# Convergence of an integral involving Hermite polynomials

• May 16th 2012, 02:11 PM
Babou
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.
• May 17th 2012, 01:04 AM
girdav
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 $\displaystyle t\geq 0$.
• May 17th 2012, 10:34 AM
Babou
Re: Convergence of an integral involving Hermite polynomials
Dear girdav,

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 $\displaystyle \geq$.