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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.

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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

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up vote 6 down vote accepted

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 $I_0(2) \approx 2.27959$.

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I am interested in other cases for my series where the power of the gamma function being taken is any natural number. Are there generalizations of the modified Bessel function for these cases, or do I get something different entirely? –  Samuel Reid Jun 14 '13 at 0:50
    
Likely something related to an exponential, but no, I am not aware of such a generalization. I recall an exercise in Bender & Orszag which looked at asymptotic developments of such a power of factorials in a series; I'll try to dredge that up. –  Ron Gordon Jun 14 '13 at 0:55
    
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
    
@HansEngler: That is wonderful! Do you have a reference or a proof of that fact? –  Samuel Reid Jun 14 '13 at 2:01
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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

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