Barnes G-function

The Barnes G-function, denoted G(z), is an entire function on the complex plane that generalizes the Euler Gamma function \Gamma(z) as the second-order case of the multiple Gamma functions, often referred to as the double Gamma function. It satisfies the functional equation G(z+1) = \Gamma(z) G(z) for \Re(z) > 0, with the normalization G(1) = 1, and is uniquely determined by this relation along with its Weierstrass canonical product form:
G(z+1) = (2\pi)^{z/2} \, e^{-(1+\gamma)z^2/2 - z/2} \prod_{k=1}^\infty \left(1 + \frac{z}{k}\right)^k e^{-z + z^2/(2k)},
where \gamma \approx 0.57721 is the Euler-Mascheroni constant.^[1]^[2] This function provides an analytic continuation of the product of factorials, defined for positive integers n as G(n+1) = \prod_{k=1}^{n-1} k!, a close relative of the superfactorial \sf(n) = \prod_{k=1}^n k!, to the complex domain, and it arises naturally in the evaluation of infinite products and determinants in number theory.^[1] Introduced by the Australian mathematician Ernest William Barnes in his 1899 paper "The theory of the G-function" in the Quarterly Journal of Mathematics, the Barnes G-function built upon earlier work by Glaisher and Kinkelin on multiple factorials and was further developed in Barnes's 1904 paper "On the theory of the multiple Gamma function" in the Transactions of the Cambridge Philosophical Society.^[2] Its asymptotic behavior is captured by a Stirling-type approximation:
\log G(x+1) \sim \frac{x^2}{2} \log x - \frac{3x^2}{4} + \frac{x}{2} \log(2\pi) + \zeta'(-1) + O\left(\frac{\log x}{x}\right),
which involves the Riemann zeta function derivative at -1 and connects to the Glaisher-Kinkelin constant A = e^{1/12 - \zeta'(-1)} \approx 1.282427, a fundamental constant in analytic number theory.^[1]^[2] The function has zeros at the non-positive integers and its logarithmic derivative is related to the digamma function and the Hurwitz zeta function.^[1] Notable applications include the computation of Barnes multiple zeta functions, regularized determinants of Laplacians on spheres, and evaluations in statistical mechanics, such as the partition function of the Potts model.^[1]^[2] Further generalizations to higher-order multiple Gamma functions G_n(z) extend its utility in multivariate analysis and special function theory.^[2]

Definition and Basic Properties

Functional Equation and Normalization

The Barnes G-function G(z) is defined as an entire function of the complex variable z satisfying the functional equation

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

where \Gamma(z) denotes the Euler gamma function.^[3] This equation, introduced by Barnes in his study of multiple gamma functions, generalizes the defining recurrence of the gamma function itself, which obeys \Gamma(z+1) = z \Gamma(z) with the normalization \Gamma(1) = 1. To ensure uniqueness among entire functions satisfying the recurrence, the Barnes G-function is normalized by the condition G(1) = 1.^[3] For positive integers, the functional equation yields an iterative relation G(n+1) = \Gamma(n) G(n). Starting from G(1) = 1 and using \Gamma(n) = (n-1)! for positive integers n, repeated application produces the finite product form

G(n+1) = \prod_{k=1}^{n-1} (k!),

which aligns with the superfactorial interpretation of the G-function at integer arguments.^[3]^[1]

Values at Integers and Special Points

The Barnes G-function G(z) has zeros at all non-positive integers, that is, G(n) = 0 for every integer n \leq 0.^[4] This property arises from the functional equation G(z+1) = \Gamma(z) G(z) combined with the poles of the gamma function \Gamma(z) at non-positive integers and the normalization G(1) = 1.^[4] For positive integers, the values are given explicitly by G(1) = 1 and, for n \geq 2,

G(n) = \prod_{k=1}^{n-2} k!.

^[4] This finite product of factorials provides a direct computational method for integer arguments. Representative small values include:

G(2) = 1 (empty product),
G(3) = 1! = 1,
G(4) = 1! \cdot 2! = 2,
G(5) = 1! \cdot 2! \cdot 3! = 12,
G(6) = 1! \cdot 2! \cdot 3! \cdot 4! = 288.

