A General Multiplication Theorem for Multivariate Hermite Polynomials

TL;DR

This paper extends the multiplication theorem of univariate Hermite polynomials to multivariate cases, enabling new applications and simplifying computations involving multivariate Hermite polynomials.

Contribution

It generalizes the multiplication theorem for univariate Hermite polynomials to multivariate Hermite polynomials and derives related results for inner-product applications.

Findings

Generalized multiplication theorem for multivariate Hermite polynomials

Derived multiplication theorem for univariate polynomials with inner-products

Facilitated new computational methods for multivariate Hermite polynomials

Abstract

The multiplication theorem for univariate Hermite polynomials $H_k(\lambda x)$ is well-known. In this paper we generalize this result to multivariate Hermite polynomials ${\rm H}_{\bf k}({\mathbf{\Lambda}}{\bf x};{\mathbf{\Sigma}})$, and use this result to derive a multiplication theorem for univariate polynomials applied to inner-products $H_k({\mathbf{\lambda}}^{\rm T} {\bf x})$.