# Multiple gamma function

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{{Short description|Generalization of the Euler gamma function and the Barnes G-function}}
{{For|derivatives of the log of the gamma function |polygamma function}}
alt=Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
In mathematics, the '''multiple gamma function''' <math>\Gamma_N</math> is a generalization of the Euler [gamma function](/source/gamma_function) and the [Barnes G-function](/source/Barnes_G-function). The double gamma function was studied by {{harvtxt|Barnes|1901}}. At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in {{harvtxt|Barnes|1904}}.

Double gamma functions <math>\Gamma_2</math> are closely related to the [q-gamma function](/source/q-gamma_function), and triple gamma functions <math>\Gamma_3</math> are related to the [elliptic gamma function](/source/elliptic_gamma_function).

==Definition==

For <math>\Re a_i>0</math>, let 

:<math>\Gamma_N(w\mid a_1,\ldots,a_N) = \exp\left(\left.\frac{\partial}{\partial s} \zeta_N(s,w \mid a_1, \ldots, a_N) \right|_{s=0} \right)\ ,</math>

where <math>\zeta_N</math> is the [Barnes zeta function](/source/Barnes_zeta_function). (This differs by a constant from Barnes's original definition.)

==Properties==

Considered as a [meromorphic function](/source/meromorphic_function) of <math>w</math>, <math>\Gamma_N(w\mid a_1,\ldots,a_N)</math> has no zeros. It has poles at <math> w= -\sum_{i=1}^N n_ia_i </math>for non-negative integers <math>n_i</math>. These poles are simple unless some of them coincide. Up to multiplication by the exponential of a polynomial, <math>\Gamma_N(w\mid a_1,\ldots,a_N)</math> is the unique meromorphic function of finite order with these [zeros and poles](/source/zeros_and_poles).
*<math>\Gamma_0(w\mid) = \frac{1}{w}\ ,</math>
*<math>\Gamma_1(w\mid a) = \frac{a^{a^{-1}w-\frac12}}{\sqrt{2\pi}} \Gamma\left(a^{-1} w\right)\ , </math>
*<math>\Gamma_N(w\mid a_1,\ldots,a_N)=\Gamma_{N-1}(w\mid a_1,\ldots,a_{N-1})\Gamma_N(w+a_N\mid a_1,\ldots,a_N)\ .</math>

In the case of the double Gamma function, the asymptotic behaviour for <math>w\to \infty</math> is known, and the leading factor is<ref name="ak22"/>
:<math>
\Gamma_2(w|a_1,a_2)\ \underset{w\to \infty}{\sim}\ w^{\frac{w^2}{2a_1a_2}} \quad \text{for}\quad \left\{\begin{array}{l} \frac{a_1}{a_2}\in\mathbb{C}\backslash(-\infty,0]\ ,
\\ w \in \mathbb{C}\backslash \left(\mathbb{R}_+a_1+\mathbb{R}_+a_2\right)\ . \end{array}\right.
</math>

==Infinite product representation==

The multiple gamma function has an [infinite product](/source/infinite_product) representation that makes it manifest that it is meromorphic, and that also makes the positions of its poles manifest. In the case of the double gamma function, this representation is <ref name="Spreafico">{{cite journal |last1=Spreafico |first1=Mauro |date=2009 |title= On the Barnes double zeta and gamma functions|journal= Journal of Number Theory|volume= 129|issue= 9|pages= 2035–2063|doi= 10.1016/j.jnt.2009.03.005|doi-access= free}}</ref>
:<math>
\Gamma_2(w\mid a_1,a_2) = \frac{e^{\lambda_1w +\lambda_2 w^2}}{w} \prod_{\begin{array}{c} (n_1,n_2)\in\mathbb{N}^2\\ (n_1,n_2)\neq (0,0)\end{array}} \frac{e^{\frac{w}{n_1a_1+n_2a_2}- \frac12 \frac{w^2}{(n_1a_1+n_2a_2)^2}}}{1+\frac{w}{n_1a_1+n_2a_2}}\ ,
</math>
where we define the <math>w</math>-independent coefficients
:<math>
\lambda_1 = -\underset{s=1}{\operatorname{Res}_0}\zeta_2(s,0\mid a_1,a_2)\ ,
</math>
:<math>
\lambda_2 = \frac12\underset{s=2}{\operatorname{Res}_0}\zeta_2(s,0\mid a_1,a_2) + \frac12 \underset{s=2}{\operatorname{Res}_1}\zeta_2(s,0\mid a_1,a_2)\ ,
</math>
where <math> \underset{s=s_0}{\operatorname{Res}_n} f(s) = \frac{1}{2\pi i}\oint_{s_0} (s-s_0)^{n-1} f(s) \, ds</math> is an <math>n</math>-th order [residue](/source/Residue_(complex_analysis)) at <math>s_0</math>.

