the [**renormalized directional MZV**]{} by $$\zeta\lp \wvec{\vec{s}}{\vec{r}}\rp = \lim_{\vep\to 0} \uni{Z}_+\lp \wvec{\vec{s}}{\vec{r}};\vep\rp .
\mlabel{eq:dmzv}$$ Here $\vec{r}$ is called the [**direction vector**]{}.

As a consequence of Corollary , we have

The renormalized directional MZVs satisfies the quasi-shuffle relation $$\zeta\lp \wvec{\vec{s}}{\vec{r}}\rp \zeta\lp \wvec{\vec{s}\,'}{\vec{r}\,'}\rp
= \zeta\lp \wvec{\vec{s}}{\vec{r}} \msh \wvec{\vec{s}\,'}{\vec{r}\,'} \rp.
\mlabel{eq:dqshuf}$$ Here the right hand side is defined in the same way as in Eq. ().

We next give an explicit formula for the renormalized directional .

[Let $\Pi_k$ be the set of ordered partitions (compositions) of $k$, consisting of ordered sequences $(i_1,\cdots,i_p)$ such that $i_1+\cdots+i_p=k$. For $1\leq j\leq p$, define the partial sum $I_j=i_1+\cdots+i_{j}$ with the convention that $I_0=0$. The [**partition vectors**]{} of $\vec{s}\in \RR^k$ from the ordered partition $(i_1,\cdots,i_p)$ are the vector $\vec{s}^{(j)}:=(s_{I_{j-1}+1},\cdots,s_{I_{j}})$, $1\leq j\leq p$. ]{}

Let $P: \CC[T][[\vep,\vep^{-1}]\to \CC[T][\vep^{-1}]$ be the Rota–Baxter operator in Eq. (). Denote $\check{P}=-P$ and $\tilde{P}=\id-P$. For $\vec{s}=(s_1,\cdots,s_k)\in \ZZ^k$ and $\vec{r}=(r_1,\cdots,r_k)\in \NN_{>0}^k$, $$\begin{aligned}
\lefteqn{{\ }\hspace{-.7cm} \uni{Z}_-\lp \wvec{\vec s}{\vec r};\vep\rp
= \hspace{-.5cm}
\sum_{(i_1,\cdots,i_p)\in \Pi_k}
\check{P}\llp
    \llp \cdots \check{P}\llp\check{P}\llp \uni{Z}\lp \wvec{\vec s ^{(1)}}{\vec r ^{(1)}};\vep\rp\rrp\,
    \uni{Z}\lp\wvec{\vec s^{(2)}}{\vec r ^{(2)}};\vep\rp \rrp \cdots\rrp
    \uni{Z}\lp\wvec{\vec s^{(p)}}{\vec r ^{(p)}};\vep\rp \rrp. } \notag \\
\lefteqn{{\ }\hspace{-.7cm}  \uni{Z}_+\lp \wvec{\vec s}{\vec r};\vep\rp
= \hspace{-.5cm}
\sum_{(i_1,\cdots,i_p)\in
\Pi_k}\tilde{P}\llp
    \llp \cdots \check{P}\llp\check{P}\llp \uni{Z}\lp \wvec{\vec s ^{(1)}}{\vec r 