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 $\eta$.



Convolution of a Hermite polynomial with Gaussian kernel
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}


