Abstract

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.

Highlights

  • Published: 4 July 2021Special functions are crucial in several disciplines such as mathematical physics and numerical analysis

  • 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

  • We show that the moments coefficients involve a hypergeometric function of the type 4 F3 (1), which can be summed in closed analytical formulas for special choices of the involved parameters

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Summary

Introduction

Special functions are crucial in several disciplines such as mathematical physics and numerical analysis. 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. The approaches followed were based on expressing products of hypergeometric functions in terms of a single generalized hypergeometric function using some suitable transformation formulas; the current article deals with some general linearization formulas.

Some Elementary Properties of the Classical Jacobi Polynomials and Their
New Moments Formulas of the Shifted Normalized Jacobi Polynomials
A New Approach for Solving Jacobi Linearization Problem via Moments Formulas
Numerical Application on the Non-Linear Riccati Equation
Tau Algorithm for the Non-Linear Riccati Differential Equation
Numerical Tests
Conclusions
Full Text
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