For arbitrary $a,b,c$, does the series $$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$ i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything known? I'm aware that it is very ugly and I am willing to go to some lengths to meet the thing halfway.
I know, for example, that if $c=0$ the series reduces in essence to a Jacobi theta function, $\theta_3$, which requires $\operatorname{Im}(b)>0$ to make sense. In my application $b$ is real, which doesn't by itself doom the series but does make it very difficult for it to converge. On the other hand, I'm happy to take a distribution-valued function, which is essentially equivalent to saying I'm happy to assume $\operatorname{Im}(b)>0$ if needed.
Away from the theta-function case, though, as soon as $c$ steps away from zero (which I do need to do), every resource I've tried goes belly-up.
Is there some known function which encapsulates this behaviour? Failing that, I'd also be interested in applications where this or similar objects came up and people were forced to just deal with it directly.