A General Multiplication Theorem for Multivariate Hermite Polynomials

TL;DR

The paper addresses a general multiplication theorem for multivariate Hermite polynomials, extending the classical univariate result to $H_{\mathbf{k}}(\boldsymbol{x};\boldsymbol{\Sigma})$ with a linear argument $\boldsymbol{\Lambda}^{\top}\boldsymbol{x}$. It develops a generating-function framework and uses Kronecker-power algebra to derive a finite-sum expansion $H_{\mathbf{k}}(\boldsymbol{\Lambda}^{\top}\boldsymbol{x};\boldsymbol{\Sigma}) = \sum_{\mathbf{q}} T_{\mathbf{k},\mathbf{q}}(\boldsymbol{\Sigma}\boldsymbol{\Lambda}^{\top}\boldsymbol{\Upsilon}^{-1}; \boldsymbol{\Sigma},\boldsymbol{\Upsilon}) H_{\mathbf{q}}(\boldsymbol{x};\boldsymbol{\Upsilon})$, with the coefficients $T_{\mathbf{k},\mathbf{q}}$ expressed through Kronecker powers, vec operators, and related tensor constructions. The work also presents practical special cases, including $\boldsymbol{\Sigma}=\sigma^2\mathbf{I}$ and $\boldsymbol{\Upsilon}=\sigma^2\mathbf{I}$, reduces to the standard univariate multiplication theorem in the appropriate limit, and connects to probabilists' and physicists' Hermite families for $\sigma^2=1$ and $\sigma^2=1/2$. These results provide a versatile tool for Hermite-transform analyses of multivariate Gaussian structures and projections.

Abstract

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