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:

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

References

• Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 76 (508), The Royal Society: 127–144, doi:10.1098/rspa.1905.0011, ISSN 0950-1207, JSTOR 92601
• 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, MR 2128719
• Ruijsenaars, S. N. M. (1997), "First order analytic difference equations and integrable quantum systems", Journal of Mathematical Physics, 38 (2): 1069–1146, doi:10.1063/1.531809, ISSN 0022-2488, MR 1434226