Asymptotics for the \({}_ 4F_ 3\) polynomials
type: Item
instance of: scholarly article
title: Asymptotics for the \({}_ 4F_ 3\) polynomials
zbMATH Open document ID: 0729.33006
DOI: 10.1016/0021-9045(91)90056-G
author: J. A. Wilson
published in: Journal of Approximation Theory
publication date: 1991-01-01T00:00:00Z
review text: About 15 years ago, Wilson showed that the 6-j symbols of angular momentum theory could be transformed into a set of polynomials orthogonal with respect to a measure on a finite set of points. For different choices of the parameters in these orthogonal polynomials the orthogonality is on \([0,\infty)\) with respect to a positive absolutely continuous measure, or such a measure plus a finite number of discrete masses for x negative. The absolutely continuous part of the measure decays like \(\exp [-kx^{1/2}]\). Here the asymptotics of the polynomials is found. Regular oscillations occur, but they are stretched out by replacing n by ln n, unlike the n which occurs for Jacobi polynomials (on a finite interval), or \(n^{1/2}\) for Laguerre polynomials (on \([0,\infty)\) but the weight function decays like exp(-x)). This interesting and important result is first found by giving different representations for these polynomials and then using a nice convexity argument. The asymptotic formula is then used to obtain an equiconvergence theorem with respect to a Dirichlet series. Finally, q- versions of these results are obtained.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42A10
zbMATH DE Number: 4204800
zbMATH Keywords: Wilson polynomials
zbMATH Keywords: asymptotics
zbMATH Keywords: equiconvergence theorem
reviewed by: R. Askey
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4204800