# Sum of Infinite Series with the Gamma Function

I am computing the volume of an infinite family of polytopes and have run into the following sum, which I am not sure how to evaluate, as it looks similar to the Riemann zeta function, except with the gamma function being summed over instead of a regular integer $n$. That is, $$\sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^2}$$ Has anyone seen this sum before, know any properties of it, what other functions it is related to, or what the sum converges to? I am also interested in what this sum is equal to for all other natural numbers in the power of the gamma function, not just 2.

• It might be helpful that $\Gamma(n)=(n-1)!$. – Clayton Jun 14 '13 at 0:47
• @Clayton: I realize this, I ran into it by factoring an $(n-1)!$ out of something. – Samuel Reid Jun 14 '13 at 0:49

$$I_0(x) = \sum_{n=0}^{\infty} \frac{(x/2)^{2 n}}{(n!)^2}$$
where $I_0(x)$ is the modified Bessel function of the first kind of zero order. Then your sum is equal to $I_0(2) \approx 2.27959$.
• In general, $\sum_{n=1}^\infty\frac{1}{\Gamma(n)^k} = \,_0F_{k-1}(;1,1,\dots,1;1)$ is a generalized hypergeometric function evaluated at $z = 1$. – Hans Engler Jun 14 '13 at 1:58
• It's the definition. The ratio of consecutive coefficients of the power series is a rational power of the power $n$, in this case it's $n^{-k}$. See en.wikipedia.org/wiki/Generalized_hypergeometric_function . – Hans Engler Jun 14 '13 at 2:13