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Función gamma elíptica
En matemáticas, la función gamma elíptica es una generalización de la función q-gamma, que es en sí la función q-analógica de la función gamma ordinaria. Es estrechamente relacionada con una función estudiada por Jackson (1905), y puede expresarse en términos de la función triple gamma.
Traducción generada por IA. El artículo original en inglés continúa abajo.
Elliptic gamma function
From Wikipedia, the free encyclopedia
Mathematic function
In mathematics , the elliptic gamma function is a generalization of the q-gamma function , which is itself the q-analog of the ordinary gamma function . It is closely related to a function studied by Jackson (1905) , and can be expressed in terms of the triple gamma function . It is given by
Γ
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{\displaystyle \Gamma (z;p,q)=\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1}/z}{1-p^{m}q^{n}z}}.}
It obeys several identities:
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{\displaystyle \Gamma (z;p,q)={\frac {1}{\Gamma (pq/z;p,q)}}\,}
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{\displaystyle \Gamma (pz;p,q)=\theta (z;q)\Gamma (z;p,q)\,}
and
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{\displaystyle \Gamma (qz;p,q)=\theta (z;p)\Gamma (z;p,q)\,}
where θ is the q-theta function .
When
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{\displaystyle p=0}
q-Pochhammer symbol :
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{\displaystyle \Gamma (z;0,q)={\frac {1}{(z;q)_{\infty }}}.}
Define
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{\displaystyle {\tilde {\Gamma }}(z;p,q):={\frac {(q;q)_{\infty }}{(p;p)_{\infty }}}(\theta (q;p))^{1-z}\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1-z}}{1-p^{m}q^{n+z}}}.}
Then the following formula holds with
r
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q
n
{\displaystyle r=q^{n}}
Felder & Varchenko (2002) ).
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⋯
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⋯
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{\displaystyle {\tilde {\Gamma }}(nz;p,q){\tilde {\Gamma }}(1/n;p,r){\tilde {\Gamma }}(2/n;p,r)\cdots {\tilde {\Gamma }}((n-1)/n;p,r)=\left({\frac {\theta (r;p)}{\theta (q;p)}}\right)^{nz-1}{\tilde {\Gamma }}(z;p,r){\tilde {\Gamma }}(z+1/n;p,r)\cdots {\tilde {\Gamma }}(z+(n-1)/n;p,r).}
Felder, G.; Varchenko, A. (2002). "Multiplication Formulas for the Elliptic Gamma Function". arXiv :math/0212155 Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character , 76 (508), The Royal Society: 127– 144, Bibcode :1905RSPSA..76..127J , doi :10.1098/rspa.1905.0011 ISSN 0950-1207 , JSTOR 92601 Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series , Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press , ISBN 978-0-521-83357-8 MR 2128719 Ruijsenaars, S. N. M. (1997), "First order analytic difference equations and integrable quantum systems" , Journal of Mathematical Physics 38 (2): 1069– 1146, Bibcode :1997JMP....38.1069R , doi :10.1063/1.531809 , ISSN 0022-2488 , MR 1434226 Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang (2008). "A gerbe for the elliptic gamma function". Duke Mathematical Journal . 141 . arXiv :math/0601337 doi :10.1215/S0012-7094-08-14111-0 . S2CID 817920 .
