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Elliptic gamma function
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<h2 id="multiplication-formula">Multiplication Formula</h2> <p>Define </p> Γ ~ ( z ; p , q ) := ( q ; q ) ∞ ( p ; p ) ∞ ( θ ( q ; p ) ) 1 − z ∏ m = 0 ∞ ∏ n = 0 ∞ 1 − p m + 1 q n + 1 − z 1 − p m q n + z . {\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}}}.} <p>Then the following formula holds with r = q n {\displaystyle r=q^{n}} (Felder & Varchenko (2002)). </p> Γ ~ ( n z ; p , q ) Γ ~ ( 1 / n ; p , r ) Γ ~ ( 2 / n ; p , r ) ⋯ Γ ~ ( ( n − 1 ) / n ; p , r ) = ( θ ( r ; p ) θ ( q ; p ) ) n z − 1 Γ ~ ( z ; p , r ) Γ ~ ( z + 1 / n ; p , r ) ⋯ Γ ~ ( z + ( n − 1 ) / n ; p , r ) . {\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).} <ul><li>Felder, G.; Varchenko, A. (2002). "Multiplication Formulas for the Elliptic Gamma Function". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0212155">math/0212155</a>.</li> <li>Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", <i>Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character</i>, 76 (508), The Royal Society: 127–144, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1905RSPSA..76..127J">1905RSPSA..76..127J</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1098%2Frspa.1905.0011">10.1098/rspa.1905.0011</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0950-1207">0950-1207</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/92601">92601</a></li> <li>Gasper, George; Rahman, Mizan (2004), <i>Basic hypergeometric series</i>, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-83357-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2128719">2128719</a></li> <li>Ruijsenaars, S. N. M. (1997), <a href="https://ir.cwi.nl/pub/2164">"First order analytic difference equations and integrable quantum systems"</a>, <i><a href="/facts/Journal_of_Mathematical_Physics/hfGFBt5M">Journal of Mathematical Physics</a></i>, 38 (2): 1069–1146, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1997JMP....38.1069R">1997JMP....38.1069R</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1063%2F1.531809">10.1063/1.531809</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0022-2488">0022-2488</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1434226">1434226</a></li> <li>Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang (2008). "A gerbe for the elliptic gamma function". <i>Duke Mathematical Journal</i>. 141. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0601337">math/0601337</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2FS0012-7094-08-14111-0">10.1215/S0012-7094-08-14111-0</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:817920">817920</a>.</li></ul>