Multiple Gamma Functions Research Articles

The modular properties and the integral representations of the multiple elliptic gamma functions

We show the modular properties of the multiple “elliptic” gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's transformation, and that of the elliptic gamma function was provided by Felder and Varchenko. In this paper, we deal with the multiple sine functions, since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite products. We also derive integral representations of the multiple sine functions and the multiple elliptic gamma functions. We introduce correspondences between the multiple elliptic gamma functions and the multiple sine functions.

Absolute tensor products

We calculate the absolute tensor product ζ(s,Fp1)⊗···⊗ζ(s,Fpr) by observing a relation between the multiple sine function of Shintani's type and Appell's O-function, or equivalently, the multiple elliptic gamma function. This allows us to give an Euler product expression for an absolute tensor product.

Series involving the Zeta function and multiple Gamma functions

The theory of multiple Gamma functions, which was recently revived in the study of the determinants of the Laplacians, was applied in several earlier works in order to evaluate some families of series involving the Riemann Zeta function as well as to compute the determinants of the Laplacians. Here, in the present paper, the authors address the converse problem and apply various (known or new) formulas for series associated with the Zeta and related functions with a view to developing the corresponding theory of multiple Gamma functions and then using these series to compute the determinants of the Laplacians on the n-dimensional unit sphere S n ( n=5,6,7) explicitly.

Shintani–Barnes zeta and gamma functions

We show that Shintani's work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. We re-prove many of Shintani's formulas and prove several new ones. Among the latter is a generalization to the Shintani–Barnes gamma functions of Raabe's 1843 formula ∫ 0 1 log Γ(x) dx= log 2π , and a further generalization to the Shintani zeta functions. These explicit formulas can be interpreted as “vanishing period integral” side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. We also relate Barnes’ triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems.

On q-Basic Multiple Gamma Functions

We define a q-analogue of the basic multiple gamma function [Formula: see text] introduced in [17] which differs from the one defined by Barnes [1] via the zeta regularized product. We call it a q-basic multiple gamma function. Using this q-basic multiple Gamma function we introduce a q-analogue of the multiple sine function of order m + 1. We study properties of such functions from the periodicity, differential-difference equations and duplication formulas, etc. points of view.

Derivatives of Dedekind sums and their reciprocity law

In this paper derivatives of Dedekind sums are defined, and their reciprocity laws are proved. They are obtained from values at non-positive integers of the first derivatives of Barnes’ double zeta functions. As special cases, they give finite product expressions of the Stirling modular form and the double gamma function at positive rational numbers.

The double gamma function and related Pick functions

Denoting by G the Barnes double gamma function we find the Stieltjes representation of the related Pick functionz↦logG(z+1)z2Logz.As a corollary we obtain that this function has a completely monotone derivative on the positive half line.

Multiple Gamma Function, Its $q$- and Elliptic Analogue

Vigneras's multiple gamma function is introduced as a function satisfying a generalization of the Bohr-Mollerup theorem. An infinite product representation and an asymptotic expansion of the function are given. Furthermore, its q- and elliptic analogue are introduced as relevant with the defining relations of q-gamma function and of elliptic gamma function.

Corrigendum and addendum to ‘asymptotic series for double zeta, double gamma and Hecke L-functions’

Refined expressions are given for the error terms in the asymptotic expansion formulas for double zeta and double gamma functions, proved in the author's former paper [2]. Some inaccurate claims in [2] are corrected.

A certain class of series associated with the zeta function

The authors apply the theory of the double Gamma function, which was recently revived in the study of the determinants of the Laplacians, in order to give closed-form evaluations of some series involving the Zeta function. Each of these series is evaluated at different arguments from those considered in several earlier works. The possibility of numerical computations of the sums of the series considered here is also indicated.

Multiple Gamma and related functions

The authors give several new (and potentially useful) relationships between the multiple Gamma functions and other mathematical functions and constants. As by-products of some of these relationships, a classical definite integral due to Euler and other definite integrals are also considered together with closed-form evaluations of some series involving the Riemann and Hurwitz Zeta functions.

