Dear All,

I would like to know if the following integral converges, and if so, what is the easiest way to prove it?

$\displaystyle I_n = \int_{- \infty}^{+ \infty} e^{- x^2 / 2} H_n(x) dx, \forall \ n \in \mathbb{N}^{+}, $

where $\displaystyle $H_n(x)$ $ is the physicists' Hermite polynomial:

$\displaystyle H_n(x) = (-1)^n e^{x^2} \frac{d^n e^{-x^2}}{d x^n} $.

Thank you.

Regards.