Q-gamma function

In q-analog theory, the ${\displaystyle q}$-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 Jackson (1905). It is given by $${\displaystyle \Gamma _{q}(x)=(1-q)^{1-x}\prod _{n=0}^{\infty }{\frac {1-q^{n+1}}{1-q^{n+x}}}=(1-q)^{1-x}\,{\frac {(q;q)_{\infty }}{(q^{x};q)_{\infty }}}}$$ when ${\displaystyle |q|<1}$, and $${\displaystyle \Gamma _{q}(x)={\frac {(q^{-1};q^{-1})_{\infty }}{(q^{-x};q^{-1})_{\infty }}}(q-1)^{1-x}q^{\binom {x}{2}}}$$ if ${\displaystyle |q|>1}$. Here ${\displaystyle (\cdot ;\cdot )_{\infty }}$ is the infinite q-Pochhammer symbol. The ${\displaystyle q}$-gamma function satisfies the functional equation $${\displaystyle \Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)}$$ In addition, the ${\displaystyle q}$-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (Askey (1978)).
For non-negative integers n, $${\displaystyle \Gamma _{q}(n)=[n-1]_{q}!}$$ where ${\displaystyle [\cdot ]_{q}}$ is the q-factorial function. Thus the ${\displaystyle q}$-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 $${\displaystyle \lim _{q\to 1\pm }\Gamma _{q}(x)=\Gamma (x).}$$ There is a simple proof of this limit by Gosper. See the appendix of (Andrews (1986)).