup> <sup>þ</sup> *bZ*<sup>3</sup>

<sup>þ</sup> *<sup>D</sup>* � *Ayn*�<sup>1</sup> <sup>þ</sup> *Bzn*�<sup>1</sup> <sup>þ</sup> *Cxn*�<sup>1</sup>

<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*0*z*<sup>2</sup> 0

<sup>0</sup> <sup>þ</sup> *an*�<sup>1</sup>*x*0*z*<sup>2</sup>

ϵ*<sup>n</sup>*�<sup>1</sup>

<sup>0</sup>*xn*�<sup>1</sup> <sup>þ</sup> <sup>2</sup>*zn*�1*z*0*a*0*x*<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*0*xn*�1*z*<sup>2</sup>

<sup>þ</sup> *<sup>D</sup>* � *Ayn*�<sup>1</sup> <sup>þ</sup> *Bzn*�<sup>1</sup> <sup>þ</sup> *Cxn*�<sup>1</sup>

<sup>0</sup>*zn*�<sup>1</sup>*b*<sup>0</sup> ϵ*<sup>n</sup>*�<sup>1</sup>

*<sup>Z</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *aXZ*<sup>2</sup> <sup>þ</sup> *bZ*<sup>3</sup>

**Lemma 2.4** *The non inverse elements in Rn are the elements of the form* P*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>1</sup> *xi*ϵ*<sup>i</sup> where xi* ∈*<sup>n</sup>*�<sup>1</sup> *<sup>q</sup> for all* 1≤*i*≤ *n* � 1*:*

#### **Proof.**

Let *<sup>X</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *xi*ϵ*<sup>i</sup>* <sup>∈</sup>*Rn*. By lemma (2.3), *<sup>X</sup>* is invertible in *Rn* if and only if *<sup>x</sup>*<sup>0</sup> is invertible in *q:* As *<sup>q</sup>* is a field, this means *x*<sup>0</sup> 6¼ 0.

**Corollary 2.5** *The ring Rn is local, with maximal ideal In* ¼ ϵ*Rn.* Notation.

Let *k*≥2, we denote:

1.

$$\pi\_k: \begin{array}{c} R\_k \ \rightarrow \ R\_{k-1} \\ \sum\_{i=0}^{k-1} \pi\_i \mathfrak{e}^i \ \leftrightarrow \ \sum\_{i=0}^{k-2} \pi\_i \mathfrak{d}^i \end{array}$$

the projection of *Rk* on *Rk*�1.

2.

$$k^{\pi}: \begin{array}{c} R\_k \ \rightarrow \ R\_1\\ \sum\_{i=0}^{k-1} \mathbf{x}\_i \mathbf{e}^i \leftrightarrow \mathbf{x}\_0 \end{array}$$

the canonical projection of *Rk* on *R*<sup>1</sup> ¼ *q*. **Corollary 2.6** *π<sup>k</sup> et k<sup>π</sup> are two ring homomorphisms.* **Proof.** We have,

$$\begin{aligned} \pi\_k \left( \sum\_{i=0}^{k-1} \mathbf{x}\_i \boldsymbol{\epsilon}^i + \sum\_{i=0}^{k-1} \mathbf{y}\_i \boldsymbol{\epsilon}^i \right) &= \pi\_k \left( \sum\_{i=0}^{k-1} (\mathbf{x}\_i + \mathbf{y}\_i) \boldsymbol{\epsilon}^i \right) \\ &= \sum\_{i=0}^{k-2} (\mathbf{x}\_i + \mathbf{y}\_i) \boldsymbol{\delta}^i \\ &= \pi\_k \left( \sum\_{i=0}^{k-1} \mathbf{x}\_i \boldsymbol{\epsilon}^i \right) + \pi\_k \left( \sum\_{i=0}^{k-1} \mathbf{y}\_i \boldsymbol{\epsilon}^i \right) \end{aligned} \tag{14}$$

and

$$
\left(\sum\_{i=0}^{k-1} x\_i \mathbf{e}^i \right) \left(\sum\_{i=0}^{k-1} y\_i \mathbf{e}^i \right) = \sum\_{i=0}^{k-1} z\_i \mathbf{e}^i, \text{where } \mathbf{z}\_j = \sum\_{i=0}^j x\_i y\_{\,-i}.
$$

$$
\pi\_k \left(\sum\_{i=0}^{k-1} z\_i \mathbf{e}^i \right) = \sum\_{i=0}^{k-2} z\_i \mathbf{e}^i \tag{15}
$$

$$
\pi\_k \left(\sum\_{i=0}^{k-1} x\_i \mathbf{e}^i \right) \pi\_k \left(\sum\_{i=0}^{k-1} y\_i \mathbf{e}^i \right) = \sum\_{i=0}^{k-2} z\_i \mathbf{e}^i
$$

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

Note that in addition, for every *k*≥1,

$$k^{\pi} = \pi\_2 \circ \pi\_3 \circ \pi\_4 \circ \dots \circ \pi\_k \tag{16}$$

So, *π<sup>k</sup>* and *k<sup>π</sup>* are tow rings morphisms. **Theorem 2.7** *Let n*≥2 *be an integer, <sup>a</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>~</sup> <sup>þ</sup> *an*�<sup>1</sup>ϵ*<sup>n</sup>*�1, *<sup>b</sup>* <sup>¼</sup> <sup>~</sup> *<sup>b</sup>* <sup>þ</sup> *bn*�<sup>1</sup>ϵ*<sup>n</sup>*�1, *<sup>X</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>~</sup> <sup>þ</sup> *xn*�<sup>1</sup>ϵ*<sup>n</sup>*�1, *<sup>Y</sup>* <sup>¼</sup> *<sup>Y</sup>*<sup>~</sup> <sup>þ</sup> *yn*�<sup>1</sup>ϵ*<sup>n</sup>*�<sup>1</sup> *and <sup>Z</sup>* <sup>¼</sup> *<sup>Z</sup>*<sup>~</sup> <sup>þ</sup> *zn*�<sup>1</sup>ϵ*<sup>n</sup>*�<sup>1</sup> *be elements of Rn with:*

$$\mathbf{Y}^2 \mathbf{Z} = \mathbf{X}^3 + a\mathbf{X}\mathbf{Z}^2 + b\mathbf{Z}^3. \tag{17}$$

*Then*

$$\tilde{\mathbf{Y}}^2 \tilde{\mathbf{Z}} = \tilde{\mathbf{X}}^3 + \tilde{a}\tilde{\mathbf{X}}\tilde{\mathbf{Z}}^2 + \tilde{b}\tilde{\mathbf{Z}}^3 + \left[\mathbf{D} - \left(\mathbf{A}\mathbf{y}\_{n-1} + \mathbf{B}\mathbf{z}\_{n-1} + \mathbf{C}\mathbf{x}\_{n-1}\right)\right] \mathbf{e}^{n-1} \tag{18}$$

where,

$$A = \mathfrak{Z}\mathfrak{z}\_0\mathfrak{z}\_0,\tag{19}$$

$$B = y\_0^2 - \mathfrak{Z}\_0^2 b\_0 - 2\mathfrak{z}\_0 \mathfrak{a}\_0 \mathfrak{x}\_0,\tag{20}$$

$$\mathbf{C} = -\left(\mathbf{3x}\_0^2 + \mathbf{a}\_0 \mathbf{z}\_0^2\right) \tag{21}$$

and

$$D = b\_{n-1} \mathbf{z}\_0^3 + a\_{n-1} \mathbf{x}\_0 \mathbf{z}\_0^2. \tag{22}$$

**Proof.**

We have:

$$\begin{aligned} Y^2 Z &= \left(\bar{Y} + \mathbf{y}\_{n-1} \mathbf{c}^{n-1}\right)^2 \left(\bar{Z} + z\_{n-1} \mathbf{c}^{n-1}\right) \\ &= \bar{Y}^2 \bar{Z} + \left(\mathbf{y}\_0^2 z\_{n-1} + 2\mathbf{y}\_0 z\_0 \mathbf{y}\_{n-1}\right) \mathbf{c}^{n-1} \\\\ \mathbf{X}^3 &= \left(\bar{X} + \mathbf{x}\_{n-1} \mathbf{c}^{n-1}\right)^3 \\ &= \bar{\mathbf{X}}^3 + 3\mathbf{x}\_0^2 \mathbf{x}\_{n-1} \mathbf{c}^{n-1} \\\\ a \mathbf{X} \mathbf{Z}^2 &= \bar{a} \bar{\mathbf{X}} \bar{\mathbf{Z}}^2 + \left(2\mathbf{z}\_{n-1} \mathbf{z}\_0 a\_0 \mathbf{x}\_0 + a\_0 \mathbf{x}\_{n-1} \mathbf{z}\_0^2 + a\_{n-1} \mathbf{x}\_0 \mathbf{z}\_0^2\right) \mathbf{c}^{n-1} \\\\ b \mathbf{Z}^3 &= \bar{b} \bar{\mathbf{Z}}^3 + \left(b\_{n-1} \mathbf{z}\_0^3 + 3 \mathbf{z}\_0^2 \mathbf{z}\_{n-1} b\_0\right) \mathbf{c}^{n-1} \end{aligned} \tag{23}$$

If

$$X^2 Z = X^3 + a X Z^2 + b Z^3,\tag{24}$$

then, *<sup>Y</sup>*~<sup>2</sup> *<sup>Z</sup>*<sup>~</sup> <sup>¼</sup> *<sup>X</sup>*~<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*~*X*~*Z*~<sup>2</sup> þ ~ *bZ*~<sup>3</sup> þ ð3*x*<sup>2</sup> <sup>0</sup>*xn*�<sup>1</sup> <sup>þ</sup> <sup>2</sup>*zn*�1*z*0*a*0*x*<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*0*xn*�1*z*<sup>2</sup> <sup>0</sup> <sup>þ</sup> *an*�1*x*0*z*<sup>2</sup> 03*z*<sup>2</sup> <sup>0</sup>*zn*�1*b*0� *y*2 <sup>0</sup>*zn*�<sup>1</sup> � <sup>2</sup>*y*0*z*0*yn*�<sup>1</sup>Þϵ*<sup>n</sup>*�<sup>1</sup> and therefore,

$$
\tilde{\mathbf{Y}}^2 \tilde{\mathbf{Z}} = \tilde{\mathbf{X}}^3 + \tilde{a}\tilde{\mathbf{X}}\tilde{\mathbf{Z}}^2 + \tilde{b}\tilde{\mathbf{Z}}^3 + \left[\mathbf{D} - \left(\mathbf{A}\mathbf{y}\_{n-1} + \mathbf{B}\mathbf{z}\_{n-1} + \mathbf{C}\mathbf{x}\_{n-1}\right)\right] \mathbf{e}^{n-1} \tag{25}
$$

where,

$$A = \mathfrak{Z}\mathfrak{z}\_0 \mathfrak{z}\_0,\tag{26}$$

Pð Þ¼ 2½ �*e* fð Þ 0, 0, 1 , 0, 1, 1 ð Þ þ *e* , 0, 1, 0 ð Þ, 0, 1, 1 ð Þ, 0, 1, ð Þ*e* , 0, 1, 1 ð Þ þ *e* , 0, ð Þ *e*, 1 ,

ð Þ 0,*e*, 1 þ *e* , 0, 1 ð Þ þ *e*, 0 , 0, 1 ð Þ þ *e*, 1 , 0, 1 ð Þ þ *e*,*e* , 0, 1 ð Þ þ *e*, 1 þ *e* ,

ð Þ 1, 1 þ *e*, 1 , 1, 1 ð Þ þ *e*,*e* , 1, 1 ð Þ þ *e*, 1 þ *e* ,ð Þ *e*, 0, 1 ,ð Þ *e*, 0, 1 þ *e* ,ð Þ *e*, 1, 0 ,

ð Þ *e*, 1 þ *e*,*e* ,ð Þ *e*, 1 þ *e*, 1 þ *e* , 1ð Þ þ *e*, 0, 0 , 1ð Þ þ *e*, 0, 1 , 1ð Þ þ *e*, 0,*e* ,

ð Þ 1 þ *e*, 0, 1 þ *e* , 1ð Þ þ *e*, 1, 0 , 1ð Þ þ *e*, 1, 1 , 1ð Þ þ *e*, 1,*e* , 1ð Þ þ *e*, 1, 1 þ *e* ,

ð Þ 1 þ *e*,*e*, 0 , 1ð Þ þ *e*,*e*, 1 , 1ð Þ þ *e*,*e*,*e* , 1ð Þ þ *e*,*e*, 1 þ *e* , 1ð Þ þ *e*, 1 þ *e*, 0 ,

, *z*<sup>0</sup> ð Þ *be two elements in* Pð Þ 2½ �*e , then:*

ð Þ¼ 2½ �*e* f½ � 0 : 1 : 0 , 0½ � : 0 : 1 , 0½ � : 1 : 1 , 0½ � : 1 : *e* , 0½ � : 1 : 1 þ *e* , 0½ � : *e* : 1 , 1½ � : 0 : 0 ,

½ � 1 : *e* : 0 , 1½ � : *e* : 1 , 1½ � : *e* : *e* , 1½ � : *e* : 1 þ *e* , 1½ � : 1 þ *e* : 0 , 1½ � : 1 þ *e* : 1 ,

½ � 1 : 1 þ *e* : *e* , 1½ � : 1 þ *e* : 1 þ *e* , ½ � *e* : 0 : 1 , ½ � *e* : 1 : 0 , ½ � *e* : 1 : 1 , ½ � *e* : 1 : *e* ,

In this section, we study the elliptic curves defined on finite local rings *Rn* of

**<sup>E</sup>**<sup>0</sup> : *<sup>Y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*XY* <sup>þ</sup> *<sup>a</sup>*3*<sup>Y</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*2*X*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*4*<sup>X</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>6</sup> (36)

1.A projective Weierstrass equation on *Rn* is an equation of the form:

*<sup>Z</sup>* <sup>þ</sup> *<sup>a</sup>*1*XYZ* <sup>þ</sup> *<sup>a</sup>*3*YZ*<sup>2</sup> <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*2*X*<sup>2</sup>

To an affine (or projective) Weierstrass Eqs. (3) and (4), we associate the

2.A affine Weierstrass equation on *Rn* is an equation of the form:

