Consider the following (real or complex) function $f(x)$ defined for $|x|<1$ by the Taylor series:
$$f(x) = \sum_{n=0}^{\infty}x^{n^2}=1+x+x^4+x^9+x^{16}+\dots$$
Is this the Taylor expansion of a known function or a function that can be written in terms of known functions? It is obviously analytic inside the complex unit circle, and I suspect with a singularity (perhaps a pole) at $x=1$. Any ideas about this function and some of its properties?