Riemann--Hilbert analysis for Jacobi polynomials orthogonal on a single contour

  • Authors:

  • A. Martínez-Finkelshtein;R. Orive

  • Affiliations:

  • University of Almería and Instituto Carlos I de Física Teórica y Computacional, Granada University, SPAIN;University of La Laguna, Canary Islands, SPAIN

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2005

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Abstract

Classical Jacobi polynomials P"n^(^@a^,^@b^), with @a,@b-1, have a number of well-known properties, in particular the location of their zeros in the open interval (-1,1). This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters @a"n,@b"n depend on n in such a way thatlimn-~@a"nn=A,limn-~@b"nn=B,with A,B@?R. We restrict our attention to the case where the limits A,B are not both positive and take values outside of the triangle bounded by the straight lines A=0, B=0 and A+B+2=0. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential. The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift-Zhou steepest descent method based on the Riemann-Hilbert reformulation of Jacobi polynomials.