5 *:*

½ � 1 : 0 : 1 , 1½ � : 0 : *e* , 1½ � : 0 : 1 þ *e* , 1½ � : 1 : 0 , 1½ � : 1 : 1 , 1½ � : 1 : *e* , 1½ � : 1 : 1 þ *e* ,

, *y*<sup>0</sup>

ð Þ 2½ �*e contains two representatives, that is, the projective plane*

, *z*<sup>0</sup> ð Þ¼ ð Þ *x* þ *xe*, *y* þ *ye*, *z* þ *ze*

*<sup>Z</sup>* <sup>þ</sup> *<sup>a</sup>*4*<sup>X</sup>* <sup>þ</sup> *<sup>a</sup>*6*Z*<sup>3</sup> (35)

, *z*<sup>0</sup> ð Þ¼ ð Þ *x*, *y*, *z or x*<sup>0</sup>

ð Þ 1 þ *e*, 1 þ *e*, 1 , 1ð Þ þ *e*, 1 þ *e*,*e* , 1ð þ *e*, 1 þ *e*, 1 þ *e*Þg*:*

, *y*<sup>0</sup>

*Let x*ð Þ , *y*, *z and x*<sup>0</sup>

*so every class in* <sup>2</sup>

**3. Elliptic curve over** *Rn*

characteristic a prime number p;

**E** : *Y*<sup>2</sup>

where ð Þ *a*1, *a*2, *a*3, *a*4, *a*<sup>6</sup> ∈*Rn*

**3.1 Elliptic curve form**

following quantities:

**189**

2

2

*x*<sup>0</sup> : *y*<sup>0</sup> : *z*<sup>0</sup> ½ �¼ ½ � *x* : *y* : *z* ⇔ *x*<sup>0</sup>

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

, *y*<sup>0</sup>

½ � *e* : 1 : 1 þ *e* , ½ �g *e* : *e* : 1 *:*

ð Þ 2½ �*e contains exactly the following 28 elements:*

ð Þ *e*, 1, 1 ,ð Þ *e*, 1,*e* ,ð Þ *e*, 1, 1 þ *e* ,ð Þ *e*,*e*, 1 ,ð Þ *e*,*e*, 1 þ *e* ,ð Þ *e*, 1 þ *e*, 0 ,ð Þ *e*, 1 þ *e*, 1 ,

(33)

(34)

ð Þ 1, 0, 0 , 1, 0, 1 ð Þ, 1, 0, ð Þ*e* , 1, 0, 1 ð Þ þ *e* , 1, 1, 0 ð Þ, 1, 1, 1 ð Þ, 1, 1, ð Þ*e* ,

ð Þ 1, 1, 1 þ *e* , 1, ð Þ *e*, 0 , 1, ð Þ *e*, 1 , 1, ð Þ *e*,*e* , 1, ð Þ *e*, 1 þ *e* , 1, 1 ð Þ þ *e*, 0 ,

$$B = y\_0^2 - \mathfrak{Z}\_0^2 b\_0 - 2\mathfrak{z}\_0 \mathfrak{a}\_0 \mathfrak{x}\_0,\tag{27}$$

$$\mathbf{C} = -\left(\mathbf{3x}\_0^2 + \mathbf{a}\_0 \mathbf{z}\_0^2\right) \tag{28}$$

and

$$D = b\_{n-1}z\_0^3 + a\_{n-1}\varkappa\_0 z\_0^2. \tag{29}$$

#### **2.2 Primitive triples**

**Definition 2.8** *Let R be a ring. We say that an element x*ð Þ , *<sup>y</sup>*, *<sup>z</sup>* <sup>∈</sup> *<sup>R</sup>*<sup>3</sup> *is primitive if: xR* þ *yR* þ *zR* ¼ *R. The set of these primitive triplets will be denoted* Pð Þ *R .*

**Remark 2.9** *The equality xR* <sup>þ</sup> *yR* <sup>þ</sup> *zR* <sup>¼</sup> *R means that there exists* ð Þ *<sup>α</sup>*, *<sup>β</sup>*, *<sup>λ</sup>* <sup>∈</sup>*R*<sup>3</sup> *such that* 1*<sup>R</sup>* ¼ *αx* þ *βy* þ *λz.*

**Proposition 2.10** *Let R be a local ring, then x*ð Þ , *<sup>y</sup>*, *<sup>z</sup>* <sup>∈</sup> *<sup>R</sup>*<sup>3</sup> *is a primitive triple if and only if at least one of the elements x, y*, *and z is invertible in R.*

#### **Proof.**

Suppose that *x*, *y* and *z* are not invertible in *R*, then:

ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* <sup>∈</sup>M<sup>3</sup> where <sup>M</sup> is the unique maximal ideal of *<sup>R</sup>*, hence

$$\mathbf{x}R + \mathbf{y}R + \mathbf{z}R \subset \mathcal{M} \subsetneq \mathbf{R},\tag{30}$$

which contradicts that ð Þ *x*, *y*, *z* is a primitive triple.

Conversely, suppose, for example, that *x* is invertible in *R*, then *xR* ¼ *R*, so *xR* þ *yR* þ *zR* ¼ *R*.

**Remark 2.11** *If R is a field, then an element x*ð Þ , *<sup>y</sup>*, *<sup>z</sup>* <sup>∈</sup>*R*<sup>3</sup> *is primitive if and only if* ð Þ *x*, *y*, *z* 6¼ ð Þ 0, 0, 0 *.*

#### **2.3 The projective plane on a finite ring**

Let *R* is a ring. The projective plane on *R* is the set of equivalence classes of Pð Þ *R* modulo; the equivalence relation � *R* defined by:

$$\chi(\mathbf{x}\_1, y\_1, \mathbf{z}\_1) \sim R(\mathbf{x}\_2, y\_2, \mathbf{z}\_2) \Leftrightarrow \exists \lambda \in \mathbb{R}^\times \;:\; (\mathbf{x}\_2, y\_2, \mathbf{z}\_2) = \lambda(\mathbf{x}\_1, y\_1, \mathbf{z}\_1). \tag{31}$$

We denote the projective plane on *R* by <sup>2</sup> ð Þ *<sup>R</sup>* , it is the quotient set Pð Þ *<sup>R</sup>* �*<sup>R</sup>* , and we write ½ � *x* : *y* : *z* for the equivalence class of ð Þ *x*, *y*, *z* ∈Pð Þ *R* . Thus, we have:

$$\left[\mathbf{x}\_{1}:\mathbf{y}\_{1}:\mathbf{z}\_{1}\right] = \left[\mathbf{x}\_{2}:\mathbf{y}\_{2}:\mathbf{z}\_{2}\right] \Leftrightarrow \exists \lambda \in \mathbb{R}^{\times}: \mathbf{x}\_{2} = \lambda \mathbf{x}\_{1}, \mathbf{y}\_{2} = \lambda \mathbf{y}\_{1} \text{ and } \mathbf{z}\_{2} = \lambda \mathbf{z}\_{1}. \tag{32}$$

**Example 2.12** *We Consider the finite ring* 2½�¼ *e* f g *α* þ *βe=α* ∈<sup>2</sup> *andβ* ∈<sup>2</sup> , *where e is an indeterminate satisfying e*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*. The group of units for this ring is* ð Þ 2½ �*e* � ¼ f g 1, 1 þ *e .*

*As this ring is local with maximal ideal e*2½ �*<sup>e</sup> , then an element x*ð Þ , *<sup>y</sup>*, *<sup>z</sup> of* 2½ �*<sup>e</sup>* <sup>3</sup> *is non primitive if and only if x*ð Þ , *<sup>y</sup>*, *<sup>z</sup>* <sup>∈</sup>f g 0,*<sup>e</sup>* <sup>3</sup> *: As one can see, there are eight elements which are not primitive, and therefore the set* Pð Þ 2½ �*e contains* 64 � 8 ¼ 56 *primitive triples as given below:*

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

$$\begin{aligned} \mathcal{P}(\mathbb{P}\_{2}[e]) &= \{(0,0,1), (0,1,1+e), (0,1,0), (0,1,1), (0,1,e), (0,1,1+e), (0,e), 1, \\\\ &(0, e, 1+e), (0, 1+e, 0), \ (0, 1+e, 1), \ (0, 1+e, e), (0, 1+e, 1+e), \\\\ &(1, 0, 0), (1, 0, 1), (1, 0, e), (1, 0, 1+e), (1, 1, 0), (1, 1, 1), (1, 1, e), \\\\ &(1, 1, 1+e), (1, e, 0), (1, e, 1), (1, e, e), (1, e, 1+e), (1, 1+e, 0), \\\\ &(1, 1+e, 1), (1, 1+e, e), (1, 1+e, 1+e), (e, 0, 1), (e, 0, 1+e), (e, 1, 0), \\\\ &(e, 1, 1), (e, 1, e), (e, 1, 1+e), (e, e, 1), (e, e, 1+e), (e, 1+e, 0), (e, 1+e, 1), \\\\ &(1+e, 0, 1+e), (1+e, 1+e), (1+e, 0, 0), (1+e, 0, 1), (1+e, 0, e), \\\\ &(1+e, 0, 1+e), (1+e, 1, 0), (1+e, 1, 1), (1+e, 1, e), (1+e, 1, 1+e), \\\\ &(1+e, e, 0), (1+e, e, 1), (1+e, e, e), (1+e, 1+e), (1+e, 1+e, 0), \\\\ &(1+e, 1+e, 1), (1+e, 1+e, e), (1+e, 1+e, 1+e)). \end{aligned}$$

*Let x*ð Þ , *y*, *z and x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> ð Þ *be two elements in* Pð Þ 2½ �*e , then: x*<sup>0</sup> : *y*<sup>0</sup> : *z*<sup>0</sup> ½ �¼ ½ � *x* : *y* : *z* ⇔ *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> ð Þ¼ ð Þ *x*, *y*, *z or x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> ð Þ¼ ð Þ *x* þ *xe*, *y* þ *ye*, *z* þ *ze so every class in* <sup>2</sup> ð Þ 2½ �*e contains two representatives, that is, the projective plane* 2 ð Þ 2½ �*e contains exactly the following 28 elements:*

$$\mathbb{P}^{2}(\mathbb{F}\_{2}[e]) = \{ [\mathbf{0}:\mathbf{1}:\mathbf{0}], [\mathbf{0}:\mathbf{0}:\mathbf{1}], [\mathbf{0}:\mathbf{1}:\mathbf{1}], [\mathbf{0}:\mathbf{1}:e], [\mathbf{0}:\mathbf{1}:\mathbf{1}+e], [\mathbf{0}:e:\mathbf{1}], [\mathbf{1}:\mathbf{0}:\mathbf{0}] \},$$

$$[\mathbf{1}:\mathbf{0}:\mathbf{1}], [\mathbf{1}:\mathbf{0}:e], [\mathbf{1}:\mathbf{0}:\mathbf{1}+e], [\mathbf{1}:\mathbf{1}:\mathbf{0}], [\mathbf{1}:\mathbf{1}:\mathbf{1}], [\mathbf{1}:\mathbf{1}:e], [\mathbf{1}:\mathbf{1}:e], [\mathbf{1}:\mathbf{1}:\mathbf{1}], \dots],$$

$$[\mathbf{1}:e:\mathbf{0}], [\mathbf{1}:e:\mathbf{1}], [\mathbf{1}:e:e], [\mathbf{1}:e:\mathbf{1}+e], [\mathbf{1}:\mathbf{1}+e:\mathbf{0}], [\mathbf{1}:\mathbf{1}+e:\mathbf{1}],$$

$$[\mathbf{1}:\mathbf{1}+e:e], [\mathbf{1}:\mathbf{1}+e:\mathbf{1}+e], [e:\mathbf{0}:\mathbf{1}], [e:\mathbf{1}:\mathbf{0}], [e:\mathbf{1}:\mathbf{1}], [e:\mathbf{1}:e],\tag{34}$$

$$[e:\mathbf{1}:\mathbf{1}+e], [e:e:\mathbf{1}] \}.\tag{34}$$

#### **3. Elliptic curve over** *Rn*

In this section, we study the elliptic curves defined on finite local rings *Rn* of characteristic a prime number p;

1.A projective Weierstrass equation on *Rn* is an equation of the form:

$$\mathbf{E}: \quad \mathbf{Y}^2 \mathbf{Z} + a\_1 \mathbf{X} \mathbf{Y} \mathbf{Z} + a\_3 \mathbf{Y} \mathbf{Z}^2 = \mathbf{X}^3 + a\_2 \mathbf{X}^2 \mathbf{Z} + a\_4 \mathbf{X} + a\_6 \mathbf{Z}^3 \tag{35}$$

2.A affine Weierstrass equation on *Rn* is an equation of the form:

$$\mathbf{E}': \quad Y^2 + a\_1 XY + a\_3 Y = X^3 + a\_2 X^2 + a\_4 X + a\_6 \tag{36}$$

where ð Þ *a*1, *a*2, *a*3, *a*4, *a*<sup>6</sup> ∈*Rn* 5 *:*

#### **3.1 Elliptic curve form**

To an affine (or projective) Weierstrass Eqs. (3) and (4), we associate the following quantities:

$$\begin{aligned} b\_2 &= a\_1^2 + 4a\_2 \\ b\_4 &= 2a\_4 + a\_1 a\_3 \\ b\_6 &= a\_3^2 + 4a\_6 \\ b\_8 &= a\_1^2 a\_6 + 4a\_2 a\_6 - a\_1 a\_3 a\_4 + a\_2 a\_3^2 - a\_4^2 \\ c\_4 &= b\_2^2 - 24b\_4 \\ \Delta &= -b\_2^2 b\_8 - 8b\_4^3 - 27b\_6^2 + 9b\_2 b\_4 b\_6 \\ j &= \frac{c\_4^3}{\Delta} j^\circ \Delta \neq 0 \end{aligned} \tag{37}$$

**Remark 3.4** *A projective elliptic curve on a field K has one of the following normal*

*char*ð Þ **<sup>K</sup>** 6¼ 2, 3 *<sup>Y</sup>*<sup>2</sup>*<sup>Z</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*4*XZ*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*6*Z*<sup>3</sup>

**Normal form**

<sup>Δ</sup> ¼ �16 4*a*<sup>3</sup>

<sup>Δ</sup> ¼ �*a*<sup>3</sup> 4

*<sup>j</sup>* 6¼ <sup>0</sup> *<sup>Y</sup>*<sup>2</sup>*<sup>Z</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*2*X*<sup>2</sup>*<sup>Z</sup>* <sup>þ</sup> *<sup>a</sup>*6*Z*<sup>3</sup> <sup>Δ</sup> ¼ �*a*<sup>3</sup>

