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q-gamma function
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<p>In <a href="/facts/Q-analog/7jXz12AP">q-analog</a> theory, the q {\displaystyle q} -gamma function, or basic gamma function, is a generalization of the ordinary <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> closely related to the <a href="/facts/Double_gamma_function/84xVAXXu">double gamma function</a>. 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 <a href="/facts/Q-Pochhammer_symbol/keASLfpe"> q {\displaystyle q} -Pochhammer symbol</a>. 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 <a href="/facts/Bohr%25E2%2580%2593Mollerup_theorem/p5qpCz7m">Bohr–Mollerup theorem</a>, which was found by <a href="/facts/Richard_Askey/r6ScCdJa">Richard Askey</a> (Askey (1978)). </p><p>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 <a href="/facts/Q-factorial/keASLfpe"> q {\displaystyle q} -factorial</a> 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. </p><p>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).} There is a simple proof of this limit by Gosper. See the appendix of (<a href="/facts/George_Andrews_(mathematician)/x58GmSuR">Andrews</a> (1986)). </p>