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New Proofs of the Explicit Formulas of Arakawa--Kaneko Zeta Values and Kaneko--Tsumura $η$- and $ψ$- Values

Masanobu Kaneko, Weiping Wang, Ce Xu, Jianqiang Zhao

TL;DR

This work develops new integral identities for multiple polylogarithms and their level-two analogues in terms of Hurwitz-type MZVs, enabling direct proofs of explicit formulas for the Arakawa–Kaneko zeta values and the Kaneko–Tsumura $\eta$- and $\psi$-values, as well as a formula for double $T$-values. By leveraging iterated integrals, Hoffman's dual, and the A-functions, the authors connect these zeta-type values to Hurwitz-type MZVs and their duals, providing analytic proofs of established formulas and extending to conjecture-related symmetrizations. The results deepen the understanding of level-two zeta generalizations and their relationship to classical MZVs, with potential implications for the algebraic structure and interrelations among these special values. The methods offer a framework to derive explicit expansions in terms of $\zeta$ and $\zeta^\star$ evaluated at shifted arguments and to verify related conjectures in the Kaneko–Tsumura family.

Abstract

In this paper, we establish some new identities of integrals involving multiple polylogarithm functions and their level two analogues in terms of Hurwitz-type multiple zeta (star) values. Using these identities, we provide new proofs of the explicit formulas of Arakawa--Kaneko zeta values, Kaneko--Tsumura $η$- and $ψ$-values, and also give a formula for double $T$-values.

New Proofs of the Explicit Formulas of Arakawa--Kaneko Zeta Values and Kaneko--Tsumura $η$- and $ψ$- Values

TL;DR

This work develops new integral identities for multiple polylogarithms and their level-two analogues in terms of Hurwitz-type MZVs, enabling direct proofs of explicit formulas for the Arakawa–Kaneko zeta values and the Kaneko–Tsumura - and -values, as well as a formula for double -values. By leveraging iterated integrals, Hoffman's dual, and the A-functions, the authors connect these zeta-type values to Hurwitz-type MZVs and their duals, providing analytic proofs of established formulas and extending to conjecture-related symmetrizations. The results deepen the understanding of level-two zeta generalizations and their relationship to classical MZVs, with potential implications for the algebraic structure and interrelations among these special values. The methods offer a framework to derive explicit expansions in terms of and evaluated at shifted arguments and to verify related conjectures in the Kaneko–Tsumura family.

Abstract

In this paper, we establish some new identities of integrals involving multiple polylogarithm functions and their level two analogues in terms of Hurwitz-type multiple zeta (star) values. Using these identities, we provide new proofs of the explicit formulas of Arakawa--Kaneko zeta values, Kaneko--Tsumura - and -values, and also give a formula for double -values.
Paper Structure (3 sections, 6 theorems, 40 equations)

This paper contains 3 sections, 6 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. New Proofs of Arakawa--Kaneko zeta-type values
  3. A formula related to Kaneko--Tsumura's conjecture

Key Result

Theorem 2.1

For any composition ${\boldsymbol{\sl{k}}}:=(k_1,k_2,\ldots,k_r)$, $k\in \mathbb{N}$, and any $\alpha\in \mathbb{C}\backslash \mathbb{N}$, the following formulas hold: where $n:=|{\boldsymbol{\sl{k}}}|+1-\mathop{\mathrm{dep}}\limits({\boldsymbol{\sl{k}}})$.

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 1 more