Summation of $\sum_{n=0}^{\infty}a^nq^{n^2}$ https://ift.tt/eA8V8J

I am trying to find the result for the sum of the form

$\sum_{n=0}^{\infty}a^nq^{n^2}$.

The special case for $a=1$ is easily given by $\vartheta(0,q)$, where $\vartheta(z,q)$ is the third Jacobi Theta function. So, whatever the answer is, it must collapse to $\vartheta(0,q)$ for $a=1$.

I tried the approach:

$\sum_{n=0}^{\infty}\exp(2niz)q^{n^2}$, where $z=-i\frac{\ln(a)}{2}$, to make the summation look like $\vartheta(z,q)$. However, Jacobi Theta functions include summation from $-\infty$ to $\infty$, while I need it to start at 0.

I tried to use the Parseval's relation as well as several other methods but I cannot proceed any further.

Thank you.

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C.Koca