Tau-function constructions of the recurrence coefficients of orthogonal polynomials
type: Item
instance of: scholarly article
title: Tau-function constructions of the recurrence coefficients of orthogonal polynomials
zbMATH Open document ID: 0930.42012
author: Yang Chen
author: Mourad E. H. Ismail
author: Walter Van Assche
published in: Advances in Applied Mathematics
publication date: 1999-01-24T00:00:00Z
review text: In this interesting paper, the authors study the asymptotic behaviour of recurrence coefficients of orthogonal polynomials, using ideas and methods from mathematical physics. Recall that if \(w\) is a nonnegative function on the real line, that is positive on a set of positive measure, and for which all moments \(\int x^jw(x) dx\), \(j= 0,1,2,\dots\), are finite, then there is an associated system of orthonormal polynomials \(\{p_n\}^\infty_{n= 0}\), satisfying \[ \int p_n p_mw= \delta_{mn}. \] There is a three term recurrence relation \[ xp_n(x)= a_{n+ 1}p_{n+ 1}(x)+ b_n p_n(x)+ a_np_{n- 1}(x), \] where \(a_n> 0\), \(b_n\in\mathbb{R}\). The authors show how ideas from mathematical physics, involving Coulomb fluid methods, chemical potentials, \(\tau\)-functions, and other ideas, may be used to compute the behaviour of \(a_n\) as \(n\to\infty\). A key role is played by a relation which shows that the \(a_n\) may be expressed as the second derivative of a certain free energy. It is remarkable how some of the tools developed in mathematical physics, bear such a close relationship to the weighted potential theory, that has had a dramatic impact on the analysis of orthogonal polynomials in the last twenty years. Three examples are given: Firstly, Freud weights such as \(w(x)= \exp(-|x|^\alpha)\), \(\alpha> 1\), for which asymptotics for \(a_n\) are well known. Secondly, the authors consider the Erdős weight \(w(x)= \exp(-\exp(|x|^\alpha))\), and show that \(a_n= {B^2_n\over 4}(1+ O(B^{-\alpha}_n))\), where \(B_n\) is the root of the equation \(n=\sqrt{\alpha B_n^\alpha\over 2\pi}\exp(B_n^\alpha)\). This is the first time such precise asymptotics have been given for \(a_n\) for these weights. The third example is the weak exponential weight \(\exp(-c|\ln x|^\alpha)\). In all cases the authors show how the Birkhoff-Trjitzinsky method can then be used to derive asymptotics of orthonormal polynomials.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1221618
zbMATH Keywords: exponential weights
zbMATH Keywords: asymptotic behaviour of recurrence coefficients
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Freud weights
zbMATH Keywords: asymptotics
zbMATH Keywords: Erdős weight
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: aama.1997.0574
OpenAlex ID: W2026887570
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DOI: 10.1006/AAMA.1997.0574
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description: scientific article; zbMATH DE number 1221618