Congruences for central binomial sums and finite polylogarithms

Mattarei, Sandro and Tauraso, Roberto (2013) Congruences for central binomial sums and finite polylogarithms. Journal of Number Theory, 133 (1). pp. 131-157. ISSN 0022-314X

Full content URL: http://dx.doi.org/10.1016/j.jnt.2012.05.036

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Congruences for central binomial sums and finite polylogarithms

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Abstract

We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients
�binom(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 £_d(x) = sum_{k=1−1}^{p-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.

Keywords:central binomial coefficients, polylogarithms
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:18501
Deposited On:11 Dec 2015 09:33

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