Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

2002/12/02 by M. Vanlessen, Vanlessen, M.
Mathematics · #30E25 #33C10 #42C05 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #math.CA #math.CV #msc:30E25 #msc:33C10 #msc:42C05

paper · pdf · doi:10.48550/arxiv.math/0212014

31 pages, 6 figures, 21 references

arxiv created 2002/12/02 · arxiv updated 2009/11/30

Abstract

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [-1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann-Hilbert problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the Riemann-Hilbert problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the Riemann-Hilbert problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order.

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