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arXiv · 1206.0407

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

Khodabakhsh Hessami Pilehrood·Tatiana Hessami Pilehrood·Roberto Tauraso
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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 $ζ^{*}(\{2\}^a,3,\{2\}^b)$ based on a finite identity for partial sums of the zeta-star series.

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Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood, Roberto Tauraso. 2013-03-22. New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series. https://arxiv.org/abs/1206.0407

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