Tridendriform and dendriform Zeta Values from Schroeder trees

TL;DR

This paper introduces new algebraic structures called tridendriform and dendriform algebras to generalize Multiple Zeta Values, utilizing Schroeder trees to reveal new properties and computational methods for Zeta Values.

Contribution

It defines new algebraic frameworks for Zeta Values using Schroeder trees, enabling broader generalizations and algebraic interpretations of these special numbers.

Findings

New algebraic structures for Zeta Values introduced.

Generalizations of Multiple Zeta Values achieved via Schroeder trees.

Enhanced methods for computing Shintani Zeta Values.

Abstract

To build new generalisations of Multiple Zeta Values, we define new spaces of formal series and formal integrals. We show that they are tridendriform and dendriform algebras. This allows us to reinterpret the fact that Multiple Zeta Values are algebra morphisms for shuffles of words in terms of finer tridendriform and dendriform structures. Applying universal properties of Schroeder trees we obtain generalisations of Multiple Zeta Values that are algebra morphisms for associative products. Hence we find new properties of Arborified Zeta Values and state how this enables the computation of some Shintani Zeta Values.