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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 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
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1 Answer

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
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