$t$-adic symmetrization map on harmonic algebra

Abstract

Bachmann, Takeyama and Tasaka introduced a $\mathbb{Q}$-linear map $\varphi$, which we call the symmetrization map in this paper, on the harmonic algebra ${\mathfrak{H}}^1$. They calculated $\varphi (w)$ 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 $\hat{\varphi }$ and calculate $\hat{\varphi }(w)$ for an element $w$ in ${\mathfrak{H}}^1[[t]]$ constructed from the theory of $2$-colored rooted tree.

15 pages. arXiv admin note: text overlap with arXiv:2009.04112