Asymptotic expansion of orthogonal polynomials via difference equations
type: Item
instance of: scholarly article
title: Asymptotic expansion of orthogonal polynomials via difference equations
zbMATH Open document ID: 1408.41025
author: Xiao-Min Huang
author: Xiang-Sheng Wang
author: Li-Hua Cao
published in: Journal of Approximation Theory
publication date: 2019-02-07T00:00:00Z
review text: This paper deals with the asymptotic solution of the linear difference equation \[ P_{n+1}(x)+P_{n-1}(x)=(A_nx-B_n) P_n(x),\quad n\geq 1, \] where \[ A_n\sim n^{-\theta}\sum_{k=0}^\infty \frac{\alpha_k}{n^k},\quad B_n\sim \sum_{k=0}^\infty \frac{\beta_k}{n^k}\qquad(n\to\infty) \] with \(\theta>0\) and \(\alpha_0>0\). Many well-known orthogonal polynomials (such as the Hermite, Laguerre and Meixner polynomials) satisfy the above difference equation. By means of a logarithm-type procedure, the authors exploit the symmetry in the difference equation to determine an asymptotic expansion up to any order of two linearly independent solutions valid away from turning points. This procedure is combined with the method of asymptotic matching to determine the expansions in the outer and oscillatory regions. To illustrate this approach the authors use the Hermite and continuous dual Hahn polynomials as examples. In the case of the Hermite polynomials a numerical verification of the accuracy of the expansion developed is given.
reviewed by: Richard B. Paris
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 39A06
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 7015416
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: difference equations
zbMATH Keywords: asymptotic matching
MaRDI profile type: Publication
full work available at URL: j.jat.2018.10.002
OpenAlex ID: W2898580599
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Wikidata QID: Q129036384
DOI: 10.1016/J.JAT.2018.10.002
description: scientific article; zbMATH DE number 7015416