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

Γ ( z ; p , q ) = ∏ m = 0 ∞ ∏ n = 0 ∞ 1 − p m + 1 q n + 1 / z 1 − p m q n z . {\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}}.}

It obeys several identities:

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

Γ ( p z ; p , q ) = θ ( z ; q ) Γ ( z ; p , q ) {\displaystyle \Gamma (pz;p,q)=\theta (z;q)\Gamma (z;p,q)\,}

and

Γ ( q z ; p , q ) = θ ( z ; p ) Γ ( z ; p , q ) {\displaystyle \Gamma (qz;p,q)=\theta (z;p)\Gamma (z;p,q)\,}

where θ is the q-theta function.

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

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

Define

Γ ~ ( z ; p , q ) := ( q ; q ) ∞ ( p ; p ) ∞ ( θ ( q ; p ) ) 1 − z ∏ m = 0 ∞ ∏ n = 0 ∞ 1 − p m + 1 q n + 1 − z 1 − p m q n + z . {\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}}}.}

Then the following formula holds with r = q n {\displaystyle r=q^{n}} (Felder & Varchenko (2002)).

Γ ~ ( n z ; p , q ) Γ ~ ( 1 / n ; p , r ) Γ ~ ( 2 / n ; p , r ) ⋯ Γ ~ ( ( n − 1 ) / n ; p , r ) = ( θ ( r ; p ) θ ( q ; p ) ) n z − 1 Γ ~ ( z ; p , r ) Γ ~ ( z + 1 / n ; p , r ) ⋯ Γ ~ ( z + ( n − 1 ) / n ; p , r ) . {\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).}

• 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.