---
abstract: 'Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at non-positive integers since the values are usually singular. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of Ihara-Kaneko-Zagier on renormalization of MZVs with positive arguments. We further show that the important quasi-shuffle (stuffle) relation for usual MZVs remains true for the renormalized MZVs.'
address:
- 'Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102'
- 'Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany; Yangtze Center of Mathematics, Sichuan University, Chengdu, 610064, P. R. China '
author:
- Li Guo
- Bin Zhang
title: Renormalization of multiple zeta values
---

Introduction
============

Multiple zeta values (MZVs), as we know in the current literature, are defined to be the values of the multi-variable analytic function, called the [**multiple zeta function**]{}, $$\zeta(s_1,\cdots, s_k)=\sum_{n_1>\cdots>n_k>0}
    \frac{1}{n_1^{s_1}\cdots n_k^{s_k}}
\mlabel{eq:mzv}$$ at positive integers $s_1,\cdots,s_k$ with $s_1>1$. With the earliest study of MZVs went back to Euler when $k=2$, their systematic study started in early 1990s with the works of Hoffman  and Zagier . Since then MZVs and their generalizations have been studied extensively by numerous authors from different point of views with connections to arithmetic geometry, mathematical physics, quantum groups and knot theory .

In comparison, little is known about special values of multiple zeta functions at integers that are not all positive. Through the recent work of Zhao  and Akiyama-Egami-Tanigawa  (see also ), we know that $\zeta(s_1,\cdots,s_k)$ can be meromorphically continued to $\CC^k$ with singularities on the subvarieties $$s_1=1; \
s_1+s_2=2,1,0,-2,-4, \cdots; {\rm\ and\ } \mlabel{eq:pole}
\sum_{i=1}^j
s_{i} \in \ZZ_{\leq j}\ (3\leq j\leq k).$$ Thus $\zeta(s_1,\cdots,s_k)$ is undefined at most points with non-positive arguments. In , several definitions were proposed for the non-positive MZVs, that is, the values of $\zeta(s_1,\cdots,s_k)$ when $s_i$ are all non-positive. Some of them are $$\lim_{r_1\to s_1}\cdots\lim_{r_k\to s_k} \zeta(r_1,\cdots,r_k),
\lim_{r_k\to s_k}\cdots\lim_{r_1\to s_1} \zeta(r_1,\cdots,r_k),
\lim_{r\to 0} \zeta(s_1+r,\cdots,s_k+r).$$ As expected they give different values. Some good properties of the variously defined non-positive MZVs were obtained in the these papers. But they fell short of the analogous properties of the usual MZVs, especially the double shuffle relations.

In this paper, we adapt a renormalization procedure (dimensional regularization plus minimal subtraction) in quantum field theory (QFT) to define the values of multiple zeta functions $\zeta(s_1,\cdots,s_k)$ at $(s_1,\cdots,s_k)$ when $s_i$, $1\leq i\leq k$, are all non-positive or all positive, that we expect to further extend to when $s_i$ are [*arbitrary integers*]{}. For our purpose, the dimensional regularization of Feynman integrals is replaced by a regularization (or deformation) of infinite series that has occurred in the study of Todd classes for toric varieties .

The renormalization procedure of QFT was put in the framework of Hopf algebra and Rota-Baxter algebra by the recent works of Connes and Kreimer , continued in , and thus made possible for applications beyond QFT. A fundamental result in this framework is the Algebraic Birkhoff Decomposition (Theorem ). It states that for a given triple $(\calh, R, \phi)$ consisting of

-   a connected filtered Hopf algebra $\calh$,

-   a commutative Rota-Baxter algebra $R$ on which the Rota-Baxter operator $P:R\to R$ is idempotent, and

-   an algebra homomorphism $\phi: \calh\to R$,

there are unique algebra homomorphisms $
\phi_-: \calh\to \CC+P(R)$ and $\phi_+: \calh\to \CC+(\id-P)(R)$ such that $$\phi=\phi_-^{\star (-1)}\star \phi_+.
\mlabel{eq:abd}$$ Here $\star$ is the convolution product and $\phi_+$ is called the [**renormalization**]{} of $\phi$. This algebraic setup is reviewed in Section  together with a discussion of quasi-shuffle algebras.

To apply this setup to the renormalization in QFT, one takes

-   $\calh=\calh_\FG$ to be the Connes-Kreimer Hopf algebra of Feynman diagrams, parameterizing regularized Feynman integrals,

-   $R=\CC[[\vep,\vep^{-1}]$ to be the Rota-Baxter algebra of Laurent series, and

-   $\phi$ to be the the regularized Feynman rule that assigns a Feynman diagram to the Laurent series expansion of the corresponding regularized Feynman integral.

Then the renormalized values of a Feynman integral is given by $\phi_+(\Gamma)$, where $\Gamma$ is the corresponding Feynman graph, when $\vep$ approaches zero. For further details see .

To apply this setup to our study of renormalized MZVs, we similarly define

-   $\calh$ to be the quasi-shuffle Hopf algebra parameterizing regularized MZVs,

-   $R$ to be the Rota-Baxter algebra $\CC[T][[\vep,\vep^{-1}]$ of log Laurent series, and

-   $\phi$ to be the algebra homomorphism sending a symbol in $\calh$ to the Laurent series expansion of the corresponding regularized MZV.

Once these are obtained in Section , the Algebraic Birkhoff Decomposition in Eq. () applies to give the renormalization $\phi_{+}$ from which the renormalized MZVs can be derived when $\vep$ goes to zero, as in the QFT case. However, there is an important difference from the QFT renormalization: in order to equip the regularized MZVs and the corresponding Hopf algebra with a suitable algebra structure that reflects the quasi-shuffle relation of the regularized MZVs, an extra parameter vector $\vec{r}$ has to be introduced in the regularized sums in addition to $\vep$. Thus the renormalized MZVs at $\vec{s}$ from the Algebraic Birkhoff Decomposition depend on $\vec{r}$, resulting in the [**renormalized directional MZVs**]{} $\zeta(\wvec{\vec{s}}{\vec{r}})$ in Definition .

This dependency on $\vec{r}$ is removed in the following Section  in a consistent manner, giving the [**renormalized MZVs**]{} $\gzeta(\vec{s})$ in Definition . Our main result Theorem  shows that the renormalized MZVs satisfy the quasi-shuffle (or stuffle) relation, and include as special cases the MZVs defined either by convergence, by analytic continuation, or by regularization in the sense of Ihara-Kaneko-Zagier . Parts of the proof are postponed to Section  and Section . Here is the hierarchy of MZVs introduced in this paper: $$\begin{aligned}
& \mbox{formal MZVs}\ \zeta(\vec{s}) \to \mbox{directional regularized MZVs}\ Z(\wvec{\vec{s}}{\vec{r}};\vep) &\\
& \to \mbox{renormalized directional MZVs}\ \zeta(\wvec{\vec{s}}{\vec{r}}) \to \mbox{renormalized MZVs}\ \gzeta(\vec{s}) &\end{aligned}$$ The concepts of regularization and renormalization have already been introduced to the study of MZVs by Ihara-Kaneko-Zagier  to take care of the divergency of the MZVs $\zeta(s_1,\cdots,s_k)$ with $s_1=1$. As a part of their process, the natural algebra homomorphism from the the quasi-shuffle algebra for convergent MZVs to the algebra of convergent MZVs is extended to an algebra homomorphism from a larger quasi-shuffle algebra to an extension of the algebra of convergent MZVs. Thus they obtained their extended MZVs as an [**algebraic continuation**]{} (we thank Robert Sczech for suggesting this term) in the sense that their extended MZVs preserves the quasi-shuffle relation. From this point of view, we obtain our renormalized MZVs as an algebraic continuation that goes beyond theirs to cover the MZVs with all non-positive arguments. In a weak sense it covers arbitrary arguments (Definition ). More recently, Manchon and Paycha  have considered renormalization of MZVs from the point of view of Chen integrals and Chen sums of symbols using a similar renormalization approach in the spirit of Connes and Kreimer. The two approaches should be related though the exact link is still not clear.

This paper should lead to further studies of MZVs with arbitrary arguments. First we can consider questions related to the renormalization procedure, such as the renormalization of MZVs with arbitrary arguments, the dependence of renormalized MZVs on the regularization and renormalization. We would also like to study the extension of the double shuffle relation to renormalized MZVs, and the possible connection to rational associators in the sense of Drinfel’d  and DMR in the sense of Racinet . Possible arithmetic properties of these renormalized MZVs, such as the Kummer type congruences, are also interesting to investigate. Some of these directions will be pursued in future works.

[**Acknowledgements.**]{} Both authors thank the Max-Planck Institute for Mathematics in Bonn for the stimulating environment where this project was started, and thank Matilde Marcolli for encouragement. We also thank Dominique Manchon and Sylvie Paycha for communications on their papers  and for their comments on our paper. The first named author thanks NSF for support, and is indebted to Kurusch Ebrahimi-Fard and Dirk Kreimer for their collaborations on QFT that inspired the renormalization approach in this paper. Thanks also go to Herbert Gangl, Robert Sczech and Jianqiang Zhao for discussions and comments.

The algebraic setup
===================

We describe the general setup for our later applications to renormalization of MZVs. In the following an algebra means a $\bfk$-algebra where $\bfk$ is a unitary commutative ring that we usually take to be $\CC$. Denote the unit of $\bfk$ by $\bfone$.

The algebraic Birkhoff decomposition
------------------------------------

We review the algebraic framework of Connes and Kreimer for renormalization of perturbative quantum field theory.

A [**connected filtered Hopf algebra**]{} is a Hopf algebra $(H, \Delta)$ with $\bfk$-submodules $H^{(n)},\ n\geq 0$ of $H$ such that $$\begin{aligned}
&H^{(n)}\subseteq H^{(n+1)}, \quad
\cup_{n\geq 0} H^{(n)} = H, \quad
H^{(p)} H^{(q)}\subseteq H^{(p+q)}, &\\
&
\Delta(H^{(n)}) \subseteq \sum_{p+q=n} H^{(p)}\otimes H^{(q)}, \quad
H^{(0)}=\bfk {\rm \ (connectedness)}.&\end{aligned}$$ Let $\lambda\in \bfk$. A [**Rota–Baxter algebra**]{} of weight $\lambda$ is a pair $(R,P)$ where $R$ is a unitary $\bfk$-algebra and $P:R\to R$ is a linear operator such that $$P(x)P(y)=P(xP(y))+P(P(x)y)+\lambda P(xy),
\mlabel{eq:rbe}$$ for any $x,\,y\in R$. Often $\theta=-\lambda$ is used, especially in the physics literature. It follows from the definition that $P(R)$ and $(-\lambda-P)(R)$ are non-unitary subalgebras of $R$. So $\bfk+P(R)$ and $\bfk +(-\lambda-P)(R)$ are unitary subalgebras.

Let $H$ be a commutative connected filtered Hopf algebra. Let $(R,P)$ be a Rota-Baxter algebra of weight $-1$. Let $\phi: H \to R$ be an algebra homomorphism.

1.  There are algebra homomorphisms $\phi_-: H \to \bfk+P(R)$ and $\phi_+: H \to \bfk+(1-P)(R)$ with the decomposition $$\phi=\phi_-^{\star\, (-1)}\star \phi_+,
    \mlabel{eq:phidecom}$$ called the [**Algebraic Birkhoff Decomposition**]{} of $\phi$. Here $\star$ is the convolution product and $\phi_-^{\star\, (-1)}$ is the inverse of $\phi_-$ with respect to $\star$. Further, $$\phi_-(x)=-P\big(\phi(x)+\sum_{(x)} \phi_-(x')\phi(x'')\big)
    \mlabel{eq:phi-}$$ and $$\phi_+(x)=(\id-P)\big(\phi(x)+\sum_{(x)} \phi_-(x')\phi(x'')\big).
    \mlabel{eq:phi+}$$ Here we have used the notation $\Delta(x)=x\ot 1 + 1\ot x + \sum_{(x)} x'\ot x''.$

2.  If $P^2=P$, then the decomposition in Eq. () is unique.

For item (), see [@C-K1] and [@Ma Theorem II.5.1]. For item (), see [@EGK3 Theorem 3.7] where one can also find a proof of item () using Rota-Baxter algebras.

Quasi-shuffle algebras
----------------------

Let $M$ be a commutative semigroup. For each integer $k\geq 0$, let $\bfk M^k$ be the free $\bfk$-module with basis $M^k$, with the convention that $M^0=\{\bfone\}$. Let $$\calh_M=\bigcup_{k=0}^\infty \bfk\, M^k.
 \mlabel{eq:qprod}$$ Following , define the [**quasi-shuffle product**]{} $\msh$ by first taking $\bfone$ to be the multiplication identity. Next for any $m,n\geq 1$ and $\vec a:=(a_1,\cdots , a_m)\in M^{m}$ and $\vec b:=(b_1, \cdots, b_n)\in M^{n}$, denote $\vec a\,'=(a_2,\cdots,a_m)$ and $\vec b\,'=(b_2,\cdots,b_n)$. Recursively define $$\vec{a} \msh \vec{b} =\big(a_1,   \vec{a}\,'\msh \vec b\big )
  + \big(b_1, \vec a \msh \vec{b}\,' \big) \\
  + \big(a_1 b_1,   \vec{a}\,' \msh \vec{b}\,'\big)
\mlabel{eq:qshuf}$$ with the convention that $\vec{a}\,'=\bfone$ if $m=1$, $\vec{b}\,'=\bfone$ if $n=1$ and $(a_1b_1,\vec{a}\,'\msh \vec{b}\,')=(a_1b_1)$ if $m=n=1$.

Quasi-shuffle is also known as harmonic product  and coincides with the stuffle product  in the study of MZVs. Variations of the stuffle product have also appeared in  . See §  for further details. It is shown  to be the same as the mixable shuffle product  which is also called overlapping shuffles  and generalized shuffles , and can be interpreted in terms of Delannoy paths .

The following theorem is a simple generalization of [@Ho2 Theorem 2.1, 3.1] where $M$ has the extra condition of being a locally finite set to ensure the grading structure on $\calh_M$.

Let $M$ be a commutative semigroup. Equip $\calh_M$ with the submodules $\calh_M^{(n)}=\oplus_{i=0}^n \bfk\, M^i$, The quasi-shuffle product $\msh$, the deconcatenation coproduct $$\begin{aligned}
&&\Delta: \calh_M \to \calh_M \barot \calh_M,\\
 \Delta( a_1, \cdots ,  a_k)&=&1\barot (a_1,  \cdots ,  a_k)
+ \sum_{i=1}^{k-1} (a_1,  \cdots,  a_i)\barot (a_{i+1},  \cdots ,  a_k) \notag\\
&&+ (a_1,  \cdots ,  a_k) \barot 1
\mlabel{eq:coprod}\end{aligned}$$ and the projection counit $\vep: \calh_M \to \bfk$ onto the direct summand $\bfk \subseteq \calh_M$. Then $\calh_M$ is a commutative connected filtered Hopf algebra.

By the same proofs as [@Ho2 Theorem 2.1] and [@Ho2 Theorem 3.1], $\calh_M$ is a bialgebra. By the definition of $\msh$ and $\Delta$, $\calh_M$ is connected filtered with the submodules $\calh_M^{(n)}, n\geq 0$. Then $\calh_M$ is automatically a Hopf algebra by  [@F-G Proposition 5.3], for example.

We prove the following property for later applications.

For $k\geq 1$, let $\Sigma_k$ be the permutation group on $\{1,\cdots,k\}$. For $\vec{a}=(a_1,\cdots,a_k)\in M^k \subseteq \calh_M$, define $\sigma(\vec{a})=(a_{\sigma(1)},\cdots,a_{\sigma(k)})$ and define $\vec{a}^{(\Sigma_k)}=\sum_{\sigma\in \Sigma_k} \sigma(\vec{a}).$ Then for $a_{k+1}\in M$, we have $$\vec{a}^{(\Sigma_k)} \msh (a_{k+1}) = (a_1,\cdots,a_k,a_{k+1})^{(\Sigma_{k+1})} + \sum_{i=1}^k (a_1,\cdots,a_ia_{k+1},\cdots,a_{k})^{(\Sigma_k)}
\mlabel{eq:permshuf}$$ where in the sum, $\sigma(a_1,\cdots,a_ia_{k+1},\cdots,a_k)=(a_{\sigma(1)},\cdots,a_{\sigma(i)}a_{k+1},\cdots,a_{\sigma(k)}).$

By the quasi-shuffle relation in Eq. (), we have $$\begin{aligned}
\lefteqn{\vec{a}^{(\Sigma_k)} \msh (a_{k+1})= \sum_{\sigma\in
\Sigma_k}
\Big((a_{\sigma(1)},\cdots,a_{\sigma(k)},a_{k+1})}\\
&& + \sum_{i=1}^k (a_{\sigma(1)},\cdots,a_{\sigma(i)},a_{k+1},a_{\sigma(i+1)},\cdots,a_{k})
 + \sum_{i=1}^k (a_{\sigma(1)},\cdots,a_{\sigma(i)}a_{k+1},\cdots,a_{\sigma(k)})\Big),\end{aligned}$$ hence the proposition. It also follows from the Partition Identity of Hoffman  whose proof only needs the quasi-shuffle relation , or the Bohnenblust-Spitzer formula for Rota-Baxter algebras .

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

We now introduce directional regularized MZVs and the corresponding Hopf algebra. We then show that the directional regularized MZVs have Laurent series expansion with log coefficients, giving an algebra homomorphism from the Hopf algebra to Laurent series. This allows us to apply the algebraic Birkhoff decomposition in Theorem  to obtain renormalized directional MZVs.

