Linearization of the product of symmetric orthogonal polynomials
type: Item
instance of: scholarly article
title: Linearization of the product of symmetric orthogonal polynomials
zbMATH Open document ID: 0806.33003
DOI: 10.1007/BF01212564
published in: Constructive Approximation
publication date: 1994-11-08T00:00:00Z
review text: Let \((p_ n)_{n\geq 0}\) be a system of monic symmetric orthogonal polynomials defined by its three-term recurrence relation. This paper investigates an associated partial difference equation in two dimensions with solutions of the form \(p_ n(x)p_ m(x)\). To this end, an auxiliary function in four variables is introduced which may be regarded as a discrete analogue of Riemann's function for hyperbolic partial differential equations. These functions are closely related to the linearization coefficients of the products \(p_ n p_ m\), and, in fact, the author's method leads to explicit representations of the product linearization coefficients for some examples. For instance, the associated Hermite polynomials, their continuous \(q\)-analogues, and the associated continuous \(q\)-ultraspherical polynomials are treated in this paper where the linearization coefficients are \({_ 3 F_ 2}\) and \({_ 3\Phi_ 2}\) functions, respectively.
reviewed by: Michael Voit
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33D70
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 43A62
zbMATH DE Number: 672160
zbMATH Keywords: Riemann method
zbMATH Keywords: partial difference equation
zbMATH Keywords: product linearization coefficients
author: Clemens Markett
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 672160