Barnes G-function

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. [ E.W.Barnes, "The theory of the G-function", "Quarterly Journ. Pure and Appl. Math." 31 (1900), 264-314.]

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

:$G(z+1)=(2pi)^\{z/2\}\; e^\{-\; [z(z+1)+gamma\; z^2]\; /2\}prod\_\{n=1\}^infty\; left\; [left(1+frac\{z\}\{n\}\; ight)^ne^\{-z+z^2/(2n)\}\; ight]$

where γ is the Euler-Mascheroni constant.

Difference equation, functional equation and special values

The Barnes G-function satisfies the difference equation

:$G(z+1)=Gamma(z)G(z)$

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

:

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 [M. F. Vignéras, "L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL$(2,mathbb\{Z\})$", Astérisque 61, 235-249 (1979).] .

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

:$G(1-z)\; =\; G(1+z)frac\{\; 1\}\{(2pi)^z\}\; exp\; int\_0^z\; pi\; z\; cot\; pi\; z\; ,\; dz.$

Multiplication formula

Like the Gamma function, the G-function also has a multiplication formula [I. Vardi, "Determinants of Laplacians and multiple gamma functions", SIAM J. Math. Anal. 19, 493-507 (1988).] ::$G(nz)=\; K(n)\; n^\{n^\{2\}z^\{2\}/2-nz\}\; (2pi)^\{-frac\{n^2-n\}\{2\}z\}prod\_\{i=0\}^\{n-1\}prod\_\{j=0\}^\{n-1\}Gleft(z+frac\{i+j\}\{n\}\; ight)$where $K(n)$ is a constant given by:

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

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

Asymptotic Expansion

The function $log\; ,G(z+1\; )$ has the following asymptotic expansion established by Barnes::$log\; G(z+1)=frac\{1\}\{12\}\; -\; log\; A\; +\; frac\{z\}\{2\}log\; 2pi\; +left(frac\{z^2\}\{2\}\; -frac\{1\}\{12\}\; ight)log\; z\; -frac\{3z^2\}\{4\}+sum\_\{k=1\}^\{N\}frac\{B\_\{2k+2\{4kleft(k+1\; ight)z^\{2k\; +\; Oleft(frac\{1\}\{z^\{2N+2\; ight).$

Here the $B\_\{k\}$ are the Bernoulli numbers and $A$ is the Glaisher-Kinkelin constant. (Note that somewhat confusingly at the time of Barnes [ E.T.Whittaker and G.N.Watson, "A course of modern analysis", CUP.] 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