> <sup>Δ</sup> <sup>¼</sup> *<sup>a</sup>*<sup>4</sup> 3

*<sup>j</sup>* 6¼ <sup>0</sup> *<sup>Y</sup>*2*<sup>Z</sup>* <sup>þ</sup> *XYZ* <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*2*X*<sup>2</sup>

<sup>4</sup> <sup>þ</sup> <sup>27</sup>*a*<sup>2</sup> 6

*<sup>Z</sup>* <sup>þ</sup> *<sup>a</sup>*3*YZ*<sup>2</sup> <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*4*XZ*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*6*Z*<sup>3</sup>

<sup>Δ</sup> <sup>¼</sup> *<sup>a</sup>*<sup>6</sup> *<sup>j</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>Z</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*4*XZ*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*6*Z*<sup>3</sup>

*<sup>j</sup>* <sup>¼</sup> <sup>1728</sup> <sup>4</sup>*a*<sup>3</sup>

<sup>2</sup>*a*<sup>6</sup> *<sup>j</sup>* ¼ � *<sup>a</sup>*<sup>3</sup>

*<sup>Z</sup>* <sup>þ</sup> *<sup>a</sup>*6*Z*<sup>3</sup>

4 4*a*<sup>3</sup> 4þ27*a*<sup>2</sup> 6

2 *a*6

*a*6

In this subsection, we give in projective coordinates the formulas for adding the points on an elliptic curve defined by Eq. (3) on the ring *Rn*, according to the normal

Using Bosma and Lenstra's theorem see [32], we can deduce the explicit formulas for the commutative additive law of the group *Ea*ð Þ *Rn* . The results are given in the next theorems following the values of the characteristic of ring *Rn* [33–36]. Let

<sup>2</sup>*Y*<sup>1</sup> <sup>þ</sup> *aX*<sup>2</sup>

½ �þ *X*<sup>1</sup> : *Y*<sup>1</sup> : *Z*<sup>1</sup> ½ �¼ *X*<sup>2</sup> : *Y*<sup>2</sup> : *Z*<sup>2</sup> ½ � *X*<sup>3</sup> : *Y*<sup>3</sup> : *Z*<sup>3</sup> *:* (44)

<sup>2</sup>*Z*<sup>1</sup> <sup>þ</sup> *bY*2*Z*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *bY*1*Z*1*Z*<sup>2</sup>

<sup>2</sup>*Z*<sup>1</sup> <sup>þ</sup> *bY*1*Z*1*Z*<sup>2</sup>

<sup>2</sup>*Y*1*Z*<sup>1</sup> <sup>þ</sup> *aX*<sup>2</sup>

<sup>1</sup>*Z*2*:* (48)

<sup>2</sup>*Z*<sup>1</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup> 1*X*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>Y</sup>*<sup>2</sup>

<sup>1</sup>*X*2*Y*<sup>2</sup> <sup>þ</sup> *aX*1*X*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *bX*1*Z*1*Z*<sup>2</sup>

<sup>1</sup>*X*2*Z*<sup>2</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *bY*2*Z*<sup>2</sup>

<sup>1</sup>*X*2*Z*<sup>2</sup> <sup>þ</sup> *bX*1*X*<sup>2</sup>

<sup>1</sup>*Z*<sup>2</sup> <sup>þ</sup> *bX*1*Z*<sup>2</sup>

<sup>2</sup>*Z*1þ

<sup>1</sup>*X*2*Z*<sup>2</sup> <sup>þ</sup> *aX*1*X*<sup>2</sup>

<sup>2</sup>*Y*<sup>1</sup> <sup>þ</sup> *aX*<sup>2</sup>

<sup>2</sup>*Z*1*:*

<sup>2</sup> <sup>þ</sup> *abX*1*Z*1*Z*<sup>2</sup>

<sup>1</sup>*Z*<sup>2</sup> <sup>þ</sup> *bX*1*Z*1*Z*<sup>2</sup>

<sup>2</sup>*Z*1þ

<sup>1</sup>*Y*2*Z*2þ

1*X*<sup>2</sup> 2þ

<sup>2</sup>þ

(45)

(46)

<sup>2</sup>*:* (47)

*forms (Table 1):*

form.

**Table 1.**

*Elliptic curve form on a field.*

*<sup>X</sup>*<sup>3</sup> <sup>¼</sup> *<sup>X</sup>*1*Y*1*Y*<sup>2</sup>

*<sup>Y</sup>*<sup>3</sup> <sup>¼</sup> *<sup>Y</sup>*<sup>2</sup> 1*Y*<sup>2</sup>

*<sup>Z</sup>*<sup>3</sup> <sup>¼</sup> *<sup>X</sup>*<sup>2</sup>

**191**

*aX*<sup>2</sup>

*<sup>X</sup>*<sup>3</sup> <sup>¼</sup> *<sup>X</sup>*1*Y*<sup>2</sup>

*bX*1*Y*1*Z*<sup>2</sup>

*bX*1*Y*1*Z*<sup>2</sup>

*abX*2*Z*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>X</sup>*2*Y*<sup>2</sup>

<sup>1</sup>*X*2*Y*<sup>2</sup> <sup>þ</sup> *<sup>X</sup>*1*X*<sup>2</sup>

<sup>1</sup>*Y*2*Z*<sup>2</sup> <sup>þ</sup> *aX*<sup>2</sup>

*bX*1*Z*1*Z*<sup>2</sup>

**3.2 Projective coordinates and group law**

*char*ð Þ¼ **K** 3 *j* ¼ 0 *Y*<sup>2</sup>

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

*char*ð Þ¼ **K** 2 *j* ¼ 0 *Y*<sup>2</sup>

**Theorem 3.5** *[Characteristic two case]:*

<sup>1</sup>*Y*<sup>2</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup>

<sup>1</sup>*Y*<sup>2</sup> <sup>þ</sup> *aX*1*X*<sup>2</sup>

<sup>2</sup>*Y*<sup>1</sup> <sup>þ</sup> *<sup>Y</sup>*<sup>2</sup>

<sup>2</sup>*Y*1*Z*<sup>1</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>X</sup>*2*Y*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *bX*<sup>2</sup> 1*Z*2 <sup>2</sup> <sup>þ</sup> *abX*<sup>2</sup> 2*Z*2 <sup>1</sup> <sup>þ</sup> *abX*<sup>2</sup> 1*Z*2

<sup>1</sup>*Z*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *bX*2*Y*2*Z*<sup>2</sup>

*Z*2 1*Z*2 2

<sup>2</sup>*Z*<sup>1</sup> <sup>þ</sup> *<sup>X</sup>*2*Y*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *bX*2*Z*<sup>2</sup>

• *If n<sup>π</sup>*ð Þ *<sup>X</sup>*<sup>1</sup> : *<sup>n</sup><sup>π</sup>*ð Þ *<sup>Y</sup>*<sup>1</sup> : *<sup>n</sup><sup>π</sup>* <sup>½</sup> ð Þ *<sup>Z</sup>*<sup>1</sup> � ¼ *<sup>n</sup><sup>π</sup>*ð Þ *<sup>X</sup>*<sup>2</sup> : *<sup>n</sup><sup>π</sup>*ð Þ *<sup>Y</sup>*<sup>2</sup> : *<sup>n</sup><sup>π</sup>* ½ � ð Þ *<sup>Z</sup>*<sup>2</sup> , then:

<sup>1</sup> <sup>þ</sup> *<sup>X</sup>*1*X*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *bY*1*Z*<sup>2</sup>

*X*2 1*X*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *bX*<sup>2</sup>

2*Y*<sup>2</sup>

<sup>2</sup>*Y*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

<sup>1</sup>*Y*2*Z*<sup>2</sup> <sup>þ</sup> *<sup>Y</sup>*1*Y*<sup>2</sup>

<sup>1</sup>*X*2*Z*<sup>2</sup> <sup>þ</sup> *aX*1*X*<sup>2</sup>

• *If n<sup>π</sup>*ð Þ *<sup>X</sup>*<sup>1</sup> : *<sup>n</sup><sup>π</sup>*ð Þ *<sup>Y</sup>*<sup>1</sup> : *<sup>n</sup><sup>π</sup>* ½ � ð Þ *<sup>Z</sup>*<sup>1</sup> 6¼ *<sup>n</sup><sup>π</sup>*ð Þ *<sup>X</sup>*<sup>2</sup> : *<sup>n</sup><sup>π</sup>*ð Þ *<sup>Y</sup>*<sup>2</sup> : *<sup>n</sup><sup>π</sup>* ½ � ð Þ *<sup>Z</sup>*<sup>2</sup> , *then:*

<sup>1</sup>*Y*2*Z*<sup>2</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup>

<sup>1</sup>*Z*<sup>2</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *bX*<sup>2</sup> 1*Z*2

Δ is called the discriminant of **E** and *j* its *j* – invariant.

**Remark 3.1***On the field R*<sup>1</sup> ¼ *q, we denote the discriminant by* Δ<sup>0</sup> *and the j-invariant by j* <sup>0</sup>*, while on the ring Rn, n*>1 *we denote the discriminant by* Δ*<sup>ε</sup>*,*<sup>n</sup> and the j-invariant by j<sup>ε</sup>*,*<sup>n</sup>.*

*We have n<sup>π</sup>*ð Þ¼ <sup>Δ</sup>*<sup>ε</sup>*,*<sup>n</sup>* <sup>Δ</sup><sup>0</sup> *and n<sup>π</sup> <sup>j</sup> ε*,*n* <sup>¼</sup> *<sup>j</sup>* 0*.*

**Definition 3.2** *Let R be a finite ring and let a* <sup>¼</sup> ð Þ *<sup>a</sup>*1, *<sup>a</sup>*2, *<sup>a</sup>*3, *<sup>a</sup>*4, *<sup>a</sup>*<sup>6</sup> <sup>∈</sup>*R*<sup>5</sup> *: An elliptic curve on R corresponding to a, which we write Ea*ð Þ *R , is the set of zeros in the projective plane* <sup>2</sup> ð Þ *R of the Weierstrass Eq. (3), for which the discriminant* Δ *is invertible in R.*

**Remark 3.3** *According to the characteristic of the ring R; chra R*ð Þ *we have the following cases:*

1. If *char R*ð Þ 6¼ 2 and *char R*ð Þ 6¼ 3, then:

$$E\_{a,b}(\mathbb{R}) = \left\{ [\mathbf{X} : \mathbf{Y} : \mathbf{Z}] \in \mathbb{P}^2(\mathbb{R}) / Y^2 \mathbf{Z} = \mathbf{X}^3 + a \mathbf{X} \mathbf{Z}^2 + b \mathbf{Z}^3 \right\} \tag{38}$$

*for a*ð Þ , *<sup>b</sup>* <sup>∈</sup>*<sup>R</sup>* � *R, with* <sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup>*<sup>a</sup>*,*<sup>b</sup>* ¼ �16 4*a*<sup>3</sup> <sup>þ</sup> <sup>27</sup>*b*<sup>2</sup> ∈*R*�.

2. If *char R*ð Þ¼ 2, then *Ea*,*<sup>b</sup>*ð Þ *R* has one of the following forms:

$$E\_{a,b}(R) = \left\{ [X:Y:Z] \in \mathbb{P}^2(R)/Y^2Z + XYZ = X^3 + aX^2Z + bZ^3 \right\} \tag{39}$$

for ð Þ *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup>*R*<sup>2</sup> , with Δ ¼ Δ*<sup>a</sup>*,*<sup>b</sup>* ¼ *b*∈*R*�. *Or:*

$$E\_a(R) = \left\{ [X:Y:Z] \in \mathbb{P}^2(R)/Y^2Z + a\_1XYZ + a\_3YZ^2 = X^3 + a\_4XZ^2 + a\_6Z^3 \right\} \tag{40}$$

for *<sup>a</sup>* <sup>¼</sup> ð Þ *<sup>a</sup>*1, *<sup>a</sup>*3, *<sup>a</sup>*4, *<sup>a</sup>*<sup>6</sup> <sup>∈</sup>*R*4, with *<sup>a</sup>*<sup>1</sup> non invertible and

$$
\Delta = \Delta\_d = a\_1^3 \left( a\_1^3 a\_6 + a\_1^2 a\_3 a\_4 + a\_1 a\_4^2 + a\_3^3 \right) + a\_3^4 \in \mathbb{R}^\times. \tag{41}
$$

3. If *char R*ð Þ¼ 3, then *Ea*,*<sup>b</sup>*ð Þ *R* has one of the following forms:

$$E\_{a,b}(\mathbf{R}) = \left\{ [\mathbf{X} : \mathbf{Y} : \mathbf{Z}] \in \mathbb{P}^2(\mathbf{R}) / \mathbf{Y}^2 \mathbf{Z} = \mathbf{X}^3 + a\mathbf{X}^2 \mathbf{Z} + b\mathbf{Z}^3 \right\} \tag{42}$$

*for a*ð Þ , *<sup>b</sup>* <sup>∈</sup>*R*<sup>2</sup> *, with* <sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup>*<sup>a</sup>*,*<sup>b</sup>* ¼ �*a*<sup>3</sup>*b*∈*R*�. *Or:*

$$E\_{a,b}(\mathbf{R}) = \left\{ [\mathbf{X} : \mathbf{Y} : \mathbf{Z}] \in \mathbb{P}^2(\mathbf{R}) / \mathbf{Y}^2 \mathbf{Z} = \mathbf{X}^3 + a\mathbf{X}\mathbf{Z}^2 + b\mathbf{Z}^3 \right\} \tag{43}$$

for ð Þ *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup>*R*<sup>2</sup> , with <sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup>*<sup>a</sup>*,*<sup>b</sup>* ¼ �*a*<sup>3</sup> <sup>∈</sup> *<sup>R</sup>*�.


**Remark 3.4** *A projective elliptic curve on a field K has one of the following normal forms (Table 1):*

#### **Table 1.**

*Elliptic curve form on a field.*

#### **3.2 Projective coordinates and group law**

In this subsection, we give in projective coordinates the formulas for adding the points on an elliptic curve defined by Eq. (3) on the ring *Rn*, according to the normal form.

