Let $f(x)=\sum_{n\geq 0}\frac{1}{n!n!}x^{n}$, is there an explicit formula for $f(x)$?
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This is, for $x \ge 0$, a special case of the modified Bessel function of the first kind. Have a look here: http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html $$ \sum_{n = 0}^{\infty} \frac{x^{n}}{(n!)^{2}} = \mathrm{I}_0 \left(2 \sqrt{x}\right) $$
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5$\begingroup$ Here is a probabilistic interpretation of this equation: Let $X_1$ and $X_2$ be iid Poisson random variables with rate $\sqrt{x}$. Then, $\mathbb P(X_1 = X_2) = e^{-2 \sqrt{x}} I_0(2 x)$. $\endgroup$– cardinalApr 16, 2012 at 0:41