\((1,1)\)-coherent pairs on the unit circle
type: Item
instance of: scholarly article
title: \((1,1)\)-coherent pairs on the unit circle
zbMATH Open document ID: 1298.42030
DOI: 10.1155/2013/307974
published in: Abstract and Applied Analysis
publication date: 2014-06-23T00:00:00Z
review text: From the abstract: ``A pair \((\mathcal{U}, \mathcal{V})\) of Hermitian regular linear functionals on the unit circle is said to be a \((1,1)\)-coherent pair if their corresponding sequences of orthogonal polynomials, \(\{\phi_n(x)\}_{n \geq 0}\) and \(\{\psi_n(x)\}_{n \geq 0}\) satisfy \(\phi_n^{[1]}(z)+a_n \phi_{n-1}^{[1]}(z)=\psi_n(z)+b_n\psi_{n-1}(z)\), \(a_n \geq 0\), \(n \geq 1\), where \(\phi_n(z)^{[1]}=\phi_{n+1}'(z)/(n+1)\).'' The authors consider \((1,1)\) coherent pairs where \(\mathcal{U}\) is the linear functional associated with the Lebesgue measure \(d\theta/2\pi\), and when \(\mathcal{U}\) is the Bernstein-Szegő linear functional associated with the measure \(((1-|C|^2)/|1+Ce^{i\theta}|^2)(d\theta/2\pi)\). In both cases, the authors derive formulae for \(\psi_n\), \(n \geq 1\), as well as the sequence of moments for the functional \(\mathcal{V}\). The authors also consider the measures corresponding to the linear functional \(\mathcal{V}\) in each case.
reviewed by: Jeremy Wade
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C47
zbMATH DE Number: 6306641
zbMATH Keywords: coherent pairs
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Lebesgue measure
zbMATH Keywords: Bernstein-Szegő measure
Wikidata QID: Q58915895
author: Luis E. Garza
author: Francisco Marcellán
author: Natalia C. Pinzón-Cortés
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/307974
OpenAlex ID: W1499156980
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description: scientific article; zbMATH DE number 6306641