length $\ell$. Then by the quasi-shuffle relation $$\uni{Z}(\wvec{\vec s\,'}{\vec r\,'}; \vep)\uni{Z}(\wvec{1}{c};\vep)
= \uni{Z}(\wvec{1,\vec s\,'}{c,\vec r\,'})+
\sum_{j=1}^\ell \uni{Z}(\wvec{s_1,\cdots,s_j,1,s_{j+1},\cdots,s_\ell}{r_1,\cdots,r_j,c,r_{j+1},\cdots,r_\ell})
+\sum _{j=1}^\ell \uni{Z}(\wvec{\vec s\,'+ \vec{e}^{(\ell)}_j}{\vec r\,' + c\vec{e}^{(\ell)}_j}; \vep).$$ Since $s_1>1$, by the proof of Theorem  and Eq. (), we have $$\uni{Z}(\wvec{1,\vec{s}\,'}{c,\vec{r}\,'})=\zeta(\vec{s}\,')X
-\sum_{j=1}^\ell \zeta(s_1,\cdots,s_j,1,s_{j+1},\cdots,s_\ell)
-\sum _{j=1}^\ell \zeta(\vec s\,'+ \vec{e}^{(\ell)}_j) + O(\vep).$$ This complete the proof for $m=1$.

Suppose the formula has been proved for $m\geq 1$ and consider $\uni{Z}(\wvec{\vec 1_{m+1}, \vec s\,'}{c \vec 1_{m+1},\vec r\,'};\vep)$. By the quasi-shuffle relation we have $$Z(\wvec{\vec 1_m,\vec s\,'}{c\vec 1_m,\vec r\,'}; \vep)Z(\wvec{1}{c};\vep)
= (m+1)Z(\wvec{\vec 1_{m+1},\vec s\,'}{c\vec 1_{m+1}, \vec r\,'}; \vep)
+\sum _{i=1}^m Z(\wvec{\vec 1_m + \vec{e}^{(m)}_i, \vec s\,'}{c \vec 1_m + c\vec{e}^{(m)}_i, \vec r\,'};
\vep)
+\sum _{j=1}^\ell Z(\wvec{\vec 1_m, \vec s\,'+\vec{e}^{(\ell)}_j}{c\vec 1_m, \vec r \,'+c \vec{e}^{(\ell)}_j}; \vep).$$ By the induction hypothesis, all terms in the two sums on the right hand side are of the form $f(X)+O(\vep)$ with $f$ polynomials in $X$ of degree $\leq m$. Thus by Eq. () and the induction hypothesis, we obtain $$Z(\wvec{\vec 1_{m+1}, \vec s\,'}{c \vec 1_{m+1},\vec r\,'};\vep)
=P_{m+1,\vec{s}\,'}(X) + O(\vep),$$ where $P_{m+1,\vec{s}\,'}(X)$ has degree $\deg P_{m,\vec{s}\,'}+1=m+1$ and has leading coefficient the leading coefficient of $P_{m,\vec{s}\,'}(X)X/(m+1)$, which is $\zeta (\vec s\,')/(m+1)!$ if $\ell>1$ and is $1/(m+1)!$ if $\ell=0$. This completes the induction.

Let 