Symbolic summation methods and congruences involving harmonic numbers

Abstract

In this paper, we establish some combinatorial identities involving harmonic numbers via the package Sigma, by which we confirm some conjectural congruences of Z.-W. Sun. For example, for any prime p>3, we have∑k=0(p−3)/2(2kk)2(2k+1)16kHk(2)≡−7Bp−3(modp),∑k=1p−1(2kk)2k16kH2k(2)≡Bp−3(modp),∑k=1(p−1)/2(2kk)2k16k(H2k−Hk)≡−73pBp−3(modp2), where Hn(m)=∑k=1n1/km (m∈Z+={1,2,…}) is the n-th harmonic numbers of order m and Bn is the n-th Bernoulli number.