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I am looking for a closed form of the expression

$$F(x) = \sum _{n=1}^\infty x^{n^2} $$

The question arose when I attempted to prove Lagrange's four square theorem via generating functions. It doesn't seem the closed form exists, but I couldnt find any confirmation in the literature.

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Using parity to extend the summation to all integers, one can recognize in the resulting expression Jacobi theta function $\vartheta_3(z,q)=\sum_{n\in\mathbb Z}q^{n^2} e^{2ni z}$. More precisely, we have $$\sum_{n=1}^{\infty}x^{n^2}=\frac{\vartheta_3(0,x)-1}{2}.$$

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