q-gamma function
Function in q-analog theory / From Wikipedia, the free encyclopedia
In q-analog theory, the -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 Jackson (1905). It is given by
when
, and
if
. Here ;\cdot )_{\infty }}
is the infinite q-Pochhammer symbol. The
-gamma function satisfies the functional equation
In addition, the
-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (Askey (1978)).
For non-negative integers n,
where
is the q-factorial function. Thus the
-gamma function can be considered as an extension of the q-factorial function to the real numbers.
The relation to the ordinary gamma function is made explicit in the limit
There is a simple proof of this limit by Gosper. See the appendix of (Andrews (1986)).