In mathematics, the elliptic gamma function is a generalization of the

q-gamma function In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by \Gamma_q(x) = (1-q)^\prod_^\infty \frac=(1-q) ...

, which is itself the

q-analog In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q'' ...

of the ordinary

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...

. It is closely related to a function studied by , and can be expressed in terms of the triple gamma function. It is given by :

\Gamma (z;p,q) = \prod_^\infty \prod_^\infty
\frac.

It obeys several identities: :

\Gamma(z;p,q)=\frac\,

\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 In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer sym ...

: :

\Gamma(z;0,q)=\frac.

Multiplication Formula

Define :

\tilde(z;p,q):=\frac(\theta(q;p))^\prod_^\infty \prod_^\infty
\frac.

Then the following formula holds with

r=q^n

(). :

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

References

* * *{{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 The ''Journal of Mathematical Physics'' is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in Januar ...

, issn=0022-2488 , volume=38 , issue=2 , pages=1069–1146, bibcode=1997JMP....38.1069R , url=https://ir.cwi.nl/pub/2164 * A gerbe for the elliptic gamma function Gamma and related functions Q-analogs