Barnes G-Function

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The Barnes G-function is an analytic continuation of the G-function defined in the construction of the Glaisher-Kinkelin constant

G(n)=([Gamma(n)]^(n-1))/(H(n-1))
(1)

for n>0, where H(n) is the hyperfactorial, which has the special values

G(n)={0 if n=-1,-2,...; 0!1!2!...(n-2)! if n=0,1,2,...
(2)

for integer n. This function is what Sloane and Plouffe (1995) call the superfactorial, and the first few values for n=1, 2, ... are 1, 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... (OEIS A000178).

The Barnes G-function can arise in spectral functions in mathematical physics (Voros 1987).

It is implemented in the Wolfram Language as BarnesG[n]. A special version of its natural logarithm optimized for large n is implemented in the Wolfram Language as LogBarnesG[n].

The Barnes G-function for complex z may be defined by

G(z+1)=(2pi)^(z/2)e^(-[z(z+1)+gammaz^2]/2)product_(n=1)^infty[(1+z/n)^ne^(-z+z^2/(2n))],
(3)

where gamma is the Euler-Mascheroni constant (Whittaker and Watson 1990, p. 264; Voros 1987). The product can be done in closed form, yielding the identity

G(z)=(e^(1/12-zeta^'(-1,z))[Gamma(z)]^(z-1))/A
(4)

for R[z]>0, where zeta^'(-1,z) is the derivative of the Hurwitz zeta function, Gamma(z) is the gamma function, and A is the Glaisher-Kinkelin constant. Another elegant closed-form expression is given by

G(z)=(2pi)^(z/2)e^((z-1)[lnGamma(z)-z/2]-gamma_(-2)(z)),
(5)

where gamma_(-2)(z) is a polygamma function of negative order. The Barnes G-function and hyperfactorial H(z) satisfy the relation

H(z-1)G(z)=e^((z-1)lnGamma(z))
(6)

for all complex z, where lnGamma(z) is the log gamma function.

G(z) is an entire function analogous to 1/Gamma(z), except that it has order 2 instead of 1.

BarnesG

The Barnes G-function is plotted above evaluated at integers values. A slight variant of the integer-valued Barnes G-function is sometimes known as the superfactorial.

The Barnes G-function satisfies the functional equation

G(z+1)=Gamma(z)G(z),
(7)

and has the Taylor series

lnG(z+1)=1/2[ln(2pi)-1]z-(1+gamma)(z^2)/2+sum_(n=3)^infty(-1)^(n-1)zeta(n-1)(z^n)/n
(8)

in |z|<1. It also gives an analytic solution to the finite product

product_(i=1)^nGamma(k+i)=(G(n+k+1))/(G(k+1)).
(9)

The Barnes G-function has the equivalent reflection formulas

(G^'(z+1))/(G(z+1))=1/2ln(2pi)+1/2-z+z(Gamma^'(z))/(Gamma(z))
(10)
ln[(G(1-z))/(G(1+z))]=piint_0^zzcot(piz)dz-zln(2pi)
(11)
(G(1/2+z))/(G(1/2-z)) =((2pi)^z)/(Gamma(1/2+z))sqrt(pi/(cos(piz)))exp[piint_0^zztan(piz)dz]
(12)

(Voros 1987; Whittaker and Watson 1990, p. 264).

The derivative is given by

d/(dz)G(z)=G(z)[(z-1)psi_0(z)-z+1/2ln(2pi)+1/2],
(13)

where psi_0(z) is the digamma function.

A Stirling-like asymptotic series for R[z]>0 as z->infty is given by

lnG(1+z)∼z^2(1/2lnz-3/4)+1/2ln(2pi)z-1/(12)lnz+zeta^'(-1)+O(1/z)
(14)

(Voros 1987). This can be made more precise as

lnG(1+z)∼z^2(1/2lnz-3/4)+1/2ln(2pi)z-1/(12)lnz+zeta^'(-1) +sum_(k=1)^n(B_(2k+2))/(4k(k+1)z^(2k))+O(1/(z^(2n+2))),
(15)

where B_k is a Bernoulli number (Adamchik 2001b; typo corrected).

G(n) has the special values

G(1/4)=A^(-9/8)Gamma^(-3/4)(1/4)e^(3/32-K/(4pi))
(16)
G(3/4)=A^(-9/8)Gamma^(-1/4)(3/4)e^(3/32+K/(4pi))
(17)
=A^(-9/8)2^(-1/8)pi^(-1/4)Gamma^(1/4)(1/4))e^(3/32+K/(4pi))
(18)

(OEIS A087013 and A087015) for n=k/4, where Gamma(z) is the gamma function, K is Catalan's constant, A is the Glaisher-Kinkelin constant, and

G(1/2)=A^(-3/2)pi^(-1/4)e^(1/8)2^(1/24)
(19)
G(3/2)=A^(-3/2)pi^(1/4)e^(1/8)2^(1/24)
(20)
G(5/2)=A^(-3/2)pi^(3/4)e^(1/8)2^(-23/24),
(21)

(OEIS A087014, A087016, and A087017) for n=k/2, where zeta^'(-1) is the derivative of the Riemann zeta function evaluated at -1. In general, for odd n=2k+1,

G(k-1/2)=c_k(pi^((2k-3)/4)e^(1/8)2^(1/24))/(2^((k-1)(k-2)/2)A^(3/2)),
(22)

where

c_k=product_(i=1)^(k-2)(2^iGamma(1/2+i))/(sqrt(pi))
(23)

for k>1, of which the first few terms are 1, 1, 1, 3, 45, 4725, 4465125, ... (OEIS A057863).

Another G-function is defined by Erdélyi et al. (1981, p. 20) as

G(z)=psi_0(1/2+hz)-psi_0(1/2z),
(24)

where psi_0(z) is the digamma function. An unrelated pair of functions are denoted g_n and G_n and are known as Ramanujan g- and G-functions.

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