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

TL;DR

This paper explores the connection between Rota--Baxter algebras and a new class of quasi-symmetric functions based on left weak compositions, establishing isomorphisms and extending combinatorial interpretations.

Contribution

It introduces LWC quasi-symmetric functions, proves their isomorphism to free Rota--Baxter algebras, and extends combinatorial and analytical frameworks to this new setting.

Findings

LWC quasi-symmetric functions form a space isomorphic to free Rota--Baxter algebras.

Transformation formulas for LWC functions generalize classical results.

Decomposition formulas relate LWC functions to multiple zeta values and their q-analogs.

Abstract

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.