Riemann-Hilbert Analysis for Orthogonal Polynomials

Abstract

This is an introduction to the asymptotic analysis of orthogonal poly- nomials based on the steepest descent method for Riemann-Hilbert problems of Deift and Zhou. We consider in detail the polynomials that are orthogonal with respect to the modified Jacobi weight (1−x)α(1+x)βh(x) on [−1, 1] where α, β > −1 and h is real analytic and positive on [−1, 1]. These notes are based on joint work with Kenneth McLaughlin, Walter Van Assche and Maarten Vanlessen.