expected value of Hermite polynomials

Lets say we have$$ f(x)=\sum_{r=0}^\infty \frac{E(H_{e_r}(X)) (-D)^r e^{-\frac{1}{2}x^2}}{r!}$$where $X$ is some symmetric random variable $H_{e_r}(X)$ is probabilist's Hermite polynomials and $D=\frac{d}{dx}$ is deferential operator over x.

I tried to simplify or even find a bound or inequality for $f(x)$ but I couldn't find any way .I tried like below but it didn't help$$ f(x)=E\sum_{r=0}^\infty \frac{H_{e_r}(X) (-D)^r e^{-\frac{1}{2}x^2}}{r!}$$$$f(x)=E \ \left( e^{-XD-\frac{1}{2}D^2} e^{-\frac{1}{2}x^2} \right)$$

any idea to simplify or approximation is helpful.