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

${\displaystyle \Gamma (z;p,q)=\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1}/z}{1-p^{m}q^{n}z)).}$

It obeys several identities:

${\displaystyle \Gamma (z;p,q)={\frac {1}{\Gamma (pq/z;p,q)))\,}$

${\displaystyle \Gamma (pz;p,q)=\theta (z;q)\Gamma (z;p,q)\,}$

and

${\displaystyle \Gamma (qz;p,q)=\theta (z;p)\Gamma (z;p,q)\,}$

where θ is the q-theta function.

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

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

Multiplication Formula

Define

${\displaystyle {\tilde {\Gamma ))(z;p,q):={\frac {(q;q)_{\infty )){(p;p)_{\infty ))}(\theta (q;p))^{1-z}\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1-z)){1-p^{m}q^{n+z))}.}$

Then the following formula holds with ${\displaystyle r=q^{n))$ (Felder & Varchenko (2002)).

${\displaystyle {\tilde {\Gamma ))(nz;p,q){\tilde {\Gamma ))(1/n;p,r){\tilde {\Gamma ))(2/n;p,r)\cdots {\tilde {\Gamma ))((n-1)/n;p,r)=\left({\frac {\theta (r;p)}{\theta (q;p)))\right)^{nz-1}{\tilde {\Gamma ))(z;p,r){\tilde {\Gamma ))(z+1/n;p,r)\cdots {\tilde {\Gamma ))(z+(n-1)/n;p,r).}$