Original question (see also the revised, possibly simpler, version below): Let $g > 1, r > 1$ be integers. Playing around with the Verlinde formula (see below), I came across the expression $$\sum_{n=1}^{r-1} \sin(\pi n/r)^{2-2g} (e^{2\pi i n^2/r}-1).$$
My goal is to reduce the complexity in $r$ of this expression; that is, to find a closed form of the sum avoiding the dependence on $r$ in the number of summands. Is this possible? Here's a related example:
The Verlinde formula, which e.g. has applications in conformal field theory, algebraic geometry, and quantum topology, is $$(r/2)^{g-1}\sum_{n=1}^{r-1} \sin(\pi n/r)^{2-2g}.$$ In this case, one can use a trick by Szenes to reduce the complexity of the sum: The sum can be written as $$\sum_{n=1}^{r-1} f(z_n).$$ where $z_n = e^{\pi i n/r}$, for a suitable meromorphic function $f : \mathbb{C} \to \mathbb{C}$ having poles only at $1$ and $-1$. Now the trick essentially is to find a suitable meromorphic form $\mu_r$ having poles at $2r$'th roots of unity and apply the Residue Theorem to $f\mu_k$ to rewrite the above sum as $$\sum_{n=1}^{r-1} f(z_n) = -\text{Res}_{z=1} f\mu_r,$$ which then turns out to be a polynomial in $r$ of degree $2g-2$.
This trick doesn't seem to apply to my slightly more complicated sum though. Another possibility might be to somehow rewrite the sum as a Gauss sum, but that doesn't quite seem to work either.
"Revised" question: So, maybe the question above does not have a straightforward answer, but I believe it might suffice to be able to work out the following (at least, it's a similar problem). Say we just have a sum like $$\sum_{n=1}^{r-1} e^{\pi i n^2/(2r)}$$ as above (almost, anyway). Then we may apply a quadratic reciprocity theorem to simplify matters. But say now that we throw in a power of $n$ to get something like $$\sum_{n=1}^{r-1} n^k e^{\pi i n^2/(2r)},$$ for $k > 0$. Can sums like these be treated in a similar manner as the quadratic Gauss sum above (perhaps just in special cases like $k = 1$, or $k = 2$); can we somehow describe the large $r$ asymptotics? Standard tricks in this field seem to involve summation by parts and the Euler--Maclaurin formula but it doesn't seem to quite work out. For example, in the case $k = 1$, summation by parts (or elementary combinatorial considerations) will imply $$\sum_{n=1}^{r-1} n e^{\pi i n^2/(2r)} = (r-1)\sum_{n=1}^{r-1} e^{\pi in^2/(2r)} - \sum_{j=1}^{r-1}\sum_{n=1}^j e^{\pi in^2/(2r)}.$$ Now, the first term is simple to handle as mentioned above, but the second one seems to be worse. Any suggestions?