Barnes Double Gamma Function Research Articles

On the Barnes double gamma function

We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function . Second, we derive an algorithm for numerically computing the double gamma function and present its complete asymptotic expansion as . Third, we derive some new properties, including a new product identity and new representations of the gamma modular forms and .

A Quantized Riemann–Hilbert Problem in Donaldson–Thomas Theory

Abstract We introduce Riemann–Hilbert problems determined by the refined Donaldson–Thomas theory. They involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a quantum torus algebra. We study the simplest case in detail and use the Barnes double gamma function to construct a solution.

On the Colmez conjecture for non-abelian CM fields

The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field E to logarithmic derivatives of Artin L-functions at $$s=0$$ . In this paper, we prove that if F is any fixed totally real number field of degree $$[F:\mathbb {Q}] \ge 3$$ , then there are infinitely many effective, “positive density” sets of CM extensions E / F such that $$E/\mathbb {Q}$$ is non-abelian and the Colmez conjecture is true for E. Moreover, these CM extensions are explicitly constructed to be ramified at arbitrary prescribed sets of prime ideals of F. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be viewed as an explicit non-abelian Chowla–Selberg formula. Our results rely crucially on an averaged version of the Colmez conjecture which was recently proved independently by Andreatta–Goren–Howard–Madapusi Pera and Yuan–Zhang.

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR β-ENSEMBLES

In a previous work [J. Math. Phys.35 (1994) 2539–2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large s asymptotic form of the general β > 0 gap probability [Formula: see text], provided both βa/2 ∈ ℤ≥0 and 2/β ∈ ℤ+. It is shown how the details of this method can be extended to remove the requirement that 2/β ∈ ℤ+. Furthermore, a large deviation formula for the gap probability Eβ(n; (0, x); ME β, N(λaβ/2e-2βNλ)) is deduced by writing it in terms of the characteristic function of a certain linear statistic. By scaling x = s/(4N)2 and taking N → ∞, this is shown to reproduce a recent conjectured formula for [Formula: see text], βa/2 ∈ ℤ≥0, and moreover to give a prediction without the latter restriction. This extended formula, which for the constant term involves the Barnes double gamma function, is shown to satisfy an asymptotic functional equation relating the gap probability with parameters (β, n, a), to a gap probability with parameters (4/β, n′, a′), where n′ = β(n + 1)/2 - 1, a′ = β(a - 2)/2 + 2.

Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function

We introduce completely monotonic functions of order r > 0 and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.

The Remainder in Ruijsenaars’ Asymptotic Expansion of Barnes Double Gamma Function

We show that the remainder in Ruijsenaars’ asymptotic expansion of the logarithm of Barnes double gamma function gives rise to a completely monotone function. Fourier expansions of the multiple Bernoulli polynomials are also obtained.

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.

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.

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.

Uniform asymptotic expansions for the Barnes double gamma function

We develop uniform asymptotic approximations for the Barnes double gamma function for various parameter values. This involves the use of the method of matched asymptotic expansions to solve the defining functional difference equation in various asymptotic regions. The resulting solutions provide extremely accurate approximations to the double gamma function. We also give applications of our results to functional difference equations that arise in the theory of moving contact lines and acoustic wave diffraction.

Exact solution to a class of functional difference equations with application to a moving contact line flow

A new integral representation for the Barnes double gamma function is derived. This is canonical in the sense that solutions to a class of functional difference equations of first order with trigonometrical coefficients can be expressed in terms of the Barnes function. The integral representation given here makes these solutions very simple to compute. Several well-known difference equations are solved by this method, and a treatment of the linear theory for moving contact line flow in an inviscid fluid wedge is given.

Asymptotics of a τ-function arising in the two-dimensional Ising model

The short-distance assymptotics of the τ-function associated to the 2-point function of the two-dimensional Ising model is computed as a function of the integration constant defined from the long-distance behavior of the τ-function. The result is expressible in terms of the Barnes double gamma function (equivalently, the BarnesG-function).

Determinants of Laplacians

The determinant of the Laplacian on spinor fields on a Riemann surface is evaluated in terms of the value of the Selberg zeta function at the middle of the critical strip. A key role in deriving this relation is played by the Barnes double gamma function.