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

Kh. Hessami Pilehrood, T. Hessami Pilehrood, R. Tauraso

2013
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Transactions of the American Mathematical Society
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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
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... on 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. (2) ≡ 0 (mod p). Glaisher [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. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NEW PROPERTIES OF MULTIPLE HARMONIC SUMS 3133 Let 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 recurrence with 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 ≤ w ≤ 9 should be given as follows (see [31] ):

doi:10.1090/s0002-9947-2013-05980-6
fatcat:dcszdlsn7fd7lkqxmmdun7bjcy