These integer values connect the Barnes G-function to the superfactorial \sf(n), defined as the product of the first n factorials \sf(n) = \prod_{k=1}^n k!.^[4] The relation is \sf(n) = n! \, G(n+1) / G(2); since G(2) = 1, this simplifies to \sf(n) = n! \, G(n+1).^[4] At the special half-integer point z = 1/2, the value is

G\left( \frac{1}{2} \right) = A^{-3/2} \, \pi^{-1/4} \, e^{1/8} \, 2^{1/24},

where A is the Glaisher–Kinkelin constant, defined as A = \exp\left( \frac{1}{12} - \zeta'(-1) \right) \approx 1.2824271291 with \zeta(s) the Riemann zeta function; this constant emerges in asymptotic analyses involving products of factorials and multiple gamma functions.^[4]^[5]

Representations

Infinite Product Form

The Barnes G-function possesses a Weierstrass canonical product representation, which serves as its explicit form and enables analytic continuation throughout the complex plane. This representation is given by

G(1 + z) = (2\pi)^{z/2} \exp\left[-\frac{1}{2}\left(z + (1 + \gamma)z^{2}\right)\right] \prod_{k=1}^{\infty} \left(1 + \frac{z}{k}\right)^{k} \exp\left(-z + \frac{z^{2}}{2k}\right),

where \gamma denotes the Euler-Mascheroni constant.^[6] The infinite product converges uniformly on compact subsets of the complex plane, confirming that G(z) is an entire function of order 2.^[7] This product form arises from the functional equation G(z+1) = \Gamma(z) G(z) through techniques such as Hadamard canonical factorization or regularization via multiple Hurwitz zeta functions, which handle the divergent aspects of the naive product over gamma functions.^[8] For positive integers n, the values of the G-function connect to the hyperfactorial H(n) = \prod_{k=1}^{n} k^{k} via the exact relation G(n+1) = \frac{(n!)^n}{H(n)}.^[9]

Logarithmic Representations

The logarithmic representations of the Barnes G-function provide expressions for \ln G(z), which are essential for numerical computations, asymptotic approximations, and analytic continuations in the complex plane, as the function itself grows rapidly. A key logarithmic form arises from the infinite product representation of G(z+1), yielding

\ln G(z+1) = \frac{z}{2} \ln (2\pi) - \frac{1}{2} z (z+1) - \frac{\gamma}{2} z^2 + \sum_{k=1}^\infty \left[ k \ln \left(1 + \frac{z}{k}\right) - z + \frac{z^2}{2 k} \right],

where \gamma is the Euler-Mascheroni constant. This series converges for complex z avoiding the branch cut along the negative real axis and is regularized by the \gamma z^2 / 2 term to handle divergent contributions from the harmonic series in the expansion. For practical evaluation, the sum is truncated at a large k where terms become negligible, making it suitable for moderate-sized |z|. An alternative integral representation connects \ln G(z+1) directly to the logarithm of the gamma function:

\ln G(z+1) = z \ln \Gamma(z+1) - \int_0^z \ln \Gamma(t+1) \, dt + \frac{1}{2} z \ln (2\pi) - \frac{1}{2} z (z+1).

Here, the principal branches of the logarithms are taken along the positive real axis and analytically continued elsewhere; the integral can be computed numerically using quadrature methods, particularly when z is real and positive. This form highlights the cumulative nature of the G-function as a "double gamma" and aids in deriving further properties.^[3] The logarithmic derivative, \frac{d}{dz} \ln G(z) = \frac{G'(z)}{G(z)}, admits a closed form involving the digamma function \psi(z):

\frac{G'(z)}{G(z)} = \frac{1}{2} \ln (2\pi) + \frac{1}{2} - z + (z-1) \psi(z).