Using Bosma and Lenstra's theorem see [32], we can deduce the explicit formulas for the commutative additive law of the group *Ea*ð Þ *Rn* . The results are given in the next theorems following the values of the characteristic of ring *Rn* [33–36]. Let

$$[X\_1:Y\_1:Z\_1]+[X\_2:Y\_2:Z\_2]=[X\_3:Y\_3:Z\_3].\tag{44}$$

**Theorem 3.5** *[Characteristic two case]:*

$$\begin{aligned} \text{\*} \quad &f \left[ n^{\pi}(X\_1) : n^{\pi}(Y\_1) : n^{\pi}(Z\_1) \right] = \left[ n^{\pi}(X\_2) : n^{\pi}(Y\_2) : n^{\pi}(Z\_2) \right], \text{ then:} \\ \text{\*} X\_3 &= X\_1 Y\_1 Y\_2^2 + X\_2 Y\_1^2 Y\_2 + X\_2^2 Y\_1^2 + X\_1 X\_2^2 Y\_1 + a X\_1^2 X\_2 Y\_2 + a X\_1 X\_2^2 Y\_1 + a X\_1^2 X\_2^2 + \\ &b X\_1 Y\_1 Z\_2^2 + b X\_2 Y\_2 Z\_1^2 + b X\_1^2 Z\_2^2 + b Y\_1 Z\_2^2 Z\_1 + b Y\_2 Z\_1^2 Z\_2 + b X\_1 Z\_2^2 Z\_1. \\ \text{\*} Y\_3 &= Y\_1^2 Y\_2^2 + X\_2 Y\_1^2 Y\_2 + a X\_1 X\_2^2 Y\_1 + a^2 X\_1^2 X\_2^2 + b X\_1^2 X\_2 Z\_2 + b X\_1 X\_2^2 Z\_1 + \\ &b X\_1 Y\_1 Z\_2^2 + b X\_1^2 Z\_2^2 + ab X\_2^2 Z\_1^2 + ab X\_1^2 Z\_2^2 + b Y\_1 Z\_1 Z\_2^2 + b X\_1 Z\_1 Z\_2^2 + ab X\_1 Z\_1 Z\_2^2 + \\ &ab X\_2 Z\_1^2 Z\_2 + b^2 Z\_1^2 Z\_2^2 \end{aligned} \tag{46}$$

$$Z\_3 = X\_1^2 X\_2 Y\_2 + X\_1 X\_2^2 Y\_1 + Y\_1^2 Y\_2 Z\_2 + Y\_1 Y\_2^2 Z\_1 + X\_1^2 X\_2^2 + Y\_1^2 X\_2 Z\_2 + X\_1^2 Y\_2 Z\_2 + \cdots \tag{47}$$

$$a X\_1^2 Y\_2 Z\_2 + a X\_2^2 Y\_1 Z\_1 + X\_1^2 X\_2 Z\_2 + a X\_1 X\_2^2 Z\_1 + b Y\_1 Z\_1 Z\_2^2 + b Y\_2 Z\_1^2 Z\_2 + b X\_1 Z\_1 Z\_2^2. \tag{48}$$

$$\begin{aligned} \bullet \text{ } f\left[n^{\pi}(X\_1) : n^{\pi}(Y\_1) : n^{\pi}(Z\_1)\right] &\neq \left[n^{\pi}(X\_2) : n^{\pi}(Y\_2) : n^{\pi}(Z\_2)\right], \text{ then:}\\ X\_3 &= X\_1 Y\_2^2 Z\_1 + X\_2 Y\_1^2 Z\_2 + X\_1^2 Y\_2 Z\_2 + X\_2^2 Y\_1 Z\_1 + a X\_1^2 X\_2 Z\_2 + a X\_1 X\_2^2 Z\_1 + \\ b X\_1 Z\_1 Z\_2^2 + b X\_2 Z\_1^2 Z\_2. \end{aligned} \tag{48}$$

$$Y\_3 = X\_1^2 X\_2 Y\_2 + X\_1 X\_2^2 Y\_1 + Y\_1^2 Y\_2 Z\_2 + Y\_1 Y\_2^2 Z\_1 + X\_1^2 Y\_2 Z\_2 + X\_2^2 Y\_1 Z\_1 + a X\_1^2 Y\_2 Z\_2 + \dots \tag{49}$$

$$a X\_2^2 Y\_1 Z\_1 + a X\_1^2 X\_2 Z\_2 + a X\_1 X\_2^2 Z\_1 + b Y\_1 Z\_1 Z\_2^2 + b Y\_2 Z\_1^2 Z\_2 + b X\_1 Z\_1 Z\_2^2 + b X\_2 Z\_1^2 Z\_2.$$

$$\mathbf{Z}\_3 = \mathbf{X}\_1^2 \mathbf{X}\_2 \mathbf{Z}\_2 + \mathbf{X}\_1 \mathbf{X}\_2^2 \mathbf{Z}\_1 + \mathbf{Y}\_1^2 \mathbf{Z}\_2^2 + \mathbf{Y}\_2^2 \mathbf{Z}\_1^2 + \mathbf{X}\_1 \mathbf{Y}\_1 \mathbf{Z}\_2^2 + \mathbf{X}\_2 \mathbf{Y}\_2 \mathbf{Z}\_1^2 + a \mathbf{X}\_1^2 \mathbf{Z}\_2^2 + a \mathbf{X}\_2^2 \mathbf{Z}\_1^2. \tag{50}$$

**Theorem 3.6** *[Characteristic three case]:*

$$\bullet \text{ } \text{If } [n^{\pi}(X\_1) : n^{\pi}(Y\_1) : n^{\pi}(Z\_1)] = [n^{\pi}(X\_2) : n^{\pi}(Y\_2) : n^{\pi}(Z\_2)], \text{ then:} $$

$$X\_3 = Y\_1 Y\_2^2 X\_1 + Y\_1^2 Y\_2 X\_2 + 2a X\_1^2 X\_2 Y\_2 + 2a X\_1 X\_2^2 Y\_1 + 2Z\_1 Z\_2^2 ab Y\_1 + 2Z\_1^2 Z\_2 ab Y\_2. \tag{51}$$

$$\mathbf{Y}\_3 = \mathbf{Y}\_1^2 \mathbf{Y}\_2^2 + 2a^2 \mathbf{X}\_1 \mathbf{X}\_2^2 + a^2 b \mathbf{X}\_1 \mathbf{Z}\_1 \mathbf{Z}\_2^2 + a^2 b \mathbf{X}\_2 \mathbf{Z}\_1^2 \mathbf{Z}\_2. \tag{52}$$

$$Z\_3 = aX\_1X\_2(Y\_1Z\_2 + Y\_2Z\_1) + a(X\_1Y\_2 + X\_2Y\_1)(X\_1Z\_2 + X\_2Z\_1) + Y\_1Y\_2(Y\_1Z\_2 + Y\_2Z\_1). \tag{53}$$

$$X\_3 = 2X\_1Y\_2Y\_1Z\_2 + X\_1Y\_2^2Z\_1 + 2X\_2Y\_1^2Z\_2 + X\_2Y\_1Y\_2Z\_1 + 2dX\_1^2X\_2Z\_2 + aX\_1X\_2^2Z\_1.\tag{54}$$

• *If n<sup>π</sup>*ð Þ *<sup>X</sup>*<sup>1</sup> : *<sup>n</sup><sup>π</sup>*ð Þ *<sup>Y</sup>*<sup>1</sup> : *<sup>n</sup><sup>π</sup>* ½ � ð Þ *<sup>Z</sup>*<sup>1</sup> 6¼ *<sup>n</sup><sup>π</sup>*ð Þ *<sup>X</sup>*<sup>2</sup> : *<sup>n</sup><sup>π</sup>*ð Þ *<sup>Y</sup>*<sup>2</sup> : *<sup>n</sup><sup>π</sup>* ½ � ð Þ *<sup>Z</sup>*<sup>2</sup> , *then:*

$$Y\_3 = 2Y\_1^2Y\_2Z\_2 + Y\_1Y\_2^2Z\_1 + 2dX\_1X\_2Y\_1Z\_2 + dX\_1X\_2Y\_2Z\_1 + 2dX\_1^2Y\_2Z\_2 + dX\_2^2Y\_1Z\_1. \tag{55}$$

$$\mathbf{Z}\_3 = \mathbf{2}\mathbf{Y}\_1^2 \mathbf{Z}\_2^2 + \mathbf{Y}\_2^2 \mathbf{Z}\_1^2 + a\mathbf{X}\_1 \mathbf{2}\mathbf{Z}\_2^2 + 2a\mathbf{X}\_2 \mathbf{2}\mathbf{Z}\_1^2. \tag{56}$$

**Theorem 3.7** *[The case where the characteristic is different from two and from three]:*

$$\begin{aligned} \bullet \text{ } & \bullet \left[ n^{\pi}(X\_1) : n^{\pi}(Y\_1) : n^{\pi}(Z\_1) \right] = \left[ n^{\pi}(X\_2) : n^{\pi}(Y\_2) : n^{\pi}(Z\_2) \right], \text{ then:}\\ \bullet X\_3 &= Y\_1^2 X\_2 Z\_2 - Z\_1 X\_1 Y\_2^2 a (Z\_1 X\_2 + X\_1 Z\_2) (Z\_1 X\_2 - X\_1 Z\_2) + (2 Y\_1 Y\_2 - 3 b Z\_1 Z\_2)(Z\_1 X\_2 - X\_1 Z\_2) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \tag{57} \end{aligned} \tag{57}$$

$$\begin{aligned} \mathbf{Y}\_3 &= \mathbf{Y}\_1 \mathbf{Y}\_2 (\mathbf{Z}\_2 \mathbf{Y}\_1 - \mathbf{Z}\_1 \mathbf{Y}\_2) - a \left( \mathbf{X}\_1 \mathbf{Y}\_1 \mathbf{Z}\_2^2 - \mathbf{Z}\_1^2 \mathbf{X}\_2 \mathbf{Y}\_2 \right) + (-2a \mathbf{Z}\_1 \mathbf{Z}\_2 - 3 \mathbf{X}\_1 \mathbf{X}\_2)(\mathbf{X}\_2 \mathbf{Y}\_1 - \mathbf{X}\_1 \mathbf{Y}\_2) \\ &- 3b Z\_1 \mathbf{Z}\_2 (\mathbf{Z}\_2 \mathbf{Y}\_1 - \mathbf{Z}\_1 \mathbf{Y}\_2) \end{aligned}$$

(58)

*k<sup>θ</sup>* :

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

we denote, *kG* <sup>¼</sup> *<sup>k</sup><sup>θ</sup> <sup>k</sup>*�<sup>1</sup>

**Lemma 4.1** *The application*

**4.1 The morphisms** *π<sup>k</sup>* **et** *θ<sup>k</sup>*

**Proof.**

and

**Proof.** We have:

As *P*∈*Ek*

so, *zk*�<sup>1</sup> ¼ 0*:*

**193**

� � � � � �

*<sup>k</sup>*�<sup>1</sup> *<sup>q</sup>* ! *Ek*

*q* � �*:*

*<sup>π</sup><sup>k</sup>* : *Ek*

*<sup>B</sup>* <sup>¼</sup> *<sup>y</sup>*<sup>2</sup>

*F X*ð Þ¼ , *<sup>Y</sup>*, *<sup>Z</sup> <sup>Y</sup>*<sup>2</sup>

*Ker <sup>π</sup><sup>k</sup>* � � <sup>¼</sup> *<sup>P</sup>*<sup>∈</sup> *Ek*

<sup>0</sup> � <sup>3</sup>*z*<sup>2</sup>

*<sup>C</sup>* ¼ � <sup>3</sup>*x*<sup>2</sup>

*<sup>D</sup>* <sup>¼</sup> *bn*�<sup>1</sup>*z*<sup>3</sup>

*The coefficients A, B and* �*C are the partial derivatives of the function*

� �. Hence, *π<sup>k</sup>* est surjectif. *Using corollary (2.6), we deduce that π<sup>k</sup> is a group homomorphism.*

*Ker <sup>π</sup><sup>k</sup>* � � <sup>¼</sup> *<sup>l</sup>*ϵ*<sup>k</sup>*�<sup>1</sup> : <sup>1</sup> : <sup>0</sup> � �<sup>j</sup> *<sup>l</sup>*∈*<sup>q</sup>*

*<sup>a</sup>*,*<sup>b</sup>*<sup>j</sup> *<sup>π</sup><sup>k</sup>*

<sup>þ</sup> *axk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> *zk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> � �<sup>2</sup>

Then, *<sup>P</sup>* <sup>¼</sup> *xk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> : <sup>1</sup> <sup>þ</sup> *yk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> : *zk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> � � <sup>¼</sup> *xk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> : <sup>1</sup> : *zk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> � �*:*.

� � � � �

is a surjective group homomorphism.

