Q-gamma function


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 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.
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.

Transformation Properties

The -gamma function satisfies the q-analog of the Gauss multiplication formula :

Integral Representation

The -gamma function has the following integral representation :

Stirling Formula

Moak obtained the following q-analogue of the Stirling formula :
where, denotes the Heaviside step function, stands for the Bernoulli number, is the dilogarithm, and is a polynomial of degree satisfying

Raabe-type formulas

Due to I. Mező, the q-analogue of the Raabe formula exists, at least if we use the q-gamma function when. With this restriction
El Bachraoui considered the case and proved that

Special values

The following special values are known.
These are the analogues of the classical formula.
Moreover, the following analogues of the familiar identity hold true:

Matrix Version

Let be a complex square matrix and Positive-definite matrix. Then a q-gamma matrix function can be defined by q-integral:
where is the q-exponential function.

Other q-Gamma Functions

For other q-gamma functions, see Yamasaki 2006.

Numerical Computation

An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.