A General Multiplication Theorem for Multivariate Hermite Polynomials

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 $H_k({\mathbf{\lambda}}^{\rm T} {\bf x})$.