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%2B1}q^{n%2B1}/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}.

References