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.
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