Multiple $T$-values with one parameter

TL;DR

This paper develops an algebraic framework for a family of multiple T-values, unifying multiple zeta values and Kaneko-Tsumura's T-values through iterated integrals and shuffle products, with initial dimension computations.

Contribution

It introduces a new algebra of functions based on iterated integrals that deforms both multiple zeta values and T-values, providing a unified structure.

Findings

Computed initial graded dimensions of the algebra

Established the algebra as a deformation of known multiple values

Connected the algebra to existing multiple value theories

Abstract

This article introduces an algebra of functions in one variable $c$ defined by iterated integrals of two specific differential forms depending on $c$, where the product is the shuffle product. This algebra can be seen as a common deformation of multiple zeta values and of Kaneko-Tsumura's recent multiple $T$-values. The first few graded dimensions, assuming that a grading by the weight does hold, are computed.