Sum of Infinite Series with the Gamma Function

Question

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.

Answer 1

Note that

$$I_0(x) = \sum_{n=0}^{\infty} \frac{(x/2)^{2 n}}{(n!)^2}$$

where $I_0(x)$ is the modified Bessel function of the first kind of zero order. Then your sum is equal to $I_0(2) \approx 2.27959$.