A canonical family of multiple orthogonal polynomials for Nikishin systems
type: Item
instance of: scholarly article
title: A canonical family of multiple orthogonal polynomials for Nikishin systems
zbMATH Open document ID: 1200.42013
author: I. A. Rocha
author: Guillermo López Lagomasino
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-09-17T00:00:00Z
full work available at URL: http://hdl.handle.net/10016/6394
review text: Let \((\mu_1,\mu_2)=N(\sigma)\) be a Nikishin system of measures, generated by a pair of measures \(\sigma=(\sigma_1,\sigma_2)\), supported on compact disjoint intervals \(\Delta_1\), \(\Delta_2\). It is well known that if \(\sigma_k'>0\) a.e. on \(\Delta_k\), \(k=1,2\), then the sequence \(\{Q_n\}_{n\geq0}\) of monic multiple orthogonal polynomials for \(N(\sigma)\) satisfies the four-term recurrence relation with asymptotically \(2\)-periodic coefficients. The authors prove that any sequence of monic polynomials \(\{P_n\}_{n\geq0}\) defined by recurrences with \(2\)-periodic constant coefficients \[ \begin{aligned} zP_{2n}(z) &=P_{2n+1}(z)+\alpha_0 P_{2n}(z)+\beta P_{2n-1}(z)+\gamma_0 P_{2n-2}(z),\\ zP_{2n+1}(z) &=P_{2n+2}(z)+\alpha_1 P_{2n+1}(z)+\beta P_{2n}(z)+\gamma_1 P_{2n-1}(z), \end{aligned} \] \(n=0,1,\dots\), with initial conditions \(P_0=1\), \(P_{-1}=P_{-2}=0\) is the sequence of monic multiple orthogonal polynomials with respect to a certain Nikishin system \(N(\sigma)\), and study the properties of this system.
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 5787096
zbMATH Keywords: Hermite-Padé orthogonal polynomials
zbMATH Keywords: Nikishin system
zbMATH Keywords: varying measures
zbMATH Keywords: ratio asymptotic
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
OpenAlex ID: W2084042884
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DOI: 10.1016/J.JMAA.2010.07.022
description: scientific article; zbMATH DE number 5787096