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. It is given by when |q|<1, and if |q|>1. Here is the infinite q-Pochhammer symbol. The q-gamma function satisfies the functional equation In addition, the q-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey. For non-negative integers n, where [\cdot]_q is the q-factorial function. Thus the q-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 q-gamma function satisfies the q-analog of the Gauss multiplication formula :

Integral representation

The q-gamma function has the following integral representation :

Stirling formula

Moak obtained the following q-analogue of the Stirling formula (see ): where, H denotes the Heaviside step function, B_k stands for the Bernoulli number, is the dilogarithm, and p_k is a polynomial of degree k 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 |q|>1. With this restriction El Bachraoui considered the case 0<q<1 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 A be a complex square matrix and Positive-definite matrix. Then a q-gamma matrix function can be defined by q-integral: where E_q 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.