Well I found the answer to your question, it is

$$\sum_x \tan(x)=-\psi _{e^{2 i}}^{(0)}\left(x-\frac{\pi }{2}\right)+\psi _{e^{2 i}}^{(0)}\left(1-\frac{\pi }{2}\right)+ix+C$$

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