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

Abstract

In this paper we present some new binomial identities for multiple harmonic sums whose indices are the sequences ( { 1 } a , c , { 1 } b ) , (\{1\}^a,c,\{1\}^b), ( { 2 } a , c , { 2 } b ) (\{2\}^a,c,\{2\}^b) and prove a number of congruences for these sums modulo a prime p . p. The congruences obtained allow us to find nice p 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 7 and 9 9 modulo p p . As a further application we provide a new proof of Zagier’s formula for ζ ∗ ( { 2 } a , 3 , { 2 } b ) \zeta ^{*}(\{2\}^a,3,\{2\}^b) based on a finite identity for partial sums of the zeta-star series.