The Hopf algebra of directional regularized multiple zeta values
----------------------------------------------------------------

We consider the commutative semigroup $$\frakM= \{{{\wvec{s}{r}}}\ \big|\ (s,r)\in \ZZ \times \RR_{>0}\}
\mlabel{eq:mbase}$$ with the multiplication $ {\wvec{s}{r}} {\wvec{s'}{r'}}={\wvec{s+s'}{r+r'}}.$ By Theorem , $$\calh_{\frakM}:=\sum_{k\geq 0} \CC\, \frakM^k,$$ with the quasi-shuffle product $\msh$ and the deconcatenation coproduct $\Delta$, is a connected filtered Hopf algebra. The same is true with the sub-semigroup $$\frakM^-=\{{\wvec{s}{r}}\ |\ (s,r)\in \ZZ_{\leq 0} \times \RR_{>0}\}.$$ For $w_i=\wvec{s_i}{r_i}\in \frakM,\ i=1,\cdots,k$, we use the notations $$\vec{w}=(w_1,\cdots,w_k)
=\wvec{s_1,\cdots,s_n}{r_1,\cdots,r_k}=\wvec{\vec{s}}{\vec{r}},\
{\rm where\ } \vec{s}=(s_1,\cdots,s_k), \vec{r}=(r_1,\cdots,r_k).$$ For $\vep\in \CC$ with ${\rm
Re}(\vep)<0$, define the [**directional regularized MZV**]{}: $$Z(\wvec{\vec{s}}{\vec{r}};\vep)=\sum_{n_1>\cdots>n_k>0}
\frac{e^{n_1\,r_1\vep} \cdots
    e^{n_k\,r_k\vep}}{n_1^{s_1}\cdots n_k^{s_k}}
\mlabel{eq:reggmzv}$$ It converges for any $\wvec{\vec{s}}{\vec{r}}$ and is regarded as the regularization of the [**formal MZV**]{} $$\zeta (\vec{s})= \sum_{n_1>\cdots>n_k>0} \frac{1}{n_1^{s_1} \cdots
    n_k^{s_k}}
\mlabel{eq:formgmzv}$$ which converges only when $s_i>0$ and $s_1>1$. It is related to the multiple polylogarithm $${\rm Li}_{s_1,\cdots,s_k}(z_1,\cdots,z_k)=\sum_{n_1>\cdots n_k>0}
    \frac{z_1^{n_1} \cdots z_k^{n_k}}{n_1^{s_1}\cdots n_k^{s_k}}$$ by a change of variables $z_i=e^{r_i\vep}, 1\leq i\leq k$. As is well-known , the product of multiple polylogarithms as functions satisfies the quasi-shuffle (stuffle) relation of the nested sums. Therefore the product of regularized MZVs as functions also satisfies the quasi-shuffle relation: if $ \wvec{\vec{s}}{\vec{r}}\msh \wvec{\vec{s}\,'}{\vec{r}\,'}
=\sum \wvec{\vec{s}\,''}{\vec{r}\,''}$, then $$Z(\wvec{\vec{s}}{\vec{r}};\vep)Z(\wvec{\vec{s}\,'}{\vec{r}\,'};\vep)
= Z(\wvec{\vec{s}}{\vec{r}}\msh \wvec{\vec{s}\,'}{\vec{r}\,'};\vep)
:= \sum Z(\wvec{\vec{s}\,''}{\vec{r}\,''};\vep).
\mlabel{eq:qsh1}$$ We thus obtained an algebra homomorphism $$Z: \calh_\frakM \to \sum_{{\wvec{\vec{s}}{\vec{r}}\in \cup_{n\geq 0} \frakM^n}}\
    \CC\, Z(\wvec{\vec{s}}{\vec{r}};\vep),
    \wvec{\vec{s}}{\vec{r}} \mapsto Z(\wvec{\vec{s}}{\vec{r}};\vep).
\mlabel{eq:regz}$$ With this map, $\calh_\frakM$ is a parametrization of the directional regularized MZVs that also reflects their multiplication property.

Log Laurent series of directional regularized multiple zeta values
------------------------------------------------------------------

We first construct Laurent series with log coefficients. We then show that the nested sums from directional regularized MZVs are such log Laurent series.

Let $\CC\{\{\vep,\vep^{-1}\}$ be the algebra of convergent Laurent series, regarded as a subalgebra of the algebra of (germs of) complex valued functions meromorphic in a neighborhood of $\vep=0$. Take $\ln \vep$ to be analytic on $\CC\backslash (-\infty,0]$.

$\ln (-\vep) $ is transcendental over $\CC\{\{\vep,\vep^{-1}\}$.

We give a simple proof for the lack of references. Assume $\ln (-\vep) $ is algebraic over the field $\CC\{\{\vep,\vep^{-1}\}$ with the monic minimal polynomial $$\ln ^n(-\vep)+a_{n-1}(\vep)\ln ^{n-1}(-\vep)+\cdots +a_0(\vep)=0.$$ Differentiating the above equation, we have $$\sum _{i=0}^n \big(a'_i(\vep )\ln ^i(-\vep)+\frac i{\vep}a_i(\vep)\ln ^{i-1}(-\vep )\big)=0.$$ The highest power term in $\ln (-\vep)$ is $(\frac
n{\vep}+a'_{n-1}(\vep ))\ln ^{n-1}(-\vep)$. Because of the minimality, $n/\vep+a'_{n-1}(\vep )$ has to be 0, which is impossible for $a_{n-1}(\vep )\in \CC\{\{\vep,\vep^{-1}\}$.

$\CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$ is closed under the differential operator $d/d\vep$. It is also closed under the indefinite integral operator: the antiderivatives of any $f\in \CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$ are in $\CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$.

Let $f(\vep)\in \CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$. Then $$f(\vep)=\sum_{n=0}^M a_n(\vep)\ln^n (- \vep)
=\sum_{n=0}^M \big(\sum_{k\geq N_n} a_{n,k}\vep^k \ln^n(-\vep)\big)$$ with $\sum_{k\geq N_n} a_{n,k}\vep^k \in \CC\{\{\vep,\vep^{-1}\}$. For each $0\leq n\leq M$, the series for the inside sum converges absolutely and uniformly in a nonempty open interval of $\{\vep\in \CC\,\big |\, -\infty<\vep <0\}$. Thus the series can be differentiated and integrated term by term. Thus we only need to show that the derivative and anti-derivatives of $\vep^k \ln ^n (-\vep)$, $k\in \ZZ, n\in \ZZ_{\geq 0}$ are linear combinations of functions of the same form. This is easy to check for derivatives.

For anti-derivatives, we use induction on $n$. It is clear when $n=0$. The induction step follows from the integration by parts formula $$\int \vep^k \ln ^n (-\vep) d\vep = \frac{1}{k+1}\vep^{k+1} \ln ^n (-\vep)
+ \frac{n}{k+1} \int  \vep^{k} \ln ^{n-1} (-\vep) d\vep$$ when $k\neq -1$ and $ \int \frac{\ln ^n (-\vep)}{\vep}d\vep = \frac{1}{n+1} \ln ^{n+1} (-\vep)+C.$

Because of Lemma , we have $$\CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]\cong
\CC\{\{\vep,\vep^{-1}\} [T]
\hookrightarrow \CC[[\vep,\vep^{-1}][T]
\mlabel{eq:lau2}$$ sending $-\ln (-\vep)$ to $T$. Here $\CC[[\vep,\vep^{-1}][T]$ denotes the polynomial algebra over the formal Laurent series $\CC[[\vep,\vep^{-1}]$.

An element of $\CC[[\vep,\vep^{-1}] [T]$ is of the form $\sum_{n=0}^M a_n(\vep)T^n$ with $$a_n(\vep)=\sum_{k\geq N_n} a_{n,k}\vep^k \in \CC[[\vep,\vep^{-1}], 0\leq n\leq M.$$ Taking $N=\min_{0\leq n\leq M} N_n$ and letting $a_{n,k}=0$ for $N\leq k< N_n$, we have $$\sum_{n=0}^M a_n(\vep)T^n = \sum_{n=0}^M \big(\sum_{k\geq N} a_{n,k}\vep^k\big) T^n =\sum_{k\geq
N} \big(\sum_{n=0}^M a_{n,k} T^n\big) \vep^k.$$ This gives an element of the [**algebra of log Laurent series**]{} $\CC[T][[\vep,\vep^{-1}]$ with coefficients in $\CC[T]$. Combining with Eq. (), we obtain a natural algebra injection $$u: \CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]  \to \CC[T][[\vep,\vep^{-1}]
\mlabel{eq:formulau}$$ with which we identify $\CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$ as a subalgebra of $\CC[T][[\vep,\vep^{-1}]$.

For any $\vec s\in \ZZ ^k$, $\vec r \in {\mathbb}{Z}_{>0}^k$, $Z(\wvec{\vec s}{\vec r};\vep)$ is in $\CC \{\{\vep,\vep^{-1}\}[ln(-\vep)]$ and can thus be regarded as an element in $\CC[T][[\vep,\vep^{-1}]$ by Eq. (). If $\vec s$ is in ${\mathbb}{Z}_{\le 0}^k$, then $Z(\wvec{\vec s}{\vec r};\vep)$ is in $\CC\{\{\vep,\vep^{-1}\}$.

First notice that, for $r,i\in \ZZ_{>0}$, $$\sum _{n\geq i}e^{nr\vep }=\frac {1}{1-e^{r\vep}}\,e^{ir\vep}
 \mlabel{eq:geo}$$ has a Laurent series expansion at $\vep=0$. Since $Z(\wvec{s}{r};\vep)$ is uniformly convergent on compact subsets in $Re(\vep)<0$, by repeatedly differentiating Eq. (), we see that, for $s\in \ZZ_{<0}$, $$\begin{aligned}
\sum _{n\geq i}n^{-s}e^{nr\vep}
&=& r^s\Big( \sum_{p=0}^{-s} \binc{-s}{p} \big(\frac{1}{1-e^{r\vep}}\big)^{(p)}(e^{ir\vep})^{(-s-p)}\Big) \notag\\
&=&\sum _{p=0}^{-s}\binc{-s}{p}
\big(\frac{1}{1-e^{r\vep}}\big)^{(p)}
\frac{e^{ir\vep}}{r^{p}\,i^{s+p}}
\mlabel{eq:diffsum}\end{aligned}$$ has a Laurent series expansion at $\vep=0$.

Now we prove by induction on $k\geq 1$. Let $k=1$. Then $\vec s=s\in \ZZ$. The case when $s\leq 0$ follows from Eq. () with $i=1$. When $s>0$, we note that $Z'(\wvec{s}{r};\vep)=rZ(\wvec{s-1}{r};\vep)$. By Eq. (), $Z(\wvec{0}{r};\vep)$ is in $\CC\{\{\vep,\vep^{-1}\}[\ln(-\vep)]$ which is closed under integration (Lemma ). Thus $Z(\wvec{1}{r};\vep)$ is in $\CC\{\{\vep,\vep^{-1}\}[\ln(-\vep)]$ and, by an induction on $s$, the same holds for $Z(\wvec{s}{r};\vep)$ for any $s>0$.

Assume that the statements hold for $k\geq 1$ and prove for $Z(\wvec{\vec s}{\vec r}; \vep)$ with $\vec s=(s_1,\cdots,s_{k+1})$ in two cases.

[**Case 1.** ]{} Suppose $s_i\leq 0$ for some $1\leq i\leq k+1$. Then for fixed $n_{i-1}>n_{i+2}+1>0$, $$\begin{aligned}
\lefteqn{
\sum_{n_{i-1}>n_i>n_{i+1}>n_{i+2}}
\frac{e^{n_ir_i\vep+n_{i+1}r_{i+1}\vep}}{n_i^{s_i}n_{i+1}^{s_{i+1}}}}\\
&=& \sum_{n_{i-1}>n_i\geq n_{i+1}>n_{i+2}}
\frac{e^{n_ir_i\vep+n_{i+1}r_{i+1}\vep}}{n_i^{s_i}n_{i+1}^{s_{i+1}}}
-\sum_{n_{i-1}> n_i=n_{i+1}>n_{i+2}}
\frac{e^{n_ir_i\vep+n_{i+1}r_{i+1}\vep}}{n_i^{s_i}n_{i+1}^{s_{i+1}}}\\
&=& \sum_{n_{i-1}>n_{i+1}>n_{i+2}}
\frac{e^{n_{i+1}r_{i+1}\vep}}{n_{i+1}^{s_{i+1}}}
\sum_{n_i=n_{i+1}}^{n_{i-1}-1}\frac{e^{n_ir_i\vep}}{n_i^{s_i}}
-\sum_{n_{i-1}> n_{i+1}>n_{i+2}}
\frac{e^{n_{i+1}(r_i+r_{i+1})\vep}}{n_{i+1}^{s_i+s_{i+1}}}.\end{aligned}$$ Applying Eq. () to the inner sum of the first term, we have Thus Then we have The induction hypothesis applies to each term on the right hand side, completing the induction on $k$ in this case. In particular this completes the induction when $\vec{s}\in \ZZ^{k+1}_{\leq 0}$.

[**Case 2.**]{} Suppose $s_i>0$ for all $1\leq i\leq k+1$. We use induction on the sum $s:=\sum_{i=1}^{k+1} s_i$. Then $s\geq
k+1$. If $s=k+1$, then $s_i=1$ for $1\leq i\leq k+1$. Note that $$\frac {d}{d\vep } Z(\wvec{\vec s}{\vec r};\vep)=\sum r_i Z(\wvec{\vec s-\vec e _i}{\vec r};\vep),
\mlabel{eq:zdiff}$$ where $\vec e_i, 1\leq i\leq k+1$ is the $i$-th unit vector in $\ZZ ^{k+1}$. Each term on the right hand side is in $\CC\{\{\vep,\vep^{-1}\}[\ln(-\vep)]$ by Case 1. So by Lemma , $Z(\wvec{\vec s}{\vec r};\vep)$ is in $\CC[T][[\vep,\vep^{-1}]$. The inductive step follows from Eq. () and the induction assumption.

Renormalized directional MZVs
-----------------------------

Combining Eq. (), Theorem  and Eq. (), we obtain an algebra homomorphism $$\uni{Z}: \calh_\frakM \to \CC\{\{\vep,\vep^{-1}\}[\log (-\vep)] \ola{u} \CC[T][[\vep,\vep^{-1}],\ \wvec{\vec{s}}{\vec{r}} \mapsto
    u\big(Z(\wvec{\vec{s}}{\vec{r}};\vep)\big).
\mlabel{eq:zmap}$$ In the same way, $\uni{Z}$ restricts to an algebra homomorphism $$\uni{Z}: \calh_{\frakM^-} \to \CC[[\vep,\vep^{-1}].$$ For any commutative $\bfk$-algebra $K$, $K[[\vep,\vep^{-1}]$ is a Rota-Baxter algebra of weight -1 with the Rota-Baxter operator $P$ to be the projection to $\vep^{-1} K [\vep^{-1}]$: $$P\big ( \sum_{n\geq N} \alpha_k \vep^k\big )=\sum_{k\leq -1}\alpha_k \vep^k.
 \mlabel{eq:prb}$$ This can be directly verified as with the well-known case of $\CC[[\vep,\vep^{-1}]$ in .

Thus we can apply the Algebraic Birkhoff Decomposition in Theorem  and obtain

We have $$\uni{Z}=\uni{Z}_-^{-1} \star \uni{Z}_+$$ and the map $\uni{Z}_+: \calh_\frakM\to \CC[T][[\vep]]$ is an algebra homomorphism which restricts to an algebra homomorphism $\uni{Z}_+: \calh_{\frakM^-}\to \CC[[\vep]]$.

Because of Corollary , the following definition is valid.

For $\vec{s}=(s_1,\cdots,s_k)\in \ZZ^k$ and $\vec{r}=(r_1,\cdots,r_k)\in \RR_{>0}^k$, define 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 $\vec{r}=(r_1,\cdots,r_k)\in \NN_{>0}^k$, $$\begin{aligned}
\lefteqn{{\ }\hspace{-.7cm} \uni{Z}_-\lp \wvec{\vec s}{\vec r};\vep\rp
= \hspace{-.5cm}
\sum_{(i_1,\cdots,i_p)\in \Pi_k}
\check{P}\llp
    \llp \cdots \check{P}\llp\check{P}\llp \uni{Z}\lp \wvec{\vec s ^{(1)}}{\vec r ^{(1)}};\vep\rp\rrp\,
    \uni{Z}\lp\wvec{\vec s^{(2)}}{\vec r ^{(2)}};\vep\rp \rrp \cdots\rrp
    \uni{Z}\lp\wvec{\vec s^{(p)}}{\vec r ^{(p)}};\vep\rp \rrp. } \notag \\
\lefteqn{{\ }\hspace{-.7cm}  \uni{Z}_+\lp \wvec{\vec s}{\vec r};\vep\rp
= \hspace{-.5cm}
\sum_{(i_1,\cdots,i_p)\in
\Pi_k}\tilde{P}\llp
    \llp \cdots \check{P}\llp\check{P}\llp \uni{Z}\lp \wvec{\vec s ^{(1)}}{\vec r ^{(1)}};\vep\rp\rrp\,
    \uni{Z}\lp\wvec{\vec s^{(2)}}{\vec r ^{(2)}};\vep\rp \rrp \cdots\rrp
    \uni{Z}\lp\wvec{\vec s^{(p)}}{\vec r ^{(p)}};\vep\rp \rrp } \notag \\
&=& \hspace{-.7cm}
\sum_{(i_1,\cdots,i_p)\in\Pi_k}\hspace{-.6cm}
\tilde{P}\llp \uni{Z}\lp\wvec{\vec s^{(p)}}{\vec r ^{(p)}};\vep\rp
    \check{P}\llp \uni{Z}\lp\wvec{\vec s^{(p-1)}}{\vec r ^{(p-1)}};\vep\rp
    \cdots \check{P}\llp\uni{Z}\lp\wvec{\vec s ^{(1)}}{\vec r ^{(1)}};\vep\rp\rrp\,
     \cdots \rrp
    \rrp.
\mlabel{eq:plus}\end{aligned}$$ \[thm:expform\]

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
-------------------------------

For $\vec{s}\in \ZZ_{> 0}^k\cup \ZZ_{\leq 0}^k$, define the [**renormalized MZV**]{} at $\vec{s}$ to be $$\gzeta\lp \vec{s}\rp
= \lim_{\delta \to 0^+} \zeta\lp \wvec{\vec{s}}{|\vec{s}|+\delta}\rp ,
\mlabel{eq:gmzv}$$ where, for $\vec s=(s_1,\cdots,s_k)$ and $\delta\in \RR_{>0}$, we denote $|\vec{s}|=(|s_1|,\cdots,|s_k|)$ and $|\vec s|+\delta=(|s_1|+\delta,\cdots,|s_k|+\delta).$

[Theorem  below is our main theorem. It shows that our renormalized MZVs is well-defined and is compatible with known MZVs defined by either convergence, analytic continuation or the Ihara-Kaneko-Zagier regularization. It also proves that it satisfies the quasi-shuffle relation. We are optimistic that this is in fact the only definition of $\gzeta(\vec{s})$ from $\zeta(\wvec{\vec s}{\vec r})$ with these properties and will elaborate on it in a subsequent work. ]{}

The limit in Eq. () exists for any $\vec{s}=(s_1,\cdots,s_k)\in \ZZ_{> 0}^k\cup \ZZ_{\leq 0}^k$. More precisely,

1.  when $s_i$ are all positive with $s_1>1$, we have $\zeta\lp \wvec{\vec{s}}{\vec{r}}\rp =\zeta(\vec{s})$ independent of $\vec{r}\in \ZZ_{>0}^k$. In particular, $\gzeta(\vec{s})=\zeta(\vec{s})$;

2.  when $s_i$ are all positive, we have $\gzeta(\vec{s})=\zeta\lp \wvec{\vec{s}}{\vec{s}}\rp $. Further, $\gzeta(\vec{s})$ agrees with the regularized MZV $Z^*_{\vec{s}}(T)$ defined by Ihara-Kaneko-Zagier ;

3.  when $s_i$ are all negative, we have $\gzeta(\vec{s})=\zeta\lp \wvec{\vec{s}}{-\vec{s}}\rp =\disp{\lim_{\vec{r}\to -\vec{s}} \zeta\lp \wvec{\vec{s}}{\vec{r}}\rp} ;$

4.  when $s_i$ are all non-positive, we have $\gzeta(\vec{s})= \disp{\lim _{\vec r\to -\vec s}\zeta \lp \wvec{\vec s}{\vec
    r}\rp ^{(\Sigma)}}$ where the right hand side is defined in Theorem . Further, $\gzeta(\vec{s})$ agrees with $\zeta(\vec{s})$ whenever the later is defined by analytic continuation.\
    [ ${}$ ]{} Furthermore,

5.  the set $\{\gzeta(\vec{s})\big| \vec{s}\in \ZZ^k_{>0}\}$ satisfies the quasi-shuffle relation;

6.  the set $\{\gzeta(\vec{s})\big| \vec{s}\in \ZZ^k_{\leq 0}\}$ satisfies the quasi-shuffle relation.

