Quasi-symmetric functions and mod p multiple harmonic sums

Abstract

We present a number of results about (finite) multiple harmonic sums modulo a prime, which provide interesting parallels to known results about multiple zeta values (i.e., infinite multiple harmonic series). In particular, we prove a "duality" result for mod p multiple harmonic sums similar to (but distinct from) that for multiple zeta values. We also exploit the Hopf algebra structure of the quasi-symmetric functions to do calculations with multiple harmonic sums mod p, and obtain, for each weight through 9, a set of generators for the space of weight-n multiple harmonic sums mod p. When combined with recent work, the results of this paper offer significant evidence that the number of quantities needed to generate the weight-n multiple harmonic series mod p is the n-th Padovan number (OEIS sequence A000931).

25 pages; version to appear in Kyushu J. Math

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Cited by in corpus (46)

• Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I
• Cyclotomic analogues of finite multiple zeta values
• Finite real multiple zeta values generate the whole space $Z$
• Congruences arising from Apéry-type series for zeta values
• Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values
• Multiple harmonic sums and Wolstenholme's theorem
• Pro-unipotent harmonic actions and a computation of $p$-adic cyclotomic multiple zeta values
• An explicit theory of ${\pi }_1^{\mathrm{un},\mathrm{crys}}({\mathbb{P}}^1-\{0,{\mu }_N,\infty \})$ - II-3 : Sequences of multiple harmonic sums viewed as periods
• A new integral-series identity of multiple zeta values and regularizations
• Depth reductions for associators
• Congruences of alternating multiple harmonic sums
• A note on derivation relations for multiple zeta values and finite multiple zeta values
• Ohno-type relation for finite multiple zeta values
• Yamamoto's interpolation of finite multiple zeta and zeta-star values
• On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions
• Sequential periods of the crystalline Frobenius
• Restricted sum formula for finite and symmetric multiple zeta values
• Ohno type relations for classical and finite multiple zeta-star values
• Identities involving cyclic sums of regularized multiple zeta values each of depth less than $5$
• Ohno-type identities for multiple harmonic sums
• Finite and Symmetric Euler Sums and Finite and Symmetric (Alternating) Multiple $T$-Values
• On finite analogues of Euler's constant
• Supercongruences of multiple harmonic $q$-sums and generalized finite/symmetric multiple zeta values
• The $\mathbf{p}$-adic duality for the finite star-multiple polylogarithms
• Identity involving symmetric sums of regularized multiple zeta-star values
• The MHS algebra and supercongruences
• Finite multiple zeta values associated with 2-colored rooted trees
• On some congruences using multiple harmonic sums of length three and four
• A combinatorial proof of the weighted sum formula for finite and symmetric multiple zeta(-star) values
• A note on ${\mathcal{F}}_n$-multiple zeta values
• On multiple zeta values and finite multiple zeta values of maximal height
• $p$-adic multiple zeta values and $p$-adic pro-unipotent harmonic actions : summary of parts I and II
• New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series
• Weighted sum formula for multiple harmonic sums modulo primes
• Symmetric Schur multiple zeta functions
• Finite Multiple Mixed Values
• A generalized regularization theorem and Kawashima's relation for multiple zeta values
• Alternating variants of multiple poly-Bernoulli numbers and finite multiple zeta values in characteristic 0 and p
• Generating functions for Ohno type sums of finite and symmetric multiple zeta-star values
• New functional equations of finite multiple polylogarithms
• A Note on Kawashima Functions
• On the linear relations among parametrized multiple series
• $p$-adic L-functions and Classical Congruences
• Modulus p^2 congruences involving harmonic numbers
• Derivation relation for finite multiple zeta values in $\hat{\mathcal{A}}$
• Analogues of the Aoki-Ohno and Le-Murakami relations for finite multiple zeta values