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Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

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Journal d’Analyse Mathematique Aims and scope

Abstract

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsP n α n β n 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 α n β n or α n +β n are geometrically close to ℤ, part of the zeros accumulate along a different trajectory of the same quadratic differential.

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Authors and Affiliations

  1. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001, Leuven, Belgium

    A. B. J. Kuijlaars

  2. Dept. Estadistica y Matematica Aplicada, Universidad de Almeria, La Canada, 04120, Almeria, Spain

    A. Martínez-Finkelshtein

  3. Instituto Carlos I de Física Teórica y Computacional, Granada University, Granada, Spain

    A. Martínez-Finkelshtein

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  1. A. B. J. Kuijlaars
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  2. A. Martínez-Finkelshtein
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Correspondence to A. B. J. Kuijlaars.

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Kuijlaars, A.B.J., Martínez-Finkelshtein, A. Strong asymptotics for Jacobi polynomials with varying nonstandard parameters. J. Anal. Math. 94, 195–234 (2004). https://doi.org/10.1007/BF02789047

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  • DOI: https://doi.org/10.1007/BF02789047

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