# A Tricky sum to evaluate (Haldane)

I'm trying to find a way to evaluate this sum (found by Haldane in Phys. Rev. Lett. 60, 635 (1988): $$S_{pq}=\sum_{n=1}^{N-1} z^{nJ} (1-z^{n})^{p-1}(1-z^{-n})^{q-1}$$ with $$z= e^{\frac{2i\pi}{N}}$$ and $$0\leq J\leq N$$ if someone have an idea, let me know, Thanks

Hint: You can use the binomial expansion and then simplify back with the N-th roots of the unit $$\exp{(\frac{2i\pi}{N})}$$.