q-gamma function

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 Jackson (1905). It is given by

\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 $|q|<1$ , and

\Gamma _{q}(x)={\frac {(q^{-1};q^{-1})_{\infty }}{(q^{-x};q^{-1})_{\infty }}}(q-1)^{1-x}q^{\binom {x}{2}}

if $|q|>1$ . Here ${\displaystyle (\cdot$ is the infinite q-Pochhammer symbol. The $q$ -gamma function satisfies the functional equation

\Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)

In addition, the $q$ -gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (Askey (1978)).
For non-negative integers n,

\Gamma _{q}(n)=[n-1]_{q}!

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

\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)).

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