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

Sasha Berger, Aarav Chandra, Jasper Jain, Daniel Xu, Ce Xu, J. Zhao

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.

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

Abstract

Many -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 -values, and discover many new identities.

Paper Structure

This paper contains 6 sections, 15 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Multiple zeta values and Euler sums
  3. Regularizations of Euler sums and their generating functions
  4. Depth 2 weighted sum formulas
  5. Depth 3 weighted sum formulas, Part A
  6. Depth 3 weighted sum formulas, Part B

Key Result

Theorem 1.1

(=Corollary cor:KTconj) For all $w\ge 4$

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 4.1
  • Theorem 4.2
  • proof
  • Corollary 4.3
  • proof
  • Theorem 4.4
  • proof
  • Lemma 5.1
  • proof
  • ...and 18 more