I encountered a problem, where I am interested in determining (or estimating) a series of the form$$S = \sum_{k=0} ^\infty z^{k^2+ck},$$while $z\in (0,1)$ and $c>0$. The most simple estimate I can think of is using $z^{k^2+ck}<z^{ck}$, which reduces the problem to the geometric series and we obtain$$S < \sum_{k=0}^\infty z^{ck}=\sum_{k=0}^\infty (z^c)^k=\frac 1 {1-z^c}.$$However, this estimate is rather inaccurate as $k^2$ is increasing much more rapidly than $ck$. So my question is, is there anyway to calculate $S$ directly, although I doubt that. Or is there any way to calculate / estimate$$\sum_{k=0}^\infty z^{k^2}.$$Does such sequences have a special name?.
Update / Background: The problem above is just the abstraction of my original problem, to find an upper bound w.r.t. to the odd integer $N$ of$$\sum_{|k|>\frac{N-1} 2} e^{-\frac{3\pi }{2 N}k^2}=2\sum_{k=0}^\infty e^{-\frac{3\pi}{2N}\left(k+\frac{N+1}2\right)^2}=2e^{-\frac{3 \pi (N+1)^2}{8N}}\sum_{k=0}^\infty z^{k^2+(N+1)k},$$for $z=e^{-\frac{3\pi}{2N}}$. So, we can bound the sum on the right by
\begin{align*}\sum_{k=0}^\infty z^{ k^2}&=-i\int_{i-\infty}^{i+\infty}e^{- \frac{3 \pi u^2}{2N}}\cot( \pi u) du\\&\leq e^{\frac{3\pi}{2N}}\int_{-\infty}^\infty e^{-\frac{3\pi u^2}{2N}}|\cot(\pi(i+u)|du\\&\leq 1.004\; e^{\frac{3\pi}{2N}} \int _{-\infty}^\infty e^{-\frac{3\pi u^2}{2N}}du\\&=1.004\; e^{\frac{3\pi}{2N}} \sqrt{\tfrac{2N}3}, \end{align*}while the integral representation is given through the theta function and the factor 1.004, is a numerical upper bound of the cotanges on the horizontal line $u\mapsto (u+i)\pi)$.
However, numerical test suggest that this estimate is rather inaccurate. For $N = 100001$ we have$$\sum_{k=0}^\infty z^{k^2+(N+1)k}\approx 1.009$$and$$ 1.004\; e^{\frac{3\pi}{2N}} \sqrt{\tfrac{2N}3}\approx 259.25.$$Does anyone have better suggestions?