Interesting infinite series

How do find an analytic solution to this infinite series. (plug this into http://atomurl.net/math/ (latex))

\sum_{n=1}^\infty \csc^2(\omega\pi n)= \frac{A}{\pi} +B

I have attempted the following to break the series down into 4

\sum_{n=1}^\infty \csc^2(\omega\pi n)= \sum_{n=1}^\infty csch^2(i\omega\pi n)= 4\sum_{n=1}^\infty \big(e^{\pi n \big( \frac{i}{2} + \frac{ \sqrt{3} }{2} \big) }-e^{-\pi n \big( \frac{i}{2} + \frac{ \sqrt{3} }{2} \big)}\big) ^{-2}

\sum_{n=1}^\infty \big(e^{\pi n \big( \frac{i}{2} + \frac{ \sqrt{3} }{2} \big) }-e^{-\pi n \big( \frac{i}{2} + \frac{ \sqrt{3} }{2} \big)}\big) ^{-2}= \big(ie^{\pi\frac{\sqrt{3}}{2}}+ie^{-\pi\frac{\sqrt{3}}{2}}\big)^{-2}+ \big(-e^{\pi\sqrt{3}}+e^{-\pi\sqrt{3}}\big)^{-2} +\big(-ie^{3\pi\frac{\sqrt{3}}{2}}-ie^{-3\pi\frac{\sqrt{3}}{2}}\big)^{-2}+ \big(e^{2\pi\sqrt{3}}-e^{-2\pi\sqrt{3}}\big)^{-2} +...

= \sum_{n=0}^\infty \big(ie^{(4n+1)\pi\frac{\sqrt{3}}{2}}+ie^{-(4n+1)\pi\frac{\sqrt{3}}{2}}\big)^{-2} +\sum_{n=0}^\infty \big(-e^{(2n+1)π√3}+e^{-(2n+1)π√3}\big)^{-2} +\sum_{n=0}^\infty \big(-ie^{(3+4n)\pi\frac{\sqrt{3}}{2}}+-ie^{-(4n+3)\pi\frac{\sqrt{3}}{2}}\big)^{-2}+
\sum_{n=0}^\infty \big(e^{(2n)π√3}-e^{-(2n)π√3}\big)^{-2}

\sum_{n=0}^\infty \big(-e^{(4n+1)\pi\sqrt{3}}-2-e^{-(4n+1)\pi\sqrt{3}}\big)^{-1}+ \sum_{n=0}^\infty \big(e^{2(2n+1)\pi\sqrt{3}}-2+e^{-2(2n+1)\pi\sqrt{3}}\big)^{-1} + \sum_{n=0}^\infty \big(e^{(3+4n)\pi\sqrt{3}}-2+e^{-(3+4n)\pi\sqrt{3}}\big)^{-1}+ \sum_{n=1}^\infty \big(e^{4n\pi\sqrt{3}}-2+e^{-4n\pi\sqrt{3}}\big)^{-1}

I have found the sums numerically and

\sum_{n=0}^\infty \big(-e^{(4n+1)\pi\sqrt{3}}-2-e^{-(4n+1)\pi\sqrt{3}}\big)^{-1}+ \sum_{n=0}^\infty \big(e^{2(2n+1)\pi\sqrt{3}}-2+e^{-2(2n+1)\pi\sqrt{3}}\big)^{-1} + \sum_{n=0}^\infty \big(e^{(3+4n)\pi\sqrt{3}}-2+e^{-(3+4n)\pi\sqrt{3}}\big)^{-1}+ \sum_{n=1}^\infty \big(e^{4n\pi\sqrt{3}}-2+e^{-4n\pi\sqrt{3}}\big)^{-1} \approx -0.00429

The only way I know how to evaluate the sum is using the polygamma function however it does not give values for A and B. Any help would be grand!

Having been a graduate student in math, I actually feel like I could handle this.

Cheers Mate I have posted it in the Killer derivatives as well

I did sorry.. A and B are both algebraic numbers 

First off, this is by far the most challenging problem I've seen on these posts. You're definitely a little further along in your math career than many.

Secondly, this looks like something you'll want to solve with the Residue Theorem. Find a function f(z) with an infinite discrete set of singularities above the real axis. Show that |f(z)| decays at a rate of 1/|z|^2 or faster for Im(z) > 0, so the integral from 0 to pi of f(Re^{it}} dt goes to 0 as R approaches infinity. Then the sum of the residues is $\frac{1}{2\pi i}\int_{-infty}^{infty}f(x)\, dx$.

Thanks for the help mate and I will be posting problems of this calibre here (if I need any help).

I don't know how to evaluate it other than numerically.