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World Scientific Publishing, International Journal of Mathematics, 08(26), p. 1550055, 2015

DOI: 10.1142/s0129167x1550055x

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A new series for $π^3$ and related congruences

Journal article published in 2010 by Zhi-Wei Sun
This paper is available in a repository.
This paper is available in a repository.

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Abstract

Let [Formula: see text] denote the second-order harmonic number [Formula: see text] for n = 0, 1, 2, …. In this paper we obtain the following identity: [Formula: see text] We explain how we found the series and develop related congruences involving Bernoulli or Euler numbers; for example, it is shown that [Formula: see text] for any prime p > 3, where E0, E1, E2, … are Euler numbers. Motivated by the Amdeberhan–Zeilberger identity [Formula: see text], we also establish the congruence [Formula: see text] for each prime p > 3.