I would like to know if there exists an expression for the generalized alternating quadratic Gauss sum $$ g_N(a,b) = \sum_{k=0}^{N-1}(-1)^{k} e^{i2\pi\frac{ak(k+b)}{N}} $$ where $N$ is an odd prime and where $a,b \in \{1,2,\dots,N-1\}$.
When $b=0$, it is straightforward to show that $g_N(a,0)=1$, since $e^{i2\pi\frac{a(N-k)^2}{N}}=e^{i2\pi\frac{ak^2}{N}}$, for all $k=1,\dots,N-1$, while $k$ and $N-k$ have opposite parity.
When $b\ne 0$, things are a bit more complicated. I have tried to simplify the expression by completing the square in the exponent and reindexing, obtaining $$ g_N(a,b)=(-1)^{\eta b}e^{-i 2\pi a \frac{\eta^2 b^2}{N}}\sum_{k=\eta b}^{N+\eta b - 1}(-1)^k e^{i 2\pi a \frac{k^2}{N}} $$ where $\eta = (N+1)/2 \equiv 2^{-1}\pmod{N}$. Then, we can pair the terms of the sum that correspond to indices that have the same square modulo $N$: these terms will either cancel each other or combine constructively depending on the value of $b$. So, for example \begin{align*} g_N(a,1)&=(-1)^{\eta +1}e^{-i 2\pi a \frac{\eta^2}{N}}\biggl(1+2\sum_{k=1}^{(N - 1)/2}(-1)^k e^{i 2\pi a \frac{k^2}{N}}\biggr) \\ g_N(a,2)&=e^{-i 2\pi \frac{a}{N}} \\ g_N(a,3)&=(-1)^{\eta}e^{-i 2\pi a \frac{9\eta^2}{N}}\biggl(1+2\sum_{k=1}^{(N - 3)/2}(-1)^k e^{i 2\pi a \frac{k^2}{N}}\biggr) \\ g_N(a,4)&=\dots \end{align*} The question of computing the generalized alternating quadratic Gauss sum then translates into the question of computing a partial alternating quadratic Gauss sum.
I've tried to split the sum into even-indexed and odd-indexed terms, to get rid of the $(-1)^k$ term, but it doesn't seem to bring me anywhere. Similarly, writing $$ (-1)^k e^{i2\pi\frac{ak(k+b)}{N}} = e^{i\pi k\bigl(\frac{2a(k+b)}{N} + 1\bigr)} $$ doesn't seem to be of much help either, since I can't work modulo $N$ anymore.
Unfortunately, I wasn't able to find any work about this type of sums. The closest topic is the asymptotic expansion of a more general sum $$ S_n(x,\theta) = \sum_{k=1}^n e^{i \pi x k^2 + i 2 \pi \theta k} $$ for large $n$ and for $x\in(0,1)$ and $\theta\in[-1/2,1/2]$ (see, for instance, this article and references therein). Even though I could try to work out the case where $x=2a/N$ and $\theta = 1/2$, I was hoping in something less involved. Moreover, this other old paper seems to suggest that an expression exists for rational $x$, even though they don't provide it (they only deal with the case $\theta = 0$ that boils down to the classic quadratic Gauss sum, suggesting that the case $\theta = 1/2$ follow similarly, but I can't see it).
Update: As a starting point, a tight upper bound for $\lvert g_N(a,b) \rvert$ (in particular, for $\lvert g_N(a,1) \rvert$) as a function of $N$ would be enough for now.
