Related papers: Extended Shuffle Product for Multiple Zeta Values
The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work,…
It is known that there are infinitely many singularities of multiple zeta functions and the special values at non-positive integer points are indeterminate. In order to give a suitable rigorous meaning of the special values there, Furusho,…
We prove shuffle relations which relate a product of regularised integrals of classical symbols to regularised nested (Chen) iterated integrals, which hold if all the symbols involved have non-vanishing residue. This is true in particular…
We confirm a conjecture about the construction of basis elements for the multiple zeta values (MZVs) at weight 27 and weight 28. Both show as expected one element that is twofold extended. This is done with some lengthy computer algebra…
The MZV algebra is the graded algebra over ${\bold Q}$ generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the Grothendieck-Teichm\"{u}ller group. We shall show that there is a canonical…
In this article, we introduce an algebraic setup of non-strict multiple zeta values (NMZVs, for short) and prove some relations of NMZVs, which are analogous to Hoffman's relations of multiple zeta values, by using this algebraic setup of…
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…
Formal multiple zeta values allow to study multiple zeta values by algebraic methods in a way that the open question about their transcendence is circumvented. In this note we show that Hoffman's basis conjecture for formal multiple zeta…
Kaneko and Yamamoto introduced a convoluted variant of multiple zeta values (MVZs) around 2016. In this paper, we will first establish some explicit formulas involving these values and their alternating version by using iterated integrals,…
We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral…
We construct bases of quasi-symmetric functions whose product rule is given by the shuffle of binary words, as for multiple zeta values in their integral representations, and then extend the construction to the algebra of free…
In this paper, the extended double shuffle relations for interpolated multiple zeta values are established. As an application, Hoffman's relations for interpolated multiple zeta values are proved. Furthermore, a generating function for sums…
This paper is a culmination of [CM20] on the study of multiple zeta values (MZV's) over function fields in positive characteristic. For any finite place $v$ of the rational function field $k$ over a finite field, we prove that the $v$-adic…
The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the…
We explore the theory of multiple zeta values (MZVs) and some of their $q$-generalisations. Multiple zeta values are numerical quantities that satisfy several combinatorial relations over the rationals. These relations include two…
Using the concept of mixable shuffles, we formulate explicitly the quantum quasi-shuffle product, as well as the subalgebra generated by primitive elements of the quantum quasi-shuffle bialgebra. We construct a braided coalgebra structure…
The real multiple zeta values $\zeta(k_1,\ldots,k_r)$ are known to form a ${\bf Q}$-algebra; they satisfy a pair of well-known families of algebraic relations called the double shuffle relations. In order to study the algebraic properties…
Lamentably, the full analytical content of the epsilon-expansion of the master two-loop two-point function, with arbitrary self-energy insertions in 4-2epsilon dimensions, is still unknown. Here we show that multiple zeta values (MZVs) of…
We analyze certain characters of vertex algebras that can be expressed using (generalized) q-MZVs. We consider: (i) characters of vertex algebras associated to arc spaces, (ii) characters (or indices) of $\mathcal{S}$-class vertex operator…
We study an arithmetic analog of the Hall algebra of a curve, when the curve is replaced by the spectrum of the integers compactified at infinity. The role of vector bundles is played by lattices with quadratic forms. This algebra H…