In q-analog theory, the q {\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
Γ q ( x ) = ( 1 − q ) 1 − x ∏ n = 0 ∞ 1 − q n + 1 1 − q n + x = ( 1 − q ) 1 − x ( q ; q ) ∞ ( q x ; q ) ∞ {\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 | q | < 1 {\displaystyle |q|<1} , and
Γ q ( x ) = ( q − 1 ; q − 1 ) ∞ ( q − x ; q − 1 ) ∞ ( q − 1 ) 1 − x q ( x 2 ) {\displaystyle \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 {\displaystyle |q|>1} . Here ( ⋅ ; ⋅ ) ∞ {\displaystyle (\cdot ;\cdot )_{\infty }} is the infinite q {\displaystyle q} -Pochhammer symbol. The q {\displaystyle q} -gamma function satisfies the functional equation
Γ q ( x + 1 ) = 1 − q x 1 − q Γ q ( x ) = [ x ] q Γ q ( x ) {\displaystyle \Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)}
In addition, the q {\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 n} ,
Γ q ( n ) = [ n − 1 ] q ! {\displaystyle \Gamma _{q}(n)=[n-1]_{q}!}
where [ ⋅ ] q {\displaystyle [\cdot ]_{q}} is the q {\displaystyle q} -factorial function. Thus the q {\displaystyle q} -gamma function can be considered as an extension of the q {\displaystyle q} -factorial function to the real numbers. The relation to the ordinary gamma function is made explicit in the limit
lim q → 1 ± Γ q ( x ) = Γ ( x ) . {\displaystyle \lim _{q\to 1\pm }\Gamma _{q}(x)=\Gamma (x).}
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