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

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

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:

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

Multiplication Formula

Define

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

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

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