Prop. 1]):

\[prop:eDS\] We have two algebra homomorphisms: $$Z^*: (\fA_*^1,*)\lra \C[T],\quad \text{and}\quad Z^\sha: (\fA_\sha^1,\sha)\lra \C[T]$$ which are uniquely determined by the properties that they both extend the evaluation map $Z:\fA^0\lra \C$ by sending $b_0=y_{1,0}$ to $T$.

In order to establish the crucial relation between $Z^*$ and $Z^\sha$ we can adopt the machinery in [@IKZ]. For any $(\bs|\bi)=(s_1,\dots,s_n|i_1,\dots,i_n)$ where $i_j$’s are integers and $s_j$’s are positive integers, let the image of the corresponding words in $\fA^1$ under $Z^*$ and $Z^\sha$ be denoted by $Z_{(\bs|\bi)}^*(T)$ and $Z_{(\bs|\bi)}^\sha(T)$ respectively. For example, $$\begin{aligned}
 TL_N(2|3)=&Z_{(1|0)}^*(T)Z_{(2|3)}^*(T)=Z^*(y_{1,0}*y_{2,3})\\
 =&Z_{(1,2|0,3) }^*(T)+Z_{(2,1|3,3) }^*(T)+Z_{(3|3)}^*(T),\end{aligned}$$ while $$\begin{aligned}
 TL_N(2|3)=&Z_{(1|0)}^\sha(T)Z_{(2|3)}^\sha(T)=Z^\sha(y_{1,0}\sha
 y_{2,3})=Z^\sha(b_0\sha ab_3)\\
 =&Z_{(1,2|0,3)}^\sha(T)+Z_{(2,1|0,3)}^\sha(T) +Z_{(2,1|3,0)}^\sha(T)  .\end{aligned}$$ Hence we find the following RDS by the next Theorem: $$L_N(2,1|3,0) +L_N(3|3)=L_N(2,1|3,N-3)+L_N(2,1|0,3).$$

\[thm:RDSoverC\] Define a $\C$-linear map $\rho:\C[T]\to \C[T]$ by $$\rho(e^{Tu})=\exp\left(\sum_{n=2}^\infty\frac{(-1)^n}{n}\zeta(n)u^n\right)e^{Tu},\qquad |u|<1.$$ Then for any index set $(\bs|\bi)$ we have $$Z_{(\bs|\bi)}^\sha(T)= \rho(Z_{(\bs|\bi)}^*(T)).$$

This is a the generalization of [@IKZ Thm. 1] to the higher level MPV cases. The proof is essentially the same. One may compare Cor. 2.24 in [@Rac]. The above steps can be easily transformed to computer codes which are used in our MAPLE programs.

Finite and regularized double shuffle relations (FDS & RDS) {#sec:RDS}
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It is generally believed that all the linear relations between MZVs 