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 , and can be expressed in terms of the triple gamma function. It is given by
:<math>\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}. </math>
It obeys several identities:
:<math>\Gamma(z;p,q)=\frac{1}{\Gamma(pq/z; p,q)}\,</math>
:<math>\Gamma(pz;p,q)=\theta (z;q) \Gamma (z; p,q)\,</math>
and
:<math>\Gamma(qz;p,q)=\theta (z;p) \Gamma (z; p,q)\,</math>
where θ is the q-theta function.
When <math>p=0</math>, it essentially reduces to the infinite q-Pochhammer symbol:
:<math>\Gamma(z;0,q)=\frac{1}{(z;q)_\infty}.</math>
Multiplication Formula
Define
:<math>\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.</math>
Then the following formula holds with <math>r=q^n</math> ().
:<math>\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).</math>