Solution to Sum of Exponential Squared Series

[m00se]

I know:

\sum_{n=0}^\infty \frac{x^n}{n!}=e^x

However, is there a similar solution for:

\sum_{n=0}^\infty \left(\frac{x^n}{n!}\right)^2Thanks 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.

 

[Gerenuk]

I'm afraid for that you need the Bessel function. I looked up the series and the answer is
\sum_{k=0}^\infty \frac{x^{2k}}{k!(k+n)!}=x^{-n}I_n(2x)
I suppose the function I_n is
http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html