Sum of finite series of $\exp(an^2 + bn + c))$

Question

Is there a way to simplify that sum to an expression without actual performing the summation, similar to the formula for calculating the sum of a (finite) geometric series?

$\sum_{n=0}^{N-1} \exp(an^2 + bn + c)$

Note that $a\neq0$

Answer 1

This is the Jacobi theta function, which knows no closed form. For instance, $\displaystyle\sum_{n=-\infty}^\infty a^{n^2}=\theta_3(0,a)$.

Answer 2

I suspect the answer is no. Firstly, the class of formulae that we can sum is quite small, and the vast majority of them have no general closed form. Secondly, Wolfram does not know how to sum $a^{n^2}$ which is quite discouraging. Thirdly, using discrete calculus reveals a very erratic rate of growth. It's rare that closed forms exist that neither wolfram nor discrete calc can detect.