Let $H_n$ be the $n$th probabilistic Hermite polynomial of degree n and $\eta = exp(-x^2/2)/\sqrt(2 \pi)$ be the standard Gaussian density.

I would like to compute the integral $f_n(x) = \int H_n(x - z) \eta(z) dz$. Any hope to get a closed form expression?

Some ideas:

1. The $n$th Hermite polynomial $H_n$ can be related to the n-th derivative of $\eta$

2. This integral is a convolution of $H_n$ with the standard Gaussian kernel $\eta$.

Let $H_n$ be the $n$th probabilistic Hermite polynomial of degree n and $\eta = \exp(-x^2/2)/\sqrt(2 \pi)$ be the standard Gaussian density.

I would like to compute the integral $f_n(x) = \int H_n(x - z) \eta(z) dz$. Any hope to get a closed form expression?

Some ideas:

1. The $n$th Hermite polynomial $H_n$ can be related to the n-th derivative of $\eta$

2. This integral is a convolution of $H_n$ with the standard Gaussian kernel $\eta$.