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I encounter a sum similar to the Jacobi Theta function except there is a lower bound $-J$ with $J\geq 0$: \begin{equation} f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n. \end{equation} My question is whether the sum has any simple relation to a well studied function (like theta functions). Secondly although the modular property of Theta function seems to be partially broken by the lower bound $-J$, does $f(q,x)$ still have any relation to modular forms?

Thanks in advance for answering my question.

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    $\begingroup$ This is an incomplete theta function, it lacks most properties of the complete version. For $|y|<1, |x|< 1$ $f(q,xy)=\int_0^1 \theta(q,x e^{-2i \pi t}) \frac{y^{-J}e^{ -2 \pi J i t}}{1-ye^{2 i\pi it}}dt$ $\endgroup$
    – reuns
    Commented Mar 25, 2019 at 20:47

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