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.

Mattarei, S., Tauraso, R. (2013). Congruences for central binomial sums and finite polylogarithms. JOURNAL OF NUMBER THEORY, 133(1), 131-157 [10.1016/j.jnt.2012.05.036].

Congruences for central binomial sums and finite polylogarithms

TAURASO, ROBERTO
2013-01-01

Abstract

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.
2013
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Congruences; Bernoulli numbers; Binomial coefficients; Polylogarithm; Finite sums
Mattarei, S., Tauraso, R. (2013). Congruences for central binomial sums and finite polylogarithms. JOURNAL OF NUMBER THEORY, 133(1), 131-157 [10.1016/j.jnt.2012.05.036].
Mattarei, S; Tauraso, R
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/90115
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