Global asymptotics of orthogonal polynomials associated with | x |2 α e- Q (x)

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Author(s)

  • R. Wong
  • L. Zhang

Detail(s)

Original languageEnglish
Pages (from-to)723-765
Journal / PublicationJournal of Approximation Theory
Volume162
Issue number4
Publication statusPublished - Apr 2010

Abstract

In this paper, we consider the asymptotics of polynomials orthogonal with respect to the weight function w (x) = | x |2 α e- Q (x), α > - frac(1, 2), where Q (x) = ∑k = 0 2 m qk xk, q2 m > 0, m > 0 is a polynomial of degree 2 m. Globally uniform asymptotic expansions are obtained for z in four regions. These regions together cover the whole complex z-plane. Due to the singularity of | x |2 α, the expansion in the region containing the origin involves Bessel functions. We also study the asymptotic behavior of the leading coefficients and the recurrence coefficients of these polynomials. Our approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou [P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Asymptotics for the mKdV equation, Ann. of Math. 137 (1993) 295-368]. © 2009 Elsevier Inc. All rights reserved.

Research Area(s)

  • Airy functions, Bessel functions, Global asymptotics, Orthogonal polynomials, Riemann-Hilbert problems

Citation Format(s)

Global asymptotics of orthogonal polynomials associated with | x |2 α e- Q (x). / Wong, R.; Zhang, L.
In: Journal of Approximation Theory, Vol. 162, No. 4, 04.2010, p. 723-765.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review