Earlier I had asked a question about the series: $$ \sum_{n=1}^{\infty} \sum_{m=1}^\infty \frac{1}{{n^2 +m^2}}. $$ Which looks like some kind of double series version of the Basel problem. Much to my disappointment this series was shown to diverge here (The Double Basel Problem). There could still be a series worthy of being called the Double Basel Problem. $$ \sum_{n=1}^{\infty} \sum_{m=1}^\infty \frac{1}{{(n^2 +m^2)^2}}. $$ This double series is certainly convergent; however evaluating the sum involves tackling some series I have never seen before. Using the same kind of trick from the other problem and differentiating, this double series can be reduced into: $$ \sum_{n=1}^{\infty} \frac{\pi^2}{4n^2} {csch}^2\pi n + \frac{\pi}{4n^3} \coth \pi n - \frac{1}{2n^4} $$
Has anyone seen series like the first two?
How do we even begin to make progress on this kind of thing?
Thanks for all of your help!
