Elliptic gamma function
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 , and can be expressed in terms of the triple gamma function. It is given by
It obeys several identities:
and
where θ is the q-theta function.
When
, it essentially reduces to the infinite q-Pochhammer symbol:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the elliptic gamma function is a generalization of the q-Gamma function
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...
, which is itself the q-analog
Q-analog
Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, 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 q → 1...
of the ordinary Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. It is closely related to a function studied by , and can be expressed in terms of the triple gamma function. It is given by
It obeys several identities:
and
where θ is the q-theta function.
When
, it essentially reduces to the infinite q-Pochhammer symbol: