Gaussian like sum with binomial coefficients

Question

Is there any way to calculate expressions of the form

$$ f(q)=\sum_{n=0}^N\binom{n}{k}\exp({2\pi i\tfrac{n}{N}\cdot q}) $$

where $q\in\mathbb{Q}$?

It reminds me quite alot of Gauss sums, but I don't know much about those.

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Addendum: Please find my more general question here: Geometric sum with binomial coefficient

Answer 1

I presume $k$ is a positive integer, $N \ge k$, and the terms for $n < k$ are taken as $0$.

Maple says $$ \sum_{n=k}^N {n \choose k} r^n = \frac{r^k}{(1-r)^{k+1}} - {N+1 \choose k} r^{N+1} {}_2F_1(1,N+2; N-k+2; r) $$