Proof of Kaneko--Tsumura Conjecture on Triple T-Values

TL;DR

This paper proves a conjecture by Kaneko and Tsumura on triple T-values, generalizes weighted sum formulas to Euler sums, and uncovers new identities using generating functions.

Contribution

It confirms the Kaneko--Tsumura conjecture on triple T-values and extends weighted sum formulas to Euler sums through generating function techniques.

Findings

Confirmed the Kaneko--Tsumura conjecture on triple T-values.

Re-proved known formulas using generating functions.

Discovered numerous new identities involving Euler sums.

Abstract

Many $\mathbb{Q}$-linear relations exist between multiple zeta values, the most interesting of which are various weighted sum formulas. In this paper, we generalized these to Euler sums and some other variants of multiple zeta values by considering the generating functions of the Euler sums. Through this approach we are able to re-prove a few known formulas, confirm a conjecture of Kaneko and Tsumura on triple $T$-values, and discover many new identities.