The items of this theorem will be proved in the rest of this paper.

() is a restatement of Theorem . () is Theorem  combined with Proposition . () and the first statement of () are contained in Corollary . The second statement of () is Proposition . () is just Corollary . () is just Theorem .

Renormalized multiple zeta values with positive arguments
---------------------------------------------------------

We first take care of the easy case when MZVs are define by the convergence of the nested sums.

Let $\vec{s}=(s_1,\cdots,s_k)$ with positive integers $s_1,\cdots,s_k$ and $s_1>1$. We have $\zeta(\wvec{\vec{s}}{\vec{r}})=\zeta(\vec{s})$, independent of the choice of $\vec{r}\in \ZZ_{>0}^k$. In particular, $\gzeta(\vec{s})=\zeta(\vec{s})$.

For such an $\vec{s}$, $Z(\wvec{\vec s}{\vec r};\vep)$ is uniformly convergent in $(-\infty, 0]$, and the summands are continuous functions. So $Z(\wvec{\vec s}{\vec r};\vep)$ is continuous in $(-\infty, 0]$. Therefore, the Laurent series of $Z(\wvec{\vec s}{\vec r};\vep)$ is a power series and, by Theorem , $\zeta(\wvec{\vec{s}}{\vec{r}})=\lim_{\vep\to 0} \uni{Z}_+(\wvec{\vec{s}}{\vec{r}};\vep)
=\lim_{\vep\to 0} \uni{Z}(\wvec{\vec{s}}{\vec{r}};\vep)
= \uni{Z}(\wvec{\vec{s}}{\vec{r}};0)=\zeta(\vec{s}).$

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 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 $\vec s \in \ZZ ^k_{>0}$ 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 our definition of $\gzeta(\vec{s})$ for $s_i \geq 1,
1\leq i\leq k$, also satisfies $\gzeta(1)=T$ (by Theorem ) and the quasi-shuffle relation (by Corollary ), the later $\gzeta(\vec{s})$ must agree with the regularized MZVs $Z^\msh_{\vec{s}}(T)$.

Renormalized multiple zeta values with non-positive arguments
-------------------------------------------------------------

We study $\gzeta(\vec{s})$ when $\vec{s}\in \ZZ_{\leq 0}^k$. We then show that these values agrees with the special values of the multiple zeta functions with negative arguments defined by analytic continuation.

### The case of $\vec{s}=(0,\cdots,0)$

Let $\vec{s}=(s_1,\cdots,s_k)\in \ZZ_{\le 0}^k$ and $\vec{r}=(r_1,\cdots,r_k)\in \ZZ_{>0}^k$.

1.  For $k\ge 2$, we have

2.  If $\ell<k$ and $s_1=\cdots=s_\ell=0$, then $$Z\lp\wvec{\vec s}{\vec r}\rp= Z\lp \wvec{0,\cdots,0}{r_1,\cdots,r_\ell}\rp
        Z\lp \wvec{s_{\ell+1},s_{\ell+2},\cdots,s_k}{r_1+\cdots+r_{\ell+1},r_{\ell+2},\cdots,r_k}\rp.$$

3.  Each coefficient in the Laurent series expansion of $Z (\wvec{\vec s}{\vec r};\vep )$ is a rational function of the form $P(\vec r)/Q(\vec r)$, where $P$, $Q$ are in $\CC[r_1,\cdots,r_k]$ with no common factors, and are of the form $\Pi_{1\le j \leq k} (r_1+r_{2}+\cdots +r_j)^{c_{j}}$, $c_{j}\in
    \ZZ _{\ge 0}$.

() Since $s_1\leq 0$, we have This gives the proposition when $k=2$. In general, applying the induction hypothesis to the second $Z$-factor completes the proof.

() Applying the first equation of item () repeatedly, we have $$\begin{aligned}
\lefteqn{ Z(\wvec{\vec s}{\vec r};\vep) = Z(\wvec{0}{r_1};\vep )
Z(\wvec{s_2, s_3, \cdots s_{k}}{r_1+r_2, r_3,\cdots, r_{k}};\vep)
=\cdots} \\
&=& Z(\wvec{0}{r_1};\vep)Z(\wvec{0}{r_1+r_2};\vep)\cdots
Z(\wvec{0}{\sum_{i=1}^{\ell}r_i};\vep)
Z(\wvec{s_{\ell+1},\cdots,s_k}
    {r_{\ell+1}+\sum_{i=1}^{\ell}r_i,r_{\ell+2},\cdots,r_k};\vep).\end{aligned}$$ Then applying the second equation of item () to the product before the last factor gives item (). () From the generating series of the Bernoulli numbers $\disp{\frac{\vep}{e^\vep-1}=\sum_{i\geq 0} B_i \frac{\vep^i}{i!}}$, $$Z(\wvec{0}{1};\vep)=\sum_{n\geq 0} e^{n\vep}
= \frac{e^\vep}{1-e^\vep}= -\frac{1}{\vep}\frac {-\vep}{e^{-\vep}-1}
    = -\frac{1}{\vep}+ \sum_{i\geq 0}\zeta(-i) \frac{\vep^{i}}{i!}
    \mlabel{eq:zeta}$$ since $B_0=1$ and $\zeta(-i)=(-1)^i\frac{B_{i+1}}{i+1}$ for $i\geq
0$. For $s\in \ZZ_{<0}$, we have $$Z(\wvec{s}{1};\vep)=\sum_{n\geq 1}{n^{-s}} {e^{n\vep}}
= \frac{d^{-s}}{d\vep} \big(\frac{e^\vep}{1-e^\vep}\big)$$ which converges uniformly on any compact subset in ${\rm
Re}(\vep)<0$. So its Laurent series expansion at $\vep=0$ is obtained by termwise differentiating Eq. (), yielding $$Z(\wvec{s}{1};\vep)=(-1)^{s-1}(-s)!\,\vep^{s-1}+\sum _{j=0}^{\infty} \zeta
(s-j)\frac {\vep^j}{j!}. \mlabel{eq:zreg}$$ Then for $r\in \ZZ _{>0}$, we have $$Z(\wvec{s}{r};\vep)=(-1)^{s-1}(-s)!(r\vep)^{s-1}+\sum _{j=0}^{\infty} \zeta
(s-j)\frac {(r\vep)^j}{j!}
\mlabel{eq:zreg2}$$ Then item () follows from item ().

Let $\Sigma_k$ denote the symmetric group on $k$ letters. For $\sigma\in \Sigma_k$ and $\vec{r}=(r_1,\cdots,r_k)$, denote $\sigma(\vec{r})=(r_{\sigma(1)},\cdots,r_{\sigma(k)})$ and $f(\vec r)^{(\Sigma_k)}=\sum _{\sigma \in \Sigma_k} f(\sigma (\vec r))$.

Let $k\geq 1$ and $\vec{0}_k=(0,\cdots,0)\in \ZZ^k$. Then $\zeta\lp\wvec{\vec 0_k}{\vec r}\rp^{(\Sigma_k)}$ is independent of the choice of $\vec{r}\in \RR_{>0}^k$ and $\gzeta (\vec 0_k)
=\frac 1{k!}\zeta\lp\wvec{\vec 0_k}{\vec r}\rp^{(\Sigma_k)}.$

This is proved by induction on $k\geq 1$. For $k=1$, by Eq. () we have $$\zeta(\wvec{0}{r})= \tilde{P}\big(Z(\wvec{0}{r})\big)\big|_{\vep=0}=\zeta (0)$$ independent of $r>0$. Thus $\gzeta(0)$ is defined and the proposition holds.

In general, by Proposition  and Corollary , $$\zeta\lp\wvec{\vec 0_{k-1}}{r_1, \cdots, r_{k-1}}\rp^{(\Sigma_{k-1})}\zeta\lp\wvec{0}{r_k}\rp
=\zeta\lp\wvec{\vec 0_k}{\vec r}\rp^{(\Sigma_{k})}
+\sum _{i=1}^{k-1}\zeta\lp\wvec{\vec 0_{k-1}}{r_1, \cdots r_i',
\cdots, r_{k-1}}\rp^{(\Sigma_{k-1})}$$ where $r_i'=r_i+r_k$. So by the induction hypothesis, $\zeta \lp\wvec{\vec 0_k}{\vec r}\rp^{(\Sigma_{k})}$ is independent of $\vec{r}$. In particular, taking $\vec{r}=(\delta,\cdots,\delta)\in \RR_{>0}^k$, we have $$\frac{1}{k!}\, \zeta\lp\wvec{\vec{0_k}}{\vec{r}}\rp^{(\Sigma_k)} = \zeta\lp\wvec{\vec{0}_k}{\vec{r}}\rp= \gzeta(\vec{0}_k).
\mbox{\proofend}$$ It will be proved in Theorem  that $\gzeta(\vec{0}_k), k\geq 1,$ satisfy the quasi-shuffle relation. It is in fact the only way to define $\gzeta(\vec{0}_k), k\geq 1,$ with $\gzeta(0)=\zeta(0)$ [@Gu3 Theorem 1.1].

### The general case of $\vec s \in \ZZ _{\le 0}^k$

[Let $\vec{s}\in \ZZ_{\leq 0}^k$. Suppose $s_i=0$ exactly for $k_j'\leq i\leq k_j''$ with $k_j'\leq k_j''$ for $1\leq j\leq q$ and $k_j''<k_{j+1}'-1$ for $1\leq j\leq q-1$. Then $(s_{k_j'},\cdots,s_{k_j''})$ are the longest consecutive zero strings, called the [**zero clusters**]{}. So (with the possibility of $k'_1=1$ or $k''_q=k$) $$\vec{s}=(s_1,\cdots,s_{k_1'-1},\!\!\!\!\underbrace{0,\cdots,0}_{k_1''-k_1'+1-{\rm terms}}\!\!\!\! , s_{k_1''+1},\cdots,s_{k_2'-1},
\cdots, \!\!\!\! \underbrace{0,\cdots,0}_{k_q''-k_q'+1-{\rm terms}}\!\!\!\! ,s_{k_q''+1},\cdots,s_k ).
$$ For each $1\leq i\leq q$, let $\vec{k}^{(i)}=(k_i',\cdots,k_i'')$ and let $\Sigma_{\vec{k}^{(i)}}$ be the permutation group of $\vec{k}^{(i)}$, naturally a subgroup of $\Sigma_k$. Define the subgroup $$\Sigma(\vec s)=\Sigma_{\vec{k}^{(1)}} \times \cdots \times \Sigma_{\vec{k}^{(q)}} \subseteq \Sigma_k.$$ ]{}

So for $\sigma=(\sigma_1,\cdots,\sigma_q)\in  \Sigma(\vec{s})$ with each $\sigma_i\in \Sigma_{\vec{k}^{(i)}}$, $1\leq i\leq q$, $\sigma(\vec{r})$ is obtained by $\sigma_i$ permuting $r_{k_i'},\cdots,r_{k_i''}$ and leaving the other entries fixed. Define $$\gzeta\lp\wvec{\vec{s}}{\vec{r}}\rp^{({\Sigma(\vec{s})})}=\sum_{\sigma\in {\Sigma(\vec{s})}}
\gzeta \lp\wvec{\vec{s}}{\sigma (\vec{r})}\rp,\quad
\uni{Z}_+(\wvec{\vec s}{\vec r}; \vep)^{({\Sigma(\vec{s})})} =\sum_{\sigma\in {\Sigma(\vec{s})}}
\uni{Z}_+(\wvec{\vec s}{\sigma (\vec{r})};\vep).$$

The limit $\disp{\lim _{\vec r\to -\vec s}\zeta\lp \wvec{\vec s}{\vec r}\rp^{({\Sigma(\vec{s})})}}$ exists.

We first give some applications of the theorem and defer its proof to Section .

For $\vec s \in \ZZ ^k_{\le 0}$, $\gzeta(\vec s)$ is well-defined and $$\gzeta (\vec s)=\frac 1{|{\Sigma(\vec{s})}|}\lim _{\vec r\to -\vec s} \zeta\lp \wvec{\vec s}{\vec r}\rp^{({\Sigma(\vec{s})})}.$$ If in addition $\vec s$ does not have consecutive zeros, then $$\gzeta(\vec{s})=\zeta \lp \wvec{\vec{s}}{-\vec{s}}\rp
=\lim_{\vec{r}\to -\vec{s}} \zeta\lp\wvec{\vec{s}}{\vec{r}}\rp.$$

Taking the limit in Theorem  when $\vec
r$ approaches $\vec s$ along the path $\vec r=\vec
s+\delta$, $\delta \to 0$, we have $$\lim _{\vec r\to -\vec s}\zeta\lp\wvec{\vec s}{\vec
r}\rp^{({\Sigma(\vec{s})})}= \lim _{\delta\to 0} \zeta\lp\wvec{\vec s}{-\vec s+\delta}\rp^{({\Sigma(\vec{s})})}.$$ By the definition of ${\Sigma(\vec{s})}$ and our choice of $\vec r$, ${\Sigma(\vec{s})}$ permutes the components of $\vec r$ that equal $\delta$. Therefore, $$\frac{1}{|{\Sigma(\vec{s})}|}\zeta\lp\wvec{\vec s}{-\vec s+\delta}\rp^{({\Sigma(\vec{s})})}= \zeta\lp\wvec{\vec s}{-\vec s+\delta}\rp$$ giving the first limit in the corollary. The second part follows since then $\Sigma(\vec s)$ is trivial.

We give an explicit formula when $k=2$. A similar formula holds for $k>2$, expressing $\gzeta(\vec{s})$ as a polynomial in the Bernoulli numbers. As a consequence, $\gzeta(\vec{s})$ is rational.

Let $s_1$, $s_2 \le 0$, but not both zero. Then $\gzeta(0,s_2)=\zeta(0)\zeta(s_2)-\zeta(s_2-1)$, and for $s_1<0$, $$\begin{aligned}
\lefteqn{\gzeta (s_1, s_2)=\sum _{j=0}^{-s_1} \binc {-s_1}{j}
\zeta (-j)\zeta (s_1+s_2+j)-\frac{1}{1-s_1}\zeta(s_1+s_2-1)}\\
&&+\sum _{j=0}^{-s_1} \binc {-s_1}{j}\frac{(-1)^{s_1+s_2-j+1}}
{-s_1-s_2-j+1}\Big(\frac{s_1+s_2}{s_1}\Big)^{s_1+s_2+j-1}
\zeta(s_1+s_2-1).\end{aligned}$$

By Theorem , Proposition  and Eq. (), we have Since $$\begin{aligned}
 &\sum _{j=0}^{-s_1} \binc {-s_1}{j}
\frac {(-1)^{j+1}}{j+1}\Big(\frac {r_1+r_2}{r_1}\Big)^{j+1}=\frac
1{-s_1+1}\sum _{j=1}^{-s_1+1} \binc {-s_1+1}{j} (-1)^{j}\Big(\frac
{r_1+r_2}{r_1}\Big)^{j} \\
=&\frac 1{-s_1+1}\big ((1-\frac {r_1+r_2}{r_1})^{-s_1+1}-1\big
)=\frac 1{-s_1+1}(-\frac {r_2}{r_1})^{-s_1+1}-\frac 1{-s_1+1},\end{aligned}$$ the conclusion follows from Corollary .

