Expressing Sums of Finite Products of Chebyshev Polynomials of the Second Kind and of Fibonacci Polynomials by Several Orthogonal Polynomials

This paper is concerned with representing sums of the finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations, each of them is expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, which involve the hypergeometric functions 1 F 1 and 2 F 1 .

Download Full-text

Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

Symmetry ◽

10.3390/sym10110617 ◽

2018 ◽

Vol 10 (11) ◽

pp. 617 ◽

Cited By ~ 5

Author(s):

Dmitry Dolgy ◽

Dae Kim ◽

Taekyun Kim ◽

Jongkyum Kwon

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Classical Orthogonal Polynomials ◽

Linear Combinations ◽

The Third ◽

Connection Problem ◽

Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 .

Download Full-text

Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

10.20944/preprints201810.0140.v1 ◽

2018 ◽

Author(s):

Dmitry Victorovich Dolgy ◽

Dae San Kim ◽

Taekyun Kim ◽

Jongkyum Kwon

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Classical Orthogonal Polynomials ◽

Linear Combinations ◽

The Third ◽

Connection Problem ◽

Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions ${}_2 F_0, {}_2 F_1$, and ${}_3 F_2$.

Download Full-text

Studies in Sums of Finite Products of the Second, Third, and Fourth Kind Chebyshev Polynomials

Mathematics ◽

10.3390/math8020210 ◽

2020 ◽

Vol 8 (2) ◽

pp. 210

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Hyunseok Lee ◽

Jongkyum Kwon

Keyword(s):

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Linear Combinations ◽

The Third ◽

Linearization Problem ◽

Finite Products ◽

Fourth Kind ◽

Classical Linearization

In this paper, we consider three sums of finite products of Chebyshev polynomials of two different kinds, namely sums of finite products of the second and third kind Chebyshev polynomials, those of the second and fourth kind Chebyshev polynomials, and those of the third and fourth kind Chebyshev polynomials. As a generalization of the classical linearization problem, we represent each of such sums of finite products as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. These are done by explicit computations and the coefficients involve terminating hypergeometric functions 2 F 1 , 1 F 1 , 2 F 2 , and 4 F 3 .

Download Full-text

Connection Problem for Sums of Finite Products of Legendre and Laguerre Polynomials

Symmetry ◽

10.3390/sym11030317 ◽

2019 ◽

Vol 11 (3) ◽

pp. 317 ◽

Cited By ~ 5

Author(s):

Taekyun Kim ◽

Kyung-Won Hwang ◽

Dae Kim ◽

Dmitry Dolgy

Keyword(s):

Orthogonal Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Laguerre Polynomials ◽

Linear Combinations ◽

Connection Problem ◽

Finite Products

The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hypergeometric functions 1 F 1 and 2 F 1 .

Download Full-text

Sums of finite products of Chebyshev polynomials of two different types

AIMS Mathematics ◽

10.3934/math.2021722 ◽

2021 ◽

Vol 6 (11) ◽

pp. 12528-12542

Author(s):

Taekyun Kim ◽

◽

Dae San Kim ◽

Dmitry V. Dolgy ◽

Jongkyum Kwon ◽

...

Keyword(s):

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Linear Combinations ◽

The Third ◽

Fourth Type ◽

Different Types ◽

Finite Products ◽

Classical Linearization

<abstract><p>In this paper, we consider sums of finite products of the second and third type Chebyshev polynomials, those of the second and fourth type Chebyshev polynomials and those of the third and fourth type Chebyshev polynomials, and represent each of them as linear combinations of Chebyshev polynomials of all types. Here the coefficients involve some terminating hypergeometric functions $ {}_{2}F_{1} $. This problem can be viewed as a generalization of the classical linearization problems and is done by explicit computations.</p></abstract>

Download Full-text

Representation by Chebyshev Polynomials for Sums of Finite Products of Chebyshev Polynomials

Symmetry ◽

10.3390/sym10120742 ◽

2018 ◽

Vol 10 (12) ◽

pp. 742 ◽

Cited By ~ 4

Author(s):

Taekyun Kim ◽

Dae Kim ◽

Lee-Chae Jang ◽

Dmitry Dolgy

Keyword(s):

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Linear Combinations ◽

Linearization Problem ◽

Finite Products

In this paper, we consider sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the previously-studied ones. We represent each of them as linear combinations of Chebyshev polynomials of all kinds whose coefficients involve some terminating hypergeometric functions 2 F 1 . The results may be viewed as a generalization of the linearization problem, which is concerned with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. These representations are obtained by explicit computations.

Download Full-text

New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽

10.3390/math9010074 ◽

2020 ◽

Vol 9 (1) ◽

pp. 74

Author(s):

Waleed Mohamed Abd-Elhameed ◽

Afnan Ali

Keyword(s):

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Algebraic Computation ◽

Ultraspherical Polynomials ◽

Current Article ◽

Linearization Coefficients

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

Download Full-text

Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas

Advances in Difference Equations ◽

10.1186/s13662-021-03244-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Waleed M. Abd-Elhameed ◽

Youssri H. Youssri

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Connection Coefficients ◽

Current Article ◽

In Virtue Of ◽

Connection Formulas

AbstractThe principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.

Download Full-text

New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

The Scientific World JOURNAL ◽

10.1155/2014/456501 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11

Author(s):

W. M. Abd-Elhameed

Keyword(s):

Galerkin Method ◽

Boundary Value Problems ◽

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Boundary Value ◽

High Order ◽

Sixth Order ◽

Special Cases ◽

Derivatives Of

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

Download Full-text