Distribution of zeroes to generalized Hermite polynomials

Abstract

Asymptotics of the orthogonal polynomial constitute a classic analytic problem. In the paper, we find a distribution of zeroes to generalized Hermite polynomials $H_{m,n}(z)$ as $m=n$, $n\to\infty$, $z=O(\sqrt n)$. These polynomials defined as the Wronskians of classic Hermite polynomials appear in a number of mathematical physics problems as well as in the theory of random matrices. Calculation of asymptotics is based on Riemann–Hilbert problem for Painlevé IV equation which has the solutions $u(z)=-2z +\partial_z\ln H_{m,n+1}(z)/H_{m+1,n}(z)$. In this scaling limit the Riemann-Hilbert problem is solved in elementary functions. As a result, we come to analogs of Plancherel–Rotach formulas for asymptotics of classical Hermite polynomials.