On the Barnes double zeta and Gamma functions

“…Bernoulli-Barnes numbers appear in a similar fashion in relation with the Barnes ζ -function [4] ζ n (z, x; a) := ∑ m∈Z n ≥0 1 (x + m 1 a 1 + · · · + m n a n ) z defined for Re(x) > 0 and Re(z) > n and continued meromorphically to C [9,10,11,16,19,21]. Specialized at negative integers −k, the Barnes ζ -function gives…”

Section: Corollary 2 For N ≥ 3 and Odd Mmentioning

confidence: 97%

Relations for Bernoulli–Barnes numbers and Barnes zeta functions

Bayad

Beck

2014

Int. J. Number Theory

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ABSTRACT. The Barnes ζ -function is ζ n (z, x; a) := ∑ m∈Z n ≥0 1 (x + m 1 a 1 + · · · + m n a n ) z defined for Re(x) > 0 and Re(z) > n and continued meromorphically to C. Specialized at negative integers −k, the Barnes ζ -function giveswhere B k (x; a) is a Bernoulli-Barnes polynomial, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing B k (0; a) gives the BernoulliBarnes numbers. We exhibit relations among Barnes ζ -functions, Bernoulli-Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.

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“…This relation can be solved using the double-gamma function, F (z) = Γ 2 (z) with pseudo-periods ε 1 and ε 2 (see [29,31] all the necessary normal-ordering relation can now be derived, they have been gathered in appendix B.2. Adapting the method described in the appendix C of [18], they can be used to show that the expressions given in 3.2 indeed solve the equations 3.1.…”

Section: Intertwinersmentioning

confidence: 99%

“…(3.6) 5 Note however, that the asymptotic expansion at z = ∞ of the function Γ 2 (z) (see, for instance, reference [29]),…”

Section: Instanton Partition Functions and Qq-charactersmentioning

confidence: 99%

A note on the algebraic engineering of 4D N=2 super Yang–Mills theories

Bourgine

Hong

2019

Physics Letters B

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Some BPS quantities of N = 1 5D quiver gauge theories, like instanton partition functions or qq-characters, can be constructed as algebraic objects of the Ding-Iohara-Miki (DIM) algebra. This construction is applied here to N = 2 super Yang-Mills theories in four dimensions using a degenerate version of the DIM algebra. We build up the equivalent of horizontal and vertical representations, the first one being defined using vertex operators acting on a free boson's Fock space, while the second one is essentially equivalent to the action of Vasserot-Shiffmann's Spherical Hecke central algebra. Using intertwiners, the algebraic equivalent of the topological vertex, we construct a set of T -operators acting on the tensor product of horizontal modules, and the vacuum expectation values of which reproduce the instanton partition functions of linear quivers. Analysing the action of the degenerate DIM algebra on the T -operator in the case of a pure U (m) gauge theory, we further identify the degenerate version of Kimura-Pestun's quiver W-algebra as a certain limit of q-Virasoro algebra. Remarkably, as previously noticed by Lukyanov, this particular limit reproduces the Zamolodchikov-Faddeev algebra of the sine-Gordon model. ✩ Preprint number: KIAS-Q18022.

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On the Barnes double zeta and Gamma functions

Abstract: We present a complete description of the analytic properties of the Barnes double zeta and Gamma functions.

Cited by 51 publications

References 26 publications

The asymptotics of ECH capacities

The asymptotics of ECH capacities

Relations for Bernoulli–Barnes numbers and Barnes zeta functions

A note on the algebraic engineering of 4D N=2 super Yang–Mills theories

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