Continued fractions and orthogonal polynomials on the unit circle
type: Item
instance of: scholarly article
title: Continued fractions and orthogonal polynomials on the unit circle
zbMATH Open document ID: 1091.42019
author: Sergey Khrushchev
published in: Journal of Computational and Applied Mathematics
publication date: 2005-04-18T00:00:00Z
review text: This survey paper contains the notes of a series of lectures given by the author at Caltech. As indicated in the abstract, the use of continued fractions in the theory of orthogonal polynomials on the unit circle is highlighted. The paper touches on the usual subjects, including the connection between periodicity and quadratic function fields. Interesting is the result on the development of a Cauchy integral into a continued fraction. This straightforward generalization of the corresponding formula for the unit circle to the real line case is dubbed the `Leganés formula' (Theorem 26) and runs as follows. Let \(\{Q_k\}_{k\geq 1}\) be the denominators for the continued fraction \[ \int \frac {d\sigma} {t-z} =\frac{1}{b_1}{{}\atop{-}}\frac{1}{b_2}{{}\atop{-\cdots-}}\frac{1}{b_n-\int d\sigma_n/(t-z)}. \] with \[ b_k(z)=-{z\over m_k}+\ell_k,\;0<m_k = \int\, d\sigma_{k-1}, \;\ell_k= \int\,td\sigma_{k-1}, \] and \(\sigma\) a measure with finite moments on \textbf{R}. Then \[ \int\,{Q_{n-1}^2 d\sigma\over t-z} = {1\over Q_n/Q_{n-1} - \int\,d\sigma_n /(t-z)}. \]
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 40A15
zbMATH DE Number: 2157674
zbMATH Keywords: continued fraction
zbMATH Keywords: \(P\)-fraction
zbMATH Keywords: periodic fraction
zbMATH Keywords: Pell equation
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: integration in finite terms
zbMATH Keywords: Schur algorithm
zbMATH Keywords: moment problem
zbMATH Keywords: Leganés formula
MaRDI profile type: Publication
full work available at URL: j.cam.2004.02.027
OpenAlex ID: W1996247023
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DOI: 10.1016/J.CAM.2004.02.027
description: scientific article; zbMATH DE number 2157674