In q-analog

Q-analog

Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1...

 theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary Gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

 closely related to the double gamma function. It was introduced by . It is given by

where is the infinite q-Pochhammer symbol. It satisfies the functional equation

For non-negative integers n,

where is the q-factorial function. Alternatively, this can be taken as an extension of the q-factorial function to the real number system.

The relation to the ordinary gamma function is made explicit in the limit