Riemann-Hilbert problem associated with Angelesco systems
type: Item
instance of: scholarly article
title: Riemann-Hilbert problem associated with Angelesco systems
zbMATH Open document ID: 1181.41019
published in: Journal of Computational and Applied Mathematics
publication date: 2009-11-04T00:00:00Z
review text: Let \(\Delta_1=[-\lambda,-1]\) and \(\Delta_2=[1,\lambda]\) with a real number \(\lambda>1\). For \(j=1,2\), let \(h_j\) be a holomorphic function in a neighbourhood \(\mathcal V_j\) of \(\Delta_j\) that does not vanish on \(\mathcal V_j\) and is positive on \(\Delta_j\). Consider the Angelesco system \((\sigma_1,\sigma_2)\) of measures defined by \[ d\sigma_j(x)=\frac{h_j(x)dx}{\sqrt{(\lambda-|x|)(|x|-1)}},\quad x\in\Delta_j,\;j=1,2. \] The main goal of this paper is to study the strong asymptotics of the sequence of multiple orthogonal polynomials \(\{Q_{\mathbf{n}}:\;\text{\textbf{n}}=(n,n),\;n\in\mathbb Z_{+}\}\) with deg\,\(Q_{\mathbf{n}}\leq 2n\), determined by the conditions \[ \int_{\Delta_j}x^{\nu}Q_{\mathbf{n}}(x)d\sigma_j(x)=0,\quad \nu=0,\dots,n,\;j=1,2. \] The authors' methods rely upon analysing the Riemann-Hilbert problem for multiple orthogonal polynomials, which was introduced in [\textit{W. Van Assche}, \textit{J. S. Geronimo} and \textit{A. B. J. Kuijlaars}, NATO Sci. Ser. II, Math. Phys. Chem. 30, 23--59 (2001; Zbl 0997.42012)].
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A28
zbMATH DE Number: 5624871
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: multiple orthogonal polynomials
author: A. Branquinho
author: Ulises Fidalgo
author: Ana Foulquié Moreno
MaRDI profile type: Publication
full work available at URL: j.cam.2009.02.032
OpenAlex ID: W2116257903
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DOI: 10.1016/J.CAM.2009.02.032
description: scientific article; zbMATH DE number 5624871