(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that

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

with n, p, and r nonnegative integers.

Hint: Use this recurrence relation, p times:

[tex]H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)[/tex]

2. Relevant equations

Normalization:

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

3. The attempt at a solution

I've written :

[tex]H_{n+p}(x) = 2xH_{n+p-1}(x) - 2nH_{n+p-2}(x)[/tex]

I'm confused as to what the [tex]x^r[/tex] 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.

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# Homework Help: Hermite Polynomials

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