The p-adic Arakawa–Kaneko zeta functions and p-adic Lerch transcendent

TextThe Arakawa–Kaneko zeta functions interpolate the poly-Bernoulli polynomials at the negative integers, while their values at the positive integers are connected to multiple zeta values and harmonic number sums. Here we construct p-adic analogues of these functions and show that these complex formulas have good p-adic counterparts. The method of construction, summing a series of forward differences of p-adic power functions, is a p-adic adaptation of an everywhere-convergent series for the Hurwitz zeta function due to Hasse. We also apply this method to construct a p-adic Lerch transcendent. Expressions for zeta values and related constants as harmonic number series which converge in both complex and p-adic senses are a prominent feature of this approach.VideoFor a video summary of this paper, please visit http://youtu.be/Mqd6W0TW9D0.