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?

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This function has no known expression in terms of common special functions. It is called "partial theta-function", 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.

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The corresponding $\sum_{-\infty}^\infty$ would be given in terms of the theta function, as in Carlo's answer. – Gerald Edgar Feb 13 '14 at 14:41
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. – 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 $$f_C(e^{2iz})+f_C(e^{-2iz})=1+\sum_{k=-\infty}^{\infty}\frac{e^{2kiz}}{C^{k^2}}=1+\vartheta_3(z,1/C)$$

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This is NOT an expression of $f_C(u)$ in terms of theta-function. – Alexandre Eremenko Feb 13 '14 at 14:25