Global asymptotics for Laguerre polynomials with large negative parameter -- a Riemann-Hilbert approach
type: Item
instance of: scholarly article
title: Global asymptotics for Laguerre polynomials with large negative parameter -- a Riemann-Hilbert approach
zbMATH Open document ID: 1173.41013
published in: The Ramanujan Journal
publication date: 2008-09-12T00:00:00Z
review text: Using the Riemann-Hilbert approach introduced by \textit{P. Deift} and \textit{X. Zhou} [Ann. Math. (2) 137, No. 2, 295--368 (1993; Zbl 0771.35042)], the asymptotic behaviour of the Laguerre polynomials \(L^{(\alpha_n)}_n(nz)\) as \(n\to\infty\), where \(\alpha_1\) is a sequence of negative numbers and \(-\alpha_n/n\) tends to a limit \(A> 1\) as \(n\to\infty\). An asymptotic expansion is obtained, which is uniformly valid in the upper half plane \(\mathbb C_+= \{z: \text{Im\,}z\geq 0\}\). A corresponding expansion is also obtained for the lower half plane \(\mathbb C_-= \{z:\text{Im\,}Z\geq 0\}\). The two expansions hold, in particular, in a region in the complex plane containing the curve \(\Gamma\), on which these polynomials are orthogonal.
reviewed by: C. L. Parihar
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 5343451
zbMATH Keywords: uniform asymptotics
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: Riemann-Hilbert problems
zbMATH Keywords: zeros of polynomials
author: Dan Dai
author: Roderick Wong
MaRDI profile type: Publication
full work available at URL: s11139-007-9108-7
OpenAlex ID: W2042534681
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DOI: 10.1007/S11139-007-9108-7
description: scientific article; zbMATH DE number 5343451