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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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  • Kh. Hessami Pilehrood
  • Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4R2
  • Email: hessamik@gmail.com
  • T. Hessami Pilehrood
  • Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4R2
  • Address at time of publication: Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia, Canada B3H 3C3
  • Email: hessamit@gmail.com
  • R. Tauraso
  • Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, 00133 Roma, Italy
  • Email: tauraso@mat.uniroma2.it
  • Received by editor(s): July 11, 2012
  • Received by editor(s) in revised form: October 4, 2012
  • Published electronically: October 2, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3131-3159
  • MSC (2010): Primary 11A07, 11M32; Secondary 11B65, 11B68
  • DOI: https://doi.org/10.1090/S0002-9947-2013-05980-6
  • MathSciNet review: 3180742