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. Up to elementary factors, it is a special case of the double gamma function.

Formally, the Barnes G-function is defined (in the form of a Weierstrass product) as

$G(z+1)=(2\pi)^{z/2} \exp(-(z(z+1)+\gamma z^2)/2)\ \times\ \prod_{n=1}^\infty \left[\left(1+\frac{z}{n}\right)^n \exp(-z+z^2/(2n))\right],$

where γ is the Euler–Mascheroni constant, exp(x) = e^x, and ∏ is capital pi notation.

Difference equation, functional equation and special values

The Barnes G-function satisfies the difference equation

G(z + 1) = Γ(z)G(z)

with normalisation G(1) = 1. The difference equation implies that G takes the following values at integer arguments:

$G(n)=\begin{cases} 0&\text{if }n=0,-1,-2,\dots\\ \prod_{i=0}^{n-2} i!&\text{if }n=1,2,\dots\end{cases}$

and thus

$G(n)=\frac{(\Gamma(n))^{n-1}}{K(n)}$

where Γ denotes the Gamma function and K denotes the K-function. The difference equation uniquely defines the G function if the convexity condition: $\frac{d^3}{dx^3}G(x)\geq 0$ is added.

The difference equation for the G function and the functional equation for the Gamma function yield the following functional equation for the G function, originally proved by Hermann Kinkelin:

$G(1-z) = G(1+z)\frac{ 1}{(2\pi)^z} \exp \int_0^z \pi x \cot \pi x \, dx.$

Multiplication formula

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

$G(nz)= K(n) n^{n^{2}z^{2}/2-nz} (2\pi)^{-\frac{n^2-n}{2}z}\prod_{i=0}^{n-1}\prod_{j=0}^{n-1}G\left(z+\frac{i+j}{n}\right)$

where K(n) is a constant given by:

$K(n)= e^{-(n^2-1)\zeta^\prime(-1)} \cdot n^{\frac{5}{12}}\cdot(2\pi)^{(n-1)/2}\,=\, (Ae^{-\frac{1}{12}})^{n^2-1}\cdot n^{\frac{5}{12}}\cdot (2\pi)^{(n-1)/2}.$

Here $\zeta^\prime$ is the derivative of the Riemann zeta function and A is the Glaisher–Kinkelin constant.

Asymptotic expansion

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

$\log G(z+1)=\frac{1}{12}~-~\log A~+~\frac{z}{2}\log 2\pi~+~\left(\frac{z^2}{2} -\frac{1}{12}\right)\log z~-~\frac{3z^2}{4}~+~ \sum_{k=1}^{N}\frac{B_{2k + 2}}{4k\left(k + 1\right)z^{2k}}~+~O\left(\frac{1}{z^{2N + 2}}\right).$

Here the B_k are the Bernoulli numbers and A is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes the Bernoulli number B_2k would have been written as ( − 1)^{k + 1}B_k, but this convention is no longer current.) This expansion is valid for z in any sector not containing the negative real axis with | z | large.

References

• Askey, R.A.; Roy, R. (2010), "Barnes G-function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/5.17