In mathematics, the Barnes G-function 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 mathematician Ernest William Barnes. It can be written in terms of the double gamma function.

Formally, the Barnes G-function is defined in the following Weierstrass product form:

is the Euler–Mascheroni constant, exp(x) = ex denotes multiplication (capital pi notation).

The integral representation, which may be deduced from the relation to the double gamma function, is

As an entire function is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.

Functional equation and integer arguments

The Barnes G-function satisfies the functional equation

. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:

takes the following values at integer arguments:

is the superfactorial.

and thus

denotes the gamma function denotes the K-function.

uniquely defines the Barnes G-function if the convexity condition,

is added. Additionally, the Barnes G-function satisfies the duplication formula,

1. ,

is the Glaisher–Kinkelin constant.

Characterisation

Similar to the Bohr–Mollerup theorem for the gamma function

.

Reflection formula

The difference equation for the G-function, in conjunction with the functional equation for the gamma function, can be used to obtain the following reflection formula for the Barnes G-function (originally proved by Hermann Kinkelin):

The log-tangent integral on the right-hand side can be evaluated in terms of the Clausen function, as is shown below:

, an integration by parts gives

gives

The Clausen function – of second order – has the integral representation

, the absolute value sign within the integrand can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the log-tangent integral, the following relation clearly holds:

Thus, after a slight rearrangement of terms, the proof is complete:

gives the equivalent form:

Adamchik (2003) has given an equivalent form of the reflection formula, but with a different proof.

in the previous reflection formula gives, after some simplification, the equivalent formula shown below

(involving Bernoulli polynomials):

Taylor series expansion

By Taylor's theorem, and considering the logarithmic derivatives of the Barnes function, the following series expansion can be obtained:

is the Riemann zeta function:

Exponentiating both sides of the Taylor expansion gives:

Comparing this with the Weierstrass product form of the Barnes function gives the following relation:

Multiplication formula

Like the gamma function, the G-function also has a multiplication formula:

is a constant given by:

is the derivative of the Riemann zeta function is the Glaisher–Kinkelin constant.

Absolute value

. From this relation and by the above presented Weierstrass product form one can show that

, then the below formula is valid instead:

for arbitrary real y.

Asymptotic expansion

The logarithm of G(z + 1) has the following asymptotic expansion, as established by Barnes:

are the Bernoulli numbers is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes the Bernoulli number large.

Relation to the log-gamma integral

The parametric log-gamma can be evaluated in terms of the Barnes G-function:

The proof is somewhat indirect, and involves first considering the logarithmic difference of the gamma function and Barnes G-function:

where

is the Euler–Mascheroni constant.

Taking the logarithm of the Weierstrass product forms of the Barnes G-function and gamma function gives:

A little simplification and re-ordering of terms gives the series expansion:

Finally, take the logarithm of the Weierstrass product form of the gamma function to obtain:

Equating the two evaluations completes the proof:

then,

Taking the logarithm of both sides introduces the analog of the Digamma function,

where

with Taylor series

References