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Difficult series

  1. Dec 10, 2009 #1
    I know:

    [tex]\sum_{n=0}^\infty \frac{x^n}{n!}=e^x [/tex]

    However, is there a similar solution for:

    [tex]\sum_{n=0}^\infty \left(\frac{x^n}{n!}\right)^2 [/tex]


    Thanks in advance; I'm not very good at this kind of maths (I teach statistics :devil:), and I've been struggling with this one for a while.
     
    Last edited: Dec 10, 2009
  2. jcsd
  3. Dec 10, 2009 #2
    I'm afraid for that you need the Bessel function. I looked up the series and the answer is
    [tex]\sum_{k=0}^\infty \frac{x^{2k}}{k!(k+n)!}=x^{-n}I_n(2x)[/tex]
    I suppose the function I_n is
    http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html
     
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