^{(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 \\
&=& \hspace{-.7cm}
\sum_{(i_1,\cdots,i_p)\in\Pi_k}\hspace{-.6cm}
\tilde{P}\llp \uni{Z}\lp\wvec{\vec s^{(p)}}{\vec r ^{(p)}};\vep\rp
    \check{P}\llp \uni{Z}\lp\wvec{\vec s^{(p-1)}}{\vec r ^{(p-1)}};\vep\rp
    \cdots \check{P}\llp\uni{Z}\lp\wvec{\vec s ^{(1)}}{\vec r ^{(1)}};\vep\rp\rrp\,
     \cdots \rrp
    \rrp.
\mlabel{eq:plus}\end{aligned}$$ \[thm:expform\]

This follows from Eq. () and () by induction on $k$. There is nothing to prove when $k=1$. Assuming the formulas for $\uni{Z}_-$ and $\uni{Z}_+$ are true for $k\leq n$. Then by Eq. (), $$\uni{Z}_-\lp\wvec{s_1,\cdots,s_{n+1}}{r_1,\cdots,r_{n+1}}\rp
=\check{P}\Big (\uni{Z}\lp \wvec{s_1,\cdots,s_{n+1}}{r_1,\cdots,r_{n+1}}\rp
+\sum_{j=1}^k \uni{Z}_-\lp\wvec{s_1,\cdots,s_j}{r_1,\cdots,r_j}\rp
\uni{Z}\lp\wvec{s_{j+1},\cdots,s_{n+1}}{r_{j+1},\cdots,r_{n+1}}\rp\Big ).$$ Now the formula for $\uni{Z}_-$ follows by applying the induction hypothesis to the $\uni{Z}_-$ factors in the sum and using the fact that any ordered partition of $(1,\cdots,n+1)$ is either the one block partition $(n+1)$ or $(i_1,\cdots,i_p,n+1-j), 1\leq j\leq n,$ with $(i_1,\cdots,i_p)$ an ordered partition of $(1,\cdots,j)$. Then the first formula for $\uni{Z}_+$ follows from Eq. (). The second formula for $\uni{Z}_+$ is just to put the $\uni{Z}$-factors to the front of $\check{P}(x)$ instead of after it.

Renormalized multiple zeta values
=================================

We now use the renormalized directional MZVs defined in Eq. () to obtain [**renormalized MZVs**]{}. Here we will focus on two cases, when the arguments are either all positive or all non-positive.

The main definition and theorem