*have AYk*�<sup>1</sup> þ *BZk*�<sup>1</sup> þ *CXk*�<sup>1</sup> ¼ *Dmod p, with*

calculated starting from *x*0, *y*0, *z*<sup>0</sup>

the existence of *xk*�<sup>1</sup> : *yk*�<sup>1</sup> : *zk*�<sup>1</sup>

**Lemma 4.2** *For all k*≥ 2,

*<sup>a</sup>*,*<sup>b</sup>*, we have

*zk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> <sup>¼</sup> *xk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> � �<sup>3</sup>

¼ 0,

ð Þ *<sup>x</sup>*1, *<sup>x</sup>*2, *:*…, *xk*�<sup>1</sup> <sup>↦</sup> <sup>P</sup>*<sup>k</sup>*�<sup>1</sup>

*a*,*b*

*<sup>a</sup>*,*<sup>b</sup>* ! *Ek*�<sup>1</sup> *<sup>π</sup>k*ð Þ *<sup>a</sup>* ,*πk*ð Þ *<sup>b</sup>*

½ � *X* : *Y* : *Z* ↦ ½ � *πk*ð Þ *X* : *πk*ð Þ *Y* : *πk*ð Þ *Z*

*π<sup>k</sup> is well defined because π<sup>k</sup> is a morphism of rings. According to theorem (2.7), we*

<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*0*z*<sup>2</sup> 0

<sup>0</sup> <sup>þ</sup> *an*�<sup>1</sup>*x*0*z*<sup>2</sup>

*<sup>i</sup>*¼<sup>1</sup> *xi*ϵ*<sup>i</sup>* : <sup>1</sup> :

h i

P*<sup>k</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>3</sup> *zi*ϵ*<sup>i</sup>*

*A* ¼ 2*y*0*z*0, (65)

<sup>0</sup>*b*<sup>0</sup> � 2*z*0*a*0*x*0, (66)

� � (67)

*<sup>Z</sup>* � *<sup>X</sup>*<sup>3</sup> � *<sup>a</sup>*0*XZ*<sup>2</sup> � *<sup>b</sup>*0*Z*<sup>3</sup> (69)

� �*:* (70)

ð Þ¼ *<sup>P</sup>* ½ � <sup>0</sup> : <sup>1</sup> : <sup>0</sup> � �*:* (71)

<sup>þ</sup> *b zk*�<sup>1</sup>ϵ*<sup>k</sup>*�<sup>1</sup> � �<sup>3</sup>

(72)

� �, which are not all equal to zero and deducing

<sup>0</sup>*:* (68)

(63)

(64)

$$\mathbf{Z}\_3 = (\mathbf{Z}\_1 \mathbf{Y}\_2 + \mathbf{Z}\_2 \mathbf{Y}\_1)(\mathbf{Z}\_2 \mathbf{Y}\_1 - \mathbf{Z}\_1 \mathbf{Y}\_2) + (3 \mathbf{X}\_1 \mathbf{X}\_2 + a \mathbf{Z}\_1 \mathbf{Z}\_2)(\mathbf{Z}\_1 \mathbf{X}\_2 - \mathbf{X}\_1 \mathbf{Z}\_2) \tag{59}$$

$$\begin{aligned} \bullet \, \, ^\pi f \left[ n^\pi(X\_1) : n^\pi(Y\_1) : n^\pi(Z\_1) \right] &\neq \left[ n^\pi(X\_2) : n^\pi(Y\_2) : n^\pi(Z\_2) \right], \, \, \text{then:}\\ X\_3 &= \left( Y\_1 Y\_2 - 6bZ\_1 Z\_2 \right) (X\_2 Y\_1 + X\_1 Y\_2) + \left( a^2 Z\_1 Z\_2 - 2aX\_1 X\_2 \right) (Z\_1 Y\_2 + Z\_2 Y\_1) \\ &- 3b \left( X\_1 Y\_1 Z\_2^2 + Z\_1^2 X\_2 Y\_2 \right) - a \left( Y\_1 Z\_1 X\_2^2 + X\_1^2 Y\_2 Z\_2 \right) \end{aligned} \tag{60}$$

$$\begin{aligned} Y\_3 &= \, Y\_1^2 Y\_2^2 + 3a X\_1^2 X\_2^2 + \left(-a^3 - 9b^2\right) Z\_1^2 Z\_2^2 - a^2 (Z\_1 X\_2 + X\_1 Z\_2)^2 - 2a^2 Z\_1 X\_1 Z\_2 X\_2 \\ &+ (9b X\_1 X\_2 - 3ab Z\_1 Z\_2)(Z\_1 X\_2 + X\_1 Z\_2) \end{aligned} \tag{61}$$

$$\mathbf{Z}\_3 = (\mathbf{Y}\_1 \mathbf{Y}\_2 + \mathbf{3}b\mathbf{Z}\_1 \mathbf{Z}\_2)(\mathbf{Z}\_1 \mathbf{Y}\_2 + \mathbf{Z}\_2 \mathbf{Y}\_1) + (\mathbf{3}\mathbf{X}\_1 \mathbf{X}\_2 + 2a\mathbf{Z}\_1 \mathbf{Z}\_2)(\mathbf{X}\_2 \mathbf{Y}\_1 + \mathbf{X}\_1 \mathbf{Y}\_2) + \tag{62}$$
  $a(\mathbf{X}\_1 \mathbf{Y}\_1 \mathbf{Z}\_2^2 + \mathbf{Z}^1 \mathbf{X}\_2 \mathbf{Y}\_2)$ .

### **4. Elliptic curve on** *Rn* **where** *char*ð Þ **Rn** 6¼ **2, 3**

The objective of this chapter is to study elliptic curves defined by a Weierstrass equation with coefficients in a ring *Rn* such that *char*ð Þ **Rn** 6¼ 2, 3. We denote it by *En <sup>a</sup>*,*<sup>b</sup>:* Let

$$\begin{array}{c} \mathbb{R}\_{\theta}: \begin{array}{c} \mathbb{F}\_{q}^{k-1} \to \begin{array}{c} E\_{a,b}^{k} \\ (\mathbb{x}\_{1}, \mathbb{x}\_{2}, \dots, \mathbb{x}\_{k-1}) \end{array} \mapsto \begin{bmatrix} \sum\_{i=1}^{k-1} \mathbb{x}\_{i} \mathbf{e}^{i} : \mathbb{1} : \sum\_{i=3}^{k-1} \mathbb{z}\_{i} \mathbf{e}^{i} \end{bmatrix} \end{array} \tag{63}$$

$$\pi^k: \begin{array}{l} E\_{a,b}^k \longrightarrow E\_{\pi\_k(a), \pi\_k(b)}^{k-1} \\ [X:Y:Z] \mapsto [\pi\_k(X):\pi\_k(Y):\pi\_k(Z)] \end{array} \tag{64}$$

$$A = \mathfrak{Z}\mathfrak{z}\_0\mathfrak{z}\_0,\tag{65}$$

$$B = y\_0^2 - \mathfrak{Z}\_0^2 b\_0 - 2\mathfrak{z}\_0 \mathfrak{a}\_0 \mathfrak{x}\_0,\tag{66}$$

$$\mathbf{C} = -\left(\mathbf{3x}\_0^2 + a\_0 x\_0^2\right) \tag{67}$$

$$D = b\_{n-1}z\_0^3 + a\_{n-1} \varkappa\_0 z\_0^2. \tag{68}$$

$$F(X,Y,Z) = Y^2 Z - X^3 - a\_0 X Z^2 - b\_0 Z^3 \tag{69}$$

$$\operatorname{Ker}(\pi^k) = \{ \left[ l \epsilon^{k-1} : \mathbf{1} : \mathbf{0} \right] \mid \ l \in \mathbb{F}\_q \}. \tag{70}$$

$$\text{Ker}\left(\pi^{k}\right) = \left\{ P \in E\_{a,b}^{k} \,|\,\,\,\pi^{k}(P) = \left[\mathbf{0} : \mathbf{1} : \mathbf{0}\right] \,\, \right\}.\tag{71}$$

$$\begin{split} \mathbf{z}\_{k-1}\mathbf{e}^{k-1} &= \left(\mathbf{x}\_{k-1}\mathbf{e}^{k-1}\right)^{3} + a\mathbf{x}\_{k-1}\mathbf{e}^{k-1} \left(\mathbf{z}\_{k-1}\mathbf{e}^{k-1}\right)^{2} + b\left(\mathbf{z}\_{k-1}\mathbf{e}^{k-1}\right)^{3} \\ &= \mathbf{0}, \end{split} \tag{72}$$

This yields *Ker <sup>π</sup><sup>k</sup>* <sup>¼</sup> *<sup>l</sup>*ϵ*<sup>k</sup>*�<sup>1</sup> : <sup>1</sup> : <sup>0</sup> <sup>j</sup> *<sup>l</sup>*∈*<sup>q</sup>* . **Lemma 4.3** *The application*

$$\theta\_k: \begin{array}{rcl} \mathbb{F}\_q & \to & E\_{a,b}^k \\\\ l & \mapsto \ [l\epsilon^{k-1}: \mathbf{1}: \mathbf{0}] \end{array} \tag{73}$$

**Proof.** ∀*P*∈*Ek*

**Proof.**

Then

*<sup>a</sup>*,*<sup>b</sup>*, *NP*∈*kG*, so *pNP* ¼ ½ � 0 : 1 : 0 *:*.

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

Let *P*∈*kG*, we have *pP* ¼ ½ � 0 : 1 : 0 *:*

Hence, there is a unique homomorphism

which makes the diagram(d) commutative.

*a*0,*b*0 *:*

*<sup>a</sup>*0,*b*<sup>0</sup> , then there exists *<sup>P</sup>*<sup>0</sup> <sup>∈</sup> *Ek*

such that *πkosk* ¼ *idE*<sup>1</sup>

**Proof.** Let *N*<sup>0</sup>

Let *P*∈*E*<sup>1</sup>

**195**

**Lemma 4.6** *If p do not divide N, then there exists a unique homomorphism*

for which the following diagram is commutative; (named Diagram(d)).

*<sup>a</sup>*0,*b*<sup>0</sup> ! *<sup>E</sup><sup>k</sup>*

*<sup>a</sup>*,*<sup>b</sup>* (79)

*kG* ⊂*ker p*ð Þ ½ � *:* (80)

1 � *NN*<sup>0</sup> ½ �¼ ½ �*t o p*½ �*:* (83)

*<sup>a</sup>*,*<sup>b</sup>* such that *π<sup>k</sup> P*<sup>0</sup> ð Þ¼ *P:* So,

*<sup>a</sup>*,*<sup>b</sup>* (81)

*<sup>a</sup>*,*<sup>b</sup>* (82)

*<sup>a</sup>*,*<sup>b</sup>* (84)

*vk* : *E*<sup>1</sup>

*vk* : *E*<sup>1</sup>

*sk* : *E*<sup>1</sup>

∈ ℤ as it exists *t*∈ ℤ checking 1 � *NN*<sup>0</sup> ¼ *tp:* Then,

According to Lemma (4.6), there is a unique homomorphism

*sk* : *E*<sup>1</sup>

**Theorem 4.7** *If p do not divide N, then there exists a unique homomorphism*

*<sup>a</sup>*0,*b*<sup>0</sup> ! *<sup>E</sup><sup>k</sup>*

*<sup>a</sup>*0,*b*<sup>0</sup> ! *Ek*

*<sup>a</sup>*0,*b*<sup>0</sup> ! *Ek*

which makes the following diagram commutative; (named Diagram(d')):

is an injective group homomorphism. **Proof.** The application *θ<sup>k</sup>* is injective by construction.

Let *l*ϵ*<sup>k</sup>*�<sup>1</sup> : 1 : 0 and *h*ϵ*<sup>k</sup>*�<sup>1</sup> : 1 : 0 be two elements in *E<sup>k</sup> <sup>a</sup>*,*<sup>b</sup>*, then:

$$k^{\pi} \left(l\epsilon^{k-1}\right) = k^{\pi} \left(h\epsilon^{k-1}\right)$$

$$k^{\pi}(\mathbf{1}) = k^{\pi}(\mathbf{1})\tag{74}$$

$$k^{\pi}(\mathbf{0}) = k^{\pi}(\mathbf{0}).$$

so, using theorem (3.7),

$$X\_3 = (l+h)e^{k-1}$$

$$Y\_3 = \mathbf{1} \tag{75}$$

$$Z\_3 = \mathbf{0}.$$

This yields

$$
\theta\_k(l+h) = \theta\_k(l) + \theta\_k(h). \tag{76}
$$

Thus, *θ<sup>k</sup>* is an injective group homomorphism.

#### **4.2 Main applications**

In this subsection, we consider a prime *p* which does not divide *N*, where *<sup>N</sup>* <sup>¼</sup> ♯*E*<sup>1</sup> *<sup>k</sup><sup>π</sup>* ð Þ *<sup>a</sup>* ,*k<sup>π</sup>* ð Þ *<sup>b</sup> :*

**Corollary 4.4** *Let P*∈*E<sup>k</sup> <sup>a</sup>*,*b, then*

$$NP = [\mathbf{0} : \mathbf{1} : \mathbf{0}] \Leftrightarrow P \in E^1\_{k^x(a), k^x(b)}.\tag{77}$$

#### **Proof.**

If *P*∈*E*<sup>1</sup> *<sup>k</sup><sup>π</sup>* ð Þ *<sup>a</sup>* ,*k<sup>π</sup>* ð Þ *<sup>b</sup>* , then *NP* <sup>¼</sup> ½ � <sup>0</sup> : <sup>1</sup> : <sup>0</sup> *:* Let *<sup>P</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> <sup>þ</sup> *<sup>X</sup>* : *<sup>y</sup>*<sup>0</sup> <sup>þ</sup> *<sup>Y</sup>* : *<sup>z</sup>*<sup>0</sup> <sup>þ</sup> *<sup>Z</sup>* <sup>∈</sup>*Ek <sup>a</sup>*,*<sup>b</sup>* and *Q* ¼ *x*<sup>0</sup> : *y*<sup>0</sup> : *z*<sup>0</sup> ∈*E*<sup>1</sup> *<sup>k</sup><sup>π</sup>* ð Þ *<sup>a</sup>* ,*k<sup>π</sup>* ð Þ *<sup>b</sup> :* If *NP* ¼ ½ � 0 : 1 : 0 , then *N P*ð Þ¼ � *Q* ½ � 0 : 1 : 0 *:* So, *P* � *Q* ¼ *kθ*ð Þ *l*1, *l*2, *:*…, *lk*�<sup>1</sup> *:* We deduce that *Nli* � 0 ½ � *p* , *i* ¼ 1, 2, …, *k* � 1, where *pgcd N*ð Þ¼ , *p* 1, which proves that *li* ¼ 0 et *P* ¼ *Q:*

**Corollary 4.5**

$$\forall P \in E\_{a,b}^k, \text{we have } pNP = [0:1:0]. \tag{78}$$

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

#### **Proof.**

∀*P*∈*Ek <sup>a</sup>*,*<sup>b</sup>*, *NP*∈*kG*, so *pNP* ¼ ½ � 0 : 1 : 0 *:*. **Lemma 4.6** *If p do not divide N, then there exists a unique homomorphism*

$$
\upsilon\_k: E^1\_{a\_0, b\_0} \longrightarrow E^k\_{a, b} \tag{79}
$$

for which the following diagram is commutative; (named Diagram(d)).

#### **Proof.**

Let *P*∈*kG*, we have *pP* ¼ ½ � 0 : 1 : 0 *:*

Then

$$k\_G \subset \ker([p]).\tag{80}$$

Hence, there is a unique homomorphism

$$
\upsilon\_k: E^1\_{a\_0, b\_0} \longrightarrow E^k\_{a, b} \tag{81}
$$

which makes the diagram(d) commutative.

