Any chance to sum this series? ($a,b,c$ are real)
Sum[((a^2 + (b + m)^2) + (c + n)^2)^(-1), {n, -∞, ∞}, {m, -∞, ∞}]
Or even a more general one?
Sum[((a^2 + (b + d m)^2) + (c + e n)^2)^(-1), {n, -∞, ∞}, {m, -∞, ∞}]
I don't know if it can help, but the sum of
Sum[((a^2 + (b + n)^2))^(-1), {n, -∞, ∞}]
is
$$\frac{\pi \sinh (2 \pi a)}{a (\cosh (2 \pi a)-\cos (2 \pi b))}.$$
Thanks for any hint!