Bachmann, Takeyama and Tasaka introduced a $\mathbb{ Q}$-linear map $\phi$, which we call the symmetrization map in this paper, on the harmonic algebra $\mathfrak{ H}^1$. They calculated $\phi(w)$ explicitly for an element $w$ in $\mathfrak{ H}^1$ related to the multiple zeta values of Mordell--Tornheim type. In this paper, we introduce its $t$-adic generalization $\widehat{ \phi}$ and calculate $\widehat{ \phi}(w)$ for an element $w$ in $\mathfrak{ H}^1[[t]]$ constructed from the theory of $2$-colored rooted tree.
Categories: math.NT math.RA