Is there a closed form to the following sum: $\sum_{n=0}^{\infty}a^nq^{n(n1)/2}$ for all $a>0$ and $0\lt q\lt 1$ ?

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. 


There is a wellestablished name for this series; it is called a partial theta function. So if by "closed form", you mean "expression in terms of objects that are interesting enough that people have a name for them", the answer is yes. For more information on the partial theta function, search arXiv for a start. 

