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? 



What is known about this power series?
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.