In the following table, the element in row $s_1$ and column $s_2$ is $\gzeta(-s_1,-s_2)$, $1\leq s_1\leq 7,\ 1\leq s_2\leq 8$. It can be seen that elements on each of the even numbered subdiagonal lines are equal, and that for $s_1=s_2$ even, $\gzeta(s_1,s_2)=0$. They both follow from Eq. (). But the second one follows readily from the quasi-shuffle relation in Theorem .(): $ 2\gzeta(s_1,s_1)=\zeta(s_1)\zeta(s_1)-\zeta(2s_1)=0.$

[$$\hspace{-1.5cm}
\begin {array} {ccccccc}
\frac 1{288}& -\frac 1{240}& \frac {83}{64512}& \frac 1{504}& -\frac
{3925}{2239488}& -\frac 1{480}& \frac {342884347}{99656663040}
\medskip
\\
-\frac 1{240}& 0& \frac 1{504}& -\frac {319}{437400}&
-\frac 1{480}& \frac {2494519}{1362493440}& \frac 1{264}
\medskip
\\
-\frac {71}{35840}& \frac 1{504}&
\frac 1{28800}& -\frac 1{480}& \frac {114139507}{139519328256}&
\frac 1{264}& -\frac {313042283533}{93600000000000}
\medskip
\\
\frac 1{504}& \frac {319}{437400}& -\frac 1{480}& 0&
\frac 1{264}& -\frac {41796929201}{26873437500000}& -\frac
{691}{65520}
\medskip
\\
\frac {32659}{15676416}&
-\frac 1{480}& -\frac {21991341}{25836912640}&
\frac 1{264}& \frac 1{127008}& -\frac {691}{65520}& \frac {26194796926873}{5884626295848960}
\medskip\\
-\frac 1{480}& -\frac {2494519}{1362493440}& \frac 1{264}& \frac
{41796929201}{26873437500000}&
-\frac {691}{65520}& 0& \frac 1{24}
\medskip
\\
-\frac {75497471}{19931332608}& \frac 1{264}& \frac
{316292283533}{93600000000000}& -\frac {691}{65520}& -\frac
{36808933898915}{8238476814188544}& \frac 1{24}& \frac 1{115200}
\medskip
\\
\frac 1{264}& \frac
{16608667097}{2879296875000}& -\frac {691}{65520}& -\frac
{4607695}{491051484}& \frac 1{24}& \frac
{63967403428993199}{3561322226607185040}& -\frac {3617}{16320}
\end {array}$$ ]{}

### Compatibility with multiple zeta values defined by analytic continuation

We recall that the multiple zeta function $\zeta(s_1,\cdots,s_k)$ has analytic continuation to $\CC^k$ with singularities on the subvarieties in Eq. ().

For $(s_1,\cdots,s_k)\in \ZZ_{\leq 0}^k$, if $\zeta(s_1,\cdots,s_k)$ is well-defined by the analytic continuation, then it agrees with $\gzeta(s_1,\cdots,s_k)$.

When $k=1$, by Eq. () for $s\leq 0$, $$\gzeta(s)= \tilde{P}\big(Z(\wvec{s}{-s})\big)\big|_{\vep=0}=\zeta (s).$$ For $k=2$, by Eq. (), exactly when $s_1+s_2$ is negative and odd, the zeta values $\zeta(s_1,s_2)$ is defined by analytic continuation and thus agrees with the iterated limit $\disp{ \lim_{z_2\to s_2} \lim_{z_1\to s_1} \zeta(z_1,z_2)}$ defined in , Eq. (3). Note that our order of arguments in the definition of multiple zeta functions is opposite to their order. So $\zeta(z_1,z_2)$ here is $\zeta(z_2,z_1)$ in their paper. Thus our order of limits here is also opposite to their order.

For $n\geq 0$ and $q\geq 1$, let $(n)_q=n(n+1)\cdots (n+q-1)$. Then by Eq. (15) in : Since $s_1+s_2$ is negative and odd, $s_1+s_2-1$ is negative and even. Hence the first term is zero. Further, for $1\leq q\leq -s_1$, either $-q$ or $s_1+s_2+q$ is negative and even. Thus the sum also vanishes, leaving $\zeta(s_1,s_2)= -\zeta(s_1+s_2)/2$. By the same argument, from Corollary , we have $$\gzeta(s_1,s_2)=\zeta(0)\zeta(s_1+s_2)
=-\frac{\zeta(s_1+s_2)}{2}=\zeta(s_1,s_2).
\mlabel{eq:negodd}$$ By Eq. (), for $k\geq 3$, $\zeta(s_1,\cdots,s_k)$ is not defined by analytic continuation for any non-positive integers $s_1,\cdots,s_k$. Thus we have completed the proof.

The proof of Theorem 
=====================

Reduction to Proposition 
-------------------------

With the notations in Theorem , define $$\uni{Z}^m_+\lp \wvec{\vec s}{\vec r};\vep\rp
= \hspace{-.7cm}
\sum_{(i_1,\cdots,i_p)\in\Pi_k}\hspace{-.6cm}
\tilde{P}\llp \uni{Z}\lp\wvec{\vec s^{(p)}}{\vec r ^{(p)}};\vep\rp
    \check{P}\llp \uni{Z}\lp\wvec{\vec s^{(p-1)}}{\vec r ^{(p-1)}};\vep\rp
    \cdots \check{P}\llp \uni{Z}\lp\wvec{\vec s^{(2)}}{\vec r ^{(2)}};\vep\rp
    \check{P}\llp \vep^m\uni{Z}\lp\wvec{\vec s ^{(1)}}{\vec r ^{(1)}};\vep\rp\rrp\,
    \rrp \cdots \rrp
    \rrp.$$ Then by Theorem , we have $\uni{Z}_+^0\lp \wvec{\vec s}{\vec r};\vep\rp
=\uni{Z}_+\lp \wvec{\vec s}{\vec r};\vep\rp.$ For $\sigma\in \Sigma_k$, let $\sigma(\vec{s})^{(j)}$, $1\leq j\leq p$, be the partition vectors of $\sigma(\vec{s})=(s_{\sigma(1)},\cdots,s_{\sigma(k)})$ from the ordered partition $(i_1,\cdots,i_p)$ in Definition . Then $$\begin{aligned}
\lefteqn{ \uni{Z}^m_+(\wvec{\vec{s}}{\vec{r}};\vep)^{(\sigma)}:=\uni{Z}^m_+(\wvec{\vec{s}}{\sigma(\vec{r})};\vep)}
\mlabel{eq:msum}
\\
&=& \hspace{-.7cm}
\sum_{(i_1,\cdots,i_p)\in\Pi_k}\hspace{-.6cm}
\tilde{P}\llp \uni{Z}\lp\wvec{\vec s^{(p)}}{\sigma(\vec r) ^{(p)}};\vep\rp
        \cdots \check{P}\llp \uni{Z}\lp\wvec{\vec s^{(2)}}{\sigma(\vec r) ^{(2)}};\vep\rp
    \check{P}\llp \vep^m\uni{Z}\lp\wvec{\vec s ^{(1)}}{\sigma(\vec r) ^{(1)}};\vep\rp\rrp\,
    \rrp \cdots \rrp.
\notag\end{aligned}$$ Define $$\uni{Z}^m_+(\wvec{\vec s}{\vec r};\vep)^{(\Sigma(\vec{s}))}
=\sum_{\sigma\in \Sigma(\vec{s})} \uni{Z}^m_+(\wvec{\vec s}{\vec
r};\vep)^{(\sigma)}$$ By Proposition .() and Theorem , each coefficient in the Laurent series expansion of $\uni{Z}^m_+(\wvec{\vec s}{\vec r};\vep)$, and thus of $\uni{Z}^m_+(\wvec{\vec s}{\vec r};\vep)^{(\Sigma(\vec{s}))}$, is a rational function $P(\wvec{\vec s}{\vec r})/Q(\wvec{\vec s}{\vec r})\in \CC(\vec{s},\vec{r})$ with $P,Q\in \CC[\vec{s},\vec{r}]$. We can assume that $P$ and $Q$ have no common factors. We call this coefficient [**ordinary**]{} at $\vec{r}=-\vec{s}$ if $Q(-\vec s)\not= 0$. We say that $ \uni {Z} _+^m\lp\wvec{\vec s}{\vec r}\rp^{(\Sigma(\vec{s}))}$ is [**ordinary**]{} if every coefficient of its Laurent series is ordinary.

Let $\sigma\in \Sigma(\vec{s})$. Let $P(\vec{r})/Q(\vec{r})$ be a coefficient of the Laurent series of $\uni{Z}_+^m(\wvec{\vec s}{\vec r})^{(\Sigma(\vec{s}))}$. The following statements are equivalent.

1.  $P(\vec{r})/Q(\vec{r})$ is ordinary at $\vec{r}=-\vec{s}$.

2.  $\disp{\lim_{\vec r\to -\vec s} P(\vec{r})/Q(\vec{r})}$ exists.

3.  $Q(\vec{r})$ does not have a linear factor $r_{k_1}+\cdots+r_{k_t}$ such that $\{k_1,\cdots,k_t\}$ is a subset of $\vec{k}^{(j)}$ for some $1\leq j\leq q$ in Definition .

() $\Leftrightarrow$ () holds for any rational functions, and () $\Rightarrow$ () is clear since if $\{k_1,\cdots,k_t\}$ is a subset of some $\vec{k}^{(j)}$, then $s_{k_1}+\cdots+s_{k_t}=0$.

() $\Rightarrow$ (): Suppose $P(\vec{r})/Q(\vec{r})$ is not ordinary at $\vec{r}=-\vec{s}$. Then by Proposition .() and Theorem , $Q(\vec{r})$ has a factor $r_{k_1}+\cdots+r_{k_t}$ with $s_{k_1}=\cdots s_{k_t}=0$. Since the denominator of a sum of fractions is a factor of the product of the denominators of the fractions, this factor is a factor in the denominator $Q_\sigma(\vec{r})$ of a coefficient of the Laurent series of $\uni{Z}_+^m(\wvec{\vec s}{\sigma(\vec r)})$ for a $\sigma\in
\Sigma(\vec{s})$. If this $\sigma$ is $\id$, then by Proposition .() and Theorem , $r_{k_1}+\cdots+r_{k_t}=r_i+\cdots+r_{i+t}$ for some $i$. Thus from $s_{r_{k_1}}=\cdots=s_{r_{k_t}}=0$, $\{r_{k_1},\cdots,r_{k_t}\}=\{r_i,\cdots,r_i+t\}$ is a subset of $\vec{k}^{(j)}$ for some $1\leq j\leq q$ with the notation of Definition . If $\sigma\neq \id$, then $r_{k_1}+\cdots+r_{k_t}=r_{\sigma(i)}+\cdots+r_{\sigma(i+t)}$ for some $i$. Since $\sigma\in \Sigma(\vec{s})$ permutes the components of $\vec{k}^{(j)}$ among themselves, $\{r_{\sigma(i)},\cdots,r_{\sigma(i+t)}\}$ is still a subset of $\vec{k}^{(j)}$. Thus in any case, $Q(\vec{r})$ has a linear factor $r_{k_1}+\cdots+r_{k_t}$ such that $\{k_1,\cdots,k_t\}$ is a subset of $\vec{k}^{(j)}$ for some $1\leq j\leq q$.

Thus to prove Theorem , we just apply the following Proposition  to the case when $m=0$ and then let $\vep$ go to $0$. Note that even though we only need $m=0$ for Theorem , we have to consider other values of $m$ for the inductive proof.

Let $\vec s \in \ZZ _{\le
0}^k$. Take $m\in \ZZ _{\le 0}$ if $s_1=0$ and take $m\in \ZZ$ if $s_1<0$. Then $ \uni {Z} _+^m\lp\wvec{\vec s}{\vec r}\rp^{(\Sigma(\vec{s}))}$ is ordinary at $\vec r=-\vec s$.

The proof of Proposition 
-------------------------

The following chart gives an outline of the proof. $$\xymatrix{
\mbox{Special case}\ar[rd] && \mbox{Lemma~\mref{lem:induc}}\ar[r]& \mbox{Case 1} \ar[rdd] & \\
& \mbox{Lemma~\mref{lem:cong}} \ar[r] & \mbox{Subcase 2.1} \ar[rd] && \\
\mbox{General case} \ar[ru] &&& \mbox{Case 2} \ar[r] & \mbox{Prop.~\mref{pp:mgood}}\\
&& \mbox{Subcase 2.2} \ar[ru] &&\\
&&& \mbox{Case 3} \ar[ruu] &
}$$

We first introduce the following notations to simplify our expressions. For any Laurent series $f$, we use $\lb f\rb$ to denote $\check {P}\lp f\rp$. Also use $\overline{\wvec{\vec{s}}{\vec{r}}}$ to denote $\uni {Z}\lp \wvec{\vec s}{\vec r};\vep \rp$ and $\overline{\vec r}$ or $\overline{r_1\cdots r_k}$ to denote $\uni{Z}\lp \wvec{\vec 0}{\vec r};\vep\rp$. With these abbreviations, we have $$\uni{Z}_+^m\lp \wvec{\vec s}{\vec r};\vep\rp
= \sum_{ (i_1,\cdots,i_p)\in \Pi_k} \tilde{P}
    \left ( \overline {\wvec{\vec s^{(p)}}{\vec r ^{(p)}}}
    \lb\cdots \lb \overline {\wvec{\vec s^{( 2 )}}{\vec r ^{( 2)}}}\lb
    \vep ^m \overline {\wvec{\vec s ^{(1)}}{ \vec r ^{( 1 )}}}\rb\,
    \rb \cdots\rb
     \right ).$$

We now prove Proposition  by induction on $k$. The case when $k=1$ is clear by Eq. (). Assume that the proposition is true for vectors with length $\le k$ and let $\vec s\in \ZZ _{\le 0}^{k+1}$. We will separately consider three cases with $$\mbox{\bf Case 1: when $s_1<0$; Case 2: $s_1=0$ but $\vec{s}\neq 0$ and
Case 3: $\vec{s}=0$}.$$ [**Case 1: assume $s_1<0$.** ]{} Then by our choice, $m$ is in $\ZZ $. Let $\vec s=(s_1, \vec s\,')$, $\vec r=(r_1, \vec
r\,')$. We clearly have the disjoint union $$\begin{aligned}
\Pi_{k+1}&=&\{(1,i_1,\cdots,i_p)\ \big|\ (i_1,\cdots,i_p)\in \Pi_k\} \notag \\
&& \cup
    \{(i_1+1,i_2,\cdots,i_p)\ \big|\ (i_1,\cdots,i_p)\in \Pi_k\}.
\mlabel{eq:ab}\end{aligned}$$ For $(i_1,\cdots,i_p)\in \Pi_k$, let $\vec{s}^{\,\prime (j)}, 1\leq j\leq p$ be the partition vectors of $\vec{s}^{\,\prime}$ from $(i_1,\cdots,i_p)$ in Definition . Similarly define $\vec{r}^{\,\prime (j)}$, $1\leq j\leq p$. Then by Eq. (), We will make use of the following lemma.

Let $\vec{s}\in \ZZ_{\leq 0}^k$, $\vec{r}\in \ZZ_{>0}^k$. Let $k'<k$ and $\vec{s}=(\vec{s}\,',\vec{s}\,''), \vec{r}=(\vec{r}\,',\vec{r}\,'')$ with $\vec{s}\,'$ and $\vec{r}\,'$ of length $k'$. Suppose a Laurent series $f(\wvec{\vec{s}\,'}{\vec{r}\,'};\vep)=\sum_{i<0} c_i\vep^i\in \CC(\vec{s}\,',\vec{r}\,')[[\vep,\vep^{-1}]$ is ordinary at $\vec{r}\,'=-\vec{s}\,'$. If $\uni{Z}_+^m(\wvec{\vec{s}\,''}{\vec{r}\,''};\vep)$ is ordinary at $\vec{r}\,''=-\vec{s}\,''$ for all $m\in \ZZ_{\leq 0}$ (resp. all $m\in \ZZ$), then $$\sum_{(i_1,\cdots,i_p)\in \Pi_{k-k'}}
    \tilde{P}\Llp \overline {\wvec{\vec s^{\,\prime\prime(p)}}{\vec r^{\,\prime\prime (p)}}}
    \lb\cdots \lb\overline {\wvec{\vec s^{\,\prime\prime (2)}}{\vec r^{\,\prime\prime (2)}}}
    \lb \vep^{m} f(\wvec{\vec{s}\,'}{\vec{r}\,'};\vep) \overline {\wvec{\vec s^{\,\prime\prime(1)}}{ \vec r^{\,\prime\prime (1)}}}
       \rb
    \rb  \cdots\rb \Rrp^{(\Sigma(\vec{s}\,''))}$$ is ordinary at $\vec{r}=-\vec{s}$ for all $m\in \ZZ_{\leq 0}$ (resp. all $m\in \ZZ$).

We have Since $i<0$, we have $m+i<0$ if $m\in \ZZ_{\leq 0}$ and $m+i\in \ZZ$ if $m\in \ZZ$. Hence each of the inner sum is ordinary at $\vec{r}\,''=-\vec{s}\,''$ and thus ordinary at $\vec r=-\vec s$ since the inner sum does not involve $\vec s\,'$ and $\vec r\,'$. By assumption each $c_i$ is ordinary at $\vec r\,'=-\vec s\,'$ and hence at $\vec r=-\vec s$ as $c_i$ does not involve $\vec s\,''$ and $\vec r\,''$. Thus the sum is ordinary at $\vec r=-\vec s$.

Back to the proof of Proposition  in Case 1, since $s_1<0$, by Eq. (), $\big\langle \vep ^m \overline {\wvec{s_1}{r_1}} \big\rangle =\check{P}(\vep^m \uni{Z}(\wvec{s_1}{r_1})$ is ordinary at $r_1=-s_1$. Therefore the first sum in Eq. () is ordinary at $\vec r=-\vec s$ by Lemma  and the induction hypothesis.

