 Reciprocal geometric infinite product

For $|q|>1$, define $S_q(z)={\prod }_{k\ge 0}(1-q^{-2k-1}z)(1-q^{-2k-1}{z}^{-1})$ on ${\mathbb{C}}^{\ast }$. The geometric decay gives locally uniform absolute tail convergence there. Its simple zeros are $q^{2k+1}$ and $q^{-2k-1}$ for $k\ge 0$, and reindexing gives $S_q(qz)=-z{S}_q(q^{-1}z)$. The variable zero is outside its domain, and the poles introduced by each reciprocal factor cannot be interpreted as a value of the product at zero.

// Destination: analysis.bigb