Asymptotic zero behavior of laguerre polynomials with negative parameter

A. B.J. Kuijlaars, K. T.R. McLaughlin

Research output: Contribution to journalArticle

32 Scopus citations

Abstract

We consider Laguerre polynomials Ln(αn) (nz) with varying negative parameters αn, such that the limit A = -limn αn/n exists and belongs to (0, 1). For A > 1, it is known that the zeros accumulate along an open contour in the complex plane. For every A ∈ (0, 1), we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit r = -lim n 1/n log[dist(αn, Z)] exists, we show that the zeros accumulate on γr ∪ [β1, β2] with β1 and β2 only depending on A. For r ∈ [0, ∞), γr is a closed loop encircling the origin, which for r = +γ, reduces to the origin. This shows a great sensitivity of the zeros to αn's proximity to the integers. We use a Riemann-Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.

Original languageEnglish (US)
Pages (from-to)497-523
Number of pages27
JournalConstructive Approximation
Volume20
Issue number4
DOIs
StatePublished - Jan 1 2004

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Keywords

  • Nonlinear steepest descent
  • Riemann-Hilbert problems
  • Sensitivity to parameter

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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