For the second term in Eq. (), for fixed $1\leq i_1\leq k$, identify an ordered partition $(i_2,\cdots,i_p)$ of $k-i_1$ with the ordered partition $(i_1,i_2,\cdots,i_p)$ of $k$, we have Let $\Sigma'=\Sigma(s_1,\vec{s}\,'^{(1)})$ and $\Sigma''=\Sigma(\vec{s}\,'^{(2)},\cdots,\vec{s}\,'^{(p)})=\Sigma(s_{i_1+1},\cdots,s_k)$. Then $\Sigma'\Sigma''=\Sigma'\times \Sigma''$ and thus is a subgroup of $\Sigma(\vec{s})$. Let $S$ be a complete set of coset representatives for the cosets $\{ \Sigma'\Sigma'' \sigma\ |\ \sigma\in \Sigma(\vec{s})\}$ of $\Sigma(\vec{s})$. So $$\begin{aligned}
&&\sum_{(i_2,\cdots,i_p)\in \Pi_{k-i_1}}
    \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime(p)}}{\vec r^{\,\prime (p)}}} \lb \cdots
    \lb \overline {\wvec{\vec s^{\,\prime(2)}}{\vec r^{\,\prime (2)}}}
    \lb \vep ^m \overline {\wvec{s_1,\vec s^{\,\prime (1)}}{r_1, \vec r ^{\,\prime (1)}}}\rb
    \rb  \cdots \rb\Big)^{(\Sigma(\vec{s}))}
\mlabel{eq:s21}
\\
&=&
\sum_{\sigma\in S} \sum_{(i_2,\cdots,i_p)\in \Pi_{k-i_1}}
    \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime(p)}}{\sigma(\vec r^{\,\prime})^{ (p)}}} \lb\cdots
    \lb \overline {\wvec{\vec s^{\,\prime(2)}}{\sigma(\vec r^{\,\prime})^{ (2)}}}
    \lb \vep ^m \overline {\wvec{s_1,\vec s^{\,\prime (1)}}{\sigma(r_1, \vec r ^{\,\prime (1)})}}\rb ^{(\Sigma')}
    \rb  \cdots \rb \Big)^{(\Sigma'')}.
\notag\end{aligned}$$

Let $\sigma\in S$, by the definition of $\Sigma(\vec{s})$, we have $\sigma(r_1,\vec{r}\,^{\prime (1)})=(r_1,r_{\sigma(2)},\cdots).$ So by Proposition , each coefficient in the Laurent series expansion of $\lb \vep ^m \overline {\wvec{s_1,\vec
s^{\,\prime (1)}}{\sigma(r_1, \vec r ^{\,\prime} (1)})}\rb
^{(\Sigma')}$ has its denominator as a product of $(r_1+r_{\sigma(j_1)}+\cdots +r_{\sigma(j_t)})$, $t\geq 0$, with $1\leq j_1,\cdots,j_t\leq i_1$. Hence the expansion is ordinary at $\sigma(\vec r\,^{\prime})^{(1)}=-\sigma(\vec s\,^{\prime})^{(1)}$ since $s_1<0$ and $s_i\leq 0$.

Further by the induction hypothesis on $k$, $$\sum_{(i_2,\cdots,i_p)\in \Pi_{k-i_1}}
    \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime(p)}}{\sigma(\vec r^{\,})^{\prime (p)}}} \lb\cdots
    \lb \vep^t \overline {\wvec{\vec s^{\,\prime(2)}}{\sigma(\vec r^{\,})^{\prime (2)}}}
    \rb  \cdots \rb \Big)^{(\Sigma'')}$$ is ordinary at $(r'_{\sigma(i_1+1)},\cdots,r'_{\sigma(k+1)})=-(s_{i_1+1},\cdots, s_{k+1})$ for $t\in \ZZ_{\leq 0}$. Thus by Lemma , the inner sum on the right hand side of Eq. () is ordinary at $\sigma(\vec r)=-\vec s$ for each $\sigma\in S$. Note that $\sigma(\vec{s})=\vec{s}$ for $\sigma\in \Sigma(\vec{s})$ and being ordinary at $\sigma(\vec{r})=-\sigma(\vec{s})$ is equivalent to being ordinary at $\vec{r}=-\vec{s}$. Hence the left hand sum is ordinary at $\vec r=-\vec s$ in Eq. () and hence in Eq. () and hence in the second term of Eq. ().

[**Case 2: assume $s_1=0$, but $\vec s\neq 0$.**]{} Then $m\le 0$. Assume $s_1=s_2=\cdots =s_\ell=0$, $s_{\ell+1}\not =0$, $\ell \le k$. Then $\vec s =(\vec s\,', \vec s\,'')$ with $\vec s\,'=\vec{0}_\ell$ and $\vec{s}\,''=(s_{\ell+1},\cdots,s_{k+1})$. Similarly denote $\vec r= (\vec r\,',\vec r\,'')$ with $\vec r\,'=(r_1,\cdots,r_\ell)$ and $\vec r\,''=(r_{\ell+1},\cdots, r_{k+1})$. In the notation of Definition , $(1,\cdots,\ell)=\vec{k}^{(1)}$.

By Lemma , in order to prove that $\uni{Z}_+^m(\wvec{\vec{s}}{\vec{r}};\vep)$ is ordinary, we only need to show that no coefficient of its Laurent series expansion has a denominator with either

1.  a [**type (i) factor:**]{} $r_{k_1}+\cdots+r_{k_t}$ where $k_1,\cdots,k_t\leq \ell$ or

2.  a [**type (ii) factor:**]{} $r_{k_1}+\cdots+r_{k_t}$ where $\{k_1,\cdots,k_t\}$ is a subset of $\vec{k}^{(j)}$ for some $2\leq j\leq q$ in Definition .

[**Subcase 2.1: there are no type (i) factors.**]{} Note that any ordered partition of $\Pi
_{k+1} $ is of the form $(i_1, i_2, \cdots, i_p, j_1, j_2, \cdots,
j_q)$ or $(i_1, i_2, \cdots, i_p+j_1, j_2, \cdots, j_q)$, with $(i_1, i_2, \cdots, i_p)\in \Pi _\ell$, $(j_1, j_2, \cdots, j_q)\in
\Pi _{k+1-\ell}$. Using the notations in Definition , let $\vec{r}^{\,\prime(1)},\cdots,\vec{r}^{\,\prime(p)}$ be the partial vectors of $\vec r\,'$ from the ordered partition $(i_1,\cdots,i_p)\in \Pi_\ell$. Similarly let $\vec{s}^{\,\prime\prime(1)},\cdots, \vec{s}^{\,\prime\prime(q)}$ (resp. $\vec{r}^{\,\prime\prime(1)},\cdots,\vec{r}^{\,\prime\prime(q)}$) be the partition vectors of $\vec s\,''$ (resp. $\vec r\,''$) from the ordered partition $(j_1,\cdots,j_q)\in \Pi_{k+1-\ell}$. Then we have $$\begin{aligned}
 \uni {Z} _+^m\lp\wvec{\vec s}{\vec r}\rp^{(\Sigma(\vec{s}))}&= & \hspace{-1cm}
 \sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell\\
(j_1, j_2, \cdots, j_q)\in \Pi _{k+1-\ell}\end{array}}}
   \hspace{-1cm} \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\lb\cdots
   \lb \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime \prime (1)}}}
   \lb \overline {\vec r^{\,\prime (p)}} \cdots
   \lb \overline {\vec r^{\,\prime (2)}}
   \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb
    \rb \cdots \rb
    \Big)^{(\Sigma(\vec{s}))}
\\
&& +  \hspace{-1.3cm}\sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell\\
(j_1, j_2, \cdots, j_q)\in \Pi _{k+1-\ell}\end{array}}}
    \hspace{-1cm} \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}} \lb \cdots
    \lb \overline {\wvec{\vec 0, \vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime (p)},\vec r^{\,\prime \prime (1)}}}\cdots
    \lb \overline {\vec r^{\,\prime (2)}}
    \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb \cdots \rb
    \Big)^{(\Sigma(\vec{s}))}\end{aligned}$$ Also define $\Sigma'=\Sigma_{\ell}=\Sigma(\vec{s}\,')$ and $\Sigma''=\Sigma(\vec{s}\,'')$ as in Definition . Then $\Sigma(\vec{s})=\Sigma_\ell \times \Sigma''$. So $$\begin{aligned}
\lefteqn{\uni {Z} _+^m\lp\wvec{\vec s}{\vec r}\rp^{(\Sigma(\vec{s}))} \hspace{-.5cm}=\hspace{-1.3cm}\sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell\\
(j_1, j_2, \cdots, j_q)\in \Pi _{k+1-\ell}\end{array}}}
    \hspace{-1cm} \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}} \lb \cdots
    \lb \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime \prime (1)}}}
    \lb \overline {\vec r^{\,\prime (p)}}\cdots
    \lb \overline {\vec r^{\,\prime (2)}}
    \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb ^{(\Sigma _\ell)}
    \rb \cdots \rb
    \Big)^{(\Sigma'')}} \notag \\
&& + \hspace{-1.3cm} \sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell\\
(j_1, j_2, \cdots, j_q)\in \Pi _{k+1-\ell}\end{array}}}
    \hspace{-1.3cm} \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}} \lb \cdots
    \lb \overline {\wvec{\vec 0, \vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime (p)},\vec r^{\,\prime \prime (1)}}}\cdots
    \lb \overline {\vec r^{\,\prime (2)}}
    \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb ^{(\Sigma _\ell)} \cdots \rb
    \Big)^{(\Sigma'')}
\mlabel{eq:mzsum}\end{aligned}$$

Now for fixed $(j_1,\cdots,j_q)\in\Pi _{k+1-\ell}$ and $\tau \in
\Sigma''$, the corresponding terms in the above summation are $$\begin{aligned}
&\sum_{\tiny{(i_1,\cdots,i_p)\in \Pi_\ell }}
    \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime \prime(q)}}{\tau(\vec r^{\,\prime \prime})^{(q)}}} \lb \cdots
    \lb \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{ (1)}}}
    \lb \overline {\vec r^{\,\prime(p)}}\cdots
    \lb \overline {\vec r^{\,\prime (2)}}
    \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb ^{(\Sigma _\ell)}\rb \cdots \rb
    \Big) & \\
&+ \sum_{\tiny{(i_1,\cdots,i_p)\in \Pi_\ell}}
    \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime \prime(q)}}{\tau(\vec r^{\,\prime \prime})^{ (q)}}} \lb \cdots
    \lb \overline {\wvec{\vec 0, \vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime (p)},\tau(\vec r^{\,\prime \prime})^{ (1)}}}\cdots
    \lb \overline {\vec r^{\,\prime (2)}}
    \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb ^{(\Sigma _\ell)} \cdots \rb
    \Big)&\\
&= \tilde{P}\Big(\overline {\wvec{\vec s^{\,\prime \prime(q)}}{\tau(\vec r^{\,\prime \prime})^{(q)}}} \lb \cdots
    \overline {\wvec{\vec s^{\,\prime \prime(2)}}{\tau(\vec r^{\,\prime \prime})^{(2)}}}
    \llb \sum_{\tiny{(i_1,\cdots,i_p)\in \Pi_\ell }}\Big (
  \overline {\wvec{\vec s^{\,\prime \prime (1)}}
  {\tau(\vec r^{\,\prime \prime})^{(1)}}}
  \lb \overline {\vec r^{\,\prime (p)}} \cdots
  \lb \overline {\vec r^{\,\prime (2)}}
  \lb \vep ^m \overline {\vec r^{\,\prime (1)}}
  \rb \rb  \cdots \rb
 & \\
  & \qquad \qquad \qquad
  +\overline {\wvec{\vec 0, \vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime (p)},
  \tau(\vec r^{\,\prime \prime})^{ (1)}}}
    \cdots
  \lb \overline {\vec r^{\,\prime (2)}}
  \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots
    \Big )^{(\Sigma _\ell)}
    \rrb  \cdots \rb
    \Big) &\end{aligned}$$ By Proposition .(), $$\overline {\wvec{\vec 0, \vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime (p)},\tau(\vec r^{\,\prime \prime})
^{(1)}}}=\overline { \vec r^{\,\prime (p)}}\overline {\wvec{\vec s^{\,\prime
\prime (1)}}{\tau(\vec r^{\,\prime \prime})^{ (1)}+(r'_{I_{p-1}+1}+\cdots +
r'_{I_p})\vec{e} _1^{(j_1)}}},$$ where $\vec{e}^{(j_1)}_1$ is the first unit vector of length $j_1$ (which is the length of $\vec{r}\,^{\prime\prime (1)}$ and $\tau(\vec{r}\,'')^{(1)}$). So the inner sum on the right hand side above becomes $$\begin{aligned}
&&\sum_{(i_1,\cdots,i_p)\in \Pi _\ell}\left (
  \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{(1)}}}
  \lb \overline {\vec r^{\,\prime (p)}} \cdots
  \lb \overline {\vec r^{\,\prime (2)}}
  \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
  \rb  \cdots \rb \right .\\
&& \qquad \left . +\,
  \overline { \vec r^{\,\prime (p)}}
  \overline {\wvec{\vec s^{\,\prime\prime (1)}}
  {\tau(\vec r^{\,\prime \prime})^{ (1)}+(r'_{I_{p-1}+1}+\cdots +r'_{I_p})\vec{e} _1^{(j_1)}}} \lb \cdots
  \lb \overline {\vec r^{\,\prime (2)}}
  \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb
    \right )^{(\Sigma _\ell)}\\
&=&\sum_{(i_1,\cdots,i_p)\in \Pi _\ell}\sum_{\sigma\in \Sigma_{\ell}} \left (
  \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{(1)}}}
  \lb \overline {\sigma(\vec r^{\,\prime})^{ (p)}} \cdots
  \lb \overline {\sigma(\vec r^{\,\prime})^{ (2)}}
  \lb \vep ^m \overline {\sigma(\vec r^{\,\prime})^{ (1)}}\rb
  \rb  \cdots \rb \right .\\
&& \qquad \left . +\,
  \overline { \sigma(\vec r^{\,\prime})^{ (p)}}
  \overline {\wvec{\vec s^{\,\prime\prime (1)}}
  {\tau(\vec r^{\,\prime \prime})^{ (1)}+(r'_{\sigma(I_{p-1}+1)}+\cdots +r'_{\sigma(I_p)})\vec{e} _1^{(j_1)}}} \lb \cdots
  \lb \overline {\sigma(\vec r^{\,\prime})^{ (2)}}
  \lb \vep ^m \overline {\sigma(\vec r^{\,\prime})^{ (1)}}\rb
    \rb  \cdots \rb
    \right )\end{aligned}$$ For a given pair $g:=(\sigma,\pi)\in \Sigma_\ell \times \Pi_\ell$, let $(g)_1$ (resp. $(g)_2$) be the first term (resp. second term) in the above sum. Thus the double sum can be simply denoted by $$\sum _{g\in \Sigma_\ell\times \Pi_\ell}((g)_1+(g)_2).$$

We denote $f \equiv _\ell g$ if no coefficient of the Laurent series expansion of $f-g$ has a denominators with a factor $r'_\ell+r'_{k_1}+\cdots +r'_{k_t}$, $t\geq 0$, with $j_1,\cdots,j_t<\ell$. This is clearly an equivalence relation.

$\disp{\sum _{g\in \Sigma_\ell \times \Pi_\ell}((g)_1+(g)_2) \equiv_\ell 0.}
$

[**A special case.**]{} We first consider the special case when $g=(\sigma,\pi)\in \Sigma_\ell\times\Pi_\ell$ is of the form $(\cdots(\ell))$, that is, $\pi$ has $(\ell)$ as the last partition factor and $\sigma(\ell)=\ell$. Denote $a= \lb \overline {\sigma(\vec r^{\,\prime})^{ (p-1)}}\cdots
  \lb \overline {\sigma(\vec r^{\,\prime})^{ (2)}}
  \lb \vep ^m \overline {\sigma(\vec r^{\,\prime})^{ (1)}}\rb
    \rb  \cdots \rb$. Then we verify that $$\begin{aligned}
(g)_1+(g)_2 &=&
  \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{(1)}}}
  \lb\bar r' _\ell  \lb a\rb \rb
    +
   \bar r' _\ell\overline {\wvec{\vec s^{\,\prime
\prime (1)}}{\tau(\vec r^{\,\prime \prime})^{ (1)}+r'_{\ell} \vec{e} _1^{(j_1)}}} \lb a \rb
\notag \\
&& \equiv _\ell
  -\overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{(1)}}}
  \lb \lb \bar r' _\ell\rb \lb a \rb  \rb
    -  \lb \bar r' _\ell\rb
    \overline {\wvec{\vec s^{\,\prime\prime (1)}}
   {\tau(\vec r^{\,\prime \prime})^{ (1)}+r'_{\ell}\vec{e} _1^{(j_1)}}}
    \lb  a \rb
\mlabel{eq:lcong}\\
&&=\overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{ (1)}}}
    \lb \bar r' _\ell\rb \lb a\rb
    -\overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{ (1)}+r'_{\ell}\vec{e} _1^{(j_1)}}}
    \lb \bar r' _\ell\rb \lb  a\rb.
\notag\end{aligned}$$ Here for the $\equiv_\ell$, note that the series expansion of $\lb \bar r'_\ell \rb + \bar r'_\ell$ is the power series part of the Laurent series of $\bar r'_\ell = \uni{Z}(\wvec{0}{r'_\ell};\vep)$, and hence by Eq. (), $r'_\ell$ does not occur in the denominators the series expansion of $\lb \bar r'_\ell \rb + \bar r'_\ell$. Then the $\equiv_\ell$ follows from the easily checked properties: if $f\equiv_\ell g$, then $ \lb f \rb \equiv \lb g\rb$, and if in addition $h\not\equiv_\ell 0$, then $fh \equiv_\ell gh$. The last equation in Eq. () holds since $\check{P}$ is an idempotent Rota-Baxter operator and hence by Eq. (), $$\lb \lb x \rb \lb y \rb \rb= \lb \lb \lp x \lb y\rb + \lb x\rb y +xy \rp \rb\rb
=\lb \lp x \lb y\rb + \lb x\rb y +xy \rp \rb = \lb x\rb \lb y \rb.$$

By Eq. (), $\lb \bar {r'_\ell}\rb =-\frac 1{r'_\ell \vep}$. Thus $$\begin{aligned}
&&\lim_{r'_\ell\to 0}\Big (\overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{(1)}}}
    \lb \bar {r'_\ell}\rb
    -\overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\tau(\vec r^{\,\prime \prime})^{ (1)}+r'_{\ell}\vec{e}_1^{(j_1)}}}
    \lb \bar {r'_\ell}\rb \Big) \\
&&    =\frac{1}{\vep} \lim_{r'_\ell\to 0} \Big(\overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{(1)}}}
    -\overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\tau(\vec r^{\,\prime \prime})^{ (1)}+r'_{\ell}\vec{e}_1^{(j_1)}}}\Big)/r'_\ell
    =\frac{1}{\vep} \frac{\partial}{\partial (r''_1)} \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{(1)}}}\end{aligned}$$ exists. Here the differentiation is taken termwise in the Laurent series. Thus $$\overline {\wvec{\vec s^{\,\prime \prime (1)}}{\tau(\vec r^{\,\prime \prime})^{(1)}}}
    \lb \bar {r'_\ell}\rb
    -\overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\tau(\vec r^{\,\prime \prime})^{ (1)}+r'_{\ell}\vec{e}_1^{(j_1)}}}
    \lb \bar {r'_\ell}\rb \equiv _\ell 0.$$

[**The general case.** ]{} We now prove the lemma in general by induction on $\ell$. If $\ell=1$, then there can be only one partition $(1)$. So the special case applies and we are done. Assume the lemma is proved for $\ell-1$ and consider the case of $\ell$.

