0

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.

refor
  • 1

1 Answers1

1

$$\sum_{j=1}^\infty r^{j(j-1)/2}=\sum_{j=1}^\infty {\sqrt r}^{\,j(j-1)}=\frac{\vartheta _2\left(0,\sqrt{r}\right)}{2 \sqrt[8]{r}}$$ where appears Jacobi theta function.

$$\sum_{j=1}^\infty \gamma^{j}r^{j(j-1)/2}= ???$$

Claude Leibovici
  • 260,315
  • Thanks @ClaudeLeibovici for the first series. It was very helpful to know that those functions correspond with the Jacobi Theta functions. However, I have a question: Is not the theta function defined as a series between $k=-\infty$ and $k=\infty$??. And in this case I have only the positive part of the series, what to do in that case? – refor Nov 30 '20 at 15:28