Convergence of an integral involving Hermite polynomials

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.

Re: Convergence of an integral involving Hermite polynomials

If $\displaystyle P$ is a polynomial of degree $\displaystyle d$, the integral $\displaystyle \int_{-\infty}^{+\infty}e^{-x^2/2}P(x)dx$ is convergent. To see that, write $\displaystyle e^t\geq \frac{t^{d+2}}{(d+2)!}$ for $\displaystyle t\geq 0$.

Re: Convergence of an integral involving Hermite polynomials

Dear girdav,

Thanks a lot for your reply. I appreciate it.

Would you mind detailing your suggestion that uses $\displaystyle e^{t} \geq \frac{t^{d+2}}{(d+2)!} $ for $\displaystyle t \geq 0 $? Indeed, the latter uses $\displaystyle t $ positive but $\displaystyle -x^2/2 $ is negative, and it seems to me easier to prove that the integral in question is finite by using $\displaystyle \leq $ rather than $\displaystyle \geq $.

Regards.

Re: Convergence of an integral involving Hermite polynomials

You get an upper bound after taking the inverse.

Re: Convergence of an integral involving Hermite polynomials

Thanks girdav. That makes sense.