In this question the summation goes from $-\infty$ to $\infty$ and the answer has a pretty "good" closed form.

Now I came across the sum $\sum_{n=1}^\infty q^{-n^2} z^n$ where $|z|<1$ and I don't know how to crack it.

Please help me!

This question has an open bounty worth +100 reputation from Mikalai Parshutsich ending in 5 hours.

The question is widely applicable to a large audience. A detailed canonical answer is required to address all the concerns.

I'm looking for a closed form that will represent this sum in terms of existing and well-known functions. If your function is not widely known I'll appreciate some links on articles where that function is studied. Thanks!

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