In mathematics, the multiple gamma function
is a generalization of the Euler
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and the
Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathemat ...
. The double gamma function was studied by . At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in .
Double gamma functions
are closely related to the
q-gamma function
In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by
\Gamma_q(x) = (1-q)^\prod_^\infty \frac=(1-q)^ ...
, and triple gamma functions
are related to the
elliptic gamma function In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by , and can be expressed in terms of the triple gamma ...
.
Definition
For
, let
:
where
is the
Barnes zeta function In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by . It is further generalized by the Shintani zeta function.
Definition
The Barnes zeta function is defined by
: \zeta_N(s,w\mid a_1,\ldots,a_N) ...
. (This differs by a constant from Barnes's original definition.)
Properties
Considered as a
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
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 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
:
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
:
It is related to the
Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathemat ...
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
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal fie ...
, with the parameter
being related to the central charge of the underlying
Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
.
In particular, the three-point function of
Liouville theory
In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation.
Liouville theory is defined for all complex values of the ...
is written in terms of the function
.
References
Further reading
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*{{Citation , last1=Ruijsenaars , first1=S. N. M. , title=On Barnes' multiple zeta and gamma functions , doi=10.1006/aima.2000.1946 , doi-access=free , mr=1800255 , year=2000 , journal=
Advances in Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes.
At the origin, the journal aimed ...
, issn=0001-8708 , volume=156 , issue=1 , pages=107–132, url=https://ir.cwi.nl/pub/2100
Gamma and related functions