Well I found the answer to your question, it is

$$\sum_x \tan(x)=ix-\psi _{e^{2 i}}^{(0)}\left(x+\frac{\pi }{2}\right)+C$$

I have verified it with difference operator and it gives tan(x). The function involved is the q-digamma function http://mathworld.wolfram.com/q-PolygammaFunction.html .

You can [verify the result yourself][1].


  [1]: http://www.wolframalpha.com/input/?i=DifferenceDelta%28ix-qpolygamma%280,x%2bPi/2,e%5E%282i%29%29,x%29