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.
Definition 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.)
Properties 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)\ ,}
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