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

[math]\displaystyle{ \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]\displaystyle{ \Gamma(z;p,q)=\frac{1}{\Gamma(pq/z; p,q)}\, }[/math]

[math]\displaystyle{ \Gamma(pz;p,q)=\theta (z;q) \Gamma (z; p,q)\, }[/math]

and

[math]\displaystyle{ \Gamma(qz;p,q)=\theta (z;p) \Gamma (z; p,q)\, }[/math]

where θ is the q-theta function.

When [math]\displaystyle{ p=0 }[/math], it essentially reduces to the infinite q-Pochhammer symbol:

[math]\displaystyle{ \Gamma(z;0,q)=\frac{1}{(z;q)_\infty}. }[/math]

Multiplication Formula

Define

[math]\displaystyle{ \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]\displaystyle{ r=q^n }[/math] ((Felder Varchenko)).

[math]\displaystyle{ \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

• 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, doi:10.1098/rspa.1905.0011, ISSN 0950-1207, Bibcode: 1905RSPSA..76..127J 
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8 
• 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, Bibcode: 1997JMP....38.1069R, https://ir.cwi.nl/pub/2164 
• Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang (2008). "A gerbe for the elliptic gamma function". Duke Mathematical Journal 141. doi:10.1215/S0012-7094-08-14111-0.