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Associated Legendre polynomials for negative order

  1. Mar 27, 2013 #1
    1. The problem statement, all variables and given/known data
    I just need to deduce the expression for the associated Legendre polynomial [itex]P_{n}^{-m}(x)[/itex] using the Rodrigues' formula

    2. Relevant equations
    Rodrigues formula reads [tex]P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^n}{dx^n}(x^2-1)^n[/tex] and knowing that [tex]P_{n}^{m}(x)=(-1)^{m}(1-x^2)^{\frac{m}{2}}\frac{d^m}{dx^m}\left[P_n(x)\right][/tex]

    3. The attempt at a solution
    Using the expressions above mentioned i get [tex]P_{n}^{m}(x)=\frac{(-1)^m}{2^{n}n!}(1-x^2)^{\frac{m}{2}}\frac{d^{n+m}}{dx^{n+m}}(x^2-1)^n[/tex]
    Then i see that -n≤m≤n, so the last expression must be useful to determine the [itex]P_{n}^{-m}(x)[/itex] polynomial, substituting m by -m, but i cannot find the relationship between the derivatives [itex]\frac{d^{n+m}}{dx^{n+m}}[/itex] and [itex]\frac{d^{n-m}}{dx^{n-m}}[/itex], that i know it must be [tex]\frac{d^{n-m}}{dx^{n-m}}(x^2-1)^n=(-1)^m\frac{(n-m)!}{(n+m)!}(1-x^2)^{m}\frac{d^{n+m}}{dx^{n+m}}(x^2-1)^n[/tex]
    Any help would be greatly appreciated.
     
  2. jcsd
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