**Theorem 4.7** *If p do not divide N, then there exists a unique homomorphism*

$$s\_k: E^1\_{a\_0, b\_0} \longrightarrow E^k\_{a, b} \tag{82}$$

such that *πkosk* ¼ *idE*<sup>1</sup> *a*0,*b*0 *:*

#### **Proof.**

Let *N*<sup>0</sup> ∈ ℤ as it exists *t*∈ ℤ checking 1 � *NN*<sup>0</sup> ¼ *tp:* Then,

$$[\mathbf{1} - \mathbf{N} \mathbf{N}'] = [t] o[p]. \tag{83}$$

According to Lemma (4.6), there is a unique homomorphism

$$
\sigma\_k: E^1\_{a\_0, b\_0} \longrightarrow E^k\_{a, b} \tag{84}
$$

which makes the following diagram commutative; (named Diagram(d')):

Let *P*∈*E*<sup>1</sup> *<sup>a</sup>*0,*b*<sup>0</sup> , then there exists *<sup>P</sup>*<sup>0</sup> <sup>∈</sup> *Ek <sup>a</sup>*,*<sup>b</sup>* such that *π<sup>k</sup> P*<sup>0</sup> ð Þ¼ *P:* So,

$$
\pi\_k \kappa\_k(P) = \pi\_k \kappa\_k \sigma\_k(P')
$$

$$
= \pi\_k([1 - NN'](P'))
$$

$$
= \pi\_k(P' - NN'P')
\tag{85}
$$

$$
= P - NN'P
$$

$$
= P.
$$

As, *pQ* ¼ ½ � 0 : 1 : 0 , then we have

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

**Corollary 4.9** *If p do not divide N, then Ek*

**Corollary 4.10** *If p do not divide N, then*

*<sup>a</sup>*,*<sup>b</sup>* ffi *CN* � *<sup>k</sup>*�<sup>1</sup>

*<sup>q</sup>* , see [27, 30, 33].

*<sup>a</sup>*,*<sup>b</sup>* ffi *=n*1 � *=n*2 � *<sup>k</sup>*�<sup>1</sup>

*<sup>a</sup>*0,*b*<sup>0</sup> ffi *CN*, with *CN* cyclic

*Ek <sup>a</sup>*,*<sup>b</sup>* ffi *<sup>E</sup>*<sup>1</sup>

We conclude,

We have, *kG* ffi *<sup>k</sup>*�<sup>1</sup>

*Ek*

*or Ek*

*E*1

*or E*1

**Corollary 4.11** *If p do not divide N, then* ffiffiffi

ffiffiffi *<sup>q</sup>* <sup>p</sup> � <sup>1</sup> � �<sup>2</sup>

According to Haas' theorem, we have:

**Proof.**

**Proof.** We have

And

**Proof.**

so

**197**

**5. Applications**

**5.1 The discrete logarithm on E***<sup>n</sup>*

*NN*<sup>0</sup>

*Q* ¼ ð Þ 1 � *tp Q* ¼ *Q* � *tpQ*

*<sup>a</sup>*,*<sup>b</sup>* ffi *<sup>E</sup>*<sup>1</sup>

*Fkof <sup>k</sup>*ð Þ¼ *P*, *Q* ð Þ *P*, *Q :* (92)

*<sup>a</sup>*0,*b*<sup>0</sup> � *<sup>k</sup>*�<sup>1</sup> *<sup>q</sup> :*

*<sup>q</sup>* , *where n*2∣ð Þ *n*<sup>1</sup> ∧ *p* � 1 *:*

*<sup>q</sup> :* (95)

*a*,*b* � �≤ ffiffiffi

*q* p (96)

*<sup>q</sup><sup>k</sup>*�<sup>1</sup> <sup>≤</sup>♯ *<sup>E</sup><sup>k</sup>*

*q<sup>k</sup>*�<sup>1</sup>

(91)

(93)

(94)

*q<sup>k</sup>*�<sup>1</sup>*:*

*<sup>q</sup>* <sup>p</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup>

*:* (97)

¼ *Q:*

*<sup>q</sup>* , *with CN cyclic*

*<sup>a</sup>*0,*b*<sup>0</sup> ffi *=n*1 � *=n*2, where *n*2∣ð Þ *n*<sup>1</sup> ∧ *p* � 1 *:*

∣*q* þ 1 � *N*∣ ≤2 ffiffiffi

*a*,*b* � �≤ ffiffiffi

In this section, we are interested in ECC using elliptic curves over the ring *Rn*.

The discrete logarithm problem that we denote DLP, (Discrete logarithm problem), is a generally difficult problem which depends on the considered group G. In many situations, due to the asymmetry existing between problems concerning the

*<sup>q</sup><sup>k</sup>*�<sup>1</sup> <sup>≤</sup> ♯ *<sup>E</sup><sup>k</sup>*

*a***,***b*

*<sup>a</sup>*0,*b*<sup>0</sup> � *<sup>k</sup>*�<sup>1</sup>

*<sup>q</sup>* <sup>p</sup> � <sup>1</sup> � �<sup>2</sup>

*<sup>q</sup>* <sup>p</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup>

**Theorem 4.8** *If p do not divide N, then Ek <sup>a</sup>*,*<sup>b</sup>* ffi *<sup>E</sup>*<sup>1</sup> *<sup>a</sup>*0,*b*<sup>0</sup> � *kG.* **Proof.** The isomorphism

$$f\_k: \begin{array}{l} E^1\_{a\_0, b\_0} \times k\_G \ \rightarrow E^k\_{a, b} \\ (P, Q) \ \leftrightarrow s\_k(P) + Q \end{array} \tag{86}$$

admits an inverse application

$$F\_k: \begin{array}{l} E\_{a,b}^k \longrightarrow E\_{a\_0, b\_0}^1 \times k\_G \\ P \mapsto (\pi\_k(P), NN'P) \end{array} . \tag{87}$$

Indeed,

$$\begin{aligned} f\_k o F\_k(P) &= f\_k \left( (\pi\_k(P), N N'P) \right) \\ &= s\_k o \pi\_k(P) + N N'P \\ &= (1 - N N')P + N N'P \\ &= P. \end{aligned} \tag{88}$$

Likewise,

$$\begin{split} F\_k \mathfrak{of}\_k(P, \mathbf{Q}) &= F\_k(\mathfrak{s}\_k(P) + \mathbf{Q}) \\ &= (\mathfrak{\kappa}\_k(\mathfrak{s}\_k(P) + \mathbf{Q}), \text{NN}'(\mathfrak{s}\_k(P) + \mathbf{Q})). \end{split} \tag{89}$$

So,

$$\begin{aligned} \pi\_k(s\_k(P) + Q) &= \pi\_k(s\_k(P)) + \pi\_k(Q) \\ &= P + [\mathbf{0} : \mathbf{1} : \mathbf{0}] \\ &= P. \end{aligned}$$

$$\begin{aligned} NN'(s\_k(P) + Q) &= NN'(s\_k(P)) + NN'Q \\ &= NN'(1 - NN')P' + NN'Q \\ &= N'tpNP' + NN'Q \\ &= [\mathbf{0} : \mathbf{1} : \mathbf{0}] + NN'Q \\ &= NN'Q. \end{aligned} \tag{90}$$

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

As, *pQ* ¼ ½ � 0 : 1 : 0 , then we have

$$\begin{aligned} \text{NN'}Q &= (\mathbb{1} - tp)Q \\ &= Q - tpQ \\ &= Q. \end{aligned} \tag{91}$$

We conclude,

$$F\_k g f\_k(P, Q) = (P, Q). \tag{92}$$

**Corollary 4.9** *If p do not divide N, then Ek <sup>a</sup>*,*<sup>b</sup>* ffi *<sup>E</sup>*<sup>1</sup> *<sup>a</sup>*0,*b*<sup>0</sup> � *<sup>k</sup>*�<sup>1</sup> *<sup>q</sup> :* **Proof.** We have, *kG* ffi *<sup>k</sup>*�<sup>1</sup> *<sup>q</sup>* , see [27, 30, 33]. **Corollary 4.10** *If p do not divide N, then*

$$\begin{aligned} \mathbf{E}\_{a,b}^{k} & \cong \mathbf{C}\_{N} \times \mathbb{F}\_{q}^{k-1}, \text{with } \mathbf{C}\_{N} \text{ cyclic} \\ & or \\ \mathbf{E}\_{a,b}^{k} & \cong \mathbb{Z}/n\_{1}\mathbb{Z} \times \mathbb{Z}/n\_{2}\mathbb{Z} \times \mathbb{F}\_{q}^{k-1}, \text{where } n\_{2} | (n\_{1} \wedge p - \mathbf{1}). \end{aligned} \tag{93}$$

**Proof.**

We have

$$\begin{aligned} E\_{a\_0, b\_0}^1 &\cong \mathbb{C}\_N \text{, with } \mathbb{C}\_N \text{ cyclic} \\ ∨ \\ E\_{a\_0, b\_0}^1 &\cong \mathbb{Z}/n\_1\mathbb{Z} \times \mathbb{Z}/n\_2\mathbb{Z}, \text{ where } n\_2 | (n\_1 \wedge p - \mathbf{1}). \end{aligned} \tag{94}$$

And

$$E\_{a,b}^{k} \cong E\_{a\_0, b\_0}^{1} \times \mathbb{F}\_q^{k-1}.\tag{95}$$

**Corollary 4.11** *If p do not divide N, then* ffiffiffi *<sup>q</sup>* <sup>p</sup> � <sup>1</sup> � �<sup>2</sup> *<sup>q</sup><sup>k</sup>*�<sup>1</sup> <sup>≤</sup>♯ *<sup>E</sup><sup>k</sup> a*,*b* � �≤ ffiffiffi *<sup>q</sup>* <sup>p</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup> *q<sup>k</sup>*�<sup>1</sup>*:* **Proof.**

According to Haas' theorem, we have:

$$|q+1-N| \le 2\sqrt{q} \tag{96}$$

so

$$\left(\left(\sqrt{q}-1\right)^{2}q^{k-1}\leq\sharp\left(E\_{a,b}^{k}\right)\leq\left(\sqrt{q}+1\right)^{2}q^{k-1}.\tag{97}$$

#### **5. Applications**

In this section, we are interested in ECC using elliptic curves over the ring *Rn*.

#### **5.1 The discrete logarithm on E***<sup>n</sup> a***,***b*

The discrete logarithm problem that we denote DLP, (Discrete logarithm problem), is a generally difficult problem which depends on the considered group G. In many situations, due to the asymmetry existing between problems concerning the

calculation of logarithms and calculation of powers which is more easier and so of great interest in cryptograph, the above mentioned makes Diffie and Hellman were the first to build a cryptosystem from this situation [37, 38].

3.Bob chooses *s* ∈ and calculates *sP:*

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

4.Alice let public *tP* and keep private *t:*

5.Bob let public *sP* and keep private *s:*

**Remark 5.3**

calculate.

Let *Pm* ∈**E***<sup>n</sup>*

**199**

3.**E***<sup>n</sup>*

generator of **E***<sup>n</sup>*

subgroup of **E***<sup>n</sup>*

2.As soon as we have a group **E***<sup>n</sup>*

*<sup>a</sup>*,*<sup>b</sup>*, is not always cyclical.

**5.4 Elliptical ElGamal cryptosystem**

1.Key generation algorithm

• *d*∈ℕ, *P*∈**E***<sup>n</sup>*

2.Encryption algorithm

• Alice Randomly chooses *k*∈;

• She also calculates *c*<sup>2</sup> ¼ *Pm* þ ½ � *k R*;

• Then, he makes public *c*1,*c*2, or *C* ¼ ð Þ *c*1;*c*<sup>2</sup> *:*

3.To decrypt received message ð Þ *c*1,*c*<sup>2</sup> , Bob calculates:

• She calculates *<sup>c</sup>*<sup>1</sup> <sup>¼</sup> ½ � *<sup>k</sup> <sup>P</sup>*∈**E***<sup>n</sup>*

4. If, Oscar (program) is giving *d*, **E***<sup>n</sup>*

6.Then, Alice and Bob build their common secret key *K* ¼ *tsP* ¼ *stP:*

1.Unlike the classic Diffie-Hellman algorithm, we do not ask that *P* be a

consider a Diffie-Hellman system on G ¼ <*P*> which is cyclic. For this construction to have a cryptographic interest, *log <sup>P</sup>*ð Þ¼ *tP t* must be not easy to

*<sup>a</sup>*,*<sup>b</sup>* be the point representing the message m, to encrypt *Pm* :

*<sup>a</sup>*,*<sup>b</sup>*, and an element *P*∈ **E***<sup>n</sup>*

*<sup>a</sup>*,*<sup>b</sup>*, *tP and sP*, then it is able to solve the

*<sup>p</sup>* of order *p* � 1, is the cyclic

*<sup>a</sup>*,*<sup>b</sup>* of finite order we can

*<sup>a</sup>*,*<sup>b</sup>:* The analogue of the subgroup <sup>∗</sup>

*<sup>a</sup>*,*<sup>b</sup>*, generated by the point *P:*

discrete elliptical logarithm problem and find *t or s:*

• Bob chooses the private key *t* ∈ known only to him.

*<sup>a</sup>*,*<sup>b</sup>*;

Now, Oscar encounters the discrete elliptic logarithm problem, because to decipher the message *Pm* he must know *t* (i.e.; calculate *t* such that *R* ¼ ½ �*t P*).

*Pm* ¼ *c*<sup>2</sup> � ½ � *k R* ¼ *c*<sup>2</sup> � ½ � *k* ½ �*t P* ¼ *c*<sup>2</sup> � ½ �*t c*1*:* (99)

*<sup>a</sup>*,*<sup>b</sup> and R* ¼ ½ �*t P* are public.

