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. 


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 $$f_C(e^{2iz})+f_C(e^{2iz})=1+\sum_{k=\infty}^{\infty}\frac{e^{2kiz}}{C^{k^2}}=1+\vartheta_3(z,1/C)$$ 

