This is quick question, hopefully. How can I evaluate $f(x) = \sum_{k=1}^{\infty}x^{k^2}$? Some power series can easily be reduced to a geometric series, taylor series etc. via term wise integration/differentiation. I want to find an expression for $f(x)$ not involving series, to be able to calculate the exact value for the sum for different $x\in (-1,1)$. I've already shown that the radius of convergence is 1, and the series looks kind of like the regular geometric series. I've tried to do some term wise integration/differentiation, which however turned out to not work very well. Perhaps this is easy, but it has been a while since I was doing these kind of problems.
Cheers!
Answer
$\displaystyle \sum_{k = 1}^{\infty}x^{k^{2}} =
-\,{1 \over 2} + {1 \over 2}\sum_{k = -\infty}^{\infty}x^{k^{2}} =
\bbox[8px,border:1px groove navy]{{\vartheta_{3}\left(0,x\right) - 1 \over 2}}$
where $\displaystyle\vartheta_{\nu}$ is a Jacobi Theta Function.
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