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:

:$$G(n)=egin{cases} 0&mbox{if }n=0,-1,-2,dots\ prod_{i=0}^{n-2} i!&mbox{if }n=1,2,dotsend{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 [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