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

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

It obeys several identities:

$\mathrm{\Gamma }(z;p,q)=\frac{1}{\mathrm{\Gamma }(pq/z;p,q)}$

$\mathrm{\Gamma }(pz;p,q)=\theta (z;q)\mathrm{\Gamma }(z;p,q)$

and

$\mathrm{\Gamma }(qz;p,q)=\theta (z;p)\mathrm{\Gamma }(z;p,q)$

where θ is the q-theta function. When $p=0$, it essentially reduces to the infinite q-Pochhammer symbol:

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

Multiplication Formula

Define

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

Then the following formula holds with $r=q^n$ (Felder & Varchenko (2002)).

$\tilde{\mathrm{\Gamma }}(nz;p,q)\tilde{\mathrm{\Gamma }}(1/n;p,r)\tilde{\mathrm{\Gamma }}(2/n;p,r)\cdots \tilde{\mathrm{\Gamma }}((n-1)/n;p,r)={\left(\frac{\theta (r;p)}{\theta (q;p)}\right)}^{nz-1}\tilde{\mathrm{\Gamma }}(z;p,r)\tilde{\mathrm{\Gamma }}(z+1/n;p,r)\cdots \tilde{\mathrm{\Gamma }}(z+(n-1)/n;p,r).$

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, 76 (508), The Royal Society: 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.