Plancherel–Rotach type asymptotic formulae for multiple orthogonal Hermite polynomials and recurrence relations

Abstract

We study the asymptotic properties of multiple orthogonal Hermite polynomials which are determined by the orthogonality relations with respect to two Hermite weights (Gaussian distributions) with shifted maxima. The starting point of our asymptotic analysis is a four-term recurrence relation connecting the polynomials with adjacent numbers. We obtain asymptotic expansions as the number of the polynomial and its variable grow consistently (the so-called Plancherel–Rotach type asymptotic formulae). Two techniques are used. The first is based on constructing expansions of bases of homogeneous difference equations, and the second on reducing difference equations to pseudodifferential ones and using the theory of the Maslov canonical operator. The results of these approaches agree.