Congruences for central binomial sums and finite polylogarithms

TextWe prove congruences, modulo a power of a prime p  , for certain finite sums involving central binomial coefficients (2kk), partly motivated by analogies with the well-known power series for (arcsinz)2 and (arcsinz)4. The right-hand sides of those congruences involve values of the finite polylogarithms £d(x)=∑k=1p−1xk/kd. 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.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=W54Ad0YFr8A.