=\lim_{\vep\to 0} \uni{Z}(\wvec{\vec{s}}{\vec{r}};\vep)
= \uni{Z}(\wvec{\vec{s}}{\vec{r}};0)=\zeta(\vec{s}).$

We now extend the last case to include the possibility of $s_1=1$ and compare it with the regularized MZVs of Ihara-Kaneko-Zagier [@IKZ].

Let $(\vec u,\vec v)$ denote the concatenation of two vectors $\vec u$ and $\vec v$.

For a log power series $f(\vep),g(\vep)\in \CC[T][[\vep]]$, denote $f(\vep)=g(\vep)+ O(\vep)$ if $g(\vep)-f(\vep)\in \vep \CC[T][[\vep]]$. Let $\vec s \in \ZZ ^k_{>0}$ be of the form $\vec{s}=(\vec{1}_{m},\vec{s}\,')$ where $m\geq 1$, $\vec 1_m =(1,1,\cdots, 1)\in \ZZ ^m$ and either $k=m$ or $s_{m+1}>1$. For $\ell=k-m$, let $\vec r\,' \in \ZZ ^\ell_{>0}$. For $c>0$, denote $X=-\ln c +T$. Then $$\uni{Z}(\wvec{\vec 1_m, \vec s\,'}{c \vec 1_m,\vec r\,'};\vep)=P_{m,\vec{s}\,'}(X)+ O(\vep),
\mlabel{eq:lead}$$ where $P_{m,\vec{s}\,'}(X)$ is a degree $m$ polynomial in $X$ with leading coefficient $\zeta (\vec s\, ')/m!$ if $\ell>0$ and $1/m!$ if $\ell=0$.

We prove by induction on $m\geq 1$. First consider $m=1$. When $\ell=0$, note that $$Z(\wvec{1}{c};\vep)=\sum_{n\geq 1} \frac
{e^{nc\vep}}{n}=-\ln(1-e^{c\vep})= -\ln c - \ln (-\vep)+\ln\big(\frac{-c\vep}{1-e^{c\vep}}\big).$$ Since $ln \frac {-\vep}{1-e^{\vep }}$ is an analytic function at $\vep =0$ with $\lim _ {\vep \to 0} \ln \frac {-\vep }{1-e^{\vep }}=0,$ we have $$\uni{Z}(\wvec{1}{c};\vep)= -\ln c +T+O(\vep)=X+O(\vep).
\mlabel{eq:lead1}$$ So Eq. () is proved for $m=1$ and $\ell=0$. When $\ell\geq 1$, let $\vec{e}_j^{(\ell)}$ be the $j$-th unit vector of 