In reviewing some calculations, I've arrived at the series:(adsbygoogle = window.adsbygoogle || []).push({});

##S(d)=-\frac 1 {d-1}+\frac 1 2 \frac{d-2}{d-3}-\frac 1 8 \frac{(d-2)(d-4)}{d-5}+\frac 1 {48} \frac{(d-2)(d-4)(d-6)}{d-7}+\dots ##

Its an infinite series but because I'm interested in its values for even ##d##s, its actually a finite series at each evaluation. Its values are:

## \begin{array}{c|ccccc} d \ \ \ &2 \ \ \ &4 \ \ \ &6 \ \ \ &8 \ \ \ &10 \ \ \ &12 \\\hline S(d) \ \ \ &-1 \ \ \ &\frac 2 3 \ \ \ &-\frac 8 {15} \ \ \ &\frac{16}{35} \ \ \ &-\frac{128}{315} \ \ \ &\frac{256}{693}\end{array} ##

But the interesting fact is that the expression ## G(d)=\frac{\Gamma(\frac d 2)\Gamma(\frac{1-d}2)}{2 \sqrt \pi} ## has the exact same values for at least the above values of ##d##. So it seems that I should be able to prove that ##S(d)=G(d)##. But I have no idea how to do it. At first I tried to somehow write the series in terms of the Hypergeometric function, but that is both impossible and useless. So, any ideas?

EDIT: Another evidence is that both ##S(d)## and ##G(d)## are singular for odd ##d##s.

Thanks

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# A Finite series and product of Gammas

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