Working with Hermite polynomials $H_n(x)$, I'm faced with the problem to interchange sums and integral in this expression
$$ \int_{\mathbb{R}} \left( \sum_n \sum_m \frac{H_n(x)}{n!}s^n \; \frac{H_m(x)}{m!}t^m \; e^{-x^2}\right )dx $$
My idea is to use twice Fubini theorem (the sum can be seen as an integral with the counting measure), but I can't prove that
$$ \int_{\mathbb{R}} \left( \sum_n \sum_m \left\lvert{ \frac{H_n(x)}{n!}s^n \; \frac{H_m(x)}{m!}t^m} \right\lvert \; e^{-x^2}\right )dx \; < +\infty$$