A new series for π3and related congruences

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.