Blumenthal's theorem for Laurent orthogonal polynomials
type: Item
instance of: scholarly article
title: Blumenthal's theorem for Laurent orthogonal polynomials
zbMATH Open document ID: 1022.42014
author: A. Sri Ranga
author: Walter Van Assche
published in: Journal of Approximation Theory
publication date: 2003-04-02T00:00:00Z
review text: The authors study polynomials which satisfy a three-term recurrence relation \(B_n(x)=(x-\beta_n)B_{n-1}(x)-\alpha_n xB_{n-2}(x)\) with positive coefficients \(\alpha_{n+1}, \beta_n\) and show that their zeros are eigenvalues of structured Hessenberg matrices. The case when \(\alpha_n \to\alpha\) and \(\beta_n \to\beta\) is analysed in more detail. Now the support of the Laurent orthogonality measure is an interval and the Laurent version of Blumenthal's theorem holds. The authors also prove the ratio asymptotics and study the asymptotic behavior for the logarithmic derivative for the polynomials \(B_n\) in case \(\alpha>0\) and \(\beta>0\).
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 47B36
zbMATH DE Number: 1891494
zbMATH Keywords: three-term recurrence relation
zbMATH Keywords: Hessenberg matrices
zbMATH Keywords: Laurent orthogonality
zbMATH Keywords: Gaussian quadrature formula
zbMATH Keywords: zero distribution
zbMATH Keywords: ratio asymptotics
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: jath.2002.3700
OpenAlex ID: W2009236436
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DOI: 10.1006/JATH.2002.3700
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description: scientific article; zbMATH DE number 1891494