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 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:
where formula_2 is the Euler–Mascheroni constant, exp("x") = "e""x" is the exponential function, and Π denotes multiplication (capital pi notation).
As an entire function, "G" 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
with normalisation "G"(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:
The functional equation implies that "G" takes the following values at integer arguments:
and thus
where formula_8 denotes the gamma function and "K" denotes the K-function. The functional equation uniquely defines the G function if the convexity condition, 
is added. Additionally, the Barnes G function satisfies the duplication formula,
Characterisation.
Similar to the Bohr-Mollerup theorem for the gamma function, for a constant formula_11, we have for formula_12
formula_13
and for formula_14
formula_15
as formula_16.
Reflection formula 1.0.
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 logtangent integral on the right-hand side can be evaluated in terms of the Clausen function (of order 2), as is shown below:
The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation formula_20 for the logcotangent integral, and using the fact that formula_21, an integration by parts gives
Performing the integral substitution formula_23 gives
The Clausen function – of second order – has the integral representation
However, within the interval formula_26, 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 logtangent integral, the following relation clearly holds:
Thus, after a slight rearrangement of terms, the proof is complete:
Using the relation formula_29 and dividing the reflection formula by a factor of formula_30 gives the equivalent form:
Ref: see Adamchik below for an equivalent form of the reflection formula, but with a different proof.
Reflection formula 2.0.
Replacing z with (1/2) − "z"" 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:
It is valid for formula_34. Here, formula_35 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:
where formula_40 is a constant given by:
Here formula_42 is the derivative of the Riemann zeta function and formula_43 is the Glaisher–Kinkelin constant.
Absolute value.
It holds true that formula_44, thus formula_45. From this relation and by the above presented Weierstrass product form one can show that
This relation is valid for arbitrary formula_47, and formula_48. If formula_49, 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:
Here the formula_52 are the Bernoulli numbers and formula_43 is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes the Bernoulli number formula_54 would have been written as formula_55, but this convention is no longer current.) This expansion is valid for formula_56 in any sector not containing the negative real axis with formula_57 large.
Relation to the Loggamma integral.
The parametric Loggamma can be evaluated in terms of the Barnes G-function (Ref: this result is found in Adamchik below, but stated without proof):
The proof is somewhat indirect, and involves first considering the logarithmic difference of the gamma function and Barnes G-function:
where
and formula_61 is the Euler–Mascheroni constant.
Taking the logarithm of the Weierstrass product forms of the Barnes 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, and integrate over the interval formula_64 to obtain:
Equating the two evaluations completes the proof:
And since formula_29 then,