Multiple $T$-values with one parameter

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