# Homework Help: Hermite Polynomials

1. Mar 28, 2007

### ultimateguy

1. The problem statement, all variables and given/known data
Show that

$$\int_{-\infty}^{\infty} x^r e^{-x^2} H_n(x) H_{n+p} dx = 0$$ if p>r and $$= 2^n \sqrt{\pi} (n+r)!$$ if p=r.

with n, p, and r nonnegative integers.
Hint: Use this recurrence relation, p times:

$$H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)$$

2. Relevant equations

Normalization:
$$\int_{-\infty}^{\infty} e^{-x^2} [H_n(x)]^2 dx = 2^n \sqrt{\pi} n!$$

3. The attempt at a solution

I've written :

$$H_{n+p}(x) = 2xH_{n+p-1}(x) - 2nH_{n+p-2}(x)$$

I'm confused as to what the $$x^r$$ in the equation is. I know that I have to use the normalization and that will give me that extra r in the factorial, but can't quite see how.

Last edited: Mar 28, 2007