On multivariatep-adicq-integrals

By using multiple p-adic q-integrals, we define the p-adic q-L-function in n-variables and the q-extension of p-adic log multiple gamma functions. From these definitions, we show that the values of the p-adic q-L-function at positive integers can be expressed in terms of the q-extension of p-adic log multiple gamma functions.

An elliptic analogue of the multiple gamma function

A hierarchy of functions including the elliptic gamma function is introduced. It can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q-analogue of the multiple gamma function. Some properties of the functions are considered.

An Asymptotic Expansion of the Double Gamma Function

The Barnes double gamma function G(z) is considered for large argument z. A new integral representation is obtained for log G(z). An asymptotic expansion in decreasing powers of z and uniformly valid for |Arg z|<π is derived from this integral. The expansion is accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is very accurate for real z. The accuracy of the error bound decreases for increasing Arg z.

Small-Scale and Mesoscale Variability of Scalars in Cloudy Boundary Layers: One-Dimensional Probability Density Functions

A key to parameterization of subgrid-scale processes is the probability density function (PDF) of conserved scalars. If the appropriate PDF is known, then grid box average cloud fraction, liquid water content, temperature, and autoconversion can be diagnosed. Despite the fundamental role of PDFs in parameterization, there have been few observational studies of conserved-scalar PDFs in clouds. The present work analyzes PDFs from boundary layers containing stratocumulus, cumulus, and cumulus-rising-into-stratocumulus clouds. Using observational aircraft data, the authors test eight different parameterizations of PDFs, including double delta function, gamma function, Gaussian, and double Gaussian shapes. The Gaussian parameterization, which depends on two parameters, fits most observed PDFs well but fails for large-scale PDFs of cumulus legs. In contrast, three-parameter parameterizations appear to be sufficiently general to model PDFs from a variety of cloudy boundary layers. If a numerical model ignores subgrid variability, the model has biases in diagnoses of grid box average liquid water content, temperature, and Kessler autoconversion, relative to the values it would obtain if subgrid variability were taken into account. The magnitude of such biases is assessed using observational data. The biases can be largely eliminated by three-parameter PDF parameterizations. Prior authors have suggested that boundary layer PDFs from short segments are approximately Gaussian. The present authors find that the hypothesis that PDFs of total specific water content are Gaussian can almost always be rejected for segments as small as 1 km.

Generalized Hölder's Theorem for Vignéras' Multiple Gamma Function

We prove that Vignéras' multiple gamma function does not satisfy any algebraic differential equation over $\mathbf{C}(z)$ by using a relation between logarithmic derivative of this function and the psi-function. Furthermore, we apply this relation to calculation of convergent factors in the Weierstrass product representation of Vignéras' multiple gamma function.

Instability of a vortex sheet leaving a right-angled corner

David Crighton had a long-standing interest in how the results of Orszag and Crow [Stud. Appl. Math. 49, 167–181 (1970)], for the instability of a vortex sheet leaving a zero-thickness trailing edge which separates a uniform stream from a fliud at rest, would be modified by creating a solid right-angled corner that restricts the stationary unperturbed flow to a quadrant. The Wiener–Hopf technique cannot be applied to this problem which, having wedge-shaped regions, is amenable to Mellin transforms. The matching conditions at the vortex sheet yield a pair of functional difference equations whose solution in terms of the Barnes double gamma function is achieved after doubling their order. Then, local series expansions valid near and far from the corner enable the appropriate pole structure to be identified and the arbitrary periodic functions to be determined. The results are applied to instability wave excitation by an acoustic wave incident on the trailing edge region.

Sums Involving the Hurwitz Zeta Function

We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers.

New series involving the zeta function

We evaluate sums of certain classes of new series involving the Riemann zeta function by using the theory of the double gamma function and a property of the gamma function. Relevant connections with various known results are also pointed out.

On Barnes' Multiple Zeta and Gamma Functions

We show how various known results concerning the Barnes multiple zeta and gamma functions can be obtained as specializations of simple features shared by a quite extensive class of functions. The pertinent functions involve Laplace transforms, and their asymptotics is obtained by exploiting this. We also demonstrate how Barnes' multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations. Both of these new approaches to the Barnes functions give rise to novel integral representations.