For $g=(\sigma,(i_1,\cdots,i_p))\in \Sigma_\ell\times \Pi_\ell$, consider the vector of partition vectors of $(\sigma(1),\cdots,\sigma(\ell))$ from $(i_1,\cdots,i_p)$, called a [**partitioned permutation**]{}, $$((\sigma(1),\cdots,\sigma(I_1)),(\sigma(I_1+1),\cdots,\sigma(I_2)),\cdots,
    (\sigma(I_{p-1}+1),\cdots,\sigma(\ell))).$$ Here $I_j=i_1+\cdots+i_j$, $1\leq j\leq p$. So $I_p=\ell$. This gives a natural 1-1 correspondence between $\Sigma_\ell\times \Pi_\ell$ and $$\Pi_\ell^\Sigma:=\{
((n_1, \cdots n_{I_1}),(n_{I_1+1}, \cdots
n_{I_2}),\cdots ,(n_{I_{p-1}+1}, \cdots, n_{\ell}))\}
\mlabel{eq:parper}$$ where $(I_1,I_2-I_1,\cdots,\ell-I_{p-1})$ is in $\Pi_\ell$ and $(n_1,\cdots,n_{\ell})$ is in $\Sigma_\ell$. We can thus identify $\Sigma_\ell\times \Pi_\ell$ with $\Pi_\ell^\Sigma$ and call $p=\length(g)$ the length of $g$.

For $1\leq p\leq \ell$, let

-   $\Sigma_{\ell,\,\leq p}\subseteq \Pi_\ell^\Sigma$ consisting of $g$ with $\length(g)\leq p$,

-   $\Sigma_{\ell,\leq p}^{(1)}$ consisting of $g\in \Sigma_{\ell,\,\leq p}$ whose last partition factor is not $(\ell)$,

-   $\Sigma_{\ell, \leq p}^{(2)}$ consisting of $g\in \Sigma_{\ell,\,\leq p}$ that do not contain $(\ell)$ as a partition factor,

-   $\Sigma_{\ell, \leq p}^{(3)}=\Sigma_{\ell,\,\leq p}^{(1)}\backslash \Sigma_{\ell,\,\leq p}^{(2)}$, that is, consisting of $g\in \Sigma_{\ell,\,\leq p}$ that do contain $(\ell)$ as a partition factor, but not as the last factor.

Similarly define $\Sigma_{\ell,=p}$ and $\Sigma_{\ell, =p}^{(i)}$ for $i=1,2,3.$ Thus $\Sigma_{\ell,\,\leq \ell}=\Sigma_\ell^\Pi$ and by the special case, we have $$\sum_{g\in \Sigma_\ell^\Pi\backslash\Sigma_{\ell,\,\leq \ell}^{(1)}} ((g)_1+(g)_2)
\equiv_\ell 0.$$ So to prove Lemma  we only need to prove $$\sum_{g\in \Sigma_{\ell,\,\leq \ell}^{(1)}} ((g)_1+(g)_2)
\equiv_\ell 0.
\mlabel{eq:cong1}$$ For this we first use the induction on $p=\length(g)$ to prove that for $(g)=(g)_1$ or $(g)_2$,

$$\sum_{g\in \Sigma_{\ell,\le p}^{(1)}}(g)
\equiv _\ell \sum _{g\in \Sigma_{\ell,=p}^{(2)}}(g)
-\sum _{i=1}^{\ell-1}\sum _{g\in \Sigma_{\ell-1,\leq p-1}}(g)^{(i)}
 \mlabel {eqn:length}$$

In the last term $(g)^{(i)}$ means replacing $(r'_1,\cdots,r'_{\ell-1})$ by $(r'_1, \cdots, r'_i+r'_\ell ,\cdots, r'_{\ell -1})$ in $(g)$.

The case of $\ell=1$ and thus $p=1$ is covered by the special case. For $\ell \ge 2$, we use induction on $p$. When $p=1$, there is only one partition. So Eq. () is an identity. If for $p$, formula () is true, then we have $$\begin{aligned}
\sum_{g\in \Sigma_{\ell,\le (p+1)}^{(1)}}(g)
&=& \sum_{g\in \Sigma_{\ell,\le p}^{(1)}}(g)
+ \sum_{g\in \Sigma_{\ell,= (p+1)}^{(1)}}(g) \mlabel{eq:gind}
\\
&\equiv _\ell& \sum _{g\in \Sigma_{\ell,=p}^{(2)}}(g)
-\sum _{i=1}^{\ell-1}\sum _{g\in \Sigma_{\ell-1, \leq p-1}}(g)^i +\sum _{g\in \Sigma_{\ell,=p+1}^{(2)}}(g)
+\sum _{g\in \Sigma_{\ell,=p+1}^{(3)}}(g).
\notag\end{aligned}$$

It is easily verified that the following relations are equivalence relations.

-   An element in $\Sigma_{\ell, =p+1}^{(3)}$ is of the form $(\cdots (\ell)(a_1,\cdots,a_j)\cdots )$ with $\{a_1,\cdots,a_j\}
    \subseteq [\ell]$. Define $g_1=(\cdots (\ell)(a_1, \cdots,
    a_j)\cdots )\sim_3 g_2$ if $g_2$ can be obtained from $g_1$ by a permutation of $(a_1,\cdots,a_j)$. Thus an equivalence class for $\sim_3$ is of the form $(\cdots(\ell)(a_1,\cdots,a_j)^{\Sigma_j}\cdots)$.

-   An element in $\Sigma_{\ell, =p}^{(2)}$ is of the form $(\cdots (a_1,
    \cdots, a_j)\cdots )$ with $\ell\in \{a_1,\cdots,a_j\}$. Define $g_1=(\cdots (a_1, \cdots, a_j)\cdots )\sim_2 g_2$, where $\ell\in
    \{a_1,\cdots,a_j\}$, if $g_2$ can be obtained from $g_1$ by by a permutation of $(a_1, \cdots a_j)$. Thus an equivalence class for $\sim_2$ is of the form $(\cdots(a_1,\cdots,a_j)^{\Sigma_{j}}\cdots)$.

-   An element in $[\ell -1]\times \Sigma_{\ell-1,=p}$ is of the form $\big(i, (\cdots (\sigma(I_{j-1}+1), \cdots,
    \sigma(I_j))\cdots)\big)$ where $\sigma\in \Sigma_{\ell-1}$, $(I_{j-1}+1,\cdots,I_j)$ is a block of an ordered partition of $\ell-1$ of length $p$ and $I_{j-1}+1\leq i\leq I_j$. Define $(i,
    (\cdots (\sigma(I_{j-1}+1),\cdots,\sigma(I_j))\cdots) \sim
    (i',g')$ if $I_{j-1}+1\leq i' \leq I_j$ and $g'$ can be obtained from $g$ by a permutation of $(\sigma(I_{j-1}+1),\cdots,
    \sigma(I_j))$. An equivalence class for $\sim$ is of the form $$\cup_{i=I_{j-1}+1}^{I_j} (i, (\cdots(\sigma(I_{j-1}+1),\cdots,\sigma(I_j))
    ^{\Sigma_{I_j-I_{j-1}+1}}, \cdots)).$$

There are obvious one-to-one correspondences between these equivalence classes Here $(a_1,\cdots,a_j)=(\sigma(I_{j-1}+1),\cdots,\sigma(I_j))$.

For $g=(\cdots (\ell)(a_1, \cdots, a_j)\cdots )
\in \Sigma_{\ell,=p+1}^{(3)}$, as in Eq. () we have $$\begin{aligned}
 (g)&=&\lb \lb \cdots \rb \bar r'_\ell \rb \overline {r'_{a_1} \cdots r'
_{a_j}}\cdots \equiv _\ell -\lb \lb \cdots \rb \lb \bar r'_\ell\rb
\rb \overline {r'_{a_1} \cdots r' _{a_j}}\cdots \\
&=&\lb \cdots \rb \lb
\bar r'_\ell \rb \overline {r'_{a_1} \cdots r' _{a_j}}\cdots \equiv
_\ell-\lb \cdots \rb \bar r'_\ell \overline {r'_{a_1} \cdots r'
_{a_j}}\cdots.\end{aligned}$$ By Proposition  and Eq. (), we have $$\bar {r'_\ell}\ \overline{r'_{a_1}r'_{a_2}\cdots r'_{a_{j}}}^{(\Sigma_{j})}
=\overline{r'_{\ell} r'_{a_1}r'_{a_2}\cdots r'_{a_{j}}}^{(\Sigma_{j+1})}+\sum
_{i=1}^{j}\overline{r'_{a_1}\cdots \tilde{r}_{a_i}'\cdots
r'_{a_j}}^{(\Sigma_{j})}$$ where $\tilde{r}_{a_i}'=r'_{a_i}+r'_\ell$. Using Eq. () and (), we obtain $$\sum _{h\sim_3 g}(h)=-\sum _{h\sim_2 \psi (g)}(h)-\sum _{h\sim
\rho (g)}(h).$$ Here, for $h=(i,g)$, $(h)=(g)^{(i)}$. Summing over all the equivalence classes, we have $$\sum _{h\in \sigma_{\ell,=p+1}^{(3)}}(h)
=-\sum _{h\in \sigma_{\ell,=p}^{(2)}}(h)
-\sum _{h\in [\ell-1]\times \Sigma_{\ell-1,=p}} (h)
=-\sum _{h\in \sigma_{\ell,=p}^{(2)}}(h)
-\sum_{i=1}^{\ell-1} \sum _{h\in \Sigma_{\ell-1,=p}} (h)^{(i)}.$$ Combining this with Eq. () gives $$\sum_{g\in \Sigma_{\ell,\le p+1}^{(1)}}(g)
\equiv _\ell  \sum _{g\in \Sigma_{\ell,=p+1}^{(2)}}(g)
-\sum _{i=1}^{\ell-1}\sum _{g\in \Sigma_{\ell-1, \leq p}}(g)^{(i)},$$ completing the inductive proof of Eq. ().

Take $p=\ell+1$ in Eq. (). Since the maximal length of an ordered partition of $\ell$ is $\ell$, we have $$\sum_{g\in \Sigma_{\ell,\le \ell+1}^{(1)}}((g)_1+(g)_2)
\equiv _\ell
-\sum _{i=1}^{\ell-1}\sum _{g\in \Sigma_{\ell-1, \leq \ell}}((g)_1^{(i)}+(g)_2^{(i)})
=-\sum _{i=1}^{\ell-1}\sum _{g\in \Pi_{\ell-1}^\Sigma}((g)_1^{(i)}+(g)_2^{(i)}).$$ Now by the induction hypothesis on $\ell$, the right hand side is $\equiv _\ell 0$. On the other hand, by its definition, $\Sigma_{\ell,\,\leq \ell+1}^{(1)}=\Sigma_{\ell,\,\leq \ell}^{(1)}.$ Therefore Eq. (), and hence Lemma , is proved.

Thus we have proved $$\begin{aligned}
&&\sum_{\Pi _\ell}\left (
  \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime \prime(1)}}}
  \lb \overline {\vec r^{\,\prime (p)}} \lb \cdots
  \lb \overline {\vec r^{\,\prime (2)}}
  \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb \rb
     \right .\\
&&  + \left .
    \overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\vec r^{\,\prime \prime (1)}+(r'_{I_{p-1}+1}+\cdots +r'_{I_p})\vec{e} _1^{(1)}}}
    \overline { \vec r^{\,\prime (p)}} \lb \cdots
    \lb\overline {\vec r^{\,\prime (2)}}
    \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \rb \right )^{(\Sigma _\ell)}\equiv _\ell0\end{aligned}$$ Because the action of $\Sigma_\ell$, the role of $\ell$ in the above expression is symmetric to any $1\leq t\leq \ell-1$. Thus no coefficient of its Laurent series has a denominator with a homogeneous linear factor $r'_{a_1}+\cdots +r'_{a_m}$ with $\{a_1,
\cdots , a_m \}\subset \{1,\cdots , \ell \}$. This completes Subcase 2.1.

[**Subcase 2.2: There are no type (ii) factors.**]{} Now for a fixed $\pi=(i_1,\cdots,i_p)\in \Pi_{\ell}$ and $\sigma\in \Sigma_\ell$, let $$\lb \overline {\sigma(\vec r^{\,\prime})^{(p)}}\cdots
    \lb \overline {\sigma(\vec r^{\,\prime})^{ (2)}}
    \lb \vep ^m \overline {\sigma(\vec r^{\,\prime})^{(1)}}
    \rb\rb  \cdots \rb
    =\sum_{i<0} c_i^{\pi,\sigma} \vep^i \in \CC(\vec{s}\,',\vec{r}\,')[[\vep,\vep^{-1}]$$ be the Laurent series expansion. Then the first sum for $\uni{Z}^m_+(\wvec{\vec{s}}{\vec{r}};\vep)^{(\Sigma(\vec{s}))}$ in Eq. () becomes $$\begin{aligned}
&& \hspace{-1cm}\sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell \\ \tiny{(j_1,\cdots,j_q)\in \Pi_{k+1-\ell}}\end{array}}}
    \hspace{-.3cm}\sum_{\sigma\in \Sigma_\ell}
     \tilde{P}\Big(
    \overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\cdots
    \lb \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime \prime (1)}}}
    \lb \overline {\sigma(\vec r^{\,\prime})^{(p)}}\cdots
    \lb \overline {\sigma(\vec r^{\,\prime})^{ (2)}}
    \lb \vep ^m \overline {\sigma(\vec r^{\,\prime})^{(1)}}\rb
    \rb  \cdots \rb \rb \cdots
    \Big)^{(\Sigma'')} \\
&=& \sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell \\ \tiny{(j_1,\cdots,j_q)\in \Pi_{k+1-\ell}}\end{array}}}
    \hspace{-.3cm}\sum_{\sigma\in \Sigma_\ell}
     \tilde{P}\Big(
    \overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\cdots
    \lb \sum_{i<0} c_i^{\pi,\sigma} \vep^i
    \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime \prime (1)}}}\rb \cdots
    \Big)^{(\Sigma'')} \\
&=& \sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell \\ \sigma\in \Sigma_\ell\end{array}}}
     \sum_{i<0} c_i^{\pi,\sigma}  \sum_{\tiny{(j_1,\cdots,j_q)\in \Pi_{k+1-\ell}}}
    \tilde{P}\Big(
    \overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\cdots \lb \vep^i
    \overline {\wvec{\vec s^{\,\prime \prime (1)}}{\vec r^{\,\prime \prime (1)}}}\rb \cdots
    \Big)^{(\Sigma'')}\end{aligned}$$ By the induction hypothesis on $k$, the inner most sum has a Laurent series expansion whose coefficients have denominators with no type (ii) factors. Since $c_i^{\pi,\sigma}$ is a rational function in $r_1,\cdots,r_\ell$, the same can be said of the whole sum.

Similarly, let $$\overline {\sigma(\vec r^{\,\prime})^{(p)}} \lb \cdots
    \lb \overline {\sigma(\vec r^{\,\prime})^{ (2)}}
    \lb \vep ^m \overline {\sigma(\vec r^{\,\prime})^{(1)}}
    \rb\rb  \cdots \rb
  =\sum_{-\infty<i} d_i^{\pi,\sigma} \vep^i \in \CC(\vec{s}\,',\vec{r}\,')[[\vep,\vep^{-1}]$$ be the Laurent series expansion. Then the second sum for $\uni{Z}^m_+(\wvec{\vec{s}}{\vec{r}};\vep)^{(\Sigma(\vec{s}))}$ in Eq. () becomes $$\begin{aligned}
&& \hspace{-1.5cm}\sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell\\
(j_1, j_2, \cdots, j_q)\in \Pi _{k+1-\ell}\end{array}}}
    \hspace{-1.3cm} \tilde{P}\Big(
    \overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\cdots
    \lb \overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\vec r^{\,\prime \prime (1)}+(r'_{I_{p-1}+1}+\cdots +r'_{I_p})\vec{e} _1^{(1)}}}
    \Big( \overline { \vec r^{\,\prime (p)}}\cdots
    \lb \overline {\vec r^{\,\prime (2)}}
    \lb \vep ^m \overline {\vec r^{\,\prime (1)}}\rb
    \rb  \cdots \Big)^{(\Sigma_\ell)}
    \rb  \cdots\Big)^{(\Sigma'')}
\\
&=& \hspace{-.8cm}\sum_{\tiny{\begin{array}{c}(i_1,\cdots,i_p)\in \Pi_\ell\\ \sigma\in \Sigma_\ell\end{array}}}
     \sum_{-\infty < i} d_i^{\pi,\sigma}\hspace{-.4cm}  \sum_{\tiny{(j_1,\cdots,j_q)\in \Pi_{k+1-\ell}}}
    \hspace{-.7cm} \tilde{P}\Big(
    \overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\cdots
    \lb \vep^i \overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\vec r^{\,\prime \prime (1)}+(r'_{\sigma(I_{p-1}+1)}+\cdots +r'_{\sigma(I_p)})\vec{e} _1^{(1)}}}
    \rb \cdots\Big)^{(\Sigma'')}\end{aligned}$$ Since $I_{p-1}+1,\cdots,I_{p}\leq \ell$ and $\sigma\in \Sigma_\ell$, we have $\sigma(I_{p-1}+1),\cdots,\sigma(I_p)\leq \ell$. Thus when $\vec r\to -\vec s$, we have $r'_{\sigma(I_{p-1}+1)},\cdots, r'_{\sigma(I_p)} \to 0$. Therefore, for the inner sum above, $$\sum_{\tiny{(j_1,\cdots,j_q)\in \Pi_{k+1-\ell}}}
    \hspace{-.7cm} \tilde{P}\Big(
    \overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\cdots
    \lb \vep^i \overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\vec r^{\,\prime \prime (1)}+(r'_{\sigma(I_{p-1}+1)}+\cdots +r'_{\sigma(I_p)})\vec{e} _1^{(1)}}}
    \rb \cdots\Big)^{(\Sigma'')}\Big |_{\vec r=-\vec s}$$ exists since it equals to $${
\sum_{\tiny{(j_1,\cdots,j_q)\in \Pi_{k+1-\ell}}}
    \hspace{-.7cm} \tilde{P}\Big(
    \overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\cdots
    \lb \vep^i \overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\vec r^{\,\prime \prime (1)}}} \rb \cdots\Big)^{(\Sigma'')}_{\Big |_{\vec r=-\vec s}}
= \hspace{-.3cm}\sum_{\tiny{(j_1,\cdots,j_q)\in \Pi_{k+1-\ell}}}
    \hspace{-.7cm} \tilde{P}\Big(
    \overline {\wvec{\vec s^{\,\prime \prime(q)}}{\vec r^{\,\prime \prime (q)}}}\cdots
    \lb \vep^i \overline {\wvec{\vec s^{\,\prime\prime (1)}}
    {\vec r^{\,\prime \prime (1)}}} \rb \cdots\Big)^{(\Sigma'')}_{\Big |_{\vec r\,''=-\vec s\,''}}
}$$ which exists by the induction hypothesis on $k$ and Lemma . Then by Lemma  again, the above inner sum is ordinary at $\vec r=-\vec s$ and hence at $\vec r\,'=-\vec s\,'$ and so is free of type (ii) factors in the denominators of it Laurent series coefficients. Therefore the whole expression has a Laurent series expansion whose coefficients do not have any denominators with a type (ii) factor since $d_i^{\pi,\sigma}$ does not involve $\vec s\,''$ and $\vec r\,''$.

