Multiple gamma function

  (Redirected from Triple gamma function)

In mathematics, the multiple gamma function 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 are closely related to the q-gamma function, and triple gamma functions are related to the elliptic gamma function.

DefinitionEdit

For  , let

 

where   is the Barnes zeta function. (This differs by a constant from Barnes's original definition.)

PropertiesEdit

Considered as a meromorphic function of  ,   has no zeros. It has poles at  for non-negative integers  . These poles are simple unless some of them coincide. Up to multiplication by the exponential of a polynomial,   is the unique meromorphic function of finite order with these zeros and poles.

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Infinite product representationEdit

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 [1]

 

where we define the  -independent coefficients

 
 

where   is an  -th order residue at  .

Reduction to the Barnes G-functionEdit

The double gamma function with parameters   obeys the relations [1]

 

It is related to the Barnes G-function by

 

The double gamma function and conformal field theoryEdit

For   and  , the function

 

is invariant under  , and obeys the relations

 

For  , it has the integral representation

 

From the function  , we define the double Sine function   and the Upsilon function   by

 

These functions obey the relations

 

plus the relations that are obtained by  . For   they have the integral representations

 
 

The functions   and   appear in correlation functions of two-dimensional conformal field theory, with the parameter   being related to the central charge of the underlying Virasoro algebra.[2] In particular, the three-point function of Liouville theory is written in terms of the function  .

ReferencesEdit

  1. ^ a b Spreafico, Mauro (2009). "On the Barnes double zeta and gamma functions". Journal of Number Theory. 129 (9): 2035–2063. doi:10.1016/j.jnt.2009.03.005.
  2. ^ Ponsot, B. Recent progress on Liouville Field Theory (Thesis). arXiv:hep-th/0301193. Bibcode:2003PhDT.......180P.

Further readingEdit