# Elliptic gamma function

In mathematics, the elliptic gamma function is a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function. It is closely related to a function studied by Jackson (1905), and can be expressed in terms of the triple gamma function. 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.

When $p=0$, it essentially reduces to the infinite q-Pochhammer symbol:

$\Gamma(z;0,q)=\frac{1}{(z;q)_\infty}.$