Quadratic relations for a q-analogue of multiple zeta values

We obtain a class of quadratic relations for a q-analogue of multiple zeta values (qMZV’s). In the limit q→1, it turns into Kawashima’s relation for multiple zeta values. As a corollary we find that qMZV’s satisfy the linear relation contained in Kawashima’s relation. In the proof we make use of a q-analogue of Newton series and Bradley’s duality formula for finite multiple harmonic q-series.     

A class of relations among multiple zeta values

We prove a new class of relations among multiple zeta values (MZV's) which contains Ohno's relation. We also give the formula for the maximal number of independent MZV's of fixed weight, under our new relations. To derive our formula for MZV's, we consider the Newton series whose values at non-negative integers are finite multiple harmonic sums.     

Duality and double shuffle relations of multiple zeta values

We prove that certain families of duality relations of the multiple zeta values (MZV's) are consequences of the extended double shuffle relations (EDSR's), thereby proving a part of the conjecture that the EDSR's give all linear relations of the MZV's.     

On the quasi-derivation relation for multiple zeta values

Recently, Masanobu Kaneko introduced a conjecture on an extension of the derivation relation for multiple zeta values. The goal of the present paper is to present a proof of the conjecture by reducing it to a class of relations for multiple zeta values studied by Kawashima. In addition, some algebraic aspects of the quasi-derivation operator on Q〈x,y〉, which was defined by modeling a Hopf algebra developed by Connes and Moscovici, will be presented.     

Relations for multiple zeta values

We prove new relations for multiple zeta values. In particular, they imply Vasil’ev’s equality and a formula for the summation of multiple zeta values of fixed weight with a constraint on the first coordinate.     

Weighted sum formula for multiple zeta values

The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case, conjectured in the early 1990s for higher dimensions and then proved by Granville and Zagier. Recently a weighted form of Euler's formula was obtained by Ohno and Zudilin. We generalize it to a weighted sum formula for multiple zeta values of all dimensions.     

Renormalization of multiple zeta values

Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at non-positive integers since the values are usually undefined. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of Ihara–Kaneko–Zagier on renormalization of MZVs with positive arguments. We further show that the important quasi-shuffle (stuffle) relation for usual MZVs remains true for the renormalized MZVs.     

Period polynomial relations between double zeta values

The even weight period polynomial relations in the double shuffle Lie algebra  $\mathfrak{ds}$ were discovered by Ihara, and completely classified in Schneps (J. Lie Theory 16(1): 19–37, 2006) by relating them to restricted even period polynomials associated to cusp forms on  $\mathrm{SL} _{2}(\mathbb{Z})$ . In an article published in the same year, Gangl et al. (Double zeta values and modular forms. Automorphic forms and zeta functions, pp. 71–106, 2006) displayed certain linear combinations of odd-component double zeta values which are equal to scalar multiples of simple zeta values in even weight, and also related them to restricted even period polynomials. In this paper, we relate the two sets of relations, showing how they can be deduced from each other by duality.     

Conical zeta values and their double subdivision relations

We introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones. The quasi-shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values. In order to achieve the closed cone subdivision relation, we also interpret linear relations among fractions as subdivisions of decorated closed cones. As a generalization of the double shuffle relation of multiple zeta values, we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values.     

An exotic shuffle relation for multiple zeta values

In this short note we will provide a new proof of the following exotic shuffle relation of multiple zeta values: This was proved by Zagier when n = 0, by Broadhurst when m = 0, and by Borwein, Bradley, and Broadhurst when m = 1. In general this was proved by Bowman and Bradley. Our new idea is to use the method of Borwein et al. to reduce the above general relation to some families of combinatorial identities which can be verified by Zeilberger’s algorithm [9, 10] that is part of the WZ method. Received: 27 November 2007 Revised: 28 June 2008     

On generating functions of multiple zeta values and generalized hypergeometric functions

Generating functions for sums of certain multiple zeta values with fixed weight, depth and i-heights are discussed. The functions are systematically expressed in terms of generalized hypergeometric functions. The expressions reproduce several known formulas for multiple zeta values as applications.     

p-adic multiple zeta values I.

Our main aim in this paper is to give a foundation of the theory of p-adic multiple zeta values. We introduce (one variable) p-adic multiple polylogarithms by Colemans p-adic iterated integration theory. We define p-adic multiple zeta values to be special values of p-adic multiple polylogarithms. We consider the (formal) p-adic KZ equation and introduce the p-adic Drinfeld associator by using certain two fundamental solutions of the p-adic KZ equation. We show that our p-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the p-adic KZ equation and our p-adic multiple zeta values appear as coefficients of the p-adic Drinfeld associator. We show various properties of p-adic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the p-adic case, via the p-adic KZ equation.     

On Mordell-Tornheim zeta values

We prove that the Mordell-Tornheim zeta value of depth can be expressed as a rational linear combination of products of the Mordell-Tornheim zeta values of lower depth than when and its weight are of different parity.

     

Periods of mirrors and multiple zeta values

In a recent paper, A. Libgober showed that the multiplicative sequence of Chern classes corresponding to the power series appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomials can be expressed in terms of multiple zeta values.

     

Multiple Dedekind zeta functions and evaluations of extended multiple zeta values

We define the number field analog of the zeta function of d-complex variables studied by Zagier in (First European Congress of Mathematics, vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhauser, Basel, 1994, pp. 497-512). We prove that in certain cases this function has a meromorphic continuation to C^d, and we identify the linear subvarieties comprising its singularities. We use our approach to meromorphic continuation to prove that there exist infinitely many values of these functions at regular points in their extended domains which can be expressed as a rational linear combination of values of the Dedekind zeta function.