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 . It is given by
when , and
if . Here is the infinite -Pochhammer symbol. The -gamma function satisfies the functional equation
In addition, the -gamma function satisfies the q-analog of the BohrâÂÂMollerup theorem, which was found by Richard Askey ().
For non-negative integers ,
where is the -factorial function. Thus the -gamma function can be considered as an extension of the -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 ().
The -gamma function satisfies the q-analog of the Gauss multiplication formula ():
The -gamma function has the following integral representation ():
Moak obtained the following q-analogue of the Stirling formula (see ):
where , denotes the Heaviside step function, stands for the Bernoulli number, is the dilogarithm, and is a polynomial of degree satisfying
Due to I. Mezà Â, the q-analogue of the Raabe formula exists, at least if we use the -gamma function when . With this restriction,
El Bachraoui considered the case and proved that
The following special values are known.
These are the analogues of the classical formula .
Moreover, the following analogues of the familiar identity hold true:
Let be a complex square matrix and positive-definite matrix. Then a -gamma matrix function can be defined by -integral:
where is the q-exponential function.
For other -gamma functions, see Yamasaki 2006.
An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.