On the duality and the derivation relations for multiple zeta values

We consider the problem of deducing the duality relation from the extended double shuffle relation for multiple zeta values. Especially we prove that the duality relation for double zeta values and that for the sum of multiple zeta values whose first components are 2’s are deduced from the derivation relation, which is known as a subclass of the extended double shuffle relation.     

Integrals over polytopes, multiple zeta values and polylogarithms, and Euler’s constant

Let T be the triangle with vertices (1, 0), (0, 1), (1, 1). We study certain integrals over T, one of which was computed by Euler. We give expressions for them both as linear combinations of multiple zeta values, and as polynomials in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler’s constant, and study integrals, one of which is the iterated Chen (Drinfeld-Kontsevich) integral, over some polytopes that are higher-dimensional analogs of T. The latter leads to a relation between certain multiple polylogarithm values and multiple zeta values.     

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.     

Generalized Jacobi–Trudi determinants and evaluations of Schur multiple zeta values

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi–Trudi formulas and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulas we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.     

A generalization of Ohnoʼs relation for multiple zeta values

In this paper, we prove that certain parametrized multiple series satisfy the same relation as Ohno?s relation for multiple zeta values. This result gives us a generalization of Ohno?s relation for multiple zeta values. By virtue of this generalization, we obtain a certain equivalence between the above relation among the parametrized multiple series and a subfamily of the relation. As applications of the above results, we obtain some results on multiple zeta values.     

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.     

An algebraic proof of the cyclic sum formula for multiple zeta values

We introduce an algebraic formulation of the cyclic sum formulas for multiple zeta values and for multiple zeta-star values. We also present an algebraic proof of cyclic sum formulas for multiple zeta values and for multiple zeta-star values by reducing them to Kawashima's relation.     

A generating function for sums of multiple zeta values and its applications

A generating function for specified sums of multiple zeta values is defined and a differential equation that characterizes this function is given. As applications, some relations for multiple zeta values over the field of rational numbers are discussed.

     

Multiple zeta values

The definition of multiple zeta values is extended in the paper. The preservation of the main properties known for multiple zeta values in the sense of their classic definition is proved.     

Multiple zeta values of fixed weight, depth, and height

We give a generating function for the sums of multiple zeta values of fixed weight, depth and height in terms of Riemann zeta values.     

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.     

The Algebra and Combinatorics of Shuffles andMultiple Zeta Values

The algebraic and combinatorial theory of shuffles, introduced by Chen and Ree, is further developed and applied to the study of multiple zeta values. In particular, we establish evaluations for certain sums of cyclically generated multiple zeta values. The boundary case of our result reduces to a former conjecture of Zagier.     

On variants of symmetric multiple zeta-star values and the cyclic sum formula

The Ramanujan Journal - The t-adic symmetric multiple zeta values were defined by Jarossay, which have been studied as a real analogue of the $${\varvec{p}}$$ -adic finite multiple zeta values. In...     

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.     

Weighted sum formulas of multiple zeta values with even arguments

Mathematische Zeitschrift - We prove a weighted sum formula of the zeta values at even arguments, and a weighted sum formula of the multiple zeta values with even arguments and its zeta-star...     

Multiple harmonic series related to multiple Euler numbers

In this paper, we define the multiple Euler numbers and consider some multiple harmonic series of Mordell-Tornheim's type, which is a partial sum of the Mordell-Tornheim zeta series defined by Matsumoto. Indeed, we prove a certain reducibility of these series as well as the multiple zeta values.     

A note on sums of products of Bernoulli numbers

In this work we obtain a new approach to closed expressions for sums of products of Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function.     

A New Shuffle Convolution for Multiple Zeta Values

Recently, interest in shuffle algebra has been renewed due to their connections with multiple zeta values. In this paper, we prove a new shuffle convolution that implies a reduction formula for the multiple zeta value z({5,1}ⁿ).Research partially supported by a grant from the Number Theory Foundation.     

Rota–Baxter algebras and left weak composition quasi-symmetric functions

Motivated by a question of Rota, this paper studies the relationship between Rota–Baxter algebras and symmetric-related functions. The starting point is the fact that the space of quasi-symmetric functions is spanned by monomial quasi-symmetric functions which are indexed by compositions. When composition is replaced by left weak composition (LWC), we obtain the concept of LWC monomial quasi-symmetric functions and the resulting space of LWC quasi-symmetric functions. In line with the question of Rota, the latter is shown to be isomorphic to the free commutative nonunitary Rota–Baxter algebra on one generator. The combinatorial interpretation of quasi-symmetric functions by P-partitions from compositions is extended to the context of left weak compositions, leading to the concept of LWC fundamental quasi-symmetric functions. The transformation formulas for LWC monomial and LWC fundamental quasi-symmetric functions are obtained, generalizing the corresponding results for quasi-symmetric functions. Extending the close relationship between quasi-symmetric functions and multiple zeta values, weighted multiple zeta values, and a q-analog of multiple zeta values are investigated, and a decomposition formula is established.     

The first derivative multiple zeta values at non-positive integers

In this article, we prove some explicit results for the first derivative multiple zeta values at non-positive integers and apply them to a certain classical problem in number theory which was studied and developed by E. Hecke, A. Fujii and K. Matsumoto. Further, we consider the relation between regular values and reverse values for the multiple zeta-function via a certain functional relation.