New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series

Abstract: In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $({1}^a,c,{1}^b),$ $({2}^a,c,{2}^b)$ and prove a number of congruences for these sums modulo a prime $p.$ The congruences obtained allow us to find nice $p$-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo $p$. Moreover, we are also able to provide a new proof of Zagier's formula for $\zeta^{*}({2}^a,3,{2}^b)$ based on a finite identity for partial sums of the zeta-star series.