Related papers: Shuffle regularization for multiple Eisenstein ser…

In this paper we consider iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$ and define a class of $\mathbb{Q}$-linear relations among them, which arises from the differential structure of the iterated integrals with respect to…

We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by developing an explicit connection between the Fourier…

It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this…

We present the $\tau$-invariant balanced quasi-shuffle algebra $\mathcal{G}^{\operatorname{f}}$, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple q-zeta values. In particular,…

We show that the most standard Eisenstein series such as $E_4(\tau)$ or $2E_2(2\tau)-E_2(\tau)$, and also the function $\theta^2(\tau)$, are in a natural way the product of two conjugate Eisenstein series of half their weight and double…

In this paper, we settle the problem posed by Singer which is on a comparison problem between the renormalized values of shuffle type and of harmonic type of multiple zeta functions.

Symmetric multiple zeta values (SMZVs) are elements in the ring of all multiple zeta values modulo the ideal generated by $\zeta(2)$ introduced by Kaneko-Zagier as counterparts of finite multiple zeta values. It is known that symmetric…

Multiple zeta values arise as special values of polylogarithms defined on Riemann surfaces of various genera. Building on the vast knowledge for classical and elliptic multiple zeta values, we explore a canonical extension of the formalism…

In this paper we shall develop a theory of (extended) double shuffle relations of Euler sums which generalizes that of multiple zeta values (see Ihara, Kaneko and Zagier, \emph{Derivation and double shuffle relations for multiple zeta…

Parametrized multiple series are generalizations of the multiple zeta values introduced by Igarashi. In this work, we completely determine all the linear relations among these parameterized multiple series. Specifically, we prove the…

In this paper, we study the multiple $L$-values and the multiple zeta values of level $N$. We set up the algebraic framework for the double shuffle relations of the multiple zeta values of level $N$. Using the regularized double shuffle…

Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle…

We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara, Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator. We apply these results to show that the full…

Recently, Maesaka, Seki and Watanabe discovered a surprising equality between multiple harmonic sums and certain Riemann sums which approximate the iterated integral expression of the multiple zeta values. In this paper, we describe the…

We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl-Kaneko-Zagier as a hybrid of classical Eisenstein series and…

Colored multiple zeta values are special values of multiple polylogarithms evaluated at Nth roots of unity. In this paper, we define both the finite and the symmetrized versions of these values and show that they both satisfy the double…

This work is an example driven overview article of recent works on the connection of multiple zeta values, modular forms and q-analogues of multiple zeta values given by multiple Eisenstein series.

We give a proof of double shuffle relations for $p$-adic multiple zeta values by developing higher dimensional version of tangential base points and discussing a relationship with two (and one) variable $p$-adic multiple polylogarithms.

We study a refinement of the symmetric multiple zeta value, called the $t$-adic symmetric multiple zeta value, by considering its finite truncation. More precisely, two kinds of regularizations (harmonic and shuffle) give two kinds of the…

We give an explicit formula for the shuffle relation in a general double shuffle framework that specializes to double shuffle relations of multiple zeta values and multiple polylogarithms. As an application, we generalize the well-known…