Elliptic gamma function explained

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, and can be expressed in terms of the triple gamma function. It is given by

\Gamma (z;p,q) = \prod ^{inf ty} _{m =0} \prod ^{inf ty (1- p ^{m+1} q ^{n+1} /z)/(1- p ^{m} q ^{n} z)} _{n =0} .

It obeys several identities:

\Gamma(z;p,q) = (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) = (1)/((z;q) _{infty}) .

Multiplication Formula

Define

\tilde{\Gamma}(z;p,q): = ((q;q) _{infty})/((p;p) _{infty}) (\theta(q;p)) ^{1-z} \prod ^{inf ty} _{m =0} \prod ^{inf ty (1- p ^{m+1} q ^{n+1-z})/(1- p ^{m} q ^{n+z})} _{n =0} . Then the following formula holds with r = q ^{n} .\tilde{\Gamma}(nz;p,q)\tilde{\Gamma}( 1 /n;p,r)\tilde{\Gamma}( 2 /n;p,r) … \tilde{\Gamma}((n -1 )/n;p,r) = \left( (\theta(r;p))/(\theta(q;p)) \right) ^{nz-1} \tilde{\Gamma}(z;p,r)\tilde{\Gamma}(z +1 /n;p,r) … \tilde{\Gamma}(z + (n -1 )/n;p,r).

References

• Felder . G. . Varchenko . A. . Multiplication Formulas for the Elliptic Gamma Function . 2002 . math/0212155.
  • • • 817920 . 10.1215/S0012-7094-08-14111-0 . A gerbe for the elliptic gamma function . 2008 . Felder . Giovanni . Henriques . André . Rossi . Carlo A. . Zhu . Chenchang . Duke Mathematical Journal . 141 . math/0601337 .