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## Meet the editor

Dr. Cheon Seoung Ryoo is a professor of Mathematics at Hannam University, South Korea. He received his PhD in Mathematics from Kyushu University, Fukuoka, Japan. Dr. Ryoo is the author of several research articles on numerical computations with guaranteed accuracy. He has also contributed works to the fields of scientific computing, analytic number theory and p-adic functional analysis. More recently, he has been working with

special functions, quantum calculus and differential equations.

Contents

Section 1

by José-Antonio de la Peña

New Aspects of Descartes' Rule of Signs by Vladimir Petrov Kostov and Boris Shapiro

Polynomials with Symmetric Zeros

by Jung Yoog Kang and Cheon Seoung Ryoo

on the Basis of the Root Locus Theory

by Ricardo Vieira

Section 2

by Nesenchuk Alla

by Pablo Olivares

Q-Tangent Polynomials

Preface III

Theory of Polynomials 1

Chapter 1 3

Chapter 2 33

Chapter 3 49

Chapter 4 69

Chapter 5 89

Applications of Polynomials 107

Chapter 6 109

Chapter 7 131

G(t)<sup>α</sup>

Cyclotomic and Littlewood Polynomials Associated to Algebras

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions of the Form F(t)<sup>x</sup>

by Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya

A Numerical Investigation on the Structure of the Zeros of the

Investigation and Synthesis of Robust Polynomials in Uncertainty

Pricing Basket Options by Polynomial Approximations

## Contents


#### Chapter 8 149

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using Paralleling-in-Order Scheme by Zheng-Yu Huang, Zheng Sun and Wei He

Preface

Polynomials are well known for their evincing properties and wide applicability in interdisciplinary areas of science. The problems arising in physical sciences and engineering are mathematically framed in terms of differential equations. Most of them can only be solved using special polynomials. Special polynomials and orthogonal polynomials provide new means of analysis for solving large classes of differential equations often encountered in physical problems. In particular, sequences of special polynomials play a fundamental role in applied mathematics. Such sequences can be described in various ways, for example, by orthogonality conditions, as solutions to differential equations, by generating functions, by recur-

Written by leading researchers and mathematicians, this book provides an overview of the current research in the field of polynomials. Topics include but are not

• The modern umbral calculus (binomial, Appell, and Sheffer polynomial

polynomials, and orthogonal polynomials of several variables

• Matrix and determinant approach to special polynomial sequences

• Orthogonal polynomials, matrix orthogonal polynomials, multiple orthogonal

• Applications of special polynomial sequences in approximation theory and in

This timely book will help fill a gap in the literature on the theory of polynomials and related fields. We hope it will promote further research and development in this

rence relations and by operational formulas.

limited to the following:

boundary value problems

• Number theory and special functions

• Fractional calculus and special functions

• Asymptotic methods in orthogonal polynomials

• Symbolic computations and special functions

sequences)

important area.
