# Discrete iterated integrals and cyclic sum formulas

Hanamichi Kawamura  Department of Mathematics, Faculty of Science Division I,
Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601,
Japan<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider a discrete version of iterated integrals by the
naive (equally divided) Riemann sum. In particular, basic three formulas for
usual iterated integrals are discretized. Moreover, we proved cyclic sum
formulas for discrete iterated integrals. They imply the cyclic sum formula
for multiple polylogarithms.

###### Key words and phrases:

iterated integral, discrete iterated integral, multiple polylogarithm, cyclic
sum formula

###### 2020 Mathematics Subject Classification:

11M32.

##  1\. Introduction

We often encounter _iterated integrals_ like

$I_{\mathrm{dch}}(1_{0};e_{a_{1}}\cdots
e_{a_{k}};(-1)_{1})=\int_{0<t_{1}<\cdots<t_{k}<1}\frac{dt_{1}}{t_{1}-a_{1}}\cdots\frac{dt_{k}}{t_{k}-a_{k}}\qquad(a_{1},\ldots,a_{k}\in\mathbb{C}\setminus(0,1)),$

in recent researches of number theory, knot theory, mathematical physics, etc.
One of most important examples of iterated integrals is the _multiple zeta
value_ , which has the expression as a nested sum other than the iterated
integral:

$(-1)^{r}I_{\mathrm{dch}}(1_{0};e_{1}e_{0}^{k_{1}-1}\cdots
e_{1}e_{0}^{k_{r}-1};(-1)_{1})=\zeta(k_{1},\ldots,k_{r})\coloneqq\sum_{0<n_{1}<\cdots<n_{r}}\frac{1}{n_{1}^{k_{1}}\cdots
n_{r}^{k_{r}}}.$ (1.1)

General iterated integrals, denoted as $I_{\gamma}(x;w;y)$, satisfy many
important formulas. The typical examples are the following.

######  Theorem 1.1.

Let $\gamma$ and $\gamma^{\prime}$ be paths. Denote by $x_{\circ}$ (resp.
$y_{\circ}$) the initial (resp. final) point of
$\circ\in\\{\gamma,\gamma^{\prime}\\}$. Assume that $y_{\gamma}$ is equal to
$x_{\gamma^{\prime}}$. Then, for $a_{1},\ldots,a_{k}\in D$ and
$w,w^{\prime}\in\mathfrak{H}_{D}$, we have

  1. (1)

(Path composition formula) $\displaystyle
I_{\gamma\gamma^{\prime}}(x_{\gamma};e_{a_{1}}\cdots
e_{a_{k}};y_{\gamma^{\prime}})=\sum_{i=0}^{k}I_{\gamma}(x_{\gamma};e_{a_{1}}\cdots
e_{a_{i}};y_{\gamma})I_{\gamma^{\prime}}(x_{\gamma^{\prime}};e_{a_{i+1}}\cdots
e_{a_{k}};y_{\gamma^{\prime}})$.

  2. (2)

(Reversal formula) $I_{\gamma}(x_{\gamma};e_{a_{1}}\cdots
e_{a_{k}};y_{\gamma})=(-1)^{k}I_{\gamma^{-1}}(y_{\gamma};e_{a_{k}}\cdots
e_{a_{1}};x_{\gamma})$.

  3. (3)

(Shuffle product formula)
$I_{\gamma}(x_{\gamma};w\mathbin{\mathcyr{sh}}w^{\prime};y_{\gamma})=I_{\gamma}(x_{\gamma};w;y_{\gamma})I_{\gamma}(x_{\gamma};w^{\prime};y_{\gamma})$.

Recently, Maesaka, Seki and Watanabe [15] discovered that:

######  Theorem 1.2 ([15, Theorem 1.3]).

Let $N,r,k_{1},\ldots,k_{r}$ be positive integers. Put $k\coloneqq
k_{1}+\cdots+k_{r}$ and define
$S\coloneqq\\{0,k_{1},k_{1}+k_{2},\ldots,k_{1}+\cdots+k_{r}\\}$. Then we have

$\sum_{\begin{subarray}{c}0=n_{0}\leq n_{1}\leq\cdots\leq n_{k}\leq
n_{k+1}=N\\\ i\in S\implies n_{i}\neq n_{i+1}\end{subarray}}\left(\prod_{i\in
S\setminus\\{k\\}}\frac{1}{N-n_{i+1}}\right)\left(\prod_{i\in\\{0,\ldots,k-1\\}\setminus
S}\frac{1}{n_{i+1}}\right)=\sum_{0<n_{1}<\cdots<n_{r}<N}\frac{1}{n_{1}^{k_{1}}\cdots
n_{r}^{k_{r}}}.$

Comparing this fact and (1.1), we can consider the left-hand side above as a