$w(t)$ and concatenation of 1-forms $f(t)$. We may think the path lies in $\C$; however, it is more revealing to use iterated integrals in $\C^\ell$ to find the analytic continuation of this function (see [@Zanamp]).

It is well-known that special values of polylogs have significant applications in arithmetic such as Zagier’s conjecture [@Zag p.622]. On the other hand, the multiple zeta values (MZV) appear naturally in the study of the fundamental group of ${{\mathbb P}}^1-\{0,1\infty\}$ which is closely related to the absolute Galois group $\operatorname{{Gal}}(\overline{\Q}/\Q)$ according the Grothendieck [@D1]. As pointed out by Goncharov, higher cyclotomy theory should study the multiple polylogs at roots of unity, not only those of the classical ones. Moreover, theoretical physicists have already found out that such values appear naturally in the study of Feynmen diagrams ([@Br1; @Br2]).

Starting from early 1990s Hoffman [@H1; @H2] has constructed some quasi-shuffle (we will call “stuffle”) algebras reflecting the essential combinatorial properties of MZVs. Recently he [@H3] extends this to incorporate the multiple polylog values (MPVs) at roots of unity, although his definition of $*$-product is different from ours. Our approach here is a quantitative comparison between the the results obtained by Racinet [@Rac] who considers MPVs from 