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

Pilehrood, Kh.; Pilehrood, T.; Tauraso, R.

doi:10.1090/S0002-9947-2013-05980-6

New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner’s series
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by Kh. Hessami Pilehrood, T. Hessami Pilehrood and R. Tauraso PDF

Trans. Amer. Math. Soc. 366 (2014), 3131-3159 Request permission

Abstract:

In this paper we present some new binomial identities 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 $p$. As a further application we 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.

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