The conditional expectation of the product of the first n-1 Hermite polynomials in a multivariate normal distribution with respect to the n-th variable. A fresh perspective on the Kibble-Slepian formula

We calculate the conditional expectation of $\prod_{j=1}^{n-1}H_{k_{j}}% (X_{j})$ given $X_{n}=z,$ if random vector $(X_{1},\ldots,X_{n})^{T}$ has multivariate normal distribution and $H_{n}(x) $ denotes $n$-th Hermite polynomial. This expectation is a polynomial in $z$ of order $\sum_{j=1}% ^{n-1}k_{j}$. Our formula has an iterative form with respect to $n$. We also present some auxiliary observations concerning the expansion of the density of the $n$-dimensional normal distribution in the series of the Hermite polynomials. Mostly concerning the properties of the coefficients of this expansion. To perform these calculations, we give a few auxiliary formulas concerning Hermite polynomials and multivariate normal distributions. We apply this result to obtain exact, simple forms of these expansions for $n=2$ and $3$, thus looking at known results from a different perspective.