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
The relation to the ordinary gamma function is made explicit in the limit
The -gamma function satisfies the q-analog of the Gauss multiplication formula (Gasper & Rahman (2004)):
The -gamma function has the following integral representation (Ismail (1981)):
Moak obtained the following q-analogue of the Stirling formula (see Moak (1984)):
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
The following special values are known.[1]
Moreover, the following analogues of the familiar identity hold true:
Let be a complex square matrix and Positive-definite matrix. Then a q-gamma matrix function can be defined by q-integral:[2]
For other q-gamma functions, see Yamasaki 2006.[3]
An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.[4]