In the course of some calculations, I came across the following powers series. For fixed $C>1$ let $$ f_C(u)=\sum_{k=0}^\infty\frac{u^k}{C^{k^2}}. $$ This series converges for all $u\in\mathbb C$, hence $f_C$ is an entire function. Can it be expressed in terms of classical special functions? Does it satisfy a differential equation? Has anbybody seen this guy somewhere else?
This function has no known expression in terms of common special functions. It is called "partial thetafunction", and there was some recent research on it: http://arxiv.org/pdf/1106.6262v1.pdf, http://arxiv.org/pdf/1106.1003.pdf, and literature cited there.

$\begingroup$ The corresponding $\sum_{\infty}^\infty$ would be given in terms of the theta function, as in Carlo's answer. $\endgroup$ – Gerald Edgar Feb 13 '14 at 14:41

$\begingroup$ And what does it tell you about $f_C(u)$? Only the functional equation $f_C(u)+f_C(u)=1+\theta_3$ whose solution is not unique. $\endgroup$ – Alexandre Eremenko Feb 14 '14 at 3:12
for $u=C$ it's an elliptic theta function,
$$f_u(u)=1+\frac{1}{2}u^{1/4}\vartheta_2(0,1/u)$$
more generally $$f_C(e^{2iz})+f_C(e^{2iz})=1+\sum_{k=\infty}^{\infty}\frac{e^{2kiz}}{C^{k^2}}=1+\vartheta_3(z,1/C)$$

1$\begingroup$ This is NOT an expression of $f_C(u)$ in terms of thetafunction. $\endgroup$ – Alexandre Eremenko Feb 13 '14 at 14:25