Barnes G-function explained

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.^[1] It can be written in terms of the double gamma function.

Formally, the Barnes G-function is defined in the following Weierstrass product form:^[2]

G(1+z)=(2\pi)^z/2\exp\left(-

	z+z^2(1+\gamma)
	2

\right)

	infty
\prod
	k=1

\left\{\left(1+

	z
	k

\right)^k\exp\left(

	z²
	2k

-z\right)\right\}

where

\gamma

is the Euler–Mascheroni constant, exp(x) = e^x is the exponential function, and

\Pi

denotes multiplication (capital pi notation).

The integral representation, which may be deduced from the relation to the double gamma function, is

logG(1+z)=

	z
	2

log(2\pi)

infty	dt
	t

+\int

\left[

1-e^-zt

2	t
	2

4\sinh

	z²
	2

e^-t-

	z
	t

\right]

As an entire function,

is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.

Functional equation and integer arguments

The Barnes G-function satisfies the functional equation

G(z+1)=\Gamma(z)G(z)

with normalization

G(1)=1

. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:

\Gamma(z+1)=z\Gamma(z).

The functional equation implies that

takes the following values at integer arguments:

G(n)=\begin{cases}

	n-2
0&ifn=0,-1,-2,...\ \prod
	i=0

i!&ifn=1,2,...\end{cases}

In particular,

G(0)=0,G(1)=1

and

G(n)=sf(n-2)

for

n\geq1

, where

is the superfactorial.

and thus

G(n)=	(\Gamma(n))^n-1
	K(n)

where

\Gamma(x)

denotes the gamma function and

denotes the K-function. In general,

K(z) G(z)=e^

for all complex

The functional equation

G(z+1)=\Gamma(z)G(z)

uniquely defines the Barnes G-function if the convexity condition,

(\forallx\geq1)

	d³
	dx³

log(G(x))\geq0

is added.^[3] Additionally, the Barnes G-function satisfies the duplication formula,^[4]

G(x)G\left(x+	1
	2

\right)²

	1
	4

G(x+1)=e

A^-3

-2x

+3x-	11
	12

x-	1
	2

\pi

G\left(2x\right)

where

is the Glaisher–Kinkelin constant.

Characterisation

Similar to the Bohr–Mollerup theorem for the gamma function, for a constant

c>0

we have for

f(x)=cG(x)

^[5]

f(x+1)=\Gamma(x)f(x)

and for

x>0

f(x+n)\sim\Gamma(x)ⁿⁿ^{x\choose

}f(n)

n\toinfty

Reflection formula

The difference equation for the G-function, in conjunction with the functional equation for the gamma function, can be used to obtain the following reflection formula for the Barnes G-function (originally proved by Hermann Kinkelin):

logG(1-z)=logG(1+z)-zlog2\pi+

	z
\int
	0

\pix\cot\pixdx.

The log-tangent integral on the right-hand side can be evaluated in terms of the Clausen function (of order 2) when

0<z<1

, as is shown below:

2\pilog\left(

	G(1-z)
	G(1+z)

\right)=2\pizlog\left(

	\sin\piz
	\pi

\right)+\operatorname{Cl}_2(2\piz)

The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation

\operatorname{Lc}(z)

for the log-cotangent integral, and using the fact that

(d/dx)log(\sin\pix)=\pi\cot\pix

, an integration by parts gives

\begin{align} \operatorname{Lc}(z)&=

	z\pi
\int
	0

x\cot\pixdx\\ &=zlog(\sin\pi

	zlog(\sin
z)-\int
	0

\pix)dx\\ &=zlog(\sin\pi

	z[log(2\sin
z)-\int
	0

\pix)-log2]dx\\ &=zlog(2\sin\pi

	zlog(2\sin
z)-\int
	0

\pix)dx. \end{align}

Performing the integral substitution

y=2\pix ⇒ dx=dy/(2\pi)

gives

zlog(2\sin\piz)-

	1
	2\pi

	2\piz
\int
	0

log\left(2\sin

	y
	2

\right)dy.

The Clausen function – of second order – has the integral representation

\operatorname{Cl}_2(\theta)=

	\theta
-\int
	0

log|2\sin

	x
	2

|dx.

However, within the interval

0<\theta<2\pi

, the absolute value sign within the integrand can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the logtangent integral, the following relation clearly holds:

\operatorname{Lc}(z)=zlog(2\sin\piz)+

	1
	2\pi

\operatorname{Cl}_2(2\piz).

