Multiple $T$-values with one parameter

Abstract

This article introduces an algebra $\mathscr{A}_{\mathsf{MTV}, c}$ 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. The algebra $\mathscr{A}_{\mathsf{MTV}, c}$ can be seen as a common deformation of multiple zeta values and of Kaneko-Tsumura's recent multiple $T$-values and it satisfies the same duality relations. Its first graded dimensions, assuming that a grading by the weight does hold, are computed, and some relations not implied by duality are found. A generating function for specific elements in this algebra is described using Gamma and hypergeometric functions.