scholarly journals Sums of finite products of Chebyshev polynomials of two different types

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>

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

Mathematics ◽  
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 .

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

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 617 ◽  
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 .

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

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 ${}_3 F_2$.

scholarly journals Representing Sums of Finite Products of Chebyshev Polynomials of Third and Fourth Kinds by Chebyshev Polynomials

2018 ◽  
Author(s):  
T. Kim ◽  
D.S. Kim ◽  
D.V. Dolgy ◽  
C.S. Ryoo
Keyword(s):  
Hypergeometric Function ◽  
Chebyshev Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Finite Products

Here we consider sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials which involve the hypergeometric function 3F2.

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

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 210 ◽  
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 .

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

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 ${}_2 F_1$.

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

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 742 ◽  
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.

scholarly journals Fourier Series for Functions Related to Chebyshev Polynomials of the First Kind and Lucas Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 276 ◽  
Author(s):  
Taekyun Kim ◽  
Dae Kim ◽  
Lee-Chae Jang ◽  
Gwan-Woo Jang
Keyword(s):  
Fourier Series ◽  
Chebyshev Polynomials ◽  
Bernoulli Polynomials ◽  
Series Expansions ◽  
Linear Combinations ◽  
Lucas Polynomials ◽  
Finite Products ◽  
Derive Fourier Series

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sums of finite products of polynomials as linear combinations of Bernoulli polynomials.

scholarly journals Representing Sums of Finite Products of Chebyshev Polynomials of the First Kind and Lucas Polynomials by Chebyshev Polynomials

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 26 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jongkyum Kwon
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Lucas Polynomials ◽  
Finite Products

In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here, the coefficients involve terminating hypergeometric functions 2F1 and these representations are obtained by explicit computations.

scholarly journals Representing Sums of Finite Products of Chebyshev Polynomials of the First Kind and Lucas Polynomials by Chebyshev Polynomials

2018 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jongkyum Kwon
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Lucas Polynomials ◽  
Finite Products

In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here the coefficients involve terminating hypergeometric functions 2F1 and these representations are obtained by explicit computations.

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