The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1]

Arno B.J. Kuijlaars, K. T.R. McLaughlin, Walter Van Assche, Maarten Vanlessen

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177 Scopus citations

Abstract

We consider polynomials that are orthogonal on [-1,1] with respect to a modified Jacobi weight (1-x)α(1+x)βh (x), with α,β>-1 and h real analytic and strictly positive on [-1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [-1,1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [-1,1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szego function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. Airy functions.

Original languageEnglish (US)
Pages (from-to)337-398
Number of pages62
JournalAdvances in Mathematics
Volume188
Issue number2
DOIs
StatePublished - Nov 10 2004

Keywords

  • Bessel functions
  • Orthogonal polynomials
  • Riemann-Hilbert problems
  • Steepest descent

ASJC Scopus subject areas

  • Mathematics(all)

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