Another representation as a product over <math>\mathbb{N}</math> leads to an algorithm for numerically computing the double Gamma function.<ref name="ak22"/>

==Reduction to the Barnes G-function==

The double gamma function with parameters <math>1,1</math> obeys the relations <ref name="Spreafico" />
:<math> \Gamma_2(w+1|1,1) = \frac{\sqrt{2\pi}}{\Gamma(w)} \Gamma_2(w|1,1) \quad , \quad \Gamma_2(1|1,1) = \sqrt{2\pi} \ . </math>
It is related to the [Barnes G-function](/source/Barnes_G-function) by 
:<math> \Gamma_2(w|\alpha,\alpha) = (2\pi)^\frac{w}{2\alpha} \alpha^{-\frac{w^2}{2\alpha^2} + \frac{w}{\alpha} - 1} G(w / \alpha)^{-1} \ . 
</math>

==The double gamma function and conformal field theory==

For <math>\Re b>0</math> and <math> Q=b+b^{-1}</math>, the function 

:<math> \Gamma_b(w) = \frac{\Gamma_2(w\mid b,b^{-1})}{\Gamma_2\left(\frac{Q}{2}\mid b,b^{-1}\right)}\ , </math>

is invariant under <math> b\to b^{-1} </math>, and obeys the relations

:<math> \Gamma_b(w+b) = \sqrt{2\pi}\frac{b^{bw-\frac12}}{\Gamma(bw)}\Gamma_b(w)\quad , \quad \Gamma_b(w+b^{-1}) = \sqrt{2\pi}\frac{b^{-b^{-1}w+\frac12}}{\Gamma(b^{-1}w)} \Gamma_b(w)\ . </math>

For <math>\Re w>0</math>, it has the integral representation <ref name="eb23"></ref>

:<math>\log\Gamma_b(w) = \int_0^\infty\frac{dt}{t}\left[\frac{e^{-wt}-e^{-\frac{Q}{2}t}}{(1-e^{-bt})(1-e^{-b^{-1}t})} -\frac{\left(\frac{Q}{2}-w\right)^2}{2}e^{-t} -\frac{\frac{Q}{2}-w}{t}\right]\ . </math>

From the function <math>\Gamma_b(w)</math>, we define the '''double Sine function''' <math>S_b(w)</math> and the '''Upsilon function''' <math>\Upsilon_b(w)</math> by

:<math> S_b(w) =\frac{\Gamma_b(w)}{\Gamma_b(Q-w)} \quad , \quad \Upsilon_b(w)=\frac{1}{\Gamma_b(w)\Gamma_b(Q-w)}\ . </math>

These functions obey the relations 

:<math> S_b(w+b) = 2\sin(\pi bw)S_b(w) \quad , \quad \Upsilon_b(w+b)=\frac{\Gamma(bw)}{\Gamma(1-bw)} b^{1-2bw}\Upsilon_b(w) \ , </math>

plus the relations that are obtained by <math>b\to b^{-1}</math>. For <math>0<\Re w<\Re Q</math> they have the integral representations

:<math> \log S_b(w) = \int_0^\infty\frac{dt}{t}\left[
\frac{ \sinh\left(\frac{Q}{2}-w\right)t}{2\sinh\left(\frac12 bt\right)\sinh\left(\frac12 b^{-1}t\right)}-\frac{Q-2w}{t}\right]\ ,</math>