Thus, after a slight rearrangement of terms, the proof is complete:

2\pilog\left(

	G(1-z)
	G(1+z)

\right)=2\pizlog\left(

	\sin\piz
	\pi

\right)+\operatorname{Cl}_2(2\piz)

Using the relation

G(1+z)=\Gamma(z)G(z)

and dividing the reflection formula by a factor of

2\pi

gives the equivalent form:

log\left(

	G(1-z)
	G(z)

\right)=zlog\left(

	\sin\piz	\right)+log\Gamma(z)+
	\pi

	1
	2\pi

\operatorname{Cl}_2(2\piz)

Adamchik (2003) has given an equivalent form of the reflection formula, but with a different proof.^[6]

Replacing

with

1/2-z

in the previous reflection formula gives, after some simplification, the equivalent formula shown below

(involving Bernoulli polynomials):

log\left(

G\left(	1
	2

+z\right)

G\left(	1
	2

-z\right)

\right)=log\Gamma\left(

	1
	2

-z\right)+B_1(z)log2\pi+

	1
	2

log2+\pi

	z
\int
	0

B_1(x)\tan\pixdx

Taylor series expansion

By Taylor's theorem, and considering the logarithmic derivatives of the Barnes function, the following series expansion can be obtained:

logG(1+z)=

	z
	2

log2\pi-\left(

	z+(1+\gamma)z²
	2

\right)+

	infty
\sum
	k=2

k	\zeta(k)
	k+1

(-1)

z^k+1.

It is valid for

0<z<1

. Here,

\zeta(x)

is the Riemann zeta function:

	infty
\zeta(s)=\sum
	n=1

	1
	n^s

Exponentiating both sides of the Taylor expansion gives:

\begin{align}G(1+z)&=\exp\left[

	z
	2

log2\pi-\left(

	z+(1+\gamma)z²
	2

\right)+

	infty
\sum
	k=2

k	\zeta(k)
	k+1

(-1)

z^k+1\right]\\ &=(2\pi)^z/2\exp\left[-

	z+(1+\gamma)z²
	2

\right]\exp

	infty
\left[\sum
	k=2

k	\zeta(k)
	k+1

(-1)

z^k+1\right].\end{align}

Comparing this with the Weierstrass product form of the Barnes function gives the following relation:

\exp

	infty
\left[\sum
	k=2

k	\zeta(k)
	k+1

(-1)

z^k+1\right]=

	infty
\prod
	k=1

\left\{\left(1+

	z
	k

\right)^k\exp\left(

	z²
	2k

-z\right)\right\}

Multiplication formula

Like the gamma function, the G-function also has a multiplication formula:^[7]

G(nz)=K(n)

	n²z²/2-nz
n

-	n^2-n	z
	2

(2\pi)

	n-1
\prod
	i=0

	n-1
\prod		G\left(z+
	j=0

	i+j
	n

\right)

where

K(n)

is a constant given by:

K(n)=

	-(n^2-1)\zeta^\prime(-1)
e

	5
	12

⋅ n

⋅ (2\pi)^(n-1)/2

-	1
	12

= (Ae

	n^2-1
)

⋅

	5
	12

⋅ (2\pi)^(n-1)/2.

Here

\zeta^\prime

is the derivative of the Riemann zeta function and

is the Glaisher–Kinkelin constant.

Absolute value

It holds true that

G(\overlinez)=\overline{G(z)}

, thus

|G(z)|^{2=G(z)G(\overline}z)

. From this relation and by the above presented Weierstrass product form one can show that

2	1+\gamma
	2

|G(x+iy)|=|G(x)|\exp\left(y

\right)\sqrt{1+

	y²
	x²

}\sqrt.This relation is valid for arbitrary

x\inR\setminus\{0,-1,-2,...\}

, and

y\inR

. If

x=0

, then the below formula is valid instead:

2	1+\gamma
	2

|G(iy)|=y\exp\left(y

infty\left(1+	y²
	k²

\right)\sqrt{\prod

k=1

\right)^k+1\exp\left(-

	y²
	k

\right)}

for arbitrary real y.

Asymptotic expansion

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

\begin{align} logG(z+1)={}&

	z²
	2

logz-

	3z²
	4

	z
	2

log2\pi-

	1
	12

logz\\ &{}+\left(

	1
	12

-logA\right)

	N
+\sum
	k=1

	B_2k	~+~O\left(
	4k\left(k+1\right)z^2k

	1
	z^2N

\right). \end{align}

Here the

B_k

are the Bernoulli numbers and

is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes ^[8] the Bernoulli number

B_2k

would have been written as

(-1)^k+1B_k

, but this convention is no longer current.) This expansion is valid for

in any sector not containing the negative real axis with

|z|

large.

