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

Bernoulli and poly-Bernoulli polynomial convolutions and identities of p-adic Arakawa–Kaneko zeta functions

“…In the case s = 1 we recover from (3.26) and (3.8) the 2-adic identity which was observed in[19] as a 2-adic analogue to the real series…”

“…When k = 1 we have ξ 1 (s, a) = sζ(s+1, a) and when s = a = 1 we have ξ k (1, 1) = ζ(k + 1) in terms of the Hurwitz zeta and Riemann zeta functions. We recently [19] constructed p-adic analogues ξ p,k (s, a) of these functions which have entirely analogous properties, and which admit p-adic interpretations of the convolution identities of this paper.…”

“…(Note: this definition disagrees with that of [18] when p = 2). This yields an internal direct product decomposition of multiplicative groups In [19] we defined p-adic Arakawa-Kaneko zeta functions ξ p,k (s, a) by the series…”

“…At the positive integers, the p-adic Arakawa-Kaneko zeta functions are given by the same series of generalized harmonic numbers as their complex counterparts [5,19], namely, the series ∞ m=0 m!P n (h equivalently we may write…”

Generalizations of poly-Bernoulli and poly-Cauchy numbers

“…These generalizations, in particular, give some symmetries for Stirling number series, and lead to a unified investigation of arithmetic and algebraic properties for poly-Bernoulli and poly-Cauchy numbers. Moreover since the multiple zeta values and the Arakawa-Kaneko zeta functions are closely related to poly-Bernoulli numbers and polynomials, inverse binomial series and Bernoulli polynomial series (see [3,10,11,32,33]), the generalizations in this paper may lead to further investigations related to various zeta functions. One of the generalizations is inspired from the polylogarithm function.…”