“Multiple gamma function”的意思、由来-开放百科全书

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Multiple gamma function

释义

Definition
Properties
Infinite product representation
Reduction to the Barnes G-function
The double gamma function and conformal field theory
References
Further reading

{{For|derivatives of the log of the gamma function |polygamma 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 {{harvtxt|Barnes|1901}}. At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in {{harvtxt|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.

Infinite product representation

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-function

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 theory

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 .

References