Sums of finite products of Chebyshev polynomials of the third and fourth kinds

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

10.20944/preprints201806.0079.v1 ◽

2018 ◽

Cited By ~ 1

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.

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Deriving the explicit formula of Chebyshev polynomials of the third kind and the fourth kind

10.1063/1.5064199 ◽

2018 ◽

Author(s):

I. P. Dewi ◽

S. Utama ◽

S. Aminah

Keyword(s):

Explicit Formula ◽

Chebyshev Polynomials ◽

The Third ◽

Fourth Kind

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Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Symmetry ◽

10.3390/sym10080354 ◽

2018 ◽

Vol 10 (8) ◽

pp. 354 ◽

Cited By ~ 3

Author(s):

Tomasz Czyżycki ◽

Jiří Hrivnák ◽

Jiří Patera

Keyword(s):

Orthogonal Polynomials ◽

Lie Algebras ◽

Chebyshev Polynomials ◽

Generating Functions ◽

Bell Polynomials ◽

The Third ◽

Weight Lattice ◽

Hybrid Character ◽

Polynomial Family

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

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Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

Mathematics ◽

10.3390/math7030224 ◽

2019 ◽

Vol 7 (3) ◽

pp. 224 ◽

Cited By ~ 15

Author(s):

Harendra Singh ◽

Rajesh Pandey ◽

Hari Srivastava

Keyword(s):

Chebyshev Polynomials ◽

Jacobi Polynomials ◽

Ritz Method ◽

Variational Problems ◽

Algebraic Equations ◽

The Third ◽

Fractional Variational Problems ◽

Nonlinear Algebraic Equations ◽

Non Linear ◽

Fourth Kind

The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature.

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