p-ADIC q-EXPANSION OF ALTERNATING SUMS OF POWERS

Let p be a fixed prime. Throughout this paper Zp, Qp, C and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Qp, cf.[1, 4, 6, 10]. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = p. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp. If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp, then we assume |q − 1|p < p − 1 p−1 , so that q = exp(x log q) for |x|p ≤ 1. Kubota and Leopoldt proved the existence of meromorphic functions, Lp(s, χ), defined over the padic number field, that serve as p-adic equivalents of the Dirichlet L-series, cf.[10, 11].