In this paper, Shanks uses the following formula: $$ \sum_{s=0}^{n-1}q^{s(2n+1)} \times \left( \prod_{k=s+1}^{n} \dfrac{1-q^{2k}}{1-q^{2k-1}}\right) = \sum_{s=1}^{2n} q^{\frac{s(s-1)}{2}}$$ to get a one line computation of the Gauss sum: $$\sum_{k=0}^{m-1} e^{\frac{2ik^2 \pi}{m}}$$ in the odd case ($m=4n+1$ or $m=4n+3$).

He proves this relation by induction which is fine with me. I was wondering however if there is a combinatorial interpretation of this formula? Perhaps using the $q$-binomial Theorem and the inverse of the $q$-Pascal matrix?

Apart from its use in Gauss sum computations, this formula can be seen as a finite version of a special case of Jacobi triple product identity. Indeed, if we identify on the left and right handsides the terms having degree $\leq 2n$ and letting $n$ go to infinity, we get:

$$\prod_{k=1}^{+ \infty} \dfrac{1-q^{2k}}{1-q^{2k-1}} = \sum_{k=1}^{+ \infty} q^{\frac{k(k-1)}{2}}.$$