The d-variate vector hermite polynomial of order k

Abstract

The multivariate Hermite polynomials occur in Edgeworth and saddle-point approximations to the distributions of random vector variables. Explicit expressions in vector notation for the multivariate Hermite polynomials of arbitrary order are given and shown to be a special case of a general addition property of Hermite polynomials. General properties of these polynomials are given in vector notation, including an integral representation, the orthogonality property, differential properties, and recurrence relations as well as inversion formulas. Various expansions of a multivariate normal density are given in terms of a multivariate normal density with different mean, different covariance, or both. In special cases these results give Mehler's formula.