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 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.

Definition

For , let

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

Properties

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.

The double gamma function and conformal field theory

For and , the function

is invariant under , and obeys the relations

For , it has the integral representation

From the function , it is possible to define the two functions

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. In particular, the three-point function of Liouville theory is written in terms of the function .

References

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