Barnes G-function

Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

The Barnes G function along part of the real axis

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. It can be written in terms of the double gamma function.

Formally, the Barnes G-function is defined in the following Weierstrass product form:

$G(1+z)=(2\pi {)}^{z/2}\exp (-\frac{z+z^2(1+\gamma )}{2})\prod _{k=1}^{\mathrm{\infty }}\{{(1+\frac{z}{k})}^k\exp (\frac{z^2}{2k}-z)\}$

where $\gamma$ is the Euler–Mascheroni constant, exp(x) = e^x is the exponential function, and Π denotes multiplication (capital pi notation).

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

$\log G(1+z)=\frac{z}{2}\log (2\pi )+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{dt}{t}[\frac{1-e^{-zt}}{4{\sinh }^2\frac{t}{2}}+\frac{z^2}{2}{e}^{-t}-\frac{z}{t}]$

As an entire function, G 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

$G(z+1)=\mathrm{\Gamma }(z)G(z)$

with normalisation G(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:

$\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(z).$

The functional equation implies that G takes the following values at integer arguments:

$G(n)=\{\begin{array}{cc}0& \text{if }n=0,-1,-2,\dots \\ \prod _{i=0}^{n-2}i!& \text{if }n=1,2,\dots \end{array}$

(in particular, $G(0)=0,G(1)=1$) and thus

$G(n)=\frac{(\mathrm{\Gamma }(n){)}^{n-1}}{K(n)}$

where $\mathrm{\Gamma }(x)$ denotes the gamma function and K denotes the K-function. The functional equation uniquely defines the Barnes G-function if the convexity condition,

$(\mathrm{\forall }x\ge 1)\frac{{\mathrm{d}}^3}{\mathrm{d}{x}^3}\log (G(x))\ge 0$

is added. Additionally, the Barnes G-function satisfies the duplication formula,

$G(x)G{(x+\frac{1}{2})}^{2}G(x+1)=e^{\frac{1}{4}}{A}^{-3}{2}^{-2x^2+3x-\frac{11}{12}}{\pi }^{x-\frac{1}{2}}G\left(2x\right)$,

where $A$ is the Glaisher–Kinkelin constant.

Characterisation

Similar to the Bohr–Mollerup theorem for the gamma function, for a constant $c>0$, we have for $f(x)=cG(x)$

$f(x+1)=\mathrm{\Gamma }(x)f(x)$

and for $x>0$

$f(x+n)\sim \mathrm{\Gamma }(x{)}^n{n}^{(\genfrac{}{}{0ex}{}{x}{2})}f(n)$

as $n\to \mathrm{\infty }$.

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):

$\log G(1-z)=\log G(1+z)-z\log 2\pi +\underset{0}{\overset{z}{\int }}\pi x\cot \pi xdx.$

The log-tangent integral on the right-hand side can be evaluated in terms of the Clausen function (of order 2), as is shown below:

$2\pi \log \left(\frac{G(1-z)}{G(1+z)}\right)=2\pi z\log \left(\frac{\sin \pi z}{\pi }\right)+{\mathrm{Cl}}_2(2\pi z)$

The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation $\mathrm{Lc}(z)$ for the log-cotangent integral, and using the fact that $(d/dx)\log (\sin \pi x)=\pi \cot \pi x$, an integration by parts gives

$\begin{array}{cc}\mathrm{Lc}(z)& =\underset{0}{\overset{z}{\int }}\pi x\cot \pi xdx\\ & =z\log (\sin \pi z)-\underset{0}{\overset{z}{\int }}\log (\sin \pi x)dx\\ & =z\log (\sin \pi z)-\underset{0}{\overset{z}{\int }}[\log (2\sin \pi x)-\log 2]dx\\ & =z\log (2\sin \pi z)-\underset{0}{\overset{z}{\int }}\log (2\sin \pi x)dx.\end{array}$

Performing the integral substitution $y=2\pi x\Rightarrow dx=dy/(2\pi )$ gives

$z\log (2\sin \pi z)-\frac{1}{2\pi }\underset{0}{\overset{2\pi z}{\int }}\log (2\sin \frac{y}{2})dy.$

