I want to know if there is a closed form for the convergence of a series of this kind:
$$\sum_{j=1}^\infty r^{j(j-1)/2}$$ given that $0<r<1$. As it is possible to see the terms of the sum are $r^{(j^2-j)/2}$, which gives problems if trying to solve by the usual way of a geometric series.
And also I have the same problem with a series which is similar to the last one:
$$\sum_{j=1}^\infty \gamma^{j}r^{j(j-1)/2}$$ I want to know the closed form for this one also, given that $0<r<1$ and $0<\gamma<1$.
I will really appreciate any idea to solve these series.
$$\sum_{j=1}^\infty r^{j(j-1)/2}=\sum_{j=1}^\infty {\sqrt r}^{\,j(j-1)}=\frac{\vartheta _2\left(0,\sqrt{r}\right)}{2 \sqrt[8]{r}}$$ where appears Jacobi theta function.
$$\sum_{j=1}^\infty \gamma^{j}r^{j(j-1)/2}= ???$$