Evaluating: $ \sum_{n=0}^{\infty}x^{n^{2}} $

Question

How do you evaluate: $\displaystyle \sum_{n=0}^{\infty}x^{n^{2}} $
Or more generally
$ \large\displaystyle \sum_{n=0}^{\infty}x^{n^{\alpha}} $
Note that: $|x| <1$

Answer 1

There is the Jacobi Theta function $$ \vartheta_3(z,q) = \sum_{n=-\infty}^\infty e^{2 i n z} q^{n^2} $$

so yours is $$ \sum_{n=0}^\infty x^{n^2} = \frac{\vartheta_3(0,x)+1}{2} $$

Of course for $a=1$ it can also be evaluated using known functions. But not for other values of $a$.