Convolution of a Hermite polynomial with Gaussian kernel

Question

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 $\eta$.

Answer 1

Using the definition of the Hermite polynomials and then integrating by parts $n$ times, we get $f_n(x)=x^n$.

Details: \begin{equation} \begin{aligned} f_n(x)&=\int(-1)^n\eta^{(n)}(x-z)\frac{\eta(z)}{\eta(x-z)}\,dz \\ &=(-1)^ne^{x^2/2}\int\eta^{(n)}(x-z)\ e^{-xz}\,dz \\ &=(-1)^{n-1}e^{x^2/2}x\int\eta^{(n-1)}(x-z)\ e^{-xz}\,dz \\ &\vdots \\ &=e^{x^2/2}x^n\int\eta(x-z)\ e^{-xz}\,dz \\ &=x^n. \end{aligned} \end{equation}