Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight

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Original languageEnglish
Pages (from-to)1-18
Journal / PublicationJournal of Approximation Theory
Online published12 Dec 2014
Publication statusPublished - Apr 2015


In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w(x) = xαe-x-t/x, x ∈. (0, ∞), > 0 and α > 0. When the matrix size → ∞, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ (0, d], > 0 fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method.

Research Area(s)

  • Asymptotics, Hankel determinants, Painlevé III equation, Perturbed Laguerre weight, Riemann-Hilbert approach