Elliptic gamma function - Alchetron, the free social encyclopedia

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 .

It obeys several identities:

Γ ( z ; p , q ) = 1 Γ ( p q / z ; p , q ) Γ ( p z ; p , q ) = θ ( z ; q ) Γ ( z ; p , q )

and

Γ ( q z ; p , q ) = θ ( z ; p ) Γ ( z ; p , q )

where θ is the q-theta function.

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

Γ ( z ; 0 , q ) = 1 ( z ; q ) ∞ .

References

Elliptic gamma function Wikipedia

(Text) CC BY-SA