Thus $\uni{Z}^m_+(\wvec{\vec{s}}{\vec{r}})^{(\Sigma(\vec{s}))}$ has no type (ii) factors, completing the proof of Case 2. [**Case 3: assume $\vec s=\vec 0$**]{}. The proof is basically the same as for Case 2 except that there is not type (ii) factors to exclude. For $\sigma\in \Sigma_{k+1}$ and $\pi=(i_1,\cdots,i_p)\in \Pi_{k+1}$, as in Eq. (), let $$g:=((n_1,\cdots,n_{I_1}),(n_{{I_1}+1},\cdots,n_{I_2}),\cdots,(n_{I_{p-1}+1},\cdots,n_{I_p}))\in \Sigma_{k+1}^\Pi$$ be the corresponding partitioned permutation. Let $$(g):= \tilde{P}\Big(\overline{\sigma(\vec{r})^{(p)}} \lb \cdots \lb\overline{\sigma(\vec{r})^{(2)}}\lb \vep^m \overline{\sigma(\vec{r})^{(1)}}\rb \rb \cdots \rb  \Big).$$ Then $ \uni{Z}^m_+(\wvec{\vec{s}}{\vec{r}};\vep)= \sum_{g\in \Sigma_{k+1}^\Pi} (g).$ We first prove the following analog of Lemma  by adapting its proof. $$\sum_{g\in \Sigma_{k+1}^\Pi} (g) \equiv _{k+1} 0.
\mlabel{eq:cong2}$$ Suppose $g$ has the last partition factor as $(k+1)$. Then $\sigma(k+1)=k+1$. Let $\lb a \rb = \lb \cdots \lb\overline{\sigma(\vec{r})^{(2)}}\lb \vep^m \overline{\sigma(\vec{r})^{(1)}}\rb \rb \cdots \rb
=\sum_{i<0} a_i \vep^i$. So $a_i$ are rational functions in $\{r_{\sigma(1)},\cdots,r_{\sigma(k)}\}
=\{r_1,\cdots,r_k\}$. Then by Eq. (), we have $$\begin{aligned}
(g)&=&\tilde{P}( \overline{r_{k+1}}\lb a\rb)\\
&=& \sum_{i<0} a_i \tilde{P}\llp\vep^i ( -(r_{k+1}\vep)^{-1}+\sum_{j=0}^\infty \zeta(-j)\frac{(r_{k+1}\vep)^j}{j!} )\rrp\\
&=& \sum_{i<0} a_i \tilde{P}\llp\vep^i (\sum_{j=0}^\infty \zeta(-j)\frac{(r_{k+1}\vep)^j}{j!} )\rrp\end{aligned}$$ since $\tilde{P}=\id-{P}$ is the projection of a Laurent series to its power series part. Thus no linear factor involving $r_{k+1}$ appears in the denominator of any coefficient of the Laurent series expansion of $(g)$.

Thus using the notation of the General Case in the proof of Lemma , we only need to prove $
\sum_{g\in \Sigma_{k+1, \leq k+1}^{(1)}} (g) \equiv _{k+1} 0.
$ For this we prove the following analog of Eq. () by using the same proof. $$\begin{aligned}
\sum_{g\in \Sigma_{k+1,\le p}^{(1)}}(g)
\equiv _{k+1} \sum _{g\in \Sigma_{{k+1},=p}^{(2)}}(g)
-\sum _{i=1}^{k}\sum _{g\in \Sigma_{k,\leq p-1}}(g)^{(i)}.
 \mlabel {eqn:length1}
\notag\end{aligned}$$ Then the rest of the proof of Lemma  carries through and gives Eq. (). Then again similar to Case 2, the symmetry of $r_1,\cdots, r_{k+1}$ in $\uni{Z}_+^m(\wvec{\vec{s}}{\vec{r}};\vep)$ shows that it is ordinary.

We have completed our inductive proof of Proposition  in all three cases.

The quasi-shuffle relation for non-positive MZVs
================================================

The purpose of this section is to prove the following theorem and thus to complete the proof of Theorem .

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

A lemma on stuffles
-------------------

For the proof of Thoerem , we use the stuffle interpretation of the quasi-shuffle product. The mathematics formulation of stuffles already appeared in Cartier’s construction of free commutative Rota-Baxter algebras  in 1972, even though stuffle was defined using the same formulation in the study of MZVs 20 years later . It is well-known in the literatures of MZVs that quasi-shuffle product is the same as the stuffle product . To see it in another way, it was proved in  that the stuffle product, in the variation of Cartier, is equivalent to the mixable shuffle product and the mixable shuffle product is shown in  to to be the same as the quasi-shuffle product.

For an integer $n\geq 1$, denote $[n]=\{1,\cdots,n\}$ which is also identified with the vector $(1,\cdots,n)$. For integers $k,\ell\geq 1$, a [**$(k,\ell)$-stuffle triple**]{} is a triple $(r,\alpha,\beta)$ in $$\frakS:= \left \{ (r,\alpha,\beta)\ \left |  \begin{array}{l}
\max(k,\ell)\leq r\leq k+\ell, \\
\alpha:[k]\to [r], \beta: [\ell]\to [r] \mbox{\ order preserving injections}, \\
\im(\alpha)\cup \im(\beta)=[r] \end{array} \right . \right \}$$ Thus for each $1\leq u\leq r$, at least one of $\alpha^{-1}(u)$ and $\beta^{-1}(u)$ is a singleton $\{w\}$ which we just write $w$. Similarly denote $$\tilde{\frakS}:= \left \{ (r,\alpha,\beta)\ \left |  \begin{array}{l}
\max(k,\ell)\leq r\leq k+\ell, \\
\alpha:[k]\to [r], \beta: [\ell]\to [r] \mbox{\ injective (might not preserve order)}, \\
\im(\alpha)\cup \im(\beta)=[r] \end{array} \right . \right \}$$

Let $\vec x=(x_1, \cdots, x_k)$ and $\vec y=(y_1,\cdots, y_\ell)$ be vectors of symbols. The [**stuffle of $\vec x$ and $\vec{y}$**]{} corresponding to $(r,\alpha,\beta)$ is defined by, in the notations of , [ $$\hspace{-.5cm}\vec z:= \Phi_{r,\alpha,\beta}(\vec{x},\vec{y})=(z_1,\cdots,z_r), \
z_u=\left \{ \begin{array}{ll}
x_{\alpha^{-1}(u)}, & \alpha^{-1}(u)\neq \emptyset, \beta^{-1}(u)=\emptyset,\\
y_{\beta^{-1}(u)}, & \alpha^{-1}(u)= \emptyset, \beta^{-1}(u) \neq \emptyset,\\
x_{\alpha^{-1}(u)}y_{\beta^{-1}(u)}, & \alpha^{-1}(u)\neq \emptyset, \beta^{-1}(u) \neq \emptyset
\end{array} \right .
\mlabel{eq:Phi}$$ ]{} With the convention that $x_{\emptyset}=y_{\emptyset}=\bfone$ and thus $x_{\alpha^{-1}(u)}y_{\beta^{-1}(u)}=x_{\alpha^{-1}(u)}$ if $\beta^{-1}(u)=\emptyset$ and $x_{\alpha^{-1}(u)}y_{\beta^{-1}(u)}=y_{\beta^{-1}(u)}$ if $\alpha^{-1}(u)=\emptyset$, we simply have $$\Phi_{r,\alpha,\beta}(\vec{x},\vec{y})=(x_{\alpha^{-1}(1)}y_{\beta^{-1}(1)},\cdots,
    x_{\alpha^{-1}(r)}y_{\beta^{-1}(r)}).
\mlabel{eq:Phi2}$$ Note that this definition makes sense even for $(r,\alpha,\beta)\in \tilde{\frakS}$. More generally, for subvectors (that is, subsequences) $\vec{k}\subv,\vec{\ell}\subv,
\vec{r}\subv=(r_{i_1},\cdots,r_{i_{p\subv}})$ of $[k],[\ell],[r]$ respectively with $\alpha(\vec{k}\subv)\cup \beta(\vec{\ell}\subv)=\vec{r}\subv$, consider the corresponding subvectors $\vec{x}\subv,\vec{y}\subv,\vec{z}\subv$ of $\vec{x},\vec{y},\vec{z}$ respectively. Define $$\Phi_{r\subv,\alpha|_{\vec{k}\subv},\beta|_{\vec{\ell}\subv}}
(\vec{x}\subv,\vec{y}\subv)
=(x_{\alpha^{-1}(r_{i_1})}y_{\beta^{-1}(r_{i_1})}, \cdots,
x_{\alpha^{-1}(r_{i_{p\subv}})}y_{\beta^{-1}(r_{i_{p\subv}})}).
\mlabel{eq:phi1}$$

1.  For $\sigma\in \Sigma_k, \tau\in \Sigma_\ell$, $ \Phi_{r,\alpha,\beta}(\sigma(\vec{x}),\tau(\vec{y}))= \Phi_{r,\alpha\circ \sigma^{-1}, \beta \circ \tau^{-1}}(\vec{x},\vec{y}).$

2.  Distinct triples $(r,\alpha,\beta)$ in $\tilde{\frakS}$ give distinct vectors $\Phi_{r,\alpha,\beta}(\vec{x},\vec{y})$.

3.  We have $ \tilde{\frakS}=\{ (r,\alpha\circ \sigma, \beta\circ \tau) |
        (r,\alpha,\beta)\in \frakS, \sigma\in \Sigma_k, \tau\in \Sigma_\ell\}$, giving natural actions of $\Sigma_k\times \Sigma_\ell$ on $\tilde{\frakS}$ and on $\{\Phi_{r,\alpha,\beta}(\vec{x},\vec{y}) \ |\ (r,\alpha,\beta)\in \tilde{\frakS}\}$. Furthermore the later action is free.

4.  Fix an $r_0$ with $\max(k,\ell)\leq r_0 \leq k+\ell$. Let $$\tilde{\frakS}_{r_0}=\{ (r,\alpha,\beta)\in\tilde{\frakS}\ |\ r=r_0\}.$$ Then for any $\pi\in \Sigma_{r_0}$, $ \pi(\Phi_{r_0,\alpha,\beta}(\vec{x},\vec{y}))
        =\Phi_{r_0, \pi^{-1}\circ\alpha, \pi^{-1}\circ\beta} (\vec{x},\vec{y}),$ giving a natural action of $\Sigma_{r_0}$ on $\tilde{\frakS}_{r_0}$. Furthermore, this action is free.

5.  For $(r,\alpha,\beta)\in \tilde{\frakS}$, denote $\Phi_{r,\alpha,\beta}(\vec{x},\vec{y})=(z_1,\cdots,z_r)$. For $1\leq r'\leq r''\leq r$, denote $\vec{r}\subv=(r',r'+1,\cdots,r'')$ and $\vec{z}\,^\sharp=(z_{r'},z_{r'+1},\cdots,z_{r''})$. Further denote $$\vec{k}\subv= \alpha^{-1}(\vec{r}\subv)=(k_{i_1},\cdots,k_{i_p}), \quad
     \vec{\ell}\subv= \beta^{-1}(\vec{r}\subv)=(\ell_{j_1},\cdots,\ell_{j_q}).$$ Then $$\vec{z}\,^\sharp=\Phi_{r\subv,\alpha|_{\vec{k}\subv}, \beta|_{\vec{\ell}\subv}}(\vec{x}\subv,\vec{y}\subv)$$ where the right hand side is defined by Eq. ().

In words, part () says that a part of a stuffle of two vectors is a stuffle of parts of the two vectors.

() follows from the definition of $\Phi$ and the bijectivity of $\sigma$ and $\tau$: $$\begin{aligned}
\Phi_{r,\alpha,\beta}(\sigma(\vec{x}),\tau(\vec{y}))
&=&(x_{\sigma(\alpha^{-1}(1))}y_{\tau(\beta^{-1}(1))}, \cdots,
x_{\sigma(\alpha^{-1}(r))}y_{\tau(\beta^{-1}(r))})\\
&=& (x_{(\alpha\circ\sigma^{-1})^{-1}(1)}y_{(\beta\circ\tau^{-1})^{-1}(1)}, \cdots,
x_{(\alpha\circ\sigma^{-1})^{-1}(r)}y_{(\beta\circ\tau^{-1})^{-1}(r)}).\end{aligned}$$ () Suppose $\Phi_{r,\alpha,\beta}(\vec{x},\vec{y})
    =\Phi_{r',\alpha',\beta'}(\vec{x},\vec{y})$ for $(r,\alpha,\beta)$ and $(r',\alpha',\beta')$ in $\tilde{\frakS}$. Then $r=r'$ by comparing the length of the vectors. Further by the equation, and the fact that the $x_i$s and $y_j$s are distinct variables, we have, for $1\leq u\leq r$, $ \alpha^{-1}(u)\neq \emptyset$ if and only if $ \alpha'{}^{-1}(u) \neq \emptyset$. This implies $\alpha=\alpha'$ since both maps are injective with the same domain and codomain. Similarly $\beta=\beta'$.

() As is well-known, for an injective $\alpha:[k]\to [r]$, let $\im (\alpha) = \{n_1<\cdots<n_k\}$ and let $\sigma(i)=\alpha^{-1}(n_i)$, $1\leq i\leq k$. Then $\sigma\in \Sigma_k$ and $\alpha\circ \sigma(i)=n_i$, $1\leq i\leq k$. Thus $\alpha\circ \sigma$ is order preserving and injective. Similarly for injective $\beta:[\ell]\to [r]$ we have $\tau\in \Sigma_{\ell}$ with $\beta\circ \tau: [\ell]\to [r]$ order preserving and injective. Then for $(r,\alpha,\beta)\in \tilde{\frakS}$ we have $(r,\alpha,\beta)=(r,(\alpha\circ \sigma)\circ \sigma^{-1},
(\beta\circ \tau)\circ \tau^{-1})$, as needed.

The natural action of $(\sigma,\tau)\in \Sigma_k\times \Sigma_\ell$ on $\{\Phi_{r,\alpha,\beta}\ |\ (r,\alpha,\beta)\in \tilde{\frakS}\}$ is given by $$\Phi_{r,\alpha,\beta}(\vec{x},\vec{y})^{(\sigma,\tau)}:
=\Phi_{r,\alpha,\beta}(\sigma(\vec{x}),\tau(\vec{y}))
=\Phi_{r,\alpha\circ \sigma^{-1},\beta\circ \tau^{-1}}(\vec{x},\vec{y})$$ by part (). So by part (), $\Phi_{r,\alpha,\beta}(\vec{x},\vec{y})^{(\sigma,\tau)}
=\Phi_{r,\alpha,\beta}(\vec{x},\vec{y})$ implies $(r,\alpha,\beta)=(r,\alpha\circ \sigma^{-1},\beta\circ \tau^{-1}).$ Thus $\alpha=\alpha\circ \sigma^{-1}$ and then $\sigma^{-1}=\id$ on $[k]$ since $\alpha$ is injective. Similarly, $\tau^{-1}=\id$ on $[\ell]$.

() Fix a $(r_0,\alpha,\beta)\in \tilde{\frakS}_{r_0}$ and let $\Phi_{r_0,\alpha,\beta}(\vec{x},\vec{y})=(z_1,\cdots,z_{r_0})$. Let $\pi\in \Sigma_{r_0}$. Then $ \pi(\Phi_{r_0,\alpha,\beta}(\vec{x},\vec{y}))
    =(z_{\pi(1)},\cdots,z_{\pi(r_0)}).$ By the definition of $\Phi_{r_0,\alpha,\beta}$ in Eq. () and the bijectivity of $\pi$, we have $$z_{\pi(u)}= x_{\alpha^{-1}(\pi(u))}y_{\beta^{-1}(\pi(u))}
= x_{(\pi^{-1}\circ\alpha)^{-1}(u)}
y_{(\pi^{-1}\circ\beta)^{-1}(u)}.$$ So $\pi(\Phi_{r_0,\alpha,\beta}(\vec{x},\vec{y}))$ is $\Phi_{r_0, \pi^{-1}\circ\alpha, \pi^{-1}\circ\beta} (\vec{x},\vec{y})$.

