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 ${\mathrm{H}}_{\mathbf{k}}(\mathbf{\Lambda }\mathbf{x};\mathbf{\Sigma })$, and use this result to derive a multiplication theorem for univariate polynomials applied to inner-products $H_k({\mathbf{\lambda }}^{\mathrm{T}}\mathbf{x})$.