Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

2003/09/27 by Arno B. J. Kuijlaars, A. B. J. Kuijlaars, Andrei Martı́nez-Finkelshtein +3
Mathematics · #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #Mathematical functions and polynomials #math.CA #math.CV #msc:33C45

paper · pdf · doi:10.48550/arxiv.math/0309443

31 pages, 12 figures. Some references added. To appear in Journal D'Analyse Mathematique

arxiv created 2004/01/10 · arxiv updated 2009/12/01

Abstract

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials P_n^{(α_n, β_n)} is studied, assuming that lim_n→∞ (α_n)/(n)=A, lim_n→∞ (β_n)/(n)=B, with A and B satisfying A > -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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