Linearization coefficients of some particular Jacobi polynomials via hypergeometric functions

Waleed M Abd-Elhameed

doi:10.1186/s13662-016-0826-2

Abstract

The main purpose of the present paper is to establish two new linearization formulas for certain Jacobi polynomials. The new established formulas are expressed in terms of terminating hypergeometric functions of the type ${}_{4}F_{3}(1)$ . In virtue of the well-known Pfaff-Saalschutz identity, or by using some computer algebra algorithms, and in particular, the algorithms of Zeilberger, Petkovsek and van Hoeij, the resulting ${}_{4}F_{3}(1)$ can be reduced for particular choices of the involved parameters. This reduction leads to obtaining several simple linearization formulas of some particular Jacobi polynomials free of any hypergeometric functions.

Highlights

Linearization problems of orthogonal polynomials in general and of Jacobi polynomials in particular are of fundamental importance
Abd-Elhameed Advances in Difference Equations (2016) 2016:91 derived in detail new linearization formulas of the third and fourth kinds of Chebyshev polynomials
They expressed these formulas in simple forms which are free of any hypergeometric functions, while Abd-Elhameed in [, ] and Abd-Elhameed et al in [ ] have developed some new linearization formulas for certain Jacobi polynomials

Summary

Introduction

Linearization problems of orthogonal polynomials in general and of Jacobi polynomials in particular are of fundamental importance. Abd-Elhameed Advances in Difference Equations (2016) 2016:91 derived in detail new linearization formulas of the third and fourth kinds of Chebyshev polynomials. They expressed these formulas in simple forms which are free of any hypergeometric functions, while Abd-Elhameed in [ , ] and Abd-Elhameed et al in [ ] have developed some new linearization formulas for certain Jacobi polynomials. Section is devoted to utilizing some symbolic computation, and in particular, the celebrated algorithms of Zeilberger, Petkovsek, and van Hoeij, for the sake of obtaining some new linearization formulas of the four kinds of Chebyshev polynomials in reduced forms. In Section , we show the importance of the developed formulas by presenting two applications of them

Some relevant properties of the classical Jacobi polynomials

Some new linearization formulas via employing computer algebra algorithms