Jacobi cubic product identity
= Jacobi cubic product identity
{c}
{title2=$\prod_{n\geq1}(1-q^n)^3=\sum_{r\geq0}(-1)^r(2r+1)q^{r(r+1)/2}$}
The odd <Dirichlet character> modulo four has $\chi_4(2r+1)=(-1)^r$. The series $B=\tfrac12\sum_n n\chi_4(n)q^{n^2/8}$ is therefore $q^{1/8}$ times the displayed sum. The <character theta proof of the eta product> gives $B=\eta^3$, proving the identity.