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

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 ζ * ({2} a , 3, {2} b ) based on a finite identity for partial sums of the zeta-star series.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3132 KH. HESSAMI PILEHROOD, T. HESSAMI PILEHROOD, AND R. TAURASOGlaisher [8] in 1900, and Lehmer [21] in 1938, proved that even the multiple harmonic sums H p−1 (m) modulo a higher power of a prime p ≥ m + 3 are related to the Bernoulli numbers:The systematic study of MZVs began in the early 1990s with the works of Hoffman [16] and Zagier [33]. The set of the MZVs has a rich algebraic structure given by the shuffle and the stuffle (harmonic shuffle or quasi-shuffle) relations. These follow from the representation of multiple zeta values in terms of iterated integrals and harmonic sums, respectively. There are many conjectures concerning multiple zeta values, and despite some recent progress, lots of open questions still remain to be answered. Licensed to New York Univ, Courant Inst. Prepared on Tue Feb 3 02:49:34 EST 2015 for download from IP 128.122.253.228. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use NEW PROPERTIES OF MULTIPLE HARMONIC SUMS 3133Let Z w denote the Q-vector space spanned by the set of multiple zeta values ζ(s 1 , . . . , s r ) with s r ≥ 2 and the total weight w = s 1 + · · · + s r , and let Z denote the Q-vector space spanned by all multiple zeta values over Q. A conjecture of Zagier [33] states that the dimension of the Q-vector space Z w is given by the Perrin numbers d w defined for w ≥ 3 by the recurrencewith the initial conditions d 0 = 1, d 1 = 0, d 2 = 1. The upper bound dim Z w ≤ d w was proved independently by Goncharov [9] and Terasoma [28].It is easy to see that the Perrin number d w is equal to the number of multiple zeta values ζ(s 1 , . . . , s r ) with s 1 + · · · + s r = w and each s j ∈ {2, 3}. While investigating the deep algebraic structure of Z, Hoffman [17] conjectured that the MZVs ζ(s 1 , . . . , s r ) of weight w with s j ∈ {2, 3} span the Q-space Z w . Very recently, this conjecture was proved using motivic ideas by Brown [4]. So the main problem which remains open is proving that the numbers ζ(s 1 , . . . , s r ) with s j ∈ {2, 3} are linearly independent over Q.According to Zagier's conjecture, a basis for Z w for 2 ≤

A Family of Multiple Harmonic Sum and Multiple Zeta Star Value Identities

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In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al [7]. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums.

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“…When s ∈ N ℓ they are called the multiple zeta value (MZV) and the multiple zeta star value (MZSV), respectively. The following is one of the main results of [7].…”

“…The key lemma and an identity family of MZSV In [7], using the corresponding identities of MHS Hessami Pilehrood et al find some new families of identities of MZSV and a new proof of the identity of Zagier [9]. We can similarly use Theorem 2.3 to obtain a new family of MZSV as follows.…”

“…In a recent paper [7] Hessami Pilehrood et al proved a few families of identities involving (alternating) multiple harmonic sums and binomial coefficients and, as applications, discovered some new congruences for multiple harmonic sums. In particular, they are able to confirm several conjectures contained in [10] posed by the second author of this paper.…”

On $$q$$ q -analogues of two-one formulas for multiple harmonic sums and multiple zeta star values

“…Zhao's proof is based on generalizations of binomial identities for finite multiple harmonic sums found in [12] to strings with repeating collections of twos and ones. In this paper, we extend this approach to q-analogues of multiple harmonic sums and, as a limit case, obtain corresponding results for q-zeta values.…”