Asymptotics of Polynomials Orthogonal With Respect to a Generalized Freud Weight With Application to Special Function Solutions of Painlev\'e-IV

Abstract

We obtain asymptotics of polynomials satisfying the orthogonality relations $$ \int_{\mathbb{R}} z^k P_n(z; t , N) \mathrm{e}^{-N \left(\frac{1}{4}z^4 + \frac{t}{2}z^2 \right)} \mathrm{d} z = 0 \quad \text{ for } \quad k = 0, 1, ..., n-1, $$ where the complex parameter $t$ is in the so-called two-cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlev\'e-IV which are indexed by a non-negative 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 non-linear steepest descent method.