Representation by several orthogonal polynomials for sums of finite products of Chebyshev polynomials of the first, third and fourth kinds

Taekyun Kim; Dae San Kim; Dmitry V. Dolgy; Dojin Kim

doi:10.1186/s13662-019-2058-8

Representation by several orthogonal polynomials for sums of finite products of Chebyshev polynomials of the first, third and fourth kinds

Taekyun Kim, Dae San Kim, Dmitry V. Dolgy, Dojin Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

The classical linearization problem concerns with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. As a generalization of this, we consider here sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the ones previously studied. We represent each of them as linear combinations of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. Here, the coefficients involve some terminating hypergeometric functions F12, F22, and F11. These representations are obtained by explicit computations.

Original language	English
Article number	110
Journal	Advances in Difference Equations
Volume	2019
Issue number	1
DOIs	https://doi.org/10.1186/s13662-019-2058-8
State	Published - 1 Dec 2019

Keywords

Chebyshev polynomials
Extended Laguerre polynomial
Gegenbauer polynomial
Hermite polynomial
Jacobi polynomial
Legendre polynomial
Sums of finite products

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10.1186/s13662-019-2058-8

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abstract = "The classical linearization problem concerns with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. As a generalization of this, we consider here sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the ones previously studied. We represent each of them as linear combinations of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. Here, the coefficients involve some terminating hypergeometric functions F12, F22, and F11. These representations are obtained by explicit computations.",

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AU - Kim, Dojin

PY - 2019/12/1

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N2 - The classical linearization problem concerns with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. As a generalization of this, we consider here sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the ones previously studied. We represent each of them as linear combinations of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. Here, the coefficients involve some terminating hypergeometric functions F12, F22, and F11. These representations are obtained by explicit computations.

AB - The classical linearization problem concerns with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. As a generalization of this, we consider here sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the ones previously studied. We represent each of them as linear combinations of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. Here, the coefficients involve some terminating hypergeometric functions F12, F22, and F11. These representations are obtained by explicit computations.

KW - Chebyshev polynomials

KW - Extended Laguerre polynomial

KW - Gegenbauer polynomial

KW - Hermite polynomial

KW - Jacobi polynomial

KW - Legendre polynomial

KW - Sums of finite products

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JF - Advances in Difference Equations

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Representation by several orthogonal polynomials for sums of finite products of Chebyshev polynomials of the first, third and fourth kinds

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