The Asymptotic Behaviour of the Hermite Polynomials

Abstract

In a recent paper, Olver (2) obtains a set of formulae that completely determine the asymptotic behaviour of the Hermite polynomials, Hn(z), as n —> ∞ and z is unrestricted. His proof depends on a technique that he has developed for discussing the asymptotics of solutions of second-order, linear, homogeneous differential equations satisfying certain conditions. We believe it fair to say that Olver's work follows the tradition of most of the major theorems of classical asymptotics. The results contained in theorems such as Watson's lemma and Perron's proof of the Method of Laplace are based on an acceptance, on an a priori basis, of the Poincaré type expansion.