Finite and symmetrized colored multiple zeta values

TL;DR

This paper introduces finite and symmetrized colored multiple zeta values, demonstrating their algebraic properties and proposing an isomorphism between their generated spaces, extending previous work on related special values.

Contribution

It defines finite and symmetrized colored multiple zeta values and provides evidence for an isomorphism between their associated spaces, generalizing prior results.

Findings

Both types satisfy double shuffle relations

Strong evidence for an isomorphism between their spaces

Generalizes previous work on multiple zeta values

Abstract

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 shuffle relations. Further, we provide strong evidence for an isomorphism connecting the two spaces generated by these two kinds of values. This is a generalization of a recent work of Kaneko and Zagier on finite and symmetrized multiple zeta values and of the second author on finite and symmetrized Euler sums.