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

Asked Mar 28 '13 at 08:32

Active Mar 28 '13 at 08:51

Viewed 482 times

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

Please advise.

edited Mar 28 '13 at 08:51

Asaf Karagila

393,674

asked Mar 28 '13 at 08:32

user58697

2,761

3

http://math.stackexchange.com/questions/314986/value-of-sum-k-1-infty-frac1k2a2 and put $a= ix$ – Cortizol Mar 28 '13 at 08:39
1

Or more directly the question 141470 – Raymond Manzoni Mar 28 '13 at 08:41

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

0 Answers0