{{Short description|Mathematic 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 {{harvtxt|Jackson|1905}}, 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> ({{harvtxt|Felder|Varchenko|2002}}). :<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> ==References== * {{cite arXiv |last1=Felder |first1=G. |last2=Varchenko |first2=A. |title=Multiplication Formulas for the Elliptic Gamma Function |date=2002 |eprint=math/0212155 }} *{{Citation | last1=Jackson | first1=F. H. | title=The Basic Gamma-Function and the Elliptic Functions | jstor=92601 | publisher=The Royal Society | year=1905 | journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character | issn=0950-1207 | volume=76 | issue=508 | pages=127–144 | doi=10.1098/rspa.1905.0011| bibcode=1905RSPSA..76..127J | doi-access=free }} *{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=Cambridge University Press | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}} *{{Citation | last1=Ruijsenaars | first1=S. N. M. | title=First order analytic difference equations and integrable quantum systems | doi=10.1063/1.531809 | mr=1434226 | year=1997 | journal=Journal of Mathematical Physics | issn=0022-2488 | volume=38 | issue=2 | pages=1069–1146| bibcode=1997JMP....38.1069R | url=https://ir.cwi.nl/pub/2164 }} *{{cite journal |s2cid=817920 |doi=10.1215/S0012-7094-08-14111-0 |title=A gerbe for the elliptic gamma function |year=2008 |last1=Felder |first1=Giovanni |last2=Henriques |first2=André |last3=Rossi |first3=Carlo A. |last4=Zhu |first4=Chenchang |journal=Duke Mathematical Journal |volume=141 |arxiv=math/0601337 }} Category:Gamma and related functions Category:Q-analogs