:<math> \log \Upsilon_b(w) = \int_0^\infty\frac{dt}{t}\left[\left(\frac{Q}{2}-w\right)^2e^{-t} -\frac{\sinh^2\frac12\left(\frac{Q}{2}-w\right)t}{\sinh\left(\frac12 bt\right)\sinh\left(\frac12 b^{-1}t\right)}\right]\ . </math>

The functions <math> \Gamma_b,S_b</math> and <math>\Upsilon_b</math> appear in correlation functions of [two-dimensional conformal field theory](/source/two-dimensional_conformal_field_theory), with the parameter <math>b</math> being related to the [central charge](/source/central_charge) of the underlying [Virasoro algebra](/source/Virasoro_algebra).<ref>{{cite thesis | last1=Ponsot | first1=B. |title=Recent progress on Liouville Field Theory | arxiv=hep-th/0301193 |bibcode=2003PhDT.......180P}}</ref> In particular, the three-point function of [Liouville theory](/source/Liouville_field_theory) is written in terms of the function <math>\Upsilon_b</math>.

==References==
{{Reflist|refs=
<ref name="ak22">{{citation
| last1=Alexanian | first1=Shahen
| last2=Kuznetsov | first2=Alexey
| title=On the Barnes double gamma function
| date=2023
| arxiv=2208.13876
| journal=Integral Transforms and Special Functions 
| volume=34
| issue=12
| pages=891–914
| doi=10.1080/10652469.2023.2238115}}</ref>
<ref name="eb23">{{cite arXiv
| last1=Eberhardt | first1=Lorenz
| title=Notes on crossing transformations of Virasoro conformal blocks
| date=2023
| eprint=2309.11540
| at=Appendix B
| class=hep-th
}}</ref>
}}

==Further reading==
*{{citation|first=E. W. |last=Barnes
 |title=   The Genesis of the Double Gamma Functions
  |journal=  Proc. London Math. Soc. |year=1899 |volume=s1-31|pages= 358&ndash;381| doi=10.1112/plms/s1-31.1.358 |url=https://zenodo.org/record/1447742
 }}
*{{Citation | last1=Barnes | first1=E. W. | title=The Theory of the Double Gamma Function | jstor=116064 | year=1899 | journal=Proceedings of the Royal Society of London | issn=0370-1662 | volume=66 | issue=424–433 | pages=265–268 | doi=10.1098/rspl.1899.0101| s2cid=186213903 }}
*{{Citation | last1=Barnes | first1=E. W. | title=The Theory of the Double Gamma Function | jstor=90809  | year=1901 | journal=Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character | issn=0264-3952 | volume=196 | issue=274–286 | pages=265–387 | doi=10.1098/rsta.1901.0006| bibcode=1901RSPTA.196..265B | doi-access= }}
*{{citation|first=E. W. |last=Barnes|title=On the theory of the multiple gamma function|journal=    Trans. Camb. Philos. Soc. |volume=19 |year=1904|pages=374–425}}
*{{Citation | last1=Friedman | first1=Eduardo | last2=Ruijsenaars | first2=Simon | title=Shintani&ndash;Barnes zeta and gamma functions | doi=10.1016/j.aim.2003.07.020 | doi-access=free | mr=2078341 | year=2004 | journal=[Advances in Mathematics](/source/Advances_in_Mathematics) | issn=0001-8708 | volume=187 | issue=2 | pages=362–395}}
*{{Citation | last1=Ruijsenaars | first1=S. N. M. | title=On Barnes' multiple zeta and gamma functions | doi=10.1006/aima.2000.1946 | doi-access=free | mr=1800255 | year=2000 | journal=[Advances in Mathematics](/source/Advances_in_Mathematics) | issn=0001-8708 | volume=156 | issue=1 | pages=107–132| url=https://ir.cwi.nl/pub/2100 }}

Category:Gamma and related functions

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Adapted from the Wikipedia article [Multiple gamma function](https://en.wikipedia.org/wiki/Multiple_gamma_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Multiple_gamma_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.