Convergence of an integral involving Hermite polynomials

**Babou** · May 16th 2012, 02:11 PM

Dear All,

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

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

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

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

Thank you.

Regards.

**girdav** · May 17th 2012, 01:04 AM

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

**Babou** · May 17th 2012, 10:34 AM

Dear girdav,

Thanks a lot for your reply. I appreciate it.

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

Regards.

**girdav** · May 17th 2012, 10:55 AM

You get an upper bound after taking the inverse.

**Babou** · May 17th 2012, 11:16 AM

Thanks girdav. That makes sense.

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