Now let $\pi, \pi'\in \Sigma_{r_0}$. By item (), $ \Phi_{r_0,\pi^{-1}\circ \alpha, \pi^{-1}\circ \beta}(\vec{x},\vec{y})
=\Phi_{r_0,\pi'{}^{-1}\circ \alpha, \pi'{}^{-1}\circ \beta}(\vec{x},\vec{y})$ implies $(r_0,\pi^{-1}\circ \alpha, \pi^{-1}\circ \beta)
    =(r_0,\pi'{}^{-1}\circ \alpha, \pi'{}^{-1}\circ \beta)$. Thus $\pi^{-1}\circ \alpha =\pi'{}^{-1}\circ \alpha$ and $\pi^{-1}\circ \beta =\pi'{}^{-1}\circ \beta$. Since $\alpha$ and $\beta$ are injective and $\im\,\alpha\cup \im\, \beta = [r_0]$. We have $\pi=\pi'$.

() This follows directly from Eq. () and Eq. ().

The proof of Theorem 
---------------------

As remarked at the beginning of this section, the quasi-shuffle product is also given by $$\vec{x} \msh \vec{y} = \sum_{(r,\alpha,\beta)\in \frakS } \Phi_{r,\alpha,\beta} (\vec{x},\vec{y}).$$ For $\vec{a}\in \ZZ^k_{\leq 0}$ and $\vec{b}\in \ZZ^\ell_{\leq 0}$, only some of the stuffles $\Phi_{r,\alpha,\beta}(\vec{a},\vec{b})$ are distinct. Denote them by $\vec{d}_j, j\in J$. So we have $ \vec{a}\msh \vec{b} =\sum_{j\in J} n_j \vec{d}_j.$ Denote $$I_j=\{ (r,\alpha,\beta) |\ \Phi_{r,\alpha,\beta}(\vec{a},\vec{b})=\vec d_j\}.$$ Define the evaluation map $$\hspace{-.45cm}f: \{ \Phi_{r,\alpha,\beta}(\vec{x},\vec{y}) | (r,\alpha,\beta)\in \frakS\}
\to \{ \Phi_{r,\alpha,\beta}(\vec{a},\vec{b}) |
(r,\alpha,\beta) \in \frakS\} = \{ \vec{d}_j | j\in J\}
\mlabel{eq:eval}$$ by sending $x_i$ to $a_i$ and $y_j$ to $b_j$. Then $ I_j =\{(r,\alpha,\beta)\ |\ \Phi_{r,\alpha,\beta}(\vec x,\vec y)\in f^{-1}(\vec{d}_j)\}$. So $|I_j|=n_j$.

Considering the actions of $\Sigma(\vec{a})\subset \Sigma_k$ and $\Sigma(\vec{b})\subset \Sigma_\ell$, we have $$\begin{aligned}
\lefteqn{\hspace{-1cm}\sum _{\sigma \in \Sigma (\vec a), \tau \in \Sigma (\vec b)}(\vec
a *\vec b, \sigma (\vec x)*\tau (\vec y))
=\sum _{\sigma\in \Sigma(\vec{a}), \tau \in \Sigma(\vec{b})}
\sum_{(r,\alpha,\beta)\in \frakS}
\big (\Phi_{r,\alpha,\beta}(\vec a, \vec b), \Phi_{r,\alpha,\beta} (\sigma(\vec x), \tau (\vec y))\big)}
\\
&=& \sum _{\sigma,\tau} \sum_{j\in J}\sum _{(r,\alpha,\beta)\in I_j}
(\vec d_j, \Phi_{r,\alpha,\beta}(\sigma(\vec x), \tau (\vec y)))\\
&=& \sum _{j\in J} \sum
_{\sigma\in \Sigma(\vec{a}), \tau\in \Sigma(\vec{b}), (r,\alpha,\beta)\in I_j} (\vec d_j, \Phi_{r,\alpha,\beta}(\sigma(\vec x), \tau (\vec y)))\end{aligned}$$ For a fixed $\vec d_j$, the directions in the inner sum are $$S_j=\{\Phi_{r,\alpha,\beta}(\sigma(\vec{x}), \tau (\vec{y}))\ | \ (r,\alpha,\beta)\in I_j, \sigma\in \Sigma(\vec{a}), \tau\in \Sigma(\vec{b}) \}.$$ By its definition, $S_j$ carries a $\Sigma (\vec a)\times \Sigma (\vec b)$ action. Also by Lemma .(), $$|S_j|=|I_j| |\Sigma (\vec a)||\Sigma (\vec b)|=n_j|\Sigma (\vec a)||\Sigma (\vec b)|.$$ We next consider the action of $\Sigma (\vec d_j)$ and prove the following lemma.

Let $r_0$ be the length of the vector $\vec{d}_j$. The free action of $\Sigma_{r_0}$ on $$\{\Phi_{r_0,\alpha,\beta}(\vec{x},\vec{y})\ |\ (r_0,\alpha,\beta)\in \tilde{\frakS}_{r_0}\}$$ defined in Lemma .() restricts to a free action of $\Sigma(\vec d_j)$ on $S_j$

Once it is shown that $S_j$ is closed under the action of $\Sigma(\vec{d}_j)$, its freeness is automatic since it is the restriction of a free action.

To prove the closeness of the action, let $\Phi_{r_0,\alpha,\beta}(\sigma(\vec{x}),\tau(\vec{y}))\in S_j$ and $\pi\in \Sigma(\vec{d}_j)$. We only need to show that there are $(r_0,\tilde{\alpha},\tilde{\beta})\in I_j$ and $(\tilde{\sigma},\tilde{\tau})\in \Sigma(\vec{a})\times
\Sigma(\vec{b})$ such that $\pi\big(\Phi_{r_0,\alpha,\beta}(\sigma(\vec{x}),\tau(\vec{y}))\big)
=\Phi_{r_0,\tilde{\alpha},\tilde{\beta}}(\tilde{\sigma}(\vec{x}), \tilde{\tau}(\vec{y})).$ By Lemma .() and (), this means $$\Phi_{r_0,\pi^{-1}\circ\alpha\circ\sigma^{-1}, \pi^{-1}\circ\beta\circ\tau^{-1}}(\vec{x},\vec{y})
=\Phi_{r_0,\tilde{\alpha}\circ \tilde{\sigma}^{-1},\tilde{\beta}\circ\tilde{\tau}^{-1}}
(\vec{x}, \vec{y}).$$ So we only need to prove $
\pi^{-1}\circ\alpha\circ\sigma^{-1} = \tilde{\alpha}\circ \tilde{\sigma}^{-1}, \quad
 \pi^{-1}\circ\beta\circ\tau^{-1}
=\tilde{\beta}\circ\tilde{\tau}^{-1}.
$ That is, to show that the following diagram commutes. $$\xymatrix{
[k] \ar[rr]^{\sigma^{-1}} \ar[d]^= && [k] \ar[rr]^\alpha &&
[r_0] \ar[d]^{\pi^{-1}} && [\ell] \ar[ll]_{\beta} &&
[\ell] \ar[ll]_{\tau^{-1}} \ar[d]^=\\
[k] \ar[rr]^{\tilde{\sigma}^{-1}} && [k] \ar[rr]^{\tilde{\alpha}} &&
[r_0] && [\ell] \ar[ll]_{\tilde{\beta}} && [\ell] \ar[ll]_{\tilde{\tau}^{-1}}
}
\mlabel{eq:sigdiag}$$ Note that the existence of such $\tilde{\sigma}$ and $\tilde{\tau}$ with $(\tilde{\sigma},\tilde{\tau})\in \Sigma_k\times \Sigma_\ell$ is already given in Lemma .(). We want to show that $(\tilde{\sigma},\tilde{\tau})$ is in $\Sigma(\vec{a})\times \Sigma(\vec{b})$ for $\pi\in \Sigma(\vec{d}_j)$.

First let us look at the action of $\pi$ “locally". Let $$\vec d_j=\Phi_{r_0,\alpha,\beta}(\vec{a},\vec{b})
=(d_1,\cdots,d_{r_0})$$ and let the sub-vector $\vec{d}\subv=(d_{r'},\cdots,d_{r''})$, $1\leq r'\leq r'' \leq r_0$, be a 0-cluster of $\vec d_j$ as defined in Definitin . Denote $\vec{r}\subv=(r',\cdots,r'')$. Let $\vec{k}\subv=\alpha^{-1}(\vec{r}\subv)$ and $\vec{\ell}\subv=\beta^{-1}(\vec{r}\subv)$. Since $\alpha$ and $\beta$ are order preserving maps between vectors, we see that $\vec{k}\subv=(k',k'+1,\cdots,k'')$ and $\vec{\ell}\subv=(\ell', \ell'+1,\cdots,\ell'')$ are sub-vectors of $[k]=(1,\cdots,k)$ and $[\ell]=(1,\cdots,\ell)$ respectively and $$\alpha\subv:=\alpha|_{\vec{k}\subv}, \quad \beta\subv:=\beta|_{\vec{\ell}\subv}$$ are order preserving maps with $\im(\alpha\subv)\cup \im(\beta\subv)=\vec{r}\subv$.

With the given $\sigma\in \Sigma(\vec a)$, $\tau\in \Sigma(\vec{b})$ and $\pi\in \Sigma_{\vec{r}\subv}\subset \Sigma(\vec{d}_j)$, the same proof for Lemma .(\[it:phi-act\]) shows that there is a bijection $\sigma\subv:\sigma(\vec{k}\subv)\to \vec{k}\subv$ and an order preserving injection $\alpha\subv: \vec{k}\subv \to \vec{r}\subv$ such that $ \pi^{-1}\circ\alpha\circ\sigma^{-1}|_{\sigma(\vec{k}\subv)}
    = \alpha\subv \circ \sigma\subv,$ that is, the left square of the following diagram is commutative $$\xymatrix{
\sigma(\vec{k}\subv) \ar[rr]^{\sigma^{-1}} \ar[d]^= && \vec{k}\subv \ar[rr]^\alpha &&
\vec{r}\subv \ar[d]^{\pi^{-1}} && \vec{\ell}\subv \ar[ll]_{\beta} &&
\tau(\vec{\ell}\subv) \ar[ll]_{\tau^{-1}} \ar[d]^=\\
\sigma(\vec{k}\subv) \ar[rr]^{\sigma\subv} && \vec{k}\subv \ar[rr]^{\alpha\subv} &&
\vec{r}\subv && \vec{\ell}\subv \ar[ll]_{\beta\subv} && \tau(\vec{\ell}\subv) \ar[ll]_{\tau\subv}
}
\mlabel{eq:sigdiag1}$$ Similarly, there is a bijection $\tau\subv:\tau(\vec{\ell}\subv)\to \vec{\ell}\subv$ and an order preserving injection $\beta\subv: \vec{\ell}\subv \to \vec{r}\subv$ such that the right square of the diagram is commutative.

Since our choice of $\pi$ is the identity when restricted to $[r_0]\backslash \vec{r}\subv$, we have the trivial commutative diagram of bijections and order preserving maps $$\xymatrix{
[k]\backslash \sigma(\vec{k}\subv) \ar[r]^{\sigma^{-1}} \ar[d]^= & [k]\backslash \vec{k}\subv \ar[r]^\alpha &
[r_0]\backslash \vec{r}\subv \ar[d]^{\pi^{-1}} & [\ell]\backslash \vec{\ell}\subv \ar[l]_{\beta} &
[\ell]\backslash \tau(\vec{\ell}\subv) \ar[l]_{\tau^{-1}} \ar[d]^=\\
[k]\backslash \sigma(\vec{k}\subv) \ar[r]^{\sigma^{-1}} & [k]\backslash \vec{k}\subv \ar[r]^{\alpha} &
[r_0]\backslash \vec{r}\subv & [\ell]\backslash \vec{\ell}\subv \ar[l]_{\beta} & [\ell]\backslash \tau(\vec{\ell}\subv) \ar[l]_{\tau^{-1}}
}
\mlabel{eq:sigdiag2}$$ Taking the union of these two diagrams, we obtain a commutative diagram () where the bijections and order preserving maps are defined by $$\tilde{\sigma}(i)=\left \{ \begin{array}{ll}
    \sigma\subv^{-1}(i), & i\in \vec{k}\subv, \\
    \sigma(i), & i\not\in \vec{k}\subv
    \end{array} \right . \quad
     \tilde{\alpha}(i)=\left \{ \begin{array}{ll}
    \alpha\subv(i), & i\in \vec{k}\subv, \\
    \alpha(i), & i \not\in \vec{k}\subv
    \end{array} \right . \quad 1 \leq i\leq k.$$ $$\tilde{\tau}(i)=\left \{ \begin{array}{ll}
    \tau\subv^{-1}(i), & i\in \vec{\ell}\subv, \\
    \tau(i), & i\not\in \vec{\ell}\subv
    \end{array} \right . \quad
     \tilde{\beta}(i)=\left \{ \begin{array}{ll}
    \beta\subv(i), & i\in \vec{\ell}\subv, \\
    \beta(i), & i \not\in \vec{\ell}\subv
    \end{array} \right . \quad 1 \leq i\leq \ell.$$

We next show that $\tilde{\sigma}\in \Sigma(\vec{a})$ and $\tilde{\tau}\in \Sigma(\vec{b})$. Let $\vec{z}\subv=(z_{r'},\cdots,z_{r''})$ be the the sub-vector of $\Phi_{r_0,\alpha,\beta}(\vec{x},\vec{y})
=(z_1,\cdots,z_{r_0})$ corresponding to $\vec{r}\subv$. Similarly, denote $$\vec{x}\subv=(x_{k'},\cdots,x_{k''}),
\vec{y}\subv=(y_{\ell'},\cdots,y_{\ell''}),
\vec{a}\subv=(a_{k'},\cdots,a_{k''}),
\vec{b}\subv=(b_{\ell'},\cdots,b_{\ell''}).$$ By Lemma .(), we have $$\vec{z}\subv = \Phi_{r\subv,\alpha\subv,\beta\subv}(\vec{x}\subv,\vec{y}\subv),
\quad
\vec{d}\subv = \Phi_{r\subv,\alpha\subv,\beta\subv}(\vec{a}\subv,\vec{b}\subv).$$ Here $r\subv=r''-r'+1$.

Under the evaluation map (), $$f(\vec{z}\subv)
=f(\Phi_{r\subv,\alpha\subv,\beta\subv}(\vec{x}\subv, \vec{y}\subv))= \Phi_{r\subv,\alpha\subv,\beta\subv}(\vec{a}\subv, \vec{b}\subv)= \vec{d}\subv=\vec{0},$$ and $f(x_i)=a_i, f(y_j)=b_j, f(x_iy_j)=a_i+b_j$ are non-positive numbers or their sums. So we must have $\vec{a}\subv=\vec{0}$ and $\vec{b}\subv=\vec{0}.$ Thus $\vec{a}\subv$ (resp. $\vec{b}\subv$) is a part of a zero cluster of $\vec{a}$ (resp. $\vec{b}$). Thus $\vec{k}\subv\subseteq \vec{k}^{(i)}\subseteq [k]$ where $\vec{k}^{(i)}$ is the index set of a zero cluster of $\vec{a}$ (see Definition ()). Since $\Sigma(\vec{a})|_{\vec{k}^{(i)}}=\Sigma_{\vec{k}^{(i)}}$, we have $\sigma(\vec{k}\subv)\subseteq \sigma(\vec{k}^{(i)})
=\vec{k}^{(i)}.$ Further, there is $\tilde{\sigma}'\in \Sigma_{\vec{k}^{(i)}}$ such that $\tilde{\sigma}'|_{\vec{k}\subv}=\sigma\subv{}^{-1}$ and $\tilde{\sigma}'|_{\vec{k}^{(i)}\backslash\vec{k}\subv}=\sigma.$ Since $\Sigma(\vec{a})=\Sigma_{\vec{k}^{(i)}}\times \Sigma''$, we further have $\tilde{\sigma}=(\tilde{\sigma}',\sigma'')\in \Sigma(\vec{a}).$ Similarly, $\tilde{\tau}\in \Sigma(\vec{b})$.

Thus $S_j$ is closed under the action of $\Sigma(\vec{r}\subv)$ and hence of $\Sigma(\vec{d}_j)$ since $\Sigma (\vec d_j)$ is the direct product of such $\Sigma(\vec{r}\subv)$ from all the zero clusters of $\vec{d}_j$.

By Lemma , there are $n_j|\Sigma (\vec a)||\Sigma (\vec b)|/|\Sigma (d_j)|$ orbits. Let $$\vec r_h, \ h=1, \cdots, n_j|\Sigma (\vec a)||\Sigma (\vec b)|/|\Sigma (d_j)|$$ be a complete set of representatives of $S_j$, then $$\sum _{\sigma \in \Sigma (\vec a), \tau \in \Sigma (\vec b)}(\vec
a *\vec b, \sigma (\vec x)*\tau (\vec y))=\sum _{j} \sum _h(\vec
d_j, \vec r_h)^{(\Sigma (\vec d_j))}$$ Thus Note that, for a fixed $h$, $\vec x \to -\vec a, \vec y\to -\vec b$ means $\vec r_h \to -\vec d_j$. Then $$\lim _{\vec x \to -\vec a, \vec y\to -\vec b}
\zeta\lp\wvec{\vec d_j}{\vec r_h}\rp^{(\Sigma (\vec d_j))}
= \lim _{r_h\to -\vec d_j}
\zeta\lp\wvec{\vec d_j}{\vec r_h}\rp^{(\Sigma (\vec d_j))}.$$ So by Theorem , we have $$\gzeta(\vec a)\gzeta(\vec b)=\frac 1{|\Sigma (\vec a)||\Sigma
(\vec b)|}\sum _{j} n_j|\Sigma (\vec a)||\Sigma (\vec b)|\gzeta
(\vec d_j)=\sum _j n_j\gzeta (\vec d_j)=\gzeta (\vec a *\vec b).
$$ This completes the proof of Theorem .

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