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).
-gamma function satisfies the functional equation
-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (Askey (1978)).
For non-negative integers
-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 (Andrews (1986)).
-gamma function satisfies the q-analog of the Gauss multiplication formula (Gasper & Rahman (2004)):
-gamma function has the following integral representation (Ismail (1981)):
sin ( π z )
Moak obtained the following q-analogue of the Stirling formula (see Moak (1984)):
log ( 2 π ) +
denotes the Heaviside step function,
stands for the Bernoulli number,
is a polynomial of degree
Due to I. Mező, the q-analogue of the Raabe formula exists, at least if we use the
El Bachraoui considered the case
The following special values are known.
These are the analogues of the classical formula
Moreover, the following analogues of the familiar identity
hold true:
be a complex square matrix and positive-definite matrix.
-gamma matrix function can be defined by
is the q-exponential function.
-gamma functions, see Yamasaki 2006.
[3] An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.