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Trying to prove an identity

  1. Dec 5, 2003 #1
    Hi everyone!
    I'm trying to prove an identity, and it's driving me insane.

    Let [tex]J_p(x) = \sum_{n=0}^{\infty} \left(-1\right)^n\frac{x^{2n+p}}{2^{2n+p}n!(n+p)!}[/tex]

    Show that
    [tex]\frac{d}{dx}(x^{-p}J_p(x)) = x^{-p}J_{p+1}(x)[/tex]

    I get the left part to

    And the right part to

    This is incorrect, since they are not equal. Please, please help me! I can post my calculations if that would help.

    Thanks in advance,
  2. jcsd
  3. Dec 5, 2003 #2


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    Staff Emeritus
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    You might notice that the lower index on
    is incorrect. The first term will be n= 1, not n=0.

    Try changing the index from n to j= n-1 so that the sum will be
    from j=0 to infinity.

    If j= n-1 then n= j+1 so, for example, the power of x, 2n-1, becomes 2(j+1)-1= 2j+1. Make that change for every n in the formula.

    Of course, since n (or j) is a "dummy" variable, you could then just replace "j" with "n" or change the n in the second sum you have with j in order to compare them.
  4. Dec 8, 2003 #3
    That's it!!
    I had a hard time realizing that that you could just change the index. But I realize that "infinity - 1" is, obviously, infity. Hince the delay...

    Thanks for helping me!
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