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Difficult series

by m00se
Tags: difficult, series
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Dec10-09, 11:55 AM
P: 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 ), and I've been struggling with this one for a while.
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Dec10-09, 01:43 PM
P: 1,060
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

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