Is there a closed form to the following sum: $\sum_{n=0}^{\infty}a^nq^{n(n1)/2}$ for all $a>0$ and $0\lt q\lt 1$ ?
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There will not be a closed form for this without some special function. The reason is that there would then be a closed form for the Jacobi theta function, without special functions. Let your function be $F(a,q)$, then $F(e^{2\pi i z+\pi i \tau},e^{2\pi i \tau})$ is $\sum_{n=0}^\infty e^{\pi i n^2 \tau+2\pi n z}$ so $\vartheta(z;\tau)=F(e^{2\pi i z+\pi i \tau},e^{2\pi i \tau})+F(e^{2\pi i z+\pi i \tau},e^{2\pi i \tau})1$. I don't know if your function can be written in terms of some alreadynamed special function. If it were it would have to be something somehow related to the Jacobi theta function. 

