Multiple gamma function

In mathematics, the multiple gamma function Γ N {\displaystyle \Gamma _{N}} is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by Barnes (1901). At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in Barnes (1904).

Double gamma functions Γ 2 {\displaystyle \Gamma _{2}} are closely related to the q-gamma function, and triple gamma functions Γ 3 {\displaystyle \Gamma _{3}} are related to the elliptic gamma function.

For ℜ a i > 0 {\displaystyle \Re a_{i}>0} , let

Γ N ( w ∣ a 1 , … , a N ) = exp ⁡ ( ∂ ∂ s ζ N ( s , w ∣ a 1 , … , a N ) | s = 0 )   , {\displaystyle \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)\ ,}

where ζ N {\displaystyle \zeta _{N}} is the Barnes zeta function. (This differs by a constant from Barnes's original definition.)

Considered as a meromorphic function of w {\displaystyle w} , Γ N ( w ∣ a 1 , … , a N ) {\displaystyle \Gamma _{N}(w\mid a_{1},\ldots ,a_{N})} has no zeros. It has poles at w = − ∑ i = 1 N n i a i {\displaystyle w=-\sum _{i=1}^{N}n_{i}a_{i}} for non-negative integers n i {\displaystyle n_{i}} . These poles are simple unless some of them coincide. Up to multiplication by the exponential of a polynomial, Γ N ( w ∣ a 1 , … , a N ) {\displaystyle \Gamma _{N}(w\mid a_{1},\ldots ,a_{N})} is the unique meromorphic function of finite order with these zeros and poles.

• Γ 0 ( w ∣ ) = 1 w   , {\displaystyle \Gamma _{0}(w\mid )={\frac {1}{w}}\ ,}
• Γ 1 ( w ∣ a ) = a a − 1 w − 1 2 2 π Γ ( a − 1 w )   , {\displaystyle \Gamma _{1}(w\mid a)={\frac {a^{a^{-1}w-{\frac {1}{2}}}}{\sqrt {2\pi }}}\Gamma \left(a^{-1}w\right)\ ,}
• Γ N ( w ∣ a 1 , … , a N ) = Γ N − 1 ( w ∣ a 1 , … , a N − 1 ) Γ N ( w + a N ∣ a 1 , … , a N )   . {\displaystyle \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})\ .}

In the case of the double Gamma function, the asymptotic behaviour for w → ∞ {\displaystyle w\to \infty } is known, and the leading factor is^[1]

Γ 2 ( w | a 1 , a 2 )   ∼ w → ∞   w w 2 2 a 1 a 2 for { a 1 a 2 ∈ C ∖ ( − ∞ , 0 ]   , w ∈ C ∖ ( R + a 1 + R + a 2 )   . {\displaystyle \Gamma _{2}(w|a_{1},a_{2})\ {\underset {w\to \infty }{\sim }}\ w^{\frac {w^{2}}{2a_{1}a_{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.}

The multiple gamma function has an 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 ^[2]

Γ 2 ( w ∣ a 1 , a 2 ) = e λ 1 w + λ 2 w 2 w ∏ ( n 1 , n 2 ) ∈ N 2 ( n 1 , n 2 ) ≠ ( 0 , 0 ) e w n 1 a 1 + n 2 a 2 − 1 2 w 2 ( n 1 a 1 + n 2 a 2 ) 2 1 + w n 1 a 1 + n 2 a 2   , {\displaystyle \Gamma _{2}(w\mid a_{1},a_{2})={\frac {e^{\lambda _{1}w+\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_{1}a_{1}+n_{2}a_{2}}}-{\frac {1}{2}}{\frac {w^{2}}{(n_{1}a_{1}+n_{2}a_{2})^{2}}}}}{1+{\frac {w}{n_{1}a_{1}+n_{2}a_{2}}}}}\ ,}

where we define the w {\displaystyle w} -independent coefficients

λ 1 = − Res 0 s = 1 ζ 2 ( s , 0 ∣ a 1 , a 2 )   , {\displaystyle \lambda _{1}=-{\underset {s=1}{\operatorname {Res} _{0}}}\zeta _{2}(s,0\mid a_{1},a_{2})\ ,}

