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 , 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}.