Operational Calculus with Hermite Polynomials and a Quite General Gaussian Integral

Abstract

Starting from two mutually alternative definitions of Hermite polynomials, we derive new relations between these polynomials for arbitrarily stretched arguments. Furthermore, some operational identities with Hermite polynomials are derived and proved by complete induction. The main result is a quite general integral with an arbitrary Gaussian distribution and the product of two Hermite polynomials with general linear arguments which generalize almost all known special integrals of such kind contained in the most comprehensive tables of integrals. Furthermore, some multi-dimensional integrals with Gaussian distributions are derived. Special representations are developed for two-dimensional integrals in complex and complex-conjugate variables which are important for the treatment of the phase space of harmonic oscillators.