Abstract. We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the well-known Broadhurst-Zagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst [4, 5]. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given. 1 Research partially supported by NSF grant DMS-9705782. 2

Abstract. We define a class of expressions for the multiple zeta function, and show how to determine whether an expression in the class vanishes identically. The class of such identities, which we call partition identities, is shown to coincide with the class of identities that can be derived as a consequence of the stuffle multiplication rule for multiple zeta values. 1.

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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: H 0 → R from a graded rational vector space H 0 generated by the “admissible words ” of the noncommutative polynomial algebra Q〈x,y〉. Now H 0 admits two (commutative) products making ζ a homomorphism–the shuffle product and the “harmonic ” product. The latter makes H 0 a subalgebra of the algebra QSym of quasi-symmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y 〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series. 1

Values of Euler-Zagier's multiple zeta function at non-positive integers are studied, especially at (0; 0; : : : ; n) and ( n; 0; : : : ; 0). Further we prove a symmetric formula among values at non-positive integers.

Abstract. In this paper we shall define the renormalization of the multiple q-zeta values (MqZV) which are special values of multiple q-zeta functions ζq(s1,...,sd) when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang [12] on the renormalization of Euler-Zagier multiple zeta values. We show that our renormalization process produces the same values if the MqZVs are well-defined originally and that these renormalizations of MqZV satisfy the q-stuffle relations if we use shifted-renormalizations for all divergent ζq(s1,..., sd) (i.e., s1 ≤ 1). Moreover, when q ↑ 1 our renormalizations agree with those of Guo and Zhang.

RÉSUMÉ. Nous prouvons que toute somme de Mordell-Tornheim avec des arguments entiers positifs peut s’écrire comme une combinaison linéaire rationnelle de valeurs prises par des fonctions multi-zêta ayant le même poids et la même profondeur. Selon un résultat de Tsumura, il s’ensuit que toute somme de Mordell-Tornheim ayant un poids et une profondeur de parité différente peut s’exprimer comme une combinaison linéaire rationnelle de produits de valeurs prises par des fonctions multi-zêta de profondeur plus petite. ABSTRACT. We prove that any Mordell-Tornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, it follows that any Mordell-Tornheim sum with weight and depth of opposite parity can be expressed as a rational linear combination of products of multiple zeta values of lower depth. 1.

Abstract. We introduce signed q-analogs of Tornheim’s double series and evaluate them in terms of double q-Euler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series and correct some mistakes in the literature. 1.

character on the quasi-symmetric functions

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