This relation follows from differentiating the functional equation G(z+1) = \Gamma(z) G(z) and integrating the resulting recurrence for the derivative, providing an efficient way to compute derivatives without evaluating G(z) separately. It also satisfies the recurrence \frac{G'(z+1)}{G(z+1)} = \frac{G'(z)}{G(z)} + \psi(z).^[10] These logarithmic forms enable accurate computation of \ln G(z) for complex z. For \operatorname{Re}(z) \geq 3/2, high-precision approximations use Padé approximants to auxiliary functions in the integral representation, achieving relative errors below $10^{-30}; extension to the full complex plane employs the reflection formula to map points outside this region. Such methods are implemented in numerical libraries for applications in special function evaluations and random matrix theory.^[6]

Identities

Reflection Formula

The reflection formula for the Barnes G-function relates its values at complementary points in a manner analogous to the reflection formula for the gamma function. It is given by

\log G(1 - z) = \log G(z + 1) - z \log (2\pi) + \int_{0}^{z} \pi x \cot(\pi x) \, dx,

where the integral represents the log-tangent integral and the equation holds in the complex plane away from the branch cut along the negative real axis.^[4] This identity is derived from the functional equation G(z+1) = \Gamma(z) G(z) combined with the reflection formula for the digamma function, \psi(1 - z) = \psi(z) + \pi \cot(\pi z). The logarithmic derivative of the G-function admits an expression as G'(z)/G(z) = \sum_{k=0}^{\infty} [\psi(z + k + 1) - \psi(k + 1)], allowing the reflection property of the digamma terms to be applied term by term to obtain the integral form after integration.^[4] The formula facilitates the evaluation of G(z) at non-integer points, particularly those with negative or small real part, by reflecting them to the right half-plane where the infinite product representation converges more rapidly or asymptotic approximations are applicable, often in conjunction with the functional equation to shift arguments iteratively.^[6] In the special case z = 1/2, the reflection formula simplifies and yields G(1/2) = 2^{1/24} e^{1/8} A^{-3/2} \pi^{-1/4}, where A is the Glaisher–Kinkelin constant defined by the limit

A = \lim_{n \to \infty} \frac{(2\pi)^{n/2} n^{n^2/2 + n/2 + 1/12} e^{-3n^2/4}}{G(n+1)},

linking the value to this fundamental constant in number theory and special functions.^[11]

Multiplication Formula

The multiplication theorem for the Barnes G-function generalizes the Gauss multiplication formula for the Gamma function to this higher-order analog, providing a relation for G at multiples of the argument in terms of products of G at shifted arguments. For a positive integer n, the formula follows from repeated application of the functional equation G(z+1) = \Gamma(z) G(z) combined with the Gauss multiplication theorem for the Gamma function \Gamma(n z) = (2\pi)^{(n-1)/2} n^{n z - 1/2} \Gamma(z) \prod_{k=1}^{n-1} \Gamma\left(z + \frac{k}{n}\right). This case is particularly useful for computations and can be derived directly via the Hurwitz zeta function representation of the logarithmic derivative.^[12] The multiplication formula facilitates relating values of the G-function at rational multiples of the argument, enabling evaluations at fractional points from known integer values and aiding in the analytic continuation across the complex plane. For instance, it allows expressing G at points like z/3 or z/4 in terms of products involving G(z) and shifted terms, which is valuable in number theory applications such as Barnes multiple zeta functions.

Expansions

Taylor Series Expansion

The Taylor series expansion for the logarithm of the Barnes G-function provides a useful representation for computations near the origin. Specifically, the power series for \ln G(z+1) around z = 0 is given by

\ln G(z+1) = \frac{1}{2} \left[ \ln(2\pi) - 1 \right] z - \frac{1 + \gamma}{2} z^2 + \sum_{n=3}^{\infty} (-1)^{n-1} \frac{\zeta(n-1)}{n} z^n,

where \gamma is the Euler-Mascheroni constant and \zeta(s) is the Riemann zeta function.^[9] This expansion arises from the infinite product representation of the G-function through regularization of divergent sums involving the zeta function. The coefficients in the series are expressed in terms of values of the Riemann zeta function at positive integers: for instance, \zeta(2) = \pi^2/6, \zeta(3) \approx 1.20206, and higher even values \zeta(2k) = (-1)^{k+1} B_{2k} (2\pi)^{2k} / (2 (2k)!), where B_{2k} are Bernoulli numbers. The linear term \frac{1}{2} [\ln(2\pi) - 1] z incorporates the normalization from the product form, while the quadratic term separates the regularized contribution from \zeta(1), effectively capturing the Euler-Mascheroni constant \gamma \approx 0.57721. Expanding to low orders yields explicit terms up to z^3:

\ln G(z+1) = \frac{1}{2} \left[ \ln(2\pi) - 1 \right] z - \frac{1 + \gamma}{2} z^2 + \frac{\pi^2}{18} z^3 + O(z^4).

