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

TL;DR

This paper discovers new identities and congruences for multiple harmonic sums modulo primes, leading to p-analogues of Leshchiner's series, refined generator sets, and a novel proof of Zagier's formula for specific zeta values.

Contribution

It introduces new hypergeometric identities for multiple harmonic sums and derives congruences that produce p-analogues of Leshchiner's series, refining known results and providing new proofs.

Findings

New identities for multiple harmonic sums with specific indices

Congruences modulo p leading to p-analogues of zeta series

A new proof of Zagier's formula for certain zeta-star values

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.