Multiple 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.

is the Barnes zeta function.

(This differs by a constant from Barnes's original definition.)

Considered as a meromorphic function of

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.

In the case of the double Gamma function, the asymptotic behaviour for

is known, and the leading factor is[1] 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] where we define the

-th order residue at

leads to an algorithm for numerically computing the double Gamma function.

[1] The double gamma function with parameters

obeys the relations [2] It is related to the Barnes G-function by For

, the function is invariant under

, 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

they have the integral representations The functions

appear in correlation functions of two-dimensional conformal field theory, with the parameter

being related to the central charge of the underlying Virasoro algebra.

[3] In particular, the three-point function of Liouville theory is written in terms of the function