The radius of convergence of this power series is |z| < 1, limited by the nearest pole of \ln G(z+1) at z = -1. Beyond this disk, analytic continuation can be achieved using the functional equation G(z+1) = \Gamma(z) G(z) or other representations, allowing evaluation in larger regions of the complex plane.^[9] This series facilitates numerical approximations and theoretical analysis near z = 0, with the zeta coefficients enabling connections to number-theoretic properties.

Asymptotic Expansion

The asymptotic expansion of the Barnes G-function provides a Stirling-like approximation for its logarithm as |z| \to \infty in the complex plane, enabling accurate evaluation and analysis for large arguments. This expansion incorporates the asymptotic behavior of the gamma function while adding higher-order terms specific to the multiple gamma structure of the G-function. The series is divergent but yields optimal approximations when truncated appropriately. The precise form of the expansion is given by

\ln G(z+1) \sim \frac{z^2}{4} + z \ln \Gamma(z+1) - \left( \frac{z(z+1)}{2} + \frac{1}{12} \right) \ln z - \ln A + \sum_{k=1}^\infty \frac{B_{2k+2}}{2k(2k+1)(2k+2) z^{2k}},

valid for |\mathrm{ph}\, z| < \pi - \delta with \delta > 0. Here, B_{2k+2} denote the Bernoulli numbers, which appear in the infinite sum as coefficients for the even-powered inverse terms in z, mirroring their role in the classical Stirling series but extended to capture the "double" nature of the approximation. This expansion holds particularly in the right half-plane \mathrm{Re}(z) > 0, where the principal branch of the logarithms is well-defined, and the remainder after N terms satisfies error bounds of the form O(|z|^{-2N}) for fixed N as |z| \to \infty. More refined estimates, including exponentially improved asymptotics with error terms exponentially small along certain rays (such as the positive real axis), have been established to enhance precision in numerical applications. The structure of the series connects directly to Stirling's asymptotic for \ln \Gamma(z+1), augmented by quadratic and logarithmic adjustments that reflect the iterative functional relation of the G-function, effectively providing a double-layered approximation.^[3]^[13]

Other Properties

Absolute Value

As |t| → ∞, the modulus |G(1 + it)| exhibits exponential decay with leading behavior involving terms from the asymptotic expansion of log G(z + 1) specialized to the line Re(z) = 1. This decay reflects the cumulative structure from the functional equation, being slower than that of the Gamma function. Two-sided inequalities for |G(z)| in the complex plane can be derived using the reflection formula G(z) G(1 − z) = exp[(1/2 − z) log(2π) + (z − 1/2)(γ + log(2π)) − ζ′(−1)] / [√(2π) Γ(z)], which relates the modulus in the left and right half-planes. For regions with Re(z) > 0, extensions of Batir's inequalities for the positive real line provide bounds in terms of the Gamma and digamma functions. These inequalities facilitate growth estimates, showing sub-Gaussian decay in the imaginary direction while exponential growth occurs for large positive Re(z).^[14]

Zeros and Analytic Continuation

The Barnes G-function G(z) is an entire function of order 2, meaning it is holomorphic everywhere in the complex plane and its growth is controlled by \exp(|z|^\rho) for any \rho > 2 but not for \rho < 2.^[9] Its zeros are located precisely at the non-positive integers z = 0, -1, -2, \dots, with no zeros elsewhere in the complex plane; these locations arise from the factors in its infinite product representation. At z = -n for n = 0, 1, 2, \dots, the multiplicity of the zero is n+1. Unlike the gamma function, which has simple poles at the non-positive integers, the Barnes G-function has no poles anywhere in the finite complex plane. This pole-free nature stems directly from its construction via the infinite product, which introduces zeros that precisely cancel the poles encountered when iteratively applying the functional equation G(z+1) = \Gamma(z) G(z) across the plane.^[3] The Barnes G-function is initially defined for positive real arguments via its relation to superfactorials or a finite product, but analytic continuation to the entire complex plane is accomplished using either the infinite Weierstrass-type product representation,