**Definition 5.1** *Let* G *be a finite cyclic group of order ρ and s*,*r two elements of* G*. We call discrete logarithm of base s of r*, *the only element m in* ½ � ½ � 0, *<sup>ρ</sup>* � <sup>1</sup> *such that sm* <sup>¼</sup> *<sup>r</sup>: The discrete logarithm for elliptic curves is defined in an analogous way to be, the only element m in* ½ � ½ � 0, *ρ* � 1 *such that mP* ¼ *Q, where P*, *and Q are two points of an additive subgroup* <sup>G</sup> *of* **<sup>E</sup>***<sup>n</sup> <sup>a</sup>*,*<sup>b</sup>:*

By using the isomorphism proved given in theorem (4.8), we get the results gathered in the next theorem:

**Theorem 5.2** *If p does not divide N, then.*


#### **5.2 Cryptography based on elliptic curves E***<sup>n</sup> a***,***b*

Elliptic curve cryptography (ECC) is public key cryptography, which relies on the use of curves over finite fields. Essentially, there are two families of these curves which are used in cryptography. The first uses elliptic curves on a finite field *pd* , where *p* is a large prime number. This family is the best choice for a high software level when implementing ECC. The second family uses elliptic curves on a binary field 2*<sup>d</sup>* where *d* is a large positive integer, this family is more appropriate at the material level point of view when implementing ECC. Another family which is also interesting in ECC implementations is the family of elliptic curves on the previously seen rings **Rn**. The most important advantage presented by the use of elliptic curves in cryptography (ECC) consists in the high security they provide for wireless applications compared to other asymmetric key cryptosystems, also their small key size. Indeed, a 160-bit key for (ECC) can replace a 1024-bit key for (RSA). Given *d*; a large integer, *P*∈**E***<sup>n</sup> <sup>a</sup>*,*<sup>b</sup>* and *<sup>Q</sup>* <sup>∈</sup>G⊂**E***<sup>n</sup> <sup>a</sup>*,*<sup>b</sup>:* The discrete elliptical logarithm problem (DLEP) consists in finding *k*∈ *such that Q* ¼ ½ � *k P*, where

$$[k]P = \underbrace{P + P + \cdots \cdot P}\_{k times} = kP.\tag{98}$$

This is in fact a difficult problem, whose resolution is exponential.

#### **5.3 Elliptical Diffie-Hellman cryptosystem**

Recall that Alice and Bob can publicly agree on a common secret (that we describe below).

1.They choose on a large integer *d*, **E***<sup>n</sup> <sup>a</sup>*,*<sup>b</sup> and P*∈ **E***<sup>n</sup> <sup>a</sup>*,*<sup>b</sup>:*

2.Alice chooses *t* ∈ and calculates *tP:*

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*


#### **Remark 5.3**


#### **5.4 Elliptical ElGamal cryptosystem**

Let *Pm* ∈**E***<sup>n</sup> <sup>a</sup>*,*<sup>b</sup>* be the point representing the message m, to encrypt *Pm* :

	- Bob chooses the private key *t* ∈ known only to him.
	- *d*∈ℕ, *P*∈**E***<sup>n</sup> <sup>a</sup>*,*<sup>b</sup> and R* ¼ ½ �*t P* are public.
	- Alice Randomly chooses *k*∈;
	- She calculates *<sup>c</sup>*<sup>1</sup> <sup>¼</sup> ½ � *<sup>k</sup> <sup>P</sup>*∈**E***<sup>n</sup> <sup>a</sup>*,*<sup>b</sup>*;
	- She also calculates *c*<sup>2</sup> ¼ *Pm* þ ½ � *k R*;
	- Then, he makes public *c*1,*c*2, or *C* ¼ ð Þ *c*1;*c*<sup>2</sup> *:*

3.To decrypt received message ð Þ *c*1,*c*<sup>2</sup> , Bob calculates:

$$P\_m = c\_2 - [k]R = c\_2 - [k][t]P = c\_2 - [t]c\_1. \tag{99}$$

Now, Oscar encounters the discrete elliptic logarithm problem, because to decipher the message *Pm* he must know *t* (i.e.; calculate *t* such that *R* ¼ ½ �*t P*).

#### **5.5 Coding example**

Let *<sup>d</sup>* be a positive integer, we consider the quotient ring **R2** <sup>¼</sup> 2*<sup>d</sup>* ½ � *<sup>X</sup> <sup>X</sup>*<sup>2</sup> ð Þ , where 2*<sup>d</sup>* is the finite field of order 2*<sup>d</sup>*.

Then the ring **R2** is identified with the ring 2*<sup>d</sup>* ½ � *<sup>ε</sup>* , where *<sup>ε</sup>*<sup>2</sup> <sup>¼</sup> 0, i.e.,

$$\mathbf{R\_2} = \left\{ a\_0 + a\_1 \cdot \varepsilon |a\_0, a\_1 \in \mathbb{F}\_{2^d} \right\}. \tag{100}$$

2.Example: with the same *a* and *b*;

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

The elliptic curve **E**<sup>2</sup>

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup>

<sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup>

<sup>1</sup>�, *<sup>α</sup>*<sup>2</sup>

<sup>1</sup>�, *<sup>α</sup>*<sup>2</sup>

<sup>1</sup>�, *<sup>α</sup>*<sup>2</sup>

**201**

*α*2

*a* ¼ 1 þ *α*; (108)

*ε:* (109)

(110)

(111)

*<sup>b</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup>

*<sup>a</sup>*,*<sup>b</sup>* contains 112 elements to know:

*<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup>�, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*2*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, 1 <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> ½ � *<sup>ε</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , *<sup>α</sup>*2*<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*2*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*2*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> ½ � : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* :

<sup>0</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> :

*<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>*<sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *αε* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *αε* <sup>þ</sup> <sup>1</sup> :

*<sup>ε</sup>* : *αε* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>ε</sup>* : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> :

<sup>1</sup> : <sup>1</sup>�, *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> :

*<sup>ε</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup>

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* :

<sup>1</sup>�, *αε* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , *αε* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> :

<sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> :

*<sup>ε</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup>

<sup>1</sup>�, *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup>

<sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> : *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> :

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* : <sup>1</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> :

*<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup>

<sup>1</sup>�, *αε* <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup>

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup>

*<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*<sup>2</sup>

*αε* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup>

*<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup>

*<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup>

*<sup>ε</sup>* : <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> :

*ε* :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup>

*ε* :

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* :

*<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*<sup>2</sup>

*<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , *<sup>α</sup>*<sup>2</sup>

<sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* : <sup>1</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup>

*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* :

**<sup>E</sup>***a*,*b*ð Þ¼f <sup>A</sup><sup>2</sup> *<sup>α</sup>*½ � ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* : <sup>1</sup> : <sup>0</sup> , *<sup>α</sup>*½ � ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* : <sup>1</sup> : <sup>0</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* : <sup>1</sup> : <sup>0</sup>�, *<sup>α</sup>*½ � ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* : <sup>1</sup> : <sup>0</sup> , *<sup>α</sup>*<sup>2</sup> ½ � *<sup>ε</sup>* : <sup>1</sup> : <sup>0</sup> , ½ � *αε* : <sup>1</sup> : <sup>0</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *αε* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>* ½ � ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* : <sup>1</sup> , *<sup>α</sup>*<sup>2</sup> ½ � *<sup>ε</sup>* : *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> :

We consider the elliptic curve on the ring **R2** given by the equation:

$$\mathbf{Y}^2 \mathbf{Z} + \mathbf{X} \mathbf{Y} \mathbf{Z} = \mathbf{X}^3 + a \mathbf{X}^2 \mathbf{Z} + b \mathbf{Z}^3. \tag{101}$$

where *a*, *b* in **R2** and *b* is invertible in **R2***:* Each element of **E**<sup>2</sup> *<sup>a</sup>*,*<sup>b</sup>* is of the form; ½ � *X* : *Y* : 1 or ½ � *xε* : 1 : 0 , with *x*∈2*<sup>d</sup>* . Write:

$$\mathbb{E}\_{a,b}^2 = \left\{ [\mathbf{X} : \mathbf{Y} : \mathbf{1}] \in \mathbb{P}\_2^2 | \mathbf{Y}^2 + \mathbf{X}\mathbf{Y} = \mathbf{X}^3 + a\mathbf{X}^2 + b \right\} \cup \left\{ [\mathbf{x}\varepsilon : \mathbf{1} : \mathbf{0}] | \mathbf{x} \in \mathbb{F}\_{2^d} \right\}. \tag{102}$$

Let **E**<sup>2</sup> *<sup>a</sup>*,*<sup>b</sup>* be the elliptic curve over *R*<sup>2</sup> and consider the irreducible polynomial *T X*ð Þ¼ <sup>1</sup> <sup>þ</sup> *<sup>X</sup>* <sup>þ</sup> *<sup>X</sup>*<sup>3</sup> in 2½ � *<sup>X</sup> :* Let *<sup>α</sup>* be such that *<sup>T</sup>*ð Þ¼ *<sup>α</sup>* 0 in <sup>8</sup> <sup>¼</sup> 2½ � *<sup>X</sup>* ð Þ *T X*ð Þ , then 1, *<sup>α</sup>*, *<sup>α</sup>*<sup>2</sup> ð Þ is a vector space base of <sup>8</sup> over 2*:*

$$\mathbb{F}\_8 = \left\{ 0, 1, a, a^2, a+1, a^2+a, a^2+1, a^2+a+1 \right\} \tag{103}$$

⋆ Put:

$$a = 1 + a;\tag{104}$$

$$b = \mathbf{1} + a^2 \mathbf{e}. \tag{105}$$

We have: **<sup>R</sup>**<sup>2</sup> <sup>¼</sup> 8½ � *<sup>ε</sup>* and **<sup>E</sup>**<sup>2</sup> *<sup>a</sup>*,*<sup>b</sup>* : *<sup>Y</sup>*<sup>2</sup> <sup>þ</sup> *XY* <sup>¼</sup> *<sup>X</sup>*<sup>3</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>X</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> ð Þ*<sup>ε</sup> :* Consider *P*∈**E**<sup>2</sup> *<sup>a</sup>*,*<sup>b</sup>* of order *l*, and consider the subgroup G ¼ <*P*>, generated by *P*, to encrypt and decrypt our messages.

1.Coding of elements of G

We will give a code to each element *Q* ¼ *mP*, where *m* ∈ f g 1, 2, *::*, *l* , defined as follows:

If *<sup>Q</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> <sup>þ</sup> *<sup>x</sup>*1*<sup>ε</sup>* : *<sup>y</sup>*<sup>0</sup> <sup>þ</sup> *<sup>y</sup>*1*<sup>ε</sup>* : *<sup>Z</sup>* , where *xi*, *yi* <sup>∈</sup><sup>8</sup> for *<sup>i</sup>* <sup>¼</sup> 0, 1, and *<sup>Z</sup>* <sup>¼</sup> <sup>0</sup> *or* 1, then we set:

$$
\omega\_i = c\_{0i} + c\_{1i}a + c\_{2i}a^2;\tag{106}
$$

$$y\_i = d\_{0i} + d\_{1i}a + d\_{2i}a^2,\tag{107}$$

where *<sup>α</sup>* is the primitive root of the irreducible polynomial *T X*ð Þ¼ <sup>1</sup> <sup>þ</sup> *<sup>X</sup>* <sup>þ</sup> *<sup>X</sup>*<sup>3</sup> , and *cij*, *dij* ∈2. So, we code *Q* as follows: If *Z* ¼ 1: *Q* ¼ *c*00*c*10*c*20*c*01*c*11*c*21*d*00*d*10*d*20*d*01*d*11*d*211*:* If *Z* ¼ 0: *Q* ¼ 00*c*01*c*11*c*21*d*01*d*11*d*2110000*:*

2.Example: with the same *a* and *b*;

$$a = 1 + a;\tag{108}$$

$$b = \mathbf{1} + a^2 \varepsilon. \tag{109}$$

The elliptic curve **E**<sup>2</sup> *<sup>a</sup>*,*<sup>b</sup>* contains 112 elements to know:

**<sup>E</sup>***a*,*b*ð Þ¼f <sup>A</sup><sup>2</sup> *<sup>α</sup>*½ � ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* : <sup>1</sup> : <sup>0</sup> , *<sup>α</sup>*½ � ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* : <sup>1</sup> : <sup>0</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* : <sup>1</sup> : <sup>0</sup>�, *<sup>α</sup>*½ � ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* : <sup>1</sup> : <sup>0</sup> , *<sup>α</sup>*<sup>2</sup> ½ � *<sup>ε</sup>* : <sup>1</sup> : <sup>0</sup> , ½ � *αε* : <sup>1</sup> : <sup>0</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *αε* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>* ½ � ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* : <sup>1</sup> , *<sup>α</sup>*<sup>2</sup> ½ � *<sup>ε</sup>* : *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup>�, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*2*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, 1 <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> ½ � *<sup>ε</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , *<sup>α</sup>*2*<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*2*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* ½ � <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*2*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> ½ � : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : (110) <sup>0</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>*<sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *αε* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *αε* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* : *αε* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>ε</sup>* : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup> *ε* : *α*2 *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* : <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> : <sup>1</sup>�, *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* : <sup>1</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*αε* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> : *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* : <sup>1</sup> , <sup>½</sup>*<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *αε* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , ½ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup>�, *αε* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , *αε* <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *αε* <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , 1ð Þ <sup>þ</sup> *<sup>α</sup> <sup>ε</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *αε* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup> , <sup>½</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>* : *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* : <sup>1</sup>�, *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> : *<sup>α</sup>*<sup>2</sup> *<sup>ε</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> : <sup>1</sup> , <sup>½</sup>*α*<sup>2</sup> *ε* : (111)

#### *Number Theory and Its Applications*

$$\begin{split} & [(a^2+1)\epsilon+1:1], [a^2\epsilon+a: (a^2+a)\epsilon+a^2+a: 1], [1+a: (a^2+1)\epsilon+a^2: \\ & [1], [(a^2+a+1)\epsilon+1+a: (a^2+a)\epsilon+a^2+a+1: 1], [a^2\epsilon+a^2: (a^2+a)\epsilon+1: \\ & [1], [a^2+1: (1+a)\epsilon+a: 1], [(a^2+1)\epsilon+a^2+a: a^2\epsilon: 1], [a^2\epsilon+1+a: a^2: \\ & [1], [0:1:0], [a:a\epsilon+a^2: 1], [\epsilon+a: a\epsilon+a^2: 1], [(a^2+a+1)\epsilon+a: a\epsilon+a^2: \\ & [1], [(a^2+a)\epsilon+a^2+a+1: \epsilon+a^2+a: 1], [a^2\epsilon+a^2+a: (a^2+a)\epsilon+a^2+a: \\ & [1], [a\epsilon+a^2: (a^2+a+1)\epsilon+1: 1], [\epsilon:1:0], [(a^2+1)\epsilon+a^2: (a^2+1)\epsilon+1: 1] \end{split} \tag{112}$$

**5.6 Encryption and decryption procedures**

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

110101111100011010000000010000

• Decryption of this message:

110100110101

**6. Conclusion**

is: hello to abdelhakim.

… and we can quote here:

Proposition 3.12.

**Acknowledgements**

communication.

