Elliptic gamma function
# 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

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

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:

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

## Multiplication Formula

Define

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

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

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