G(z+1) = (2\pi)^{z/2} \exp\left( -\frac{1}{2} z(z+1) - \frac{\gamma z^2}{2} \right) \prod_{k=1}^\infty \left[ \left(1 + \frac{z}{k}\right)^k \exp\left( -z + \frac{z^2}{2k} \right) \right],

which converges uniformly on compact sets and defines the function holomorphically everywhere, or by stepping across the plane using the functional equation starting from values on the positive real axis. The functional equation itself facilitates continuation by relating values in adjacent strips, ensuring no singularities are introduced beyond the inherent zeros. As |z| \to \infty with |\arg z| \leq \pi - \delta for fixed \delta > 0, the growth of the G-function is given asymptotically by

|G(z)| \sim \exp\left( \frac{|z|^2}{2} (\ln |z| - 1) \right),

which follows from the leading terms in its full asymptotic expansion for \ln G(z+1),

\ln G(z+1) \sim \frac{1}{4}z^2 + z \ln \Gamma(z+1) - \left( \frac{1}{2}z(z+1) + \frac{1}{12} \right) \ln z - \ln A + \sum_{k=1}^\infty \frac{B_{2k+2}}{(2k)(2k+1)(2k+2) z^{2k}},

where A is the Glaisher-Kinkelin constant and B_m are Bernoulli numbers; substituting the Stirling approximation for \ln \Gamma(z+1) yields the dominant behavior. This exponential growth of order 2 underscores the function's classification and distinguishes it from the gamma function, which grows like \exp(|z| (\ln |z| - 1)).^[9]

Relations to Other Functions

Relation to the Gamma Function

The Barnes G-function satisfies the functional equation G(z+1) = \Gamma(z) \, G(z), with the normalization condition G(1) = 1. This recurrence relation directly links the G-function to the Gamma function, allowing iterative computation and highlighting its role as a generalization.^[3] By iteratively applying this relation, the G-function can be expressed for positive integers n \geq 2 as G(n) = \prod_{k=1}^{n-2} k!, a finite product of factorials equivalent to shifted Gamma function values at positive integers. This structure positions the G-function as a "higher-order" analog of the Gamma function, extending its multiplicative properties to a broader class of special functions.^[3]^[15] Unlike the Gamma function, which exhibits simple poles at the non-positive integers z = 0, -1, -2, \dots, the G-function is an entire function of order two with simple zeros precisely at these points, avoiding poles through the accumulation of zeros that balance the recurrence.^[16]^[3] The G-function was introduced by Ernest William Barnes in 1904 as a key component of his theory of multiple gamma functions, generalizing the classical Gamma function to higher dimensions via successive applications of such recurrences.^[17]

Relation to the Log-Gamma Integral

The Barnes G-function admits an integral characterization in terms of the logarithm of the gamma function, providing an alternative definition to its product form. Specifically,

\ln G(z+1) = z \ln \Gamma(z+1) - \int_0^z \ln \Gamma(t+1) \, dt + \frac{1}{2} z \ln (2\pi) - \frac{1}{2} z (z+1),

where the logarithms take their principal values on the positive real axis and are analytically continued via continuity.^[3] This expression holds for complex z in suitable domains, reflecting the entire nature of G(z). This relation derives from the functional equation G(z+1) = \Gamma(z) G(z), which upon taking logarithms yields \ln G(z+1) - \ln G(z) = \ln \Gamma(z). Integrating both sides from 1 to z+1 and incorporating the digamma function \psi(w) = \frac{d}{dw} \ln \Gamma(w), whose integral representation is \psi(z+1) = \int_0^\infty \left( \frac{e^{-t}}{t} - \frac{e^{-(z+1)t}}{1 - e^{-t}} \right) dt, leads to the log-gamma integral form after accounting for boundary terms and normalization G(1) = 1. The focus remains on the cumulative effect of the log-gamma, avoiding direct use of the digamma integral for the derivation. Combined with the normalization condition G(1) = 1, this integral representation uniquely determines the Barnes G-function as an entire function of order 2, equivalently to its Weierstrass product form \frac{1}{G(z+1)} = e^{A z^2 + B z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^n e^{-z + z^2/(2n)}, where A and B are constants related to the Riemann zeta function.^[3] Numerically, this form facilitates computation of G(z) in vertical strips of the complex plane, such as |\operatorname{Re} z - k| < 1/2 for integer k, by leveraging efficient algorithms for \ln \Gamma(w) and quadrature for the integral, which converges rapidly due to the smoothness of \ln \Gamma(t+1).[https://www.sciencedirect.com/science/article/pii/S0010465503004983) This approach is particularly advantageous for high-precision evaluations where the product form may suffer from overflow or underflow.

