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:

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