Elliptic gamma function

Last updated February 28, 2023

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 }}}.

Multiplication Formula

Define

{\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}}}.

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).

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References

Felder, G.; Varchenko, A. (2002). "Multiplication Formulas for the Elliptic Gamma Function". arXiv: math/0212155 .
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, The Royal Society, 76 (508): 127–144, Bibcode:1905RSPSA..76..127J, 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, Bibcode:1997JMP....38.1069R, doi:10.1063/1.531809, ISSN 0022-2488, MR 1434226
Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang (2008). "A gerbe for the elliptic gamma function". Duke Mathematical Journal. 141. arXiv: math/0601337 . doi:10.1215/S0012-7094-08-14111-0. S2CID 817920.

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