Journal d’Analyse Mathematique

, Volume 94, Issue 1, pp 195–234

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Article

DOI: 10.1007/BF02789047

Cite this article as:
Kuijlaars, A.B.J. & Martínez-Finkelshtein, A. J. Anal. Math. (2004) 94: 195. doi:10.1007/BF02789047

Abstract

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsPnαnβn are studied, assuming that
$$\mathop {\lim }\limits_{n \to \infty } \frac{{\alpha _n }}{n} = A, \mathop {\lim }\limits_{n \to \infty } \frac{{\beta _n }}{n} = B,$$
(1)
withA andB satisfyingA>−1,B>−1,A+B<−1. The asymptotic analysis is based on the non-Hermitian orthogonality of these polynomials and uses the Deift/Zhou steepest descent analysis for matrix Riemann-Hilbert problems. As a corollary, asymptotic zero behavior is derived. We show that in a generic case, the zeros distribute on the set of critical trajectories Γ of a certain quadratic differential according to the equilibrium measure on Γ in an external field. However, when either αnβn or αnn are geometrically close to ℤ, part of the zeros accumulate along a different trajectory of the same quadratic differential.

Copyright information

© The Hebrew University Magnes Press 2004

Authors and Affiliations

  • A. B. J. Kuijlaars
    • 1
  • A. Martínez-Finkelshtein
    • 2
    • 3
  1. 1.Department of MathematicsKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.Dept. Estadistica y Matematica AplicadaUniversidad de AlmeriaAlmeriaSpain
  3. 3.Instituto Carlos I de Física Teórica y ComputacionalGranada UniversityGranadaSpain

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