Barnes G-function

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In mathematics, the Barnes G-function (typically denoted 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.^[1]

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

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

where γ is the Euler-Mascheroni constant.

1 Difference equation, functional equation and special values
2 Multiplication formula
3 Asymptotic Expansion
4 References

[edit] 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:

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

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^[2].

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:

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

[edit] Multiplication formula

Like the Gamma function, the G-function also has a multiplication formula^[3]:

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

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.

[edit] Asymptotic Expansion

The function $\log \,G(z+1 )$ has the following asymptotic expansion 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 ^[4] the Bernoulli number $B 2 k$ 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.

[edit] References

^ E.W.Barnes, "The theory of the G-function", Quarterly Journ. Pure and Appl. Math. 31 (1900), 264-314.
^ M. F. Vignéras, L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL $(2,\mathbb{Z})$ , Astérisque 61, 235-249 (1979).
^ I. Vardi, Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal. 19, 493-507 (1988).
^ E.T.Whittaker and G.N.Watson, "A course of modern analysis", CUP.