Symmetries of Bernoulli polynomial series and Arakawa–Kaneko zeta functions

TextThe 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.VideoFor a video summary of this paper, please visit http://youtu.be/0vQqfgrkX2k.