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

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 .

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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 ${}_3 F_2$.

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

Mathematics ◽

10.3390/math6100210 ◽

2018 ◽

Vol 6 (10) ◽

pp. 210 ◽

Cited By ~ 13

Author(s):

Taekyun Kim ◽

Dae Kim ◽

Jongkyum Kwon ◽

Dmitry Dolgy

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Classical Orthogonal Polynomials ◽

Linear Combinations ◽

Fibonacci Polynomials ◽

Finite Products

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 .

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

10.20944/preprints201809.0258.v1 ◽

2018 ◽

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Jongkyum Kwon ◽

Dmitry V. Dolgy

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Classical Orthogonal Polynomials ◽

Linear Combinations ◽

Fibonacci Polynomials ◽

Finite Products

This paper is concerned with representing sums of 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 ${}_2 F_1$.

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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 .

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On the Complex Zeros of Some Families of Orthogonal Polynomials

Abstract and Applied Analysis ◽

10.1155/2010/263860 ◽

2010 ◽

Vol 2010 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Eugenia N. Petropoulou

Keyword(s):

Asymptotic Behavior ◽

Orthogonal Polynomials ◽

Jacobi Polynomials ◽

Laguerre Polynomials ◽

The Real ◽

Ultraspherical Polynomials ◽

Complex Zeros

The complex zeros of the orthogonal Laguerre polynomials for , ultraspherical polynomials for , Jacobi polynomials for , , , orthonormal Al-Salam-Carlitz II polynomials for , , and -Laguerre polynomials for , are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.

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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>

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Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II. Associated big q-Jacobi polynomials

Journal of Mathematical Analysis and Applications ◽

10.1016/0022-247x(92)90360-p ◽

1992 ◽

Vol 171 (2) ◽

pp. 477-497 ◽

Cited By ~ 10

Author(s):

D.P. Gupta ◽

M.E.H. Ismail ◽

D.R. Masson

Keyword(s):

Orthogonal Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Basic Hypergeometric Functions ◽

Contiguous Relations

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Representing by Orthogonal Polynomials for Sums of Finite Products of Fubini Polynomials

Mathematics ◽

10.3390/math7040319 ◽

2019 ◽

Vol 7 (4) ◽

pp. 319 ◽

Cited By ~ 1

Author(s):

Dae Kim ◽

Dmitry Dolgy ◽

Dojin Kim ◽

Taekyun Kim

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Connection Problem ◽

Finite Products

In the classical connection problem, it is dealt with determining the coefficients in the expansion of the product of two polynomials with regard to any given sequence of polynomials. As a generalization of this problem, we will consider sums of finite products of Fubini polynomials and represent these in terms of orthogonal polynomials. Here, the involved orthogonal polynomials are Chebyshev polynomials of the first, second, third and fourth kinds, and Hermite, extended Laguerre, Legendre, Gegenbauer, and Jabcobi polynomials. These representations are obtained by explicit computations.

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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.

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