scholarly journals Linearization coefficients of some particular Jacobi polynomials via hypergeometric functions

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Waleed M Abd-Elhameed
Keyword(s):  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Linearization Coefficients

scholarly journals New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽  
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.

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

Mathematics ◽  
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.

scholarly journals New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

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.

scholarly journals Operational Methods in the Study of Sobolev-Jacobi Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 124 ◽  
Author(s):  
Nicolas Behr ◽  
Giuseppe Dattoli ◽  
Gérard Duchamp ◽  
Silvia Penson
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Hermite Polynomials ◽  
Umbral Calculus ◽  
Generalized Hypergeometric Functions ◽  
Normal Ordering ◽  
Gamma Functions ◽  
Image Technique ◽  
Operational Methods

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of Sobolev-Jacobi (SJ) polynomials explicitly.

Explicit Formulas for the Associated Jacobi Polynomials and Some Applications

1987 ◽  
Vol 39 (4) ◽  
pp. 983-1000 ◽  
Author(s):  
Jet Wimp
Keyword(s):  
Weight Function ◽  
Closed Form ◽  
Recurrence Relation ◽  
Hypergeometric Functions ◽  
Continued Fraction ◽  
Jacobi Polynomials ◽  
Main Diagonal ◽  
Discrete Point ◽  
Explicit Formulas ◽  
Explicit Representations

In this paper we determine closed-form expressions for the associated Jacobi polynomials, i.e., the polynomials satisfying the recurrence relation for Jacobi polynomials with n replaced by n + c, for arbitrary real c ≧ 0. One expression allows us to give in closed form the [n — 1/n] Padé approximant for what is essentially Gauss' continued fraction, thus completing the theory of explicit representations of main diagonal and off-diagonal Padé approximants to the ratio of two Gaussian hypergeometric functions and their confluent forms, an effort begun in [2] and [19]. (We actually give only the [n — 1/n] Padé element, although other cases are easily constructed, see [19] for details.)We also determine the weight function for the polynomials in certain cases where there are no discrete point masses. Concerning a weight function for these polynomials, so many writers have obtained so many partial results that our formula should be considered an epitome rather than a real discovery, see the discussion in Section 3.

scholarly journals Applications of the Generalized Kummer’s Summation Theorem to Transformation Formulas and Generating Functions

2018 ◽  
Vol 3 (2) ◽  
pp. 331-338 ◽  
Author(s):  
Ahmed Ali Atash ◽  
Hussein Saleh Bellehaj
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
General Transformation ◽  
Summation Theorem ◽  
Transformation Formulas

AbstractIn this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions K5 and K12 by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.

A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials

1981 ◽  
Vol 33 (4) ◽  
pp. 915-928 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Hypergeometric Function ◽  
Jacobi Polynomials ◽  
Defining Relation ◽  
Linearization Problem ◽  
Image Position ◽  
Linearization Coefficients ◽  
Negative Representation

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:(1.1)Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion(1.2)

Linearization of the Product of Jacobi Polynomials. III

1971 ◽  
Vol 23 (2) ◽  
pp. 332-338 ◽  
Author(s):  
Richard Askey ◽  
George Gasper
Keyword(s):  
Banach Algebra ◽  
Jacobi Polynomials ◽  
Expansion Coefficients ◽  
Linearization Coefficients

In a series of papers [1; 2; 3; 4] the operation of linearizing the product of two Jacobi polynomials Pn(α, β)(x), α, β > –1, has been investigated and the existence of a natural Banach algebra associated with the linearization coefficients has been proven. This was proven for α + β + 1 ≧ 0 in [3] and for a slightly larger region in [4]. It was shown in [4] that such a Banach algebra does not exist for . The method used in [1; 3; 4] was to prove the non-negativity of the expansion coefficients from which the existence of the Banach algebra easily follows. However, as shown in [4], the coefficients for a subset of can be negative infinitely often and so a different method must be used for these values of α and β.

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 .

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