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

${\mathrm{Cl}}_2(\theta )=-\underset{0}{\overset{\theta }{\int }}\log |2\sin \frac{x}{2}|dx.$

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:

$\mathrm{Lc}(z)=z\log (2\sin \pi z)+\frac{1}{2\pi }{\mathrm{Cl}}_2(2\pi z).$

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

$2\pi \log \left(\frac{G(1-z)}{G(1+z)}\right)=2\pi z\log \left(\frac{\sin \pi z}{\pi }\right)+{\mathrm{Cl}}_2(2\pi z).◻$

Using the relation $G(1+z)=\mathrm{\Gamma }(z)G(z)$ and dividing the reflection formula by a factor of $2\pi$ gives the equivalent form:

$\log \left(\frac{G(1-z)}{G(z)}\right)=z\log \left(\frac{\sin \pi z}{\pi }\right)+\log \mathrm{\Gamma }(z)+\frac{1}{2\pi }{\mathrm{Cl}}_2(2\pi z)$

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

Replacing z with Template:Sfrac − z in the previous reflection formula gives, after some simplification, the equivalent formula shown below (involving Bernoulli polynomials):

$\log \left(\frac{G(\frac{1}{2}+z)}{G(\frac{1}{2}-z)}\right)=\log \mathrm{\Gamma }(\frac{1}{2}-z)+B_1(z)\log 2\pi +\frac{1}{2}\log 2+\pi \underset{0}{\overset{z}{\int }}{B}_1(x)\tan \pi xdx$

Taylor series expansion

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

$\log G(1+z)=\frac{z}{2}\log 2\pi -\left(\frac{z+(1+\gamma )z^2}{2}\right)+\sum _{k=2}^{\mathrm{\infty }}(-1{)}^k\frac{\zeta (k)}{k+1}{z}^{k+1}.$

It is valid for $0<z<1$. Here, $\zeta (x)$ is the Riemann zeta function:

$\zeta (s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}.$

Exponentiating both sides of the Taylor expansion gives:

$\begin{array}{cc}G(1+z)& =\exp [\frac{z}{2}\log 2\pi -\left(\frac{z+(1+\gamma )z^2}{2}\right)+\sum _{k=2}^{\mathrm{\infty }}(-1{)}^k\frac{\zeta (k)}{k+1}{z}^{k+1}]\\ & =(2\pi {)}^{z/2}\exp [-\frac{z+(1+\gamma )z^2}{2}]\exp [\sum _{k=2}^{\mathrm{\infty }}(-1{)}^k\frac{\zeta (k)}{k+1}{z}^{k+1}].\end{array}$

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

$\exp [\sum _{k=2}^{\mathrm{\infty }}(-1{)}^k\frac{\zeta (k)}{k+1}{z}^{k+1}]=\prod _{k=1}^{\mathrm{\infty }}\{{(1+\frac{z}{k})}^k\exp (\frac{z^2}{2k}-z)\}$

Multiplication formula

Like the gamma function, the G-function also has a multiplication formula:

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

where $K(n)$ is a constant given by:

$K(n)=e^{-(n^2-1){\zeta }^{\prime }(-1)}\cdot n^{\frac{5}{12}}\cdot (2\pi {)}^{(n-1)/2}=(A{e}^{-\frac{1}{12}}{)}^{n^2-1}\cdot n^{\frac{5}{12}}\cdot (2\pi {)}^{(n-1)/2}.$

Here ${\zeta }^{\prime }$ is the derivative of the Riemann zeta function and $A$ is the Glaisher–Kinkelin constant.

