Congruences for central binomial sums and finite polylogarithms

Text. We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients ((2k)(k)), partly motivated by analogies with the well-known power series for (arcsin z)(2) and (arcsin z)(4). The right-hand sides of those congruences involve values of the finite polylogarithms L-d(x) = Sigma(p-1)(k=1) x(k)/k(d). Exploiting the available functional equations for the latter we compute those values, modulo the required powers of p, in terms of familiar quantities such as Fermat quotients and Bernoulli numbers. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/warch?v-W54Ad0YFr8A. (C) 2012 Elsevier Inc. All rights reserved.