λ 2 = 1 2 Res 0 s = 2 ζ 2 ( s , 0 ∣ a 1 , a 2 ) + 1 2 Res 1 s = 2 ζ 2 ( s , 0 ∣ a 1 , a 2 )   , {\displaystyle \lambda _{2}={\frac {1}{2}}{\underset {s=2}{\operatorname {Res} _{0}}}\zeta _{2}(s,0\mid a_{1},a_{2})+{\frac {1}{2}}{\underset {s=2}{\operatorname {Res} _{1}}}\zeta _{2}(s,0\mid a_{1},a_{2})\ ,}

where Res n s = s 0 f ( s ) = 1 2 π i ∮ s 0 ( s − s 0 ) n − 1 f ( s ) d s {\displaystyle {\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} is an n {\displaystyle n} -th order residue at s 0 {\displaystyle s_{0}} .

Another representation as a product over N {\displaystyle \mathbb {N} } leads to an algorithm for numerically computing the double Gamma function.^[1]

The double gamma function with parameters 1 , 1 {\displaystyle 1,1} obeys the relations ^[2]

Γ 2 ( w + 1 | 1 , 1 ) = 2 π Γ ( w ) Γ 2 ( w | 1 , 1 ) , Γ 2 ( 1 | 1 , 1 ) = 2 π   . {\displaystyle \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 }}\ .}

It is related to the Barnes G-function by

Γ 2 ( w | α , α ) = ( 2 π ) w 2 α α − w 2 2 α 2 + w α − 1 G ( w / α ) − 1   . {\displaystyle \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}\ .}

For ℜ b > 0 {\displaystyle \Re b>0} and Q = b + b − 1 {\displaystyle Q=b+b^{-1}} , the function

Γ b ( w ) = Γ 2 ( w ∣ b , b − 1 ) Γ 2 ( Q 2 ∣ b , b − 1 )   , {\displaystyle \Gamma _{b}(w)={\frac {\Gamma _{2}(w\mid b,b^{-1})}{\Gamma _{2}\left({\frac {Q}{2}}\mid b,b^{-1}\right)}}\ ,}

is invariant under b → b − 1 {\displaystyle b\to b^{-1}} , and obeys the relations

Γ b ( w + b ) = 2 π b b w − 1 2 Γ ( b w ) Γ b ( w ) , Γ b ( w + b − 1 ) = 2 π b − b − 1 w + 1 2 Γ ( b − 1 w ) Γ b ( w )   . {\displaystyle \Gamma _{b}(w+b)={\sqrt {2\pi }}{\frac {b^{bw-{\frac {1}{2}}}}{\Gamma (bw)}}\Gamma _{b}(w)\quad ,\quad \Gamma _{b}(w+b^{-1})={\sqrt {2\pi }}{\frac {b^{-b^{-1}w+{\frac {1}{2}}}}{\Gamma (b^{-1}w)}}\Gamma _{b}(w)\ .}

For ℜ w > 0 {\displaystyle \Re w>0} , it has the integral representation ^[3]

log ⁡ Γ b ( w ) = ∫ 0 ∞ d t t [ e − w t − e − Q 2 t ( 1 − e − b t ) ( 1 − e − b − 1 t ) − ( Q 2 − w ) 2 2 e − t − Q 2 − w t ]   . {\displaystyle \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]\ .}

From the function Γ b ( w ) {\displaystyle \Gamma _{b}(w)} , we define the double Sine function S b ( w ) {\displaystyle S_{b}(w)} and the Upsilon function Υ b ( w ) {\displaystyle \Upsilon _{b}(w)} by

