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Hermite Polynomials

  1. Mar 28, 2007 #1
    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

    [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.
    Last edited: Mar 28, 2007
  2. jcsd
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