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Simplifying Root of Unity Double Summation
Good afternoon. I have a particular summation,
$$\zeta_{n,k}(N)=\frac{k!}{N^{n+1-k}}\sum_{j=0}^n\sum_{i=0}^{N-1}\binom{n}{j}w_N^{(j-k)i}$$
Here, the $w_N$ is the root of unity $w_N=e^\frac{2i\pi}{N} …