Thread: Convergence of an integral involving Hermite polynomials

1. 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?

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

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.

2. Re: Convergence of an integral involving Hermite polynomials

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 $t\geq 0$.

3. Re: Convergence of an integral involving Hermite polynomials

Dear girdav,

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.

4. Re: Convergence of an integral involving Hermite polynomials

You get an upper bound after taking the inverse.

5. Re: Convergence of an integral involving Hermite polynomials

Thanks girdav. That makes sense.