Relation to the log-gamma integral

The parametric log-gamma can be evaluated in terms of the Barnes G-function:^[9]

	z
\int
	0

log\Gamma(x)dx=

	z(1-z)	+
	2

	z
	2

log2\pi+(z-1)log\Gamma(z)-logG(z)

The proof is somewhat indirect, and involves first considering the logarithmic difference of the gamma function and Barnes G-function:

zlog\Gamma(z)-logG(1+z)

where

	1
	\Gamma(z)

=ze^\gamma

	infty
\prod
	k=1

\left\{\left(1+

	z
	k

\right)e^-z/k\right\}

and

\gamma

is the Euler–Mascheroni constant.

Taking the logarithm of the Weierstrass product forms of the Barnes G-function and gamma function gives:

\begin{align} &zlog\Gamma(z)-logG(1+z)=-zlog\left(

	1
	\Gamma(z)

\right)-logG(1+z)\\[5pt] ={}&{-z}\left[logz+\gammaz

	infty
+\sum
	k=1

\{log\left(1+

	z
	k

\right)-

	z
	k

\}\right]\\[5pt] &{}-\left[

	z
	2

log2\pi-

	z	-
	2

	z²	-
	2

	z²\gamma
	2

	infty
\sum		\{klog\left(1+
	k=1

	z
	k

\right)+

	z²
	2k

-z\}\right] \end{align}

A little simplification and re-ordering of terms gives the series expansion:

\begin{align} &

	infty
\sum
	k=1

\{(k+z)log\left(1+

	z	\right)-
	k

	z²
	2k

-z\}\\[5pt] ={}&{-z}logz-

	z
	2

log2\pi+

	z	+
	2

	z²
	2

	z²\gamma
	2

-zlog\Gamma(z)+logG(1+z) \end{align}

Finally, take the logarithm of the Weierstrass product form of the gamma function, and integrate over the interval

[0,z]

to obtain:

\begin{align} &

	z
\int		log\left(
	0

	1
	\Gamma(x)

\right)dx\\[5pt] ={}&{-(zlogz-z)}-

	z²\gamma
	2

	infty
\sum
	k=1

\{(k+z)log\left(1+

	z	\right)-
	k

	z²
	2k

-z\} \end{align}

Equating the two evaluations completes the proof:

	z
\int
	0

log\Gamma(x)dx=

	z(1-z)	+
	2

	z
	2

log2\pi+zlog\Gamma(z)-logG(1+z)

And since

G(1+z)=\Gamma(z)G(z)

then,

	z
\int
	0

log\Gamma(x)dx=

	z(1-z)	+
	2

	z
	2

log2\pi-(1-z)log\Gamma(z)-logG(z).

\psi(x)

\varphi(x)\equiv

	d
	dx

logG(x),

where ^[10]

\varphi(x)=(x-1)[\psi(x)-1]+\varphi(1), \varphi(1)=

	ln(2\pi)-1
	2

with Taylor series

\varphi(x)=\varphi(1)-(\gamma+1)(x-1)+\sum_k\ge(-1)^{k\zeta(k)(x-1)}^k.

References

E. W. . Barnes. The theory of the G-function. Q. J. Pure Appl. Math.. 31. 1900. 264–314.
Juensang. Choi. H. M.. Srivastava. Certain classes of series involving the Zeta Function. 1999. J. Math. Anal. Appl.. 231. 91–117. 10.1006/jmaa.1998.6216.
M. F.. Vignéras. L'équation fonctionelle de la fonction zêta de Selberg du groupe modulaire PSL
(2,Z)

. Astérisque. 61. 235–249. 1979.
A duplication formula for the double gamma function $Gamma_2$ . Bulletin of the Korean Mathematical Society . 1996 . 33 . 2 . 289–294 . Park . Junesang .
Book: Marichal . Jean Luc . Naim . Zenaidi. A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions . Developments in Mathematics . Springer . 10.1007/978-3-030-95088-0. 218. 2022 . 70 . 978-3-030-95087-3 .
Adamchik. Viktor S.. Contributions to the Theory of the Barnes function. 2003. math/0308086 .
I.. Vardi. Determinants of Laplacians and multiple gamma functions. SIAM J. Math. Anal.. 19. 493–507. 1988. 2 . 10.1137/0519035.
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Yury A. . Neretin. The double gamma function and Vladimar Alekseevsky. 2402.07740. 2024 . math.HO .
Merkle. Milan. Ribero Merkle. Monica Moulin. Krull's theory for the double gamma functions. Appl. Math. Comput.. 218. 2011. 3. 935–943. 2831334. 10.1016/j.amc.2011.01.090.