Relation to the Glaisher-Kinkelin Constant

The Glaisher-Kinkelin constant A, approximately 1.2824271291, is defined as A = e^{1/12 - \zeta'(-1)}, where \zeta'(s) denotes the derivative of the Riemann zeta function evaluated at s = -1. This constant arises in asymptotic analyses of products involving factorials and powers, such as the hyperfactorial function H(n) = \prod_{k=1}^n k^k. Introduced by J. W. L. Glaisher in 1894 for the asymptotic behavior of the hyperfactorial and further developed by H. Kinkelin in related contexts, the constant was later connected to the Barnes G-function through E. W. Barnes' work on multiple gamma functions in 1904. Barnes demonstrated that the G-function provides an analytic continuation and generalization that incorporates A naturally in its properties. The Glaisher-Kinkelin constant appears explicitly in the asymptotic expansion of \ln G(z+1) for large |z| with |\arg z| < \pi, where the constant term includes -\ln A:

\ln G(z+1) = \frac{z^2}{2} \log(2\pi) - \ln A - \frac{3z^2}{4} + \left( \frac{z^2}{2} - \frac{1}{12} \right) \log z + O\left( \frac{1}{z} \right).

^[18] This Stirling-like series, derived by Barnes, highlights A's role in the leading behavior of the G-function, analogous to how the Euler-Mascheroni constant appears in the gamma function's expansion. For integer arguments, the relation manifests in the limit

\lim_{n \to \infty} \frac{G(n+1) \, n^{n^2/2 + 1/12} \, (2\pi)^{n/2} \, e^{-3n^2/4}}{A} = e^{1/12},

which connects the growth of G(n+1) = \prod_{k=1}^n k! to A. This expression underscores the constant's significance in evaluating the G-function at positive integers via Barnes' integral representation and product formula.

Extensions to Multiple G-Functions

The multiple G-function provides a higher-dimensional generalization of the Barnes G-function, extending its properties to multi-variable settings in connection with the Barnes multiple gamma function. Defined for z ∈ ℝ^n, the function G_n(z) satisfies the functional equation

G_n(\mathbf{z} + \mathbf{e}_j) = \Gamma_n(\mathbf{z}) G_n(\mathbf{z})

for each standard basis vector \mathbf{e}_j (j = 1, \dots, n), where \Gamma_n(\mathbf{z}) denotes the Barnes multiple gamma function of n variables, with the normalization condition G_n(\mathbf{1}) = 1, where \mathbf{1} is the n-dimensional vector of ones.^[19] For n=1, this reduces to the standard Barnes G-function, satisfying G_1(z + 1) = \Gamma_1(z) G_1(z) with G_1(1) = 1, where \Gamma_1(z) = \Gamma(z) is the Euler gamma function.^[3] The multi-variable functional equation generalizes this by incorporating shifts along each coordinate axis, enabling product forms over successive applications of the basis shifts.^[19] Barnes introduced the underlying multiple gamma function \Gamma_n(\mathbf{z}) in 1904 as part of his development of higher-order generalizations of the gamma function, motivated by analytic continuations and zeta function theory; the multiple G-function emerges as the accompanying "exponential" component fulfilling the shift relations.^[20] These functions find applications in the theory of Barnes multiple zeta functions, which regularize infinite products and series to define \Gamma_n(\mathbf{z}) via derivatives at s=0, and in contour integral representations that express properties of \Gamma_n(\mathbf{z}), such as reflection formulas and asymptotic behaviors in multiple variables.^[2]