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

The principal objective of this article is to develop new formulas of the so-called Chebyshev polynomials of the fifth-kind. Some fundamental properties and relations concerned with these polynomials are proposed. New moments formulas of these polynomials are obtained. Linearization formulas for these polynomials are derived using the moments formulas. Connection problems between the fifth-kind Chebyshev polynomials and some other orthogonal polynomials are explicitly solved. The linking coefficients are given in forms involving certain generalized hypergeometric functions. As special cases, the connection formulas between Chebyshev polynomials of the fifth-kind and the well-known four kinds of Chebyshev polynomials are shown. The linking coefficients are all free of hypergeometric functions.

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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.

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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 .

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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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On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

The Scientific World JOURNAL ◽

10.1155/2013/657806 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Paul Barry

Keyword(s):

Orthogonal Polynomials ◽

Closed Form ◽

Chebyshev Polynomials ◽

Recurrence Relations ◽

Connection Coefficients ◽

Riordan Arrays ◽

Boubaker Polynomials ◽

Riordan Array

The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.

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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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New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽

10.3390/math9131573 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1573

Author(s):

Waleed Mohamed Abd-Elhameed ◽

Badah Mohamed Badah

Keyword(s):

Differential Equation ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Tau Method ◽

Riccati Differential Equation ◽

Shifted Jacobi Polynomials ◽

Linearization Problem ◽

Linearization Coefficients ◽

New Formula ◽

Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

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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.

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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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