Closed form of the sum $\sum_{n=1}^\infty q^{-n^2} z^n$ for $|z|<1$

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The closed form of the sum $\sum_{n=1}^\infty q^{-n^2} z^n$ for $|z|<1$ is given by:
\[
\frac{z e^{-z/\sqrt{q}}}{1 - e^{-z/\sqrt{q}}}
\]
gr.inc/question/find-the-closed-form-of-the-s … infty-q-n2-zn-where/