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

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I don't think that there is a closed form, but there is a simple continued fraction, see formula (1.1) in – Johann Cigler May 25 '12 at 15:22
Thanks, but how can I get rid of the element $(-1)^n$ appearing in this reference formula (1.1) ? – guy May 25 '12 at 19:43
In a deleted answer, "Guy" asks if there is a connection to hypergeometric series. – S. Carnahan May 27 '12 at 9:48

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 already-named special function. If it were it would have to be something somehow related to the Jacobi theta function.

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but in Jacobi theta function the sum starts from (- infinity), and in the questions it starts from 0. Does it still hold? – guy May 25 '12 at 8:03
You are corrected. I have turned around my answer to avoid that mistake. – Will Sawin May 25 '12 at 8:43

There is a well-established 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.

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