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| {{DISPLAYTITLE:''q''-gamma function}}
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| 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 {{harvtxt|Jackson|1905}}. It is given by
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| :<math>\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}
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| </math>
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| when |q|<1, and
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| : <math> \Gamma_q(x)=\frac{(q^{-1};q^{-1})_\infty}{(q^{-x};q^{-1})_\infty}(q-1)^{1-x}q^{\binom{x}{2}} </math>
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| if |q|>1. Here (·;·)<sub>∞</sub> is the infinite [[q-Pochhammer symbol]]. It satisfies the functional equation
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| :<math>\Gamma_q(x+1) = \frac{1-q^{x}}{1-q}\Gamma_q(x)=[x]_q\Gamma_q(x)
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| </math>
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| For non-negative integers ''n'',
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| :<math>\Gamma_q(n)=[n-1]_q!</math>
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| where [·]<sub>''q''</sub>! is the [[q-factorial]] function. Alternatively, this can be taken as an extension of the q-factorial function to the real number system.
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| The relation to the ordinary gamma function is made explicit in the limit
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| :<math>\lim_{q \to 1\pm} \Gamma_q(x) = \Gamma(x).</math> | |
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| A q-analogue of [[Stirling's formula]] for |q|<1 is given by
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| :<math> \Gamma_q(x) =[2]_{q^{\ }}^{\frac 12} \Gamma_{q^2}\left(\frac 12\right)(1-q)^{\frac 12-x}e^{\frac{\theta q^x}{1-q-q^x}}, \quad 0<\theta<1.</math>
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| A q-analogue of the [[multiplication formula]] for |q|<1 is given by
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| :<math> \Gamma_{q^n}\left(\frac {x}n\right)\Gamma_{q^n}\left(\frac {x+1}n\right)\cdots\Gamma_{q^n}\left(\frac {x+n-1}n\right) =[n]_q^{\frac 12-x}\left([2]_q \Gamma^2_{q^2}\left(\frac12\right)\right)^{\frac{n-1}{2}}\Gamma_q(x).</math>
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| Due to I. Mező, the q-analogue of the [[Gamma_function#Raabe.27s_formula|Raabe formula]] exists, at least if we use the q-gamma function when |q|>1. With this restriction
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| : <math> \int_0^1\log\Gamma_q(x)dx=\frac{\zeta(2)}{\log q}+\log\sqrt{\frac{q-1}{\sqrt[6]{q}}}+\log(q^{-1};q^{-1})_\infty \quad(q>1). </math>
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| ==References==
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| *{{Citation | last1=Jackson | first1=F. H. | title=The Basic Gamma-Function and the Elliptic Functions | jstor=92601 | publisher=The Royal Society | year=1905 | journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character | issn=0950-1207 | volume=76 | issue=508 | pages=127–144}}
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| *{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}}
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| *{{Citation | last1=Mansour | first1=M| title=An asymptotic expansion of the q-gamma function Γq(x)| journal=Journal of Nonlinear Mathematical Physics | volume=13 | number=4 | year=2006 | pages=479–483}}[http://staff.www.ltu.se/~norbert/home_journal/electronic/134lett2.pdf]
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| *{{Citation | last1=Mező | first1=István | title=A q-Raabe formula and an integral of the fourth Jacobi theta function | year=2012 | journal=Journal of Number Theory | volume=130 | issue=2 | pages=360-369}}
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| [[Category:Gamma and related functions]]
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| [[Category:Q-analogs]]
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| {{mathanalysis-stub}}
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