Asymptotics of polynomials orthogonal with respect to a generalized Freud weight with application to special function solutions of Painlevé‐IV

Abstract

We obtain asymptotics of polynomials satisfying the orthogonality relations where the complex parameter is in the so‐called two‐cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlevé‐IV which are indexed by a nonnegative integer and can be written in terms of parabolic cylinder functions. The proofs are based on the characterization of orthogonal polynomials in terms of a Riemann–Hilbert problem and the Deift–Zhou nonlinear steepest descent method.