On the refined Kaneko-Zagier conjecture for general integer indices

TL;DR

This paper demonstrates that the generalized refined Kaneko-Zagier conjecture for integer indices follows from the positive index case, using an inductive formula for extended multiple zeta values.

Contribution

It shows that the conjecture for general integer indices can be derived from the positive index case, simplifying the proof of the extended conjecture.

Findings

The generalized conjecture is reducible to the positive index case.

An inductive formula for extended MZVs with non-positive entries is established.

The approach simplifies understanding the algebraic structure of extended MZVs.

Abstract

The refined Kaneko-Zagier conjecture claims that the algebras spanned by two kinds of "completed" finite multiple zeta values, called $\hat{A}$- and $\hat{S}$-MZVs, are isomorphic. Recently, Komori defined $\hat{S}$-MZVs of general integer (i.e., not necessarily positive) indices, extending the existing definition for positive indices. In view of the refined Kaneko-Zagier conjecture, Komori's work suggests that these extended values are closely connected to $\hat{A}$-MZVs of general indices, which can be defined in an obvious way. In this paper, we show that the generalization of the refined Kaneko-Zagier conjecture for general integer indices is actually deduced from the conjecture for positive indices. The key ingredient is an inductive formula for $\hat{A}$-MZVs or $\hat{S}$-MZVs of indices which contain at least one non-positive entry.