Is there an explicit formula for $\sum x^n/(n!)^2$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:13:21Z http://mathoverflow.net/feeds/question/94119 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94119/is-there-an-explicit-formula-for-sum-xn-n2 Is there an explicit formula for $\sum x^n/(n!)^2$? xmchenhit 2012-04-15T14:44:41Z 2012-04-16T00:27:23Z <p>Let $f(x)=\sum_{n\geq 0}\frac{1}{n!n!}x^{n}$, is there an explicit formula for $f(x)$?</p> http://mathoverflow.net/questions/94119/is-there-an-explicit-formula-for-sum-xn-n2/94123#94123 Answer by Woett for Is there an explicit formula for $\sum x^n/(n!)^2$? Woett 2012-04-15T15:09:47Z 2012-04-15T16:53:38Z <p>This is, for $x \ge 0$, a special case of the modified Bessel function of the first kind. Have a look here: <a href="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html" rel="nofollow">http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html</a> $$\sum_{n = 0}^{\infty} \frac{x^{n}}{(n!)^{2}} = \mathrm{I}_0 \left(2 \sqrt{x}\right)$$</p>