A General Multiplication Theorem for Multivariate Hermite Polynomials

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 $H_k({\mathbfλ}^{\rm T} {\bf x})$.

💡 Research Summary

The paper “A General Multiplication Theorem for Multivariate Hermite Polynomials” extends the well‑known multiplication property of univariate Hermite polynomials to the multivariate setting. The authors start from the definition of multivariate Hermite polynomials (H_{\mathbf k}(x;\Sigma)) associated with a symmetric positive‑definite covariance matrix (\Sigma\in\mathbb R^{n\times n}). Using the generating function
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