On differential equations for orthogonal polynomials on the unit circle
type: Item
instance of: scholarly article
title: On differential equations for orthogonal polynomials on the unit circle
zbMATH Open document ID: 1172.33001
author: M. N. Rebocho
author: A. Branquinho
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-10T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/11242
review text: In the paper under review, the authors study orthogonal polynomials on the unit circle whose Carathéodory function \(F\) satisfies the differential equation \(zAF'=BF^2+CF+D\) with polynomial coefficients and \(A\neq 0\). The set of such Carathéodory functions is well-known also as the Laguerre-Hahn class on the unit circle. In the Theorems 1 and 2 the authors give a characterization of the above mentioned class. The formulation is rather long and they split the proofs in some lemmas and corollaries. In the last section, they investigate the semi-classical class on the unit circle and its characterization in terms of second order linear differential equations. More precisely, they consider Carathéodory functions associated to a measure \(\mu\) of the type \(d\mu=wd\theta+\sum_{k=1}^N\lambda_k\delta_{z_k}\), \((\lambda_k\geq 0\), \(N\in\mathbb{N})\), where \(w\) is the absolutely continuous part with respect to the Lebesgue measure \(d\theta\), and \(\delta_{z_k}\) is the Dirac measure at \(z_k\) with \(z_k\) on the unit circle for \(k=1,\dots,N\). The authors give the result of this section in Theorem 3, and they use some known lemmas to prove it.
reviewed by: Mehdi Hassani
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 5563865
zbMATH Keywords: Carathéodory function
zbMATH Keywords: Laguerre-Hahn class on the unit circle
zbMATH Keywords: orthogonal polynomials on the unit circle
zbMATH Keywords: semi-classical functions
zbMATH Keywords: measure on the unit circle
MaRDI profile type: Publication
OpenAlex ID: W2002255867
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DOI: 10.1016/J.JMAA.2009.03.008
description: scientific article; zbMATH DE number 5563865