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Legendre polynomials and binomial series

  1. Dec 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Where P_n(x) is the nth legendre polynomial, find f(n) such that
    [tex]\int_{0}^{1} P_n(x)dx = f(n) {1/2 \choose k} + g(n)[/tex]


    2. Relevant equations

    Legendre generating function:
    [tex](1 - 2xh - h^2)^{-1/2} = \sum_{n = 0}^{\infty} P_n(x)h^n[/tex]

    3. The attempt at a solution

    I'm not sure if that g(n) term is necessary.

    First I integrate both sides of the generating function on 0->1. I can then replace the (1+h^2)^1/2 term with a binomial series. I'm not sure how to cancel out the rest of the factors to solve for P_n. Any help would be appreciated, thanks.
     
    Last edited: Dec 7, 2012
  2. jcsd
  3. Dec 7, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    The generating function is
    [tex] \frac{1}{\sqrt{1-2xh + h^2}},[/tex]
    not what you wrote (see, eg., http://en.wikipedia.org/wiki/Legendre_polynomials ).

    Anyway, why try to solve for P_n? When you integrate you just need the coefficient of h^n on both sides.
     
  4. Dec 8, 2012 #3
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