12
votes
1answer
201 views

Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$

The so-called "two squares theorem" can be proven by establishing the following identity: $$\left(\sum_{n=-\infty}^\infty e^{\pi i \tau n^2}\right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi ...
4
votes
1answer
117 views

series involving $\log \left(\tanh\frac{\pi k}{2} \right)$

I found an interesting series $$\sum_{k=1}^\infty \log \left(\tanh \frac{\pi k}{2} \right)=\log(\vartheta_4(e^{-\pi}))=\log \left(\frac{\pi^{\frac{1}{4}}}{2^{\frac{1}{4}}\Gamma \left( ...
6
votes
0answers
293 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
0
votes
0answers
83 views

Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $ Is it related to a known transcendental function? From the research I did, it ...
5
votes
1answer
347 views

Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?

In my recent explorations I stumbled upon the following series $$ \vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z ...