How to calculate the value of a sum which is similar to a Jacobi theta function?

Question

I would like to know the value of the following infinite series: $$\vartheta^*_3(q,z)=\sum_{n=0}^\infty q^{n^2}e^{2niz},$$ which is very similar to the definition of $\vartheta_3$ in Wolfram. Except $\vartheta^*_3$ is the sum from $n=0$ to $\infty$ whereas $\vartheta_3$ is taken over $n=-\infty$ to $\infty$. Is there an analytic solution for this sum?

Answer 1

Uncomplete answer : The real part of the series. $$\text{Real part}\qquad Re\left(\sum_{n=-\infty}^0 q^{n^2}e^{2niz}\right)=Re\left(\sum_{n=0}^\infty q^{n^2}e^{2niz}\right)$$ $$\text{Imaginary part}\qquad Im\left(\sum_{n=-\infty}^0 q^{n^2}e^{2niz}\right)=-Im\left(\sum_{n=0}^\infty q^{n^2}e^{2niz}\right)$$ $$\vartheta_3(q,z)=\sum_{n=-\infty}^\infty q^{n^2}e^{2niz}=\sum_{n=-\infty}^0 q^{n^2}e^{2niz}+\sum_{n=0}^\infty q^{n^2}e^{2niz}-q^{0^2}e^{2*0*iz}$$ $$\vartheta_3(q,z)=2Re\left(\sum_{n=0}^\infty q^{n^2}e^{2niz}\right)- 1$$ $$Re\left(\sum_{n=0}^\infty q^{n^2}e^{2niz}\right)=\frac12\left(\vartheta_3(q,z)+1\right) $$