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Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication ∗ on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product III . We extend this commutative algebra structure to a Hopf algebra… (More)

We establish a new class of relations among the multiple zeta values ζ(k1, . . . , kl) = ∑ n1>···>nl≥1 1 n k1 1 · · ·n kl k , which we call the cyclic sum identities. These identities have an elementary proof, and imply the “sum theorem” for multiple zeta values. They also have a succinct statement in terms of “cyclic derivations” as introduced by Rota,… (More)

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of… (More)

Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ : H0 → R from a… (More)

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 harmonic sums similar to (but distinct from) that for multiple zeta values. We also exploit the… (More)

LiI(x1, . . . , xn) = AI(∞;x1, . . . , xk) are called multiple polylogarithms [2]: they generalize the classical polylogarithm Lin(x1) = A(n)(∞;x1). It is immediate from the defining equations (1) and (2) that the sums SI can be written in terms of the AI . To state the relation precisely, let C(n) be the set of compositions of n, i.e., ordered sequences… (More)

Recent work in perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms… (More)

We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in one-to-one correspondence with affine Weyl groups. For each such figure in n-dimensional Euclidean space, we construct a sequence of polynomials P/c: Rn —► Rn so that the mapping P^ is conjugate to stretching… (More)

In a recent paper, A. Libgober showed that the multiplicative sequence {Qi(c1, . . . , ci)} of Chern classes corresponding to the power series Q(z) = Γ(1 + z)−1 appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomials Qi can be expressed in terms of multiple zeta values.… (More)

- Michael E. Hoffman
- Electr. J. Comb.
- 2008

We define a homomorphism ζ from the algebra of quasi-symmetric functions to the reals which involves the Euler constant and multiple zeta values. Besides advancing the study of multiple zeta values, the homomorphism ζ appears in connection with two Hirzebruch genera of almost complex manifolds: the Γ-genus (related to mirror symmetry) and the Γ̂-genus… (More)