**203**

• The encryption of our message "for the elliptical curve", is;

(113)

(114)

01111101100110111110001101010000001010100000 11001001011010000000101111100011010100000010 10110110101001111011011111101011111000110101 00000010101011111000110111001111110011100111 11100101101001101011100110011111101101010011 10110100110101001010101101101000000101010010 10101101100000110010011110100110101101111111

01000001101011101000011101111110011000101111 10001101110011111100110110111111010100000110 10101011000101010110100110101111110011000101 00000010101101101010011101000001101011111100 1100010101100010101010110011100111101001101010

1.The generalization of Hass's theorem, corollary 4.9.

The results obtained are very important from theoretical points of view because to study an elliptic curve on a finite local ring it suffices to study these curves on finite fields, for the applications of these curves they can be applied in cryptography to reinforce security and we can use them in cryptanalysis to solve the PDL on special curves. This results are very imploring and give applications in different fields such as classical mechanics, number theory, cryptology, information theory

2.The result of the corollary 4.11, then in [24], we have the result of the

The authors thank professor Karim Mounirh for his correcting typographical, grammatical, spelling, and punctuation errors to ensure the clear and accurate

We consider: *<sup>P</sup>* <sup>¼</sup> *<sup>α</sup>* : *<sup>α</sup>* <sup>þ</sup> *<sup>α</sup>* <sup>½</sup> <sup>2</sup> <sup>þ</sup> *αε* : <sup>1</sup>� ¼ 0100000110101, then G ¼ <sup>&</sup>lt;*P*<sup>&</sup>gt; is of order 28. We attach to each point *Q* ∈G a letter of the alphabet and a code. We collect the results in the following **Table 2**:


**Table 2.** *Points coding.* *Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

#### **5.6 Encryption and decryption procedures**

• The encryption of our message "for the elliptical curve", is;

01111101100110111110001101010000001010100000 11001001011010000000101111100011010100000010 10110110101001111011011111101011111000110101 00000010101011111000110111001111110011100111 11100101101001101011100110011111101101010011 10110100110101001010101101101000000101010010 10101101100000110010011110100110101101111111 110101111100011010000000010000 (113)

• Decryption of this message:

01000001101011101000011101111110011000101111 10001101110011111100110110111111010100000110 10101011000101010110100110101111110011000101 00000010101101101010011101000001101011111100 1100010101100010101010110011100111101001101010 110100110101 (114)

is: hello to abdelhakim.

#### **6. Conclusion**

The results obtained are very important from theoretical points of view because to study an elliptic curve on a finite local ring it suffices to study these curves on finite fields, for the applications of these curves they can be applied in cryptography to reinforce security and we can use them in cryptanalysis to solve the PDL on special curves. This results are very imploring and give applications in different fields such as classical mechanics, number theory, cryptology, information theory … and we can quote here:


#### **Acknowledgements**

The authors thank professor Karim Mounirh for his correcting typographical, grammatical, spelling, and punctuation errors to ensure the clear and accurate communication.

*Number Theory and Its Applications*

**References**

586-615

[1] Boneh D, Franklin M. Identity-based encryption from the Weil pairing. SIAM Journal on Computing. 2003;**32**(3):

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

[11] Lang S. Algebraic Number Theory.

[12] Lenstra AK, Lenstra HW Jr, editors. The Development of the Number Field Sieve, Volume 1554 of Lecture Notes in Mathematics. Berlin: Springer-Verlag;

[13] Joux A. A one round protocol for tripartite Diffie-Hellman. Journal of Cryptology. 2004;**17**(4):263-276

[14] Joux A, Pierrot C. Improving the Polynomial Time Precomputation of Frobenius Representation Discrete Logarithm Algorithms Simplified Setting for Small Characteristic Finite

Fields. Springer-Verlag; 2014

[17] Menezes AJ, Okamoto T,

[15] Joux A, Vitse V. Elliptic curve discrete logarithm problem over small degree extension fields application to the static Diffie-Hellman problem on

*:* Journal of Cryptology. 2013;**26**

[16] Maurer UM, Wolf S. Diffie Hellman oracles. In: Crypto, 96, LNCS. 1109. Springer-Verlag; 1996. pp. 268-282

Vanstone SA. Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Transactions on Information Theory. 1993;**39**(5):1639-1646

[18] Popescu C. An identification scheme based on the elliptic curve discrete logarithm problem. In: Proceedings of the Fourth International Conference/ Exhibition on High Performance Computing in the Asia-Pacific Region. Vol. 2. Beijing: IEEE; 2000.

[19] Fontein F. Elliptic Curves over Rings with a Point of View on Cryptography and Factoring. Vol. 28. Company, Reading, Massachusetts:

New York: Springer; 1986

1993

**E** *q*<sup>5</sup>

(1):119-143

pp. 624-625

[2] Hoffstein J, Pipher J, Silverman JH. An Introduction to Mathematical Cryptography. Springer, New York, NJ,

USA: Undergraduate Texts in

[3] Koblitz N. Elliptic curve cryptosystems. Mathematics of Computation. 1987;**48**:203-209

[4] Diffie W, Hellman ME. New directions in cryptography. In: IEEE Transactions on Information Theory. Vol. 22. IEEE Xplore; 1976. pp. 644-654

[5] Johnson D,Menezes A, Vanstone S. The elliptic curve digital signature algorithm (ECDSA). International Journal of Information Security. 2001;**1**(1):36-63

[6] Tzer-Shyong C, Gwo-Shiuan H, Tzuoh-Pyng L, Yu-Fang C. Digital signature scheme resulted from

cryptosystem. In: Tencon '02. Proceedings. 2002 IEEE Region 10 Conference on Computers,

pp. 192-195

identification protocol by elliptic curve

Communications, Control and Power Engineering. Vol. 1. Beijing: IEEE; 2002.

[7] Shannon CE. Communication theory

[8] Shannon CE. A mathematical theory

of secrecy systems. Bell System Technical Journal. 1949;**28**(4):656-715

of communication. Bell System Technical Journal. 1948;**27**(4):379-423

University Press; 1998

**205**

[9] Beutelspacher A, Rosenbaum U. Projective Geometry: From Fondations to Application. Cambridge: Cambridge

[10] Hartshorne J. Algebraic Geometry. Vol. 52. Springer-Verlag, GTM; 1977

Mathematics; 2008

#### **Author details**

Abdelhakim Chillali\* and Lhoussain El Fadil Sidi Mohamed Ben Abdellah University, Morocco

\*Address all correspondence to: abdelhakim.chillali@usmba.ac.ma

© 2020 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.

*Elliptic Curve over a Local Finite Ring* Rn *DOI: http://dx.doi.org/10.5772/intechopen.93476*

#### **References**

[1] Boneh D, Franklin M. Identity-based encryption from the Weil pairing. SIAM Journal on Computing. 2003;**32**(3): 586-615

[2] Hoffstein J, Pipher J, Silverman JH. An Introduction to Mathematical Cryptography. Springer, New York, NJ, USA: Undergraduate Texts in Mathematics; 2008

[3] Koblitz N. Elliptic curve cryptosystems. Mathematics of Computation. 1987;**48**:203-209

[4] Diffie W, Hellman ME. New directions in cryptography. In: IEEE Transactions on Information Theory. Vol. 22. IEEE Xplore; 1976. pp. 644-654

[5] Johnson D,Menezes A, Vanstone S. The elliptic curve digital signature algorithm (ECDSA). International Journal of Information Security. 2001;**1**(1):36-63

[6] Tzer-Shyong C, Gwo-Shiuan H, Tzuoh-Pyng L, Yu-Fang C. Digital signature scheme resulted from identification protocol by elliptic curve cryptosystem. In: Tencon '02. Proceedings. 2002 IEEE Region 10 Conference on Computers, Communications, Control and Power Engineering. Vol. 1. Beijing: IEEE; 2002. pp. 192-195

[7] Shannon CE. Communication theory of secrecy systems. Bell System Technical Journal. 1949;**28**(4):656-715

[8] Shannon CE. A mathematical theory of communication. Bell System Technical Journal. 1948;**27**(4):379-423

[9] Beutelspacher A, Rosenbaum U. Projective Geometry: From Fondations to Application. Cambridge: Cambridge University Press; 1998

[10] Hartshorne J. Algebraic Geometry. Vol. 52. Springer-Verlag, GTM; 1977

[11] Lang S. Algebraic Number Theory. New York: Springer; 1986

[12] Lenstra AK, Lenstra HW Jr, editors. The Development of the Number Field Sieve, Volume 1554 of Lecture Notes in Mathematics. Berlin: Springer-Verlag; 1993

[13] Joux A. A one round protocol for tripartite Diffie-Hellman. Journal of Cryptology. 2004;**17**(4):263-276

[14] Joux A, Pierrot C. Improving the Polynomial Time Precomputation of Frobenius Representation Discrete Logarithm Algorithms Simplified Setting for Small Characteristic Finite Fields. Springer-Verlag; 2014

[15] Joux A, Vitse V. Elliptic curve discrete logarithm problem over small degree extension fields application to the static Diffie-Hellman problem on **E** *q*<sup>5</sup> *:* Journal of Cryptology. 2013;**26** (1):119-143

[16] Maurer UM, Wolf S. Diffie Hellman oracles. In: Crypto, 96, LNCS. 1109. Springer-Verlag; 1996. pp. 268-282

[17] Menezes AJ, Okamoto T, Vanstone SA. Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Transactions on Information Theory. 1993;**39**(5):1639-1646

[18] Popescu C. An identification scheme based on the elliptic curve discrete logarithm problem. In: Proceedings of the Fourth International Conference/ Exhibition on High Performance Computing in the Asia-Pacific Region. Vol. 2. Beijing: IEEE; 2000. pp. 624-625

[19] Fontein F. Elliptic Curves over Rings with a Point of View on Cryptography and Factoring. Vol. 28. Company, Reading, Massachusetts:

Diplomarbeit, Carl von Ossietzky Universität Oldenburg Diplomstudiengang Mathematik; 2005

[20] Silverman JH. Advanced topics in the arithmetic of elliptic curves. In: Volume 151 of Graduate Texts in Mathematics. Springer; 1994

[21] Silverman JH. The arithmetic of elliptic curves. In: Graduate Texts in Mathematics. Vol. 106. Springer; 1986

[22] Semaev I. New algorithm for the discrete logarithm problem on elliptic curves. Computer Science and Security. 2015; Cryptology ePrint Archive, Report 2015/310

[23] Elhassani M, Chillali A, Mouhib A. Elliptic curve and lattice cryptosystem. In: Proceedings - 2019 International Conference on Intelligent Systems and Advanced Computing Sciences. ISACS; 2019

[24] Boulbot A, Chillali A, Mouhib A. Elliptic curves over the ring R. Boletim da Sociedade Paranaense de Matematica (BSPM). 2020;**38**(3):193-201

[25] Sahmoudi M, Chillali A. Elliptic curve on a family of finite ring. WSEAS Transactions on Mathematics. 2019;**18**: 415-422

[26] Sahmoudi M, Chillali A. SCCcryptosystem on an algebraic closure ring. Journal of Discrete Mathematical Sciences and Cryptography. 2019. DOI: 10.1080/09720529.2019.1624338

[27] Chillali A. Identification methods over. In: ICACM'11 Proceedings of the 2011 International Conference on Applied & Computational Mathematics, ACM Digital Library. Recent Researches in Applied and Computational Mathematics. 2011. pp. 133-138

[28] Hassib MH, Chillali A, Elomary MA. Special ideal ring A3 and cryptography. In: Proceedings of the International Conference JNS3. IEEE; 2013. pp. 1-4

[29] Tadmori A, Chillali A, Ziane M. The binary operations calculus in **E***<sup>a</sup>*,*b*,*<sup>c</sup>:* International Journal of Mathematical Models and Methods in Applied Sciences. 2015;**9**:171-175

[30] Chillali A. The *J<sup>ε</sup>*,*<sup>n</sup>*-invariant of **E***<sup>n</sup> <sup>A</sup>*,*<sup>B</sup>:* In: Recent Advances in Computers, Communications, Applied Social Science and Mathematics -Proceedings of ICANCM'11, ICDCC'11, IC-ASSSE-DC'11. 2011

[31] Tadmori A, Chillali A, Ziane M. Elliptic curve over ring **A4** ¼ 2*<sup>d</sup>* ½ � *<sup>ε</sup>* ; *<sup>ε</sup>*<sup>4</sup> <sup>¼</sup> <sup>0</sup>*:* Applied Mathematical Sciences. 2015;**9**(35):1721-1733

[32] Bosma W, Lenstra H. Complete system of two addition laws for elliptic curved. Journal of Number Theory. 1995;**53**:229-240

[33] Chillali A. Cryptography over elliptic curve of the ring *q*½ � *<sup>ε</sup>* , *<sup>ε</sup>*<sup>4</sup> <sup>¼</sup> <sup>0</sup>*:* World Academy of Science, Engineering and Technology. 2011;**5**:917-919

[34] Hassib MH, Chillali A, Elomary MA. Elliptic curves over a chain ring of characteristic 3. Journal of Taibah University for Science. 2014;**9**(3): 276-287

[35] Tadmori A, Chillali A, Ziane M. Normal form of the elliptic curve over the finite ring. Journal of Mathematics and System Science. 2014;**4**:194-196

[36] Tadmori A, Chillali A, Ziane M. Elliptic curve over SPIR of characteristic two. In: Proceeding of the 2013 International Conference on Applied Mathematics and Computational Methodes. 2013. pp. 41-44

[37] Okamoto T, Uchiyama S. A new pubic-key cryptosystem as secure as factoring. In: Eurocrypt, 98, LNCS. 1403. Springer-Verlag; 1998. pp. 308-318

[38] Washington LC. Elliptic curves number theory and cryptography. In: Discrete Mathematics and its Applications. Chapman and Hall/CRC; 2003

## *Edited by Cheon Seoung Ryoo*

Number theory and its applications are well known for their proven properties and excellent applicability in interdisciplinary fields of science. Until now, research on number theory and its applications has been done in mathematics, applied mathematics, and the sciences. In particular, number theory plays a fundamental and important role in mathematics and applied mathematics. This book is based on recent results in all areas related to number theory and its applications.

Published in London, UK © 2020 IntechOpen © valeriebarry / iStock

Number Theory and Its Applications

Number Theory

and Its Applications

*Edited by Cheon Seoung Ryoo*