En matemàtiques , en la teoria q-anàleg , la funció q-gamma , o funció gamma bàsica , és una generalització de la funció gamma ordinària, i està molt estretament relacionada amb la funció gamma doble . Aquesta va ser introduïda per Jackson (1905) ,
Es defineix com
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{\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 }}}}
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{\displaystyle |q|<1}
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{\displaystyle \Gamma _{q}(x)={\frac {(q^{-1};q^{-1})_{\infty }}{(q^{-x};q^{-1})_{\infty }}}(q-1)^{1-x}q^{\binom {x}{2}}}
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{\displaystyle |q|>1}
∞ és el símbol q-Pochhammer infinit . Satisfà l'equació funcional
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{\displaystyle \Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)}
Per a enters no negatius n,
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{\displaystyle \Gamma _{q}(n)=[n-1]_{q}!}
on [·]q q-factorial . Alternativament, això pot ser pres com una extensió de la funció q-factorial per al sistema de nombres reals.
La relació amb la funció gamma ordinària es fa explícita en el límit
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{\displaystyle \lim _{q\to 1\pm }\Gamma _{q}(x)=\Gamma (x).}
Fórmules tipus Raabe
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Valors especials
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Són coneguts els següents valors especials:
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{\displaystyle \Gamma _{e^{-\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /16}{\sqrt {e^{\pi }-1}}}{2^{11/12}\pi ^{3/4}{\sqrt[{3}]{{\sqrt {2}}-1}}{\sqrt[{24}]{4+3{\sqrt {2}}}}}}\Gamma \left({\frac {1}{4}}\right),}
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{\displaystyle \Gamma _{e^{-2\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /8}{\sqrt {e^{2\pi }-1}}}{2{\sqrt[{8}]{2}}\pi ^{3/4}}}\Gamma \left({\frac {1}{4}}\right),}
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{\displaystyle \Gamma _{e^{-4\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /4}{\sqrt {e^{4\pi }-1}}}{2^{7/4}\pi ^{3/4}}}\Gamma \left({\frac {1}{4}}\right),}
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{\displaystyle \Gamma _{e^{-8\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /2}{\sqrt {\left({\sqrt {2}}-1\right)\left(e^{8\pi }-1\right)}}}{4{\sqrt[{4}]{2}}\pi ^{3/4}}}\Gamma \left({\frac {1}{4}}\right).}
Aquests són els anàlegs de la fórmula clàssica
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{\displaystyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}}
D'altra banda, els següents anàlegs de la identitat familiaritzada
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{\displaystyle \Gamma \left({\frac {1}{4}}\right)\Gamma \left({\frac {3}{4}}\right)={\sqrt {2}}\pi }
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{\displaystyle \Gamma _{e^{-2\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-2\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /16}\left(e^{2\pi }-1\right)\Gamma \left({\frac {1}{4}}\right)^{2}}{4\pi ^{3/2}{\sqrt[{3}]{{\sqrt {2}}-1}}{\sqrt[{24}]{8+6{\sqrt {2}}}}}},}
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{\displaystyle \Gamma _{e^{-4\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-4\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /8}\left(e^{4\pi }-1\right)\Gamma \left({\frac {1}{4}}\right)^{2}}{2^{23/8}\pi ^{3/2}}},}
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{\displaystyle \Gamma _{e^{-8\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-8\pi }}\left({\frac {3}{4}}\right)={\frac {{\sqrt {{\sqrt {2}}-1}}e^{-29\pi /4}\left(e^{8\pi }-1\right)\Gamma \left({\frac {1}{4}}\right)^{2}}{16\pi ^{3/2}}}.}
Un q-anàleg de la fórmula de Stirling per a
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{\displaystyle \Gamma _{q}(x)=[2]_{q^{\ }}^{\frac {1}{2}}\Gamma _{q^{2}}\left({\frac {1}{2}}\right)(1-q)^{{\frac {1}{2}}-x}e^{\frac {\theta q^{x}}{1-q-q^{x}}},\quad 0<\theta <1.}
Un q-anàleg de la fórmula de multiplicació per a
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{\displaystyle \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 {1}{2}}-x}\left([2]_{q}\Gamma _{q^{2}}^{2}\left({\frac {1}{2}}\right)\right)^{\frac {n-1}{2}}\Gamma _{q}(x).}
Jackson , F. H.Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character . The Royal Society, 76, 508, 1905, p. 127–144. DOI : 10.1098/rspa.1905.0011 . ISSN : 0950-1207 .Gasper , George; Rahman , Mizan. Basic hypergeometric series . 96. 2nd. Cambridge University Press , 2004. ISBN 978-0-521-83357-8 Mansour , M «An asymptotic expansion of the q-gamma function Γq(x)». Journal of Nonlinear Mathematical Physics , 13, 2006, p. 479–483. DOI : 10.2991/jnmp.2006.13.4.2 .[1] Mező , István «A q-Raabe formula and an integral of the fourth Jacobi theta function». Journal of Number Theory , 133, 2, 2012, p. 692–704. DOI : 10.1016/j.jnt.2012.08.025 .
![{\displaystyle \Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1369d42adac73e85d50ce69ed4e83fe4d9432ce7)
![{\displaystyle \Gamma _{q}(n)=[n-1]_{q}!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4be916fb8ef9c2bb7a93ee692b3719d096150dbe)
![{\displaystyle \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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18dd5281744cfa39cf8031d498765542f51ae6b7)
![{\displaystyle \Gamma _{e^{-\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /16}{\sqrt {e^{\pi }-1}}}{2^{11/12}\pi ^{3/4}{\sqrt[{3}]{{\sqrt {2}}-1}}{\sqrt[{24}]{4+3{\sqrt {2}}}}}}\Gamma \left({\frac {1}{4}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/798320a81c9c4c3397549726d4dc9d5772d9b3ef)
![{\displaystyle \Gamma _{e^{-2\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /8}{\sqrt {e^{2\pi }-1}}}{2{\sqrt[{8}]{2}}\pi ^{3/4}}}\Gamma \left({\frac {1}{4}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da096c89bdb5922d59fbef370e1a8ba4fa084b10)
![{\displaystyle \Gamma _{e^{-8\pi }}\left({\frac {1}{2}}\right)={\frac {e^{-7\pi /2}{\sqrt {\left({\sqrt {2}}-1\right)\left(e^{8\pi }-1\right)}}}{4{\sqrt[{4}]{2}}\pi ^{3/4}}}\Gamma \left({\frac {1}{4}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3957d847df1d1151f68099e2654e33d876a38003)
![{\displaystyle \Gamma _{e^{-2\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-2\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /16}\left(e^{2\pi }-1\right)\Gamma \left({\frac {1}{4}}\right)^{2}}{4\pi ^{3/2}{\sqrt[{3}]{{\sqrt {2}}-1}}{\sqrt[{24}]{8+6{\sqrt {2}}}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3c59d781764ca2f9b3f49978f966aedef8a71f)
![{\displaystyle \Gamma _{q}(x)=[2]_{q^{\ }}^{\frac {1}{2}}\Gamma _{q^{2}}\left({\frac {1}{2}}\right)(1-q)^{{\frac {1}{2}}-x}e^{\frac {\theta q^{x}}{1-q-q^{x}}},\quad 0<\theta <1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9e8d25291b18f7be7dba6c3eb740809d5ca997)
![{\displaystyle \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 {1}{2}}-x}\left([2]_{q}\Gamma _{q^{2}}^{2}\left({\frac {1}{2}}\right)\right)^{\frac {n-1}{2}}\Gamma _{q}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1dca8fdc67944b3be5b715e7d36401358f5ace)