S b ( w ) = Γ b ( w ) Γ b ( Q − w ) , Υ b ( w ) = 1 Γ b ( w ) Γ b ( Q − w )   . {\displaystyle 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)}}\ .}

These functions obey the relations

S b ( w + b ) = 2 sin ⁡ ( π b w ) S b ( w ) , Υ b ( w + b ) = Γ ( b w ) Γ ( 1 − b w ) b 1 − 2 b w Υ b ( w )   , {\displaystyle 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)\ ,}

plus the relations that are obtained by b → b − 1 {\displaystyle b\to b^{-1}} . For 0 < ℜ w < ℜ Q {\displaystyle 0<\Re w<\Re Q} they have the integral representations

log ⁡ S b ( w ) = ∫ 0 ∞ d t t [ sinh ⁡ ( Q 2 − w ) t 2 sinh ⁡ ( 1 2 b t ) sinh ⁡ ( 1 2 b − 1 t ) − Q − 2 w t ]   , {\displaystyle \log S_{b}(w)=\int _{0}^{\infty }{\frac {dt}{t}}\left[{\frac {\sinh \left({\frac {Q}{2}}-w\right)t}{2\sinh \left({\frac {1}{2}}bt\right)\sinh \left({\frac {1}{2}}b^{-1}t\right)}}-{\frac {Q-2w}{t}}\right]\ ,}

log ⁡ Υ b ( w ) = ∫ 0 ∞ d t t [ ( Q 2 − w ) 2 e − t − sinh 2 ⁡ 1 2 ( Q 2 − w ) t sinh ⁡ ( 1 2 b t ) sinh ⁡ ( 1 2 b − 1 t ) ]   . {\displaystyle \log \Upsilon _{b}(w)=\int _{0}^{\infty }{\frac {dt}{t}}\left[\left({\frac {Q}{2}}-w\right)^{2}e^{-t}-{\frac {\sinh ^{2}{\frac {1}{2}}\left({\frac {Q}{2}}-w\right)t}{\sinh \left({\frac {1}{2}}bt\right)\sinh \left({\frac {1}{2}}b^{-1}t\right)}}\right]\ .}

The functions Γ b , S b {\displaystyle \Gamma _{b},S_{b}} and Υ b {\displaystyle \Upsilon _{b}} appear in correlation functions of two-dimensional conformal field theory, with the parameter b {\displaystyle b} being related to the central charge of the underlying Virasoro algebra.^[4] In particular, the three-point function of Liouville theory is written in terms of the function Υ b {\displaystyle \Upsilon _{b}} .

• Barnes, E. W. (1899), "The Genesis of the Double Gamma Functions", Proc. London Math. Soc., s1-31: 358–381, doi:10.1112/plms/s1-31.1.358
• Barnes, E. W. (1899), "The Theory of the Double Gamma Function", Proceedings of the Royal Society of London, 66 (424–433): 265–268, doi:10.1098/rspl.1899.0101, ISSN 0370-1662, JSTOR 116064, S2CID 186213903
• Barnes, E. W. (1901), "The Theory of the Double Gamma Function", Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 196 (274–286): 265–387, Bibcode:1901RSPTA.196..265B, doi:10.1098/rsta.1901.0006, ISSN 0264-3952, JSTOR 90809
• Barnes, E. W. (1904), "On the theory of the multiple gamma function", Trans. Camb. Philos. Soc., 19: 374–425
• Friedman, Eduardo; Ruijsenaars, Simon (2004), "Shintani–Barnes zeta and gamma functions", Advances in Mathematics, 187 (2): 362–395, doi:10.1016/j.aim.2003.07.020, ISSN 0001-8708, MR 2078341
• Ruijsenaars, S. N. M. (2000), "On Barnes' multiple zeta and gamma functions", Advances in Mathematics, 156 (1): 107–132, doi:10.1006/aima.2000.1946, ISSN 0001-8708, MR 1800255