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

Published online by Cambridge University Press:  20 November 2018

Mizan Rahman
Mizan Rahman*
Affiliation:
Carleton University, Ottawa, Ontario
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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)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 915 - 928
Copyright
Copyright © Canadian Mathematical Society 1981

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