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 given by

:$$Gamma (z;p,q) = prod_{m=0}^infty prod_{n=0}^inftyfrac{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)= heta (z;q) Gamma (z; p,q),

and

:$$Gamma(qz;p,q)= heta (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}.