How to prove the Fourier Hermite series $$\int H_n(x) f(x) \phi(x)\,dx \;=\; \int f^{(n)}(x) \phi(x)\,dx \quad ?$$
Here we have the Hermite polynomial $$H_n(x)=(-1)^n e^{x^2/2} \frac{\partial^n}{\partial x^n}e^{-x^2/2},$$ and we assume also
$\quad \phi(x)=\frac{1}{\sqrt{2\pi}} e^{-x^2/2}$
$\quad f^{(n)}(x)$ is the $n$th derivative of the function $f(x)$
$\quad f(x)$ satisfies $\int f(x)^2 \phi(x)\,dx <\infty.$