Quasi-shuffle products revisited

TL;DR

This paper revisits quasi-shuffle products, extending their algebraic framework to broader contexts and applying these results to various multiple zeta value generalizations and related functions.

Contribution

It generalizes the algebraic machinery of quasi-shuffle products to encompass multiple zeta q-values and related functions, providing new algebraic formulas and insights.

Findings

Extended algebraic formulas for quasi-shuffle algebras

Applications to multiple zeta values and their generalizations

Transparent derivation of algebraic relations in extended quasi-shuffle settings

Abstract

Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zeta values and some of their analogues and generalizations. The second author, together with Kajikawa, Ohno, and Okuda, significantly extended the definition of quasi-shuffle algebras so it could be applied to multiple zeta q-values. This article extends some of the algebraic machinery of the first author's original paper to the more general definition, and uses this extension to obtain various algebraic formulas in the quasi-shuffle algebra in a transparent way. Some applications to multiple zeta values, interpolated multiple zeta values, multiple q-zeta values, and multiple polylogarithms are given.