2004/10/13 by Andrei Martı́nez-Finkelshtein, A. Martinez-Finkelshtein, Martinez-Finkelshtein, A. +2 · 1 citation
Computer Science · Mathematics · #33C45 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Numerical methods in inverse problems #math.CA #math.CV #msc:33C45
paper · pdf · doi:10.48550/arxiv.math/0410320
37 pages, 10 figures
arxiv created 2004/10/13 · openalex publication_date 2004/10/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Classical Jacobi polynomials Pn(α,β), with α, β>-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 This property is no longer valid for other values of the parameters; in general, zeros are complex.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 αn,βn depend on n in such a way that limn→∞\fracαnn=A, limn→∞\fracβnn=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.