$\vec s \in \ZZ ^k_{>0}$ with $\vec{s}=(\vec{1}_{m},\vec{s}\,')$ with $m\geq 1$ and $s_{m+1}>1$. Then $\gzeta(\vec s)=P_{m,\vec{s}\,'}(T)$, where $P_{m,\vec{s}\,'}$ is the polynomial in Lemma . Further, $\gzeta(\vec s)=\zeta\lp\wvec{\vec s}{\vec s}\rp$.

By Lemma  and Eq. (), $\gzeta\lp \wvec{\vec 1_m, \vec s\,'}{c\vec 1_m, \vec r\,'}\rp=P_{m,\vec{s}\,'}(X)$ independent of $\vec{r}\,'$. Since $\lim _{c\to 1} X=T=X\big|_{c=1},$ we obtain $$\lim_{c\to 1, \vec r\,'\to \vec s\,'} \zeta\lp \wvec{\vec 1_m, \vec s\,'}{c\vec 1_m, \vec r\,'}\rp=P_{m,\vec{s}\,'}(T)=\zeta\lp\wvec{\vec s}{\vec s}\rp. \mbox{\proofend}$$

$\gzeta(\vec{s}), \vec{s}\in \ZZ_{>0}^k$, satisfy the quasi-shuffle relation.

The subset $\{{\wvec{s}{s}} |\, s\in \ZZ_{>0}\}$ of the semigroup $\frakM$ in Eq. () is a subsemigroup and the $\CC$-space $\calh$ generated by it is a sub-algebra of the quasi-shuffle algebra $\calh_\frakM$. Thus the algebra homomorphism $\uni{Z}_+:\calh_\frakM\to
\CC[T][[\vep]]$ in Corollary  restricts to an algebra homomorphism $\uni{Z}_+:\calh\to \CC[T][[\vep]]$ with $$\uni{Z}_+\big(\lp{\wvec{s_1}{s_1}}, \cdots, {\wvec{s_k}{s_k}}\rp;0\big)=\zeta\lp\wvec{\vec{s}}{\vec{s}}\rp=\gzeta(\vec s).$$ Hence the corollary.

Let $\vec{s}=(s_1,\cdots,s_k)$ with positive integers $s_1,\cdots,s_k$ and $Z^\msh_{\vec{s}}(T)$ be the regularized MZVs of Ihara-Kaneko-Zagier. Then $\gzeta(\vec s)=Z^\msh_{\vec{s}}(T)$.

We recall [@IKZ Proposition 1] that $Z^\msh_{\vec{s}}(T)$ with $s_i\geq 1, 1\leq i\leq k$, is obtained as the unique extension of the MZVs $\zeta(\vec{s})$ with $s_i\geq 1$, $s_1>1$, such that $Z^\msh_{(1)}(T)=T$ and such that the quasi-shuffle relation still holds for $Z^*_{\vec{s}}(T)$. Since our definition of $\gzeta(\vec{s})$ agrees with $\zeta(\vec{s})$ for $s_i\geq 1$ and $s_1>1$, and 