The Cylindrical Hermite Polynomials of a Diagonal Gaussian Measure on a Hilbert Space

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the variances $c_{k}$, the coordinates $x_{k}$ and the conventions of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers; $H^{v}_{n}$ are the Hermite polynomials, and finite products are those of Finite Product Notation.

Multi-indices $\alpha\in\mathcal{A}$ and their length bounds are those of Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order.

(Cylindrical Hermite polynomials)¶ For $\alpha\in\mathcal{A}$ with length bound $n$, the cylindrical Hermite polynomial of index $\alpha$ is the function $H_{\alpha}:X\to\mathbb{R}$,

$$H_{\alpha}(x)=\prod_{k=1}^{n}H^{c_{k}}_{\alpha_{k}}(x_{k})\qquad(x\in X);$$

it does not depend on the length bound, since $H^{c_{k}}_{0}(t)=1$ for every $t\in\mathbb{R}$ by Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §low-orders.