Technical Summary: A General Multiplication Theorem for Multivariate Hermite Polynomials

Problem Statement

The multiplication theorem for univariate Hermite polynomials, which describes the expansion of $H_k(\lambda x)$ in terms of $H_{k-2i}(x)$, is a well-established result in mathematical literature. However, a generalized version of this theorem for multivariate Hermite polynomials $H_k(\Lambda x; \Sigma)$, where the argument is transformed by a matrix $\Lambda$ and the polynomials are defined with respect to a general positive definite covariance matrix $\Sigma$, was not explicitly formulated in the provided text.

The paper addresses the need to derive a multiplication theorem that relates multivariate Hermite polynomials of a linearly transformed vector to a series of multivariate Hermite polynomials of the original vector, potentially under a different covariance structure.

Methodology

The derivation relies on the generating function approach for multivariate Hermite polynomials. The author utilizes the known generating function identity:
$$\sum_{k \in \mathbb{N}^n} \frac{t^k}{k!} H_k(x; \Sigma) = \exp\left(t^T \Sigma^{-1} x - \frac{1}{2} t^T \Sigma^{-1} t\right)$$

The core of the methodology involves the following steps:

1. Substitution: The argument $x$ in the generating function is replaced by $\Lambda^T x$, where $\Lambda \in \mathbb{R}^{n \times m}$.
2. Decomposition: The resulting exponential term is factored into two parts: one resembling the generating function of Hermite polynomials with a new covariance matrix $\Upsilon$, and a residual exponential term representing the difference between the transformed and original covariance structures.
3. Re-expansion: The first part is expanded using the generating function for $H_q(x; \Upsilon)$. The second part (the residual) is expanded as a power series involving Kronecker powers and vectorization ($\text{vec}$).
4. Coefficient Matching: By equating the coefficients of the power series terms $t^{\otimes k}$ on both sides of the equation, the author isolates the relationship between the coefficients of the original polynomials $H_k(\Lambda^T x; \Sigma)$ and the target polynomials $H_q(x; \Upsilon)$.
5. Linear Algebraic Manipulation: The proof employs properties of Kronecker products, specifically the identity $(A^T b)^k = A^{\otimes k T} b^{\otimes |k|}$, to resolve the tensor structures and extract the explicit coefficients.

Key Contributions and Results

1. General Multiplication Theorem

The primary result is the derivation of the general multiplication theorem for multivariate Hermite polynomials. For symmetric, positive definite covariance matrices $\Sigma \in \mathbb{R}^{n \times n}$ and $\Upsilon \in \mathbb{R}^{m \times m}$, and any matrix $\Lambda \in \mathbb{R}^{n \times m}$, the theorem states:

$$H_k(\Lambda^T x; \Sigma) = \sum_{q \in \mathbb{N}^m: |q| \in \{|k|, |k|-2, \dots\}} T_{k,q}(\Sigma \Lambda^T \Upsilon^{-1}; \Sigma, \Upsilon) H_q(x; \Upsilon)$$

The coefficients $T_{k,q}$ are explicitly defined using Kronecker powers and vectorization:
$$T_{k,q}(\tilde{\Lambda}; \Sigma, \Upsilon) = \frac{k!}{2^i q! i!} I_n^{\otimes k T} \left( \tilde{\Lambda}^{\otimes q} \otimes \left( \text{vec}(\tilde{\Lambda} \Upsilon^{-1} \tilde{\Lambda}^T - \Sigma^{-1}) \right)^{\otimes i} \right) \Bigg|_{i = \frac{|k|-|q|}{2}}$$

2. Special Cases and Intermediate Generality

The paper derives several specific instances of the general theorem to enhance practical applicability:

• Scalar Covariance: When $\Sigma = \sigma^2 I_n$ and $\Upsilon = \sigma^2 I_m$, the coefficients simplify significantly, removing the need for general inverse covariance calculations in the Kronecker terms.
• Univariate Projections: For the case where the transformation is a vector inner product ($\Lambda^T x = \lambda^T x$ with $\lambda \in \mathbb{R}^m$), the theorem provides a series expansion for $H_k(\lambda^T x)$ in terms of products of univariate Hermite polynomials $\prod_j H_{q_j}(x_j)$.
  • For probabilists' Hermite polynomials ($\Upsilon = I$):
    $$H_{\mathbf{k}}(\lambda^T x) = \sum_{q} \frac{k!}{2^i q! i!} \lambda^q (\|\lambda\|_2^2 - 1)^i \prod_j H_{q_j}(x_j)$$
  • For physicists' Hermite polynomials ($\Upsilon = \frac{1}{2}I$):
    $$H_{\mathbf{k}}(\lambda^T x) = \sum_{q} \frac{k!}{q! i!} \lambda^q (\|\lambda\|_2^2 - 1)^i \prod_j H_{q_j}(x_j)$$
    (Note: The summation indices $i$ are constrained by $i = (k - |q|)/2$.)

3. Consistency with Known Results

The paper demonstrates that the standard univariate multiplication theorems (Equation 1 in the text) are recovered as special cases when $n=m=1$ and appropriate covariance matrices are selected.

Significance and Claims

The author states that these results were derived during the course of other research but were found to be superfluous for the original purpose. Consequently, the paper is presented primarily as a reference for other researchers who may require these specific expansions.

The paper claims the results are novel to the best of the author's knowledge. It does not propose new applications, experimental validations, or future research directions. Instead, its value lies in providing a rigorous, generalized mathematical tool for manipulating multivariate Hermite polynomials under linear transformations, particularly useful for applications involving vector projections and Hermite transforms. The work serves to fill a gap in the literature by extending a well-known univariate property to the multivariate domain with general covariance structures.