# A certain sum with q by the power of binomial (n 2)

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 math.sun.ac.za/~hproding/pdffiles/touchard-2011.pdf. –  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

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 $\vartheta(z;\tau)=F(e^{2\pi i z+\pi i \tau},e^{2\pi i \tau})+F(e^{-2\pi i z+\pi i \tau},e^{2\pi i \tau})-1$.