Elliptic gamma function

In mathematics, the elliptic gamma function is a q-series analog. It is given by

$\Gamma (z;p,q)=\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1}/z}{1-p^{m}q^{n}z}}.$

It obeys several identities:

$\Gamma (z;p,q)={\frac {1}{\Gamma (pq/z;p,q)}}\,$

$\Gamma (pz;p,q)=\theta (z;q)\Gamma (z;p,q)\,$

and

$\Gamma (qz;p,q)=\theta (z;p)\Gamma (z;p,q),\,$

where θ is the q-theta function.