of $\QX$ not beginning with $x_1$ and not ending with $x_0$. Let $\pi_\cv:\QX\ra \QX_\cv$ be the projection. Passing to the limit one get:

*([@Rac Prop.2.11])* The series $\calI_\cv:=\lim_{a\to 0^+,b\to 1^-}\pi_\cv(\calI_{[a,b]})$ is group-like in $(\CXX_\cv,\gD)$.

Let $\calI$ be the unique group-like element in $(\CXX,\gD)$ whose coefficients of $x_0$ and $x_1$ are 0 such that $\pi_\cv(\calI)=\calI_\cv$. In order to do the numerical computation we need to find out explicitly the coefficients for $\calI$. Put $$\calI=\sum_{p\in \N, ,\gs_1,\dots,\gs_p\in  \gG\cup\{0\}}
C(\gs_1,\dots,\gs_p)x_{\gs_1}\cdots x_{\gs_p}.$$

\[prop:CoeffI\] Let $p,$ $m$ and $n$ be three non-negative integers. If $p>0$ then we assume $\gs_1\ne 1$ and $\gs_p\ne 0$. Set $(\gs_1,\dots,\gs_p,\{0\}^n) =(\gs_1,\dots,\gs_q)$. Then we have $$\begin{aligned}
 \notag
     \ C(\{1\}^m,\gs_1,& \dots,\gs_p,\{0\}^n)\\
     =& \left\{
          \begin{array}{ll}
            0, {\displaystyle \phantom{\frac1m}}  & \hbox{if $mn=p=0$;}  \\
            Z\big(\pi_\bfY(x_{\gs_1}\cdots x_{\gs_p})\big) , {\displaystyle \phantom{\frac1m}}  & \hbox{if $m=n=0$;}  \\
          {\displaystyle -\frac1m \sum_{i=1}^q C(\{1\}^{m-1},\gs_1,\dots,\gs_i,1,\gs_{i+1},\dots,\gs_q),} & \hbox{if $m>0$;} \\
                      {\displaystyle -\frac1n \sum_{i=1}^p C(\gs_1,\dots,\gs_{i-1},0,\gs_i,\dots,\gs_p,\{0\}^{n-1}),} & \hbox{if $m=0,n>0$.}
          \end{array}
        \right. \label{equ:CoeffI}
 \end{aligned}$$ Here $Z$ is defined by .

This proposition provides the recursive relations we may use to compute all the coefficients of $\calI$.

Since $\calI$ is group-like we have $$\label{equ:Igam}
\gD \calI=\calI\otimes\calI.$$ The first case follows from this immediately since $C(0)=C(1)=0$. The second case is essentially the definition of $Z$. If $m>0$ then we 