Euler pentagonal identity
= Euler pentagonal identity
{c}
{title2=$\prod_{n\geq1}(1-q^n)=\sum_{r\in\mathbb Z}(-1)^r q^{r(3r+1)/2}$}
The identity holds for $|q|<1$, with both sides locally uniformly convergent. In the <character theta proof of the eta product>, pair the terms indexed by $n$ and $-n$ and then select $n=6r+1$; their <Dirichlet character> signs are $(-1)^r$. Removing $q^{1/24}$ gives the displayed product identity.