Absolute value

It holds true that $G(\bar{z})=\overline{G(z)}$, thus $|G(z){|}^2=G(z)G(\bar{z})$. From this relation and by the above presented Weierstrass product form one can show that

$|G(x+iy)|=|G(x)|\exp \left(y^2\frac{1+\gamma }{2}\right)\sqrt{1+\frac{y^2}{x^2}}\sqrt{\prod _{k=1}^{\mathrm{\infty }}{(1+\frac{y^2}{(x+k{)}^2})}^{k+1}\exp (-\frac{y^2}{k})}.$

This relation is valid for arbitrary $x\in ℝ\setminus \{0,-1,-2,\dots \}$, and $y\in ℝ$. If $x=0$, then the below formula is valid instead:

$|G(iy)|=y\exp \left(y^2\frac{1+\gamma }{2}\right)\sqrt{\prod _{k=1}^{\mathrm{\infty }}{(1+\frac{y^2}{k^2})}^{k+1}\exp (-\frac{y^2}{k})}$

for arbitrary real y.

Asymptotic expansion

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

$\begin{array}{cc}\log G(z+1)=& \frac{z^2}{2}\log z-\frac{3z^2}{4}+\frac{z}{2}\log 2\pi -\frac{1}{12}\log z\\ & +(\frac{1}{12}-\log A)+\sum _{k=1}^N\frac{B_{2k+2}}{4k(k+1)z^{2k}}+O\left(\frac{1}{z^{2N+2}}\right).\end{array}$

Here the $B_k$ are the Bernoulli numbers and $A$ is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes 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.

Relation to the log-gamma integral

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

$\underset{0}{\overset{z}{\int }}\log \mathrm{\Gamma }(x)dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi +z\log \mathrm{\Gamma }(z)-\log G(1+z)$

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

$z\log \mathrm{\Gamma }(z)-\log G(1+z)$

where

$\frac{1}{\mathrm{\Gamma }(z)}=z{e}^{\gamma z}\prod _{k=1}^{\mathrm{\infty }}\left\{(1+\frac{z}{k})e^{-z/k}\right\}$

and $\gamma$ is the Euler–Mascheroni constant.

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

$\begin{array}{cc}& z\log \mathrm{\Gamma }(z)-\log G(1+z)=-z\log \left(\frac{1}{\mathrm{\Gamma }(z)}\right)-\log G(1+z)\\ =& -z[\log z+\gamma z+\sum _{k=1}^{\mathrm{\infty }}\{\log (1+\frac{z}{k})-\frac{z}{k}\left\}\right]\\ & -[\frac{z}{2}\log 2\pi -\frac{z}{2}-\frac{z^2}{2}-\frac{z^2\gamma }{2}+\sum _{k=1}^{\mathrm{\infty }}\{k\log (1+\frac{z}{k})+\frac{z^2}{2k}-z\left\}\right]\end{array}$

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

$\begin{array}{cc}& \sum _{k=1}^{\mathrm{\infty }}\{(k+z)\log (1+\frac{z}{k})-\frac{z^2}{2k}-z\}\\ =& -z\log z-\frac{z}{2}\log 2\pi +\frac{z}{2}+\frac{z^2}{2}-\frac{z^2\gamma }{2}-z\log \mathrm{\Gamma }(z)+\log G(1+z)\end{array}$

Finally, take the logarithm of the Weierstrass product form of the gamma function, and integrate over the interval $[0,z]$ to obtain:

$\begin{array}{cc}& \underset{0}{\overset{z}{\int }}\log \mathrm{\Gamma }(x)dx=-\underset{0}{\overset{z}{\int }}\log \left(\frac{1}{\mathrm{\Gamma }(x)}\right)dx\\ =& -(z\log z-z)-\frac{z^2\gamma }{2}-\sum _{k=1}^{\mathrm{\infty }}\{(k+z)\log (1+\frac{z}{k})-\frac{z^2}{2k}-z\}\end{array}$

Equating the two evaluations completes the proof:

$\underset{0}{\overset{z}{\int }}\log \mathrm{\Gamma }(x)dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi +z\log \mathrm{\Gamma }(z)-\log G(1+z)$

And since $G(1+z)=\mathrm{\Gamma }(z)G(z)$ then,

$\underset{0}{\overset{z}{\int }}\log \mathrm{\Gamma }(x)dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi -(1-z)\log \mathrm{\Gamma }(z)-\log G(z).$

References

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