Congruences for finite triple harmonic sums

Abstract

Zhao (2003a) first established a congruence for any odd prime p>3, S(1,1,1;p)≡−2B _p−3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β,γ,p) (mod p) is considered for all positive integers α,β,γ. We refer to w=α+β+γ as the weight of the sum, and show that if w is even, S(α,β,γ,p)≡0 (mod p) for p≥w+3; if w is odd S(α,γ,γ,p)≡rB _p−w (mod p) for p≥w, here r is an explicit rational number independent of p. A congruence of Catalan number is obtained as a special case.