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

<math>Gamma (z;p,q) = prod_^infty prod_^infty

frac. </math>

It obeys several identities:

<math>Gamma(z;p,q)=frac,</math>

<math>Gamma(pz;p,q)=theta (z;q) Gamma (z; p,q),</math>

and

<math>Gamma(qz;p,q)=theta (z;p) Gamma (z; p,q),,</math>

where &theta; is the q-theta function.

When <math>p=0</math>, it essentially reduces to the infinite q-Pochhammer symbol:

<math>Gamma(z;0,q)=frac.</math>

References

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