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<title>Elliptic gamma function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Elliptic gamma function</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>elliptic gamma function</b> is a generalization of the <a href="Q-gamma_function" title="Q-gamma function">q-gamma function</a>, which is itself the <a href="Q-analog" title="Q-analog">q-analog</a> of the ordinary <a href="Gamma_function" title="Gamma function">gamma function</a>. It is closely related to a function studied by <a href="#CITEREFJackson1905">Jackson (1905)</a>, and can be expressed in terms of the <a href="Triple_gamma_function" class="mw-redirect" title="Triple gamma function">triple gamma function</a>. It is given by

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<annotation encoding="application/x-tex">{\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}}.}</annotation>

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</math></span><img src="./37683b1cb7572224134ebb657e1446f0c91fd006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.24ex; height:7.009ex;" alt="{\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}}.}" loading="lazy"></span></dd></dl>

<p>It obeys several identities:

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (z;p,q)={\frac {1}{\Gamma (pq/z;p,q)}}\,}</annotation>

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</math></span><img src="./817cc99dbf5496539cf615e861b7d89dbe89f323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.037ex; height:6.009ex;" alt="{\displaystyle \Gamma (z;p,q)={\frac {1}{\Gamma (pq/z;p,q)}}\,}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (pz;p,q)=\theta (z;q)\Gamma (z;p,q)\,}</annotation>

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</math></span><img src="./097432c9ed60f06b8209eb9e69cd0015c2b82b35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.06ex; height:2.843ex;" alt="{\displaystyle \Gamma (pz;p,q)=\theta (z;q)\Gamma (z;p,q)\,}" loading="lazy"></span></dd></dl>

<p>and

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (qz;p,q)=\theta (z;p)\Gamma (z;p,q)\,}</annotation>

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</math></span><img src="./1ccb97fa9a7747ea481ca6a80bce30510e529bd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.06ex; height:2.843ex;" alt="{\displaystyle \Gamma (qz;p,q)=\theta (z;p)\Gamma (z;p,q)\,}" loading="lazy"></span></dd></dl>

<p>where θ is the <a href="Q-theta_function" title="Q-theta function">q-theta function</a>.

</p><p>When <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0}">

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<annotation encoding="application/x-tex">{\displaystyle p=0}</annotation>

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</math></span><img src="./b3e6ac10fa45fb984d886065f959a6bdd467b5e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=0}" loading="lazy"></span>, it essentially reduces to the infinite <a href="Q-Pochhammer_symbol" title="Q-Pochhammer symbol">q-Pochhammer symbol</a>:

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (z;0,q)={\frac {1}{(z;q)_{\infty }}}.}</annotation>

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</math></span><img src="./cb6ff28c86a13d786021ea425122883b3089002c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.108ex; height:6.009ex;" alt="{\displaystyle \Gamma (z;0,q)={\frac {1}{(z;q)_{\infty }}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Multiplication_Formula">Multiplication Formula</h2></div>

<p>Define

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<annotation encoding="application/x-tex">{\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}}}.}</annotation>

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</math></span><img src="./b54d268fc662c073176ec39b10988db60e693b0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.049ex; height:6.843ex;" alt="{\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}}}.}" loading="lazy"></span></dd></dl>

<p>Then the following formula holds with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=q^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle r=q^{n}}</annotation>

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</math></span><img src="./15952af6efde4d3ff173aea6664265604496a69a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.445ex; height:2.676ex;" alt="{\displaystyle r=q^{n}}" loading="lazy"></span> (<a href="#CITEREFFelderVarchenko2002">Felder & Varchenko (2002)</a>).

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<annotation encoding="application/x-tex">{\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).}</annotation>

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</math></span><img src="./f998d356b9823a2bacc902bdb3261c8572dc7c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:120.231ex; height:6.843ex;" alt="{\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).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFFelderVarchenko2002" class="citation arxiv cs1">Felder, G.; Varchenko, A. (2002). "Multiplication Formulas for the Elliptic Gamma Function". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0212155">math/0212155</a></span>.</cite></li>

<li><cite id="CITEREFJackson1905" class="citation cs2">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>, <b>76</b> (508), The Royal Society: <span class="nowrap">127–</span>144, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1905RSPSA..76..127J">1905RSPSA..76..127J</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1905.0011">10.1098/rspa.1905.0011</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0950-1207">0950-1207</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/92601">92601</a></cite></li>

<li><cite id="CITEREFGasperRahman2004" class="citation cs2">Gasper, George; Rahman, Mizan (2004), <i>Basic hypergeometric series</i>, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-83357-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2128719">2128719</a></cite></li>

<li><cite id="CITEREFRuijsenaars1997" class="citation cs2">Ruijsenaars, S. N. M. (1997), <a rel="nofollow" class="external text" href="https://ir.cwi.nl/pub/2164">"First order analytic difference equations and integrable quantum systems"</a>, <i><a href="Journal_of_Mathematical_Physics" title="Journal of Mathematical Physics">Journal of Mathematical Physics</a></i>, <b>38</b> (2): <span class="nowrap">1069–</span>1146, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997JMP....38.1069R">1997JMP....38.1069R</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.531809">10.1063/1.531809</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2488">0022-2488</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1434226">1434226</a></cite></li>

<li><cite id="CITEREFFelderHenriquesRossiZhu2008" class="citation journal cs1">Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang (2008). "A gerbe for the elliptic gamma function". <i>Duke Mathematical Journal</i>. <b>141</b>. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0601337">math/0601337</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-08-14111-0">10.1215/S0012-7094-08-14111-0</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:817920">817920</a>.</cite></li></ul></div><div><div class="zim-footer">

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