# Curious $\sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$ identity [duplicate]

Let $$F(x) = \sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$$

It seems that for odd integer $k$ $$F\left(\frac{k}{2}\right) = \frac{2}{k^2}$$ My evidence is strictly computational, and I have no idea how to approach a proper proof. So standard questions are:

• is it a known fact
• is it a fact
• is it curious
• what is a proof strategy

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## marked as duplicate by Dominic Michaelis, Norbert, Dennis Gulko, vonbrand, Davide GiraudoMar 28 '13 at 11:00

math.stackexchange.com/questions/314986/… and put $a= ix$ –  Cortizol Mar 28 '13 at 8:39
Or more directly the question 141470 –  Raymond Manzoni Mar 28 '13 at 8:41
Thanks everyone. Sorry for being illiterate. –  user58697 Mar 28 '13 at 9:00