On three term recurrence relations for orthonormal polynomials on the unit circle
type: Item
instance of: scholarly article
title: On three term recurrence relations for orthonormal polynomials on the unit circle
zbMATH Open document ID: 0757.42008
DOI: 10.1007/BF02844470
author: C. Belingeri
author: P. E. Ricci
publication date: 1993-01-16T00:00:00Z
review text: Let \(\{P_ n(z)\}\) be the system of orthogonal polynomials on the unit circle with respect to a given weight function \(w(z)\). Further, let \(P_ n(z)=\sum^ n_{h=0}\alpha_{h,n}z^ h\) and \(k_ n=\alpha_{n,n}\). The authors prove that if \(n\geq 2\) and \(\alpha_{0,n-1}\neq 0\), then \[ A_ nP_ n(z)=(z+B_ n)P_{n-1}(z)-A_{n-1}B_ nzP_{n-2}(z), \] where \(A_ n=k_{n-1}/k_ n\), \(B_ n=(\alpha_{0,n}/\alpha_{0,n- 1})A_ n\). Analogous recurrence relations are given for the polynomials \(P^*_ n(z)=\sum^ n_{h=0}\overline\alpha_{h,n}z^{n-h}\), \(P_{-n}(z)=\sum^ n_{h=0}\overline\alpha_{h,n}z^{-h}\) and \(P^*_{-n}(z)=\sum^ n_{h=0}\alpha_{h,n}z^{n-h}\).
reviewed by: Luigi Gatteschi
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 91673
zbMATH Keywords: polynomials orthogonal on the unit circle
zbMATH Keywords: three term recurrence relation
MaRDI profile type: Publication
cites work: Q4170993
cites work: Q3483797
cites work: Q5828253
cites work: Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle
cites work: Q4038311
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 91673