Is there an analytic expression for $\sum\limits_{n=0}^{\infty}x^{n^2}$

Question

In statistical mechanics I often come across average energies of the form:

$$\begin{equation} \langle\epsilon_n\rangle=\alpha \sum_{n=0}^{\infty}n^2e^{-\alpha n^2} \end{equation}$$

where $\alpha$ is positive and by rewriting

$$\begin{equation} \langle\epsilon_n\rangle=-\alpha \partial_{\alpha}\sum_{n=0}^{\infty}e^{-\alpha n^2} \end{equation}$$

However, I cannot seem to solve these kinds of sums. Is there any way to solve the sum $$\sum_{n=0}^{\infty}x^{n^2}$$ analytically?

See this fairly similar question please. Also see Jacobi theta function. — Тyma Gaidash, Oct 25 '21 at 21:53
$$\sum_{x=1}^{\infty} (c)^{-x^2}=\frac{\sqrt{\pi}}{2 \sqrt{ln(c)}}-1/2+\frac{ \sqrt{\pi}}{\sqrt{\ln(c)}}\sum_{x=1}^{\infty}e^{\frac{-\pi^2 x^2}{\ln(c)}}$$
A horrible derivation Currently looking more into it, and gained a little bit more insight, but here's somewhat of an answer, and can be used as an easy way to get a good estimation. — Gerben, Dec 05 '21 at 00:09

score 3 · Accepted Answer · answered Oct 25 '21 at 21:57

$f(y)=\sum_{n=0}^\infty e^{-\pi yn^2}$ is a theta function, which arose long ago also in number theory, and as far as I know there is no more-elementary expression for it.

(Despite not being elementary, it does enjoy some interesting properties, useful in number theory, such as the functional equation $f(1/y)=\sqrt{y}\cdot f(y)$, proven via Poisson summation.)

Is there an analytic expression for $\sum\limits_{n=0}^{\infty}x^{n^2}$

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