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

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

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:

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

References

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR2128719