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


Authors: Kh. Hessami Pilehrood, T. Hessami Pilehrood and R. Tauraso
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
Published electronically: October 2, 2013
MathSciNet review: 3180742
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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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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-2013-05980-6
Keywords: Multiple harmonic sum, Bernoulli number, congruence, multiple zeta value
Received by editor(s): July 11, 2012
Received by editor(s) in revised form: October 4, 2012
Published electronically: October 2, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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