Strong asymptotics for orthogonal polynomials with regularly and slowly varying recurrence coefficients
type: Item
instance of: scholarly article
title: Strong asymptotics for orthogonal polynomials with regularly and slowly varying recurrence coefficients
zbMATH Open document ID: 0772.42012
author: Walter Van Assche
author: Jeffrey S. Geronimo
author: Dale T. Smith
published in: Journal of Approximation Theory
publication date: 1993-06-29T00:00:00Z
review text: Assume that the orthogonal polynomials \(p_ n(x)\), \(n=0,1,\dots\), are defined by the recursive equation \[ a_{n+1} p_{n+1}(x)+b_ n p_ n(x)+a_ n p_{n-1}(x)=xp_ n(x),\quad n=1,2,\dots, \] with \(p_{- 1}\equiv 0\) and \(p_ 1\equiv 1\) and the \(a_ n\)'s and \(b_ n\)'s are assumed to be regularly and slowly varying at infinity, i.e., there exists an increasing positive sequence \(\{\lambda_ n,\;n=0,1,\dots\}\) such that \[ \lim_{n\to\infty}{a_ n\over\lambda_ n}:= a>0,\quad\lim_{n\to\infty} {b_ n\over\lambda_ n}:= b\in\mathbb{R}, \] with \[ \lim_{n\to\infty} n\left({\lambda_{n+1}\over\lambda_ n}- 1\right):=\alpha\geq 0. \] Further it is assumed that \(a_{n+1}>0\), \(b_ n\in\mathbb{R}\), \(n=0,1,\dots\), and \[ \lim_{n\to\infty} n{a_{n+1}-a_ n\over\lambda_ n}=a\alpha,\quad\lim_{n\to\infty}n{b_{n+1}-b_ n\over \lambda_ n}=b\alpha. \] Polynomials orthogonal with respect to Freud and Erdős type weights belong to this class and have been subject of intense investigations. In the paper strong asymptotics of the contracted orthogonal polynomials \(p_ n(\lambda_ n y)\) have been investigated away from the interval of oscillation.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 222122
zbMATH Keywords: regularly and slowly varying
zbMATH Keywords: recurrence coefficients
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Freud and Erdős type weights
zbMATH Keywords: strong asymptotics
reviewed by: Herbert Stahl
MaRDI profile type: Publication
full work available at URL: jath.1993.1012
OpenAlex ID: W1973116370
DOI: 10.1006/JATH.1993.1012
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description: scientific article; zbMATH DE number 222122