Alternating Multiple $T$-Values: Weighted Sums, Duality, and Dimension Conjecture

TL;DR

This paper explores weighted sums and duality formulas of alternating multiple T-values, introduces related convoluted T-values and functions, and investigates the structure and dimension conjecture of the space they generate.

Contribution

It defines new weighted sums of AMTVs, establishes duality formulas, introduces related functions, and provides evidence for a dimension conjecture of the generated vector space.

Findings

Duality formulas for AMTVs established

Introduction of alternating convoluted T-values and $\psi$-function

Evidence supporting the tribonacci sequence dimension conjecture

Abstract

In this paper, we define some weighted sums of the alternating multiple $T$-values (AMTVs), and study several duality formulas for them by using the tools developed in our previous papers. Then we introduce the alternating version of the convoluted $T$-values and Kaneko-Tsumura $\psi$-function, which are proved to be closely related to the AMTVs. At the end of the paper, we study the $\Q$-vector space generated by the AMTVs of any fixed weight $w$ and provide some evidence for the conjecture that their dimensions $\{d_w\}_{w\ge 1}$ form the tribonacci sequence 1, 2, 4, 7, 13, ....