Symmetries of Bernoulli polynomial series and Arakawa-Kaneko zeta functions
نویسنده
چکیده
The Arakawa-Kaneko zeta functions interpolate the poly-Bernoulli numbers at the negative integers and their values at positive integers are connected to multiple zeta values. We give everywhere-convergent series expansions of these functions in terms of Bernoulli polynomials and Dirichlet series related to harmonic numbers, exhibiting their explicit analytic continuation. Differentiating the Barnes multiple zeta functions of order r with respect to their order produces Dirichlet series attached to Bernoulli polynomials. These series are invariant under an involution in which the order of the derivative is dual to the value of the first variable and the order of the zeta function is dual to the value of the second variable. This symmetry relation generalizes duality relations of Euler sums, and is featured in our series expansions of Arakawa-Kaneko zeta values.
منابع مشابه
The Arakawa–kaneko Zeta Function and Poly-bernoulli Polynomials
The purpose of this paper is to introduce a generalization of the Arakawa–Kaneko zeta function and investigate their special values at negative integers. The special values are written as the sums of products of Bernoulli and poly-Bernoulli polynomials. We establish the basic properties for this zeta function and their special values.
متن کاملRational series for multiple zeta and log gamma functions
We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-...
متن کاملOn Functions of Arakawa and Kaneko and Multiple Zeta Functions
We study for s ∈ N the functions ξk(s) = 1 Γ(s) R ∞ 0 t et−1 Lik(1−e )dt, and more generally ξk1,...,kr (s) = 1 Γ(s) R ∞ 0 t et−1 Lik1,...,kr (1 − e )dt, introduced by Arakawa and Kaneko [2] and relate them with (finite) multiple zeta functions, partially answering a question of [2]. In particular, we give an alternative proof of a result of Ohno [8].
متن کاملA note on poly-Bernoulli numbers and multiple zeta values
We review several occurrences of poly-Bernoulli numbers in various contexts, and discuss in particular some aspects of relations of poly-Bernoulli numbers and special values of certain zeta functions, notably multiple zeta values.
متن کاملq-BERNOULLI NUMBERS AND POLYNOMIALS ASSOCIATED WITH MULTIPLE q-ZETA FUNCTIONS AND BASIC L-SERIES
By using q-Volkenborn integration and uniform differentiable on Zp, we construct p-adic q-zeta functions. These functions interpolate the qBernoulli numbers and polynomials. The value of p-adic q-zeta functions at negative integers are given explicitly. We also define new generating functions of q-Bernoulli numbers and polynomials. By using these functions, we prove analytic continuation of som...
متن کامل