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Is there a closed form of the following finite sum? $$ \sum\limits_{n=0}^{N}q^{n^2} $$ where $q>0$. The only thing I found somehow related is the Jacobi $\theta$ elliptic function, but there it is an infinite sum.

Joshhh
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    If you can relate a summation to a special function, this is a good sign that no closed-form exists. Otherwise there would be no need for existence of the special function. –  Nov 19 '18 at 12:59

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