How do I find a closed form solution of the following sum? $$\sigma=\sum_{i=1}^n \gamma^{\alpha i^2 + \beta i}=\sum_{i=1}^n \gamma^{\alpha i^2}\gamma^{\beta i}$$
My attempt:
I tried using the method which would solve $\sum\gamma^{\alpha i}$ where I would do this:
$$ \begin{aligned} \sigma =&\, \gamma^{\alpha} + \gamma^{2\alpha} + \dots + \gamma^{n\alpha} \\ \gamma^\alpha \sigma =& \hspace{2.675em} \gamma^{2\alpha} + \dots + \gamma^{n\alpha} + \gamma^{(n+1)\alpha} \end{aligned} $$
However there isn't a factor which does the same trick because of the $i^2$. Is there a different method for solving this, or is there no closed form solution?
There is no "closed form" for this sum due to the $i^2$ term.