# theta function with a low bound in the sum

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

Thanks in advance for answering my question.

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