y_{s_1,i_1}y_{s_2,i_1+i_2}\cdots
 y_{s_n,i_1+i_2+\dots+i_n}.$$ We now define an algebra of words as follows:

Set $A_0=\{\bfone\}$ to be the set of the empty word. Define $\fA=\Q\langle A\rangle$ to be the graded noncommutative polynomial $\Q$-algebra generated by letters $a$ and $b_i$ for $i\equiv 0,\dots,N-1\pmod{N}$, where $A$ is a locally finite set of generators whose degree $n$ part $A_n$ consists of words (i.e., a monomial in the letters) of depth $n$. Let $\fA^0$ be the subalgebra of $\fA$ generated by words not beginning with $b_0$ and not ending with $a$. The words in $\fA^0$ are called *admissible words.*

Observe that every MPV can be expressed uniquely as an iterated integral over the closed interval $[0,1]$ of an admissible word $w$ in $\fA^0$. Then we denote this MPV by $$\label{equ:Z}
Z(w):=\int_0^1 w.$$ Therefore we have (cf. [@Rac (2.5) and (2.6)]) $$\begin{aligned}
 \label{equ:1-1LZ}
L_N(s_1,\dots,s_n|i_1,i_2,\dots,i_n)=&Z(
y_{s_1,i_1}y_{s_2,i_1+i_2}\cdots
 y_{s_n,i_1+i_2+\dots+i_n}),\\
 Z(y_{s_1,i_1}y_{s_2,i_2}\cdots
 y_{s_n,i_n})=&L_N(s_1,\dots,s_n|i_1,i_2-i_1,\dots,i_n-i_{n-1}).
 \label{equ:1-1ZL}\end{aligned}$$ For example $L_3(1,2,2|1,0,2) =Z(y_{1,1}y_{2,1}y_{2,0}).$ On the other hand, during 1960s Chen developed a theory of iterated integral which can be applied in our situation.

\[chen’s\] *([@Chen (1.5.1)])* Let $\om_i$ $(i\ge 1)$ be $\C$-valued 1-forms on a manifold $M$. For every path $p$, $$\int_p \om_1\cdots \om_r\int_p \om_{r+1}\cdots \om_{r+s}=
 \int_p (\om_1\cdots \om_r) \sha (\om_{r+1}\cdots \om_{r+s})$$ where $\sha$ is the shuffle 