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

Abstract

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.

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

“…We point out here that the main difference between our study in the current paper and the study in the recent paper [34] is that the authors in [34] investigated the linearization formula…”

“…This product formula led to some simplified linearization formulas for certain choices of the involved parameters. In [34], the authors established some specific and general linearization formulas of some classes of Jacobi polynomials based on the reduction of certain hypergeometric functions of unit arguments. For some other articles concerned with linearization problem, one can refer to [35][36][37][38][39].…”

“…The linearization coefficients of problem (2) were found in terms of a product of two terminating hypergeometric functions of the type 3 F 2 (1) (see [17]). In [34], the authors found closed forms for one or two of the appearing 3 F 2 (1) for some specific choices of the involved parameters, and therefore, some new reduced linearization formulas were developed. In the current paper, we follow two different approaches to develop some linearization formulas in the form of (1).…”

Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

“…Based on the result of Lemma 3, some manipulations lead to formula (35). Now, it is easy to deduce the high-order derivatives formula of the shifted Chebyshev polynomials of the sixth kind in terms of their original shifted ones only by replacing x by 2 x L − 1 in formula (29).…”

“…It is worthwhile to mention that the duplication, connection and linearization coefficients of Jacobi polynomials and their particular polynomials are expressed in terms of different hypergeometric functions of unit argument. These hypergeometric functions may be reduced in some cases by either standard formulas or through some suitable symbolic algorithms such as Zielberger's algorithm, see for example [33][34][35].…”