The normalized probabilist's Hermite polynomials are $\frac{1}{\sqrt{n!}}He_n(x)$, and they satisfy orthonormality. Are there any simple formulas for the following two sums,
$$\displaystyle\sum_{n=0}^\infty \frac{1}{\sqrt{n!}}He_n(x)$$
$$\displaystyle\sum_{n=0}^\infty \frac{1}{n!}He_n(x)He_n(y)$$
It is also not clear to me that the sums are even convergent.
Edit: the following formula appears in this Math Overflow post (after converting from physicists' to probabilists'):
$$\displaystyle\sum_{n=0}^\infty \frac{1}{n!}He_n(x)He_n(y)u^n = \frac{1}{\sqrt{1-u^2}}\exp\left(\frac{xy}{1+u} - \frac{u^2}{2(1-u^2)}(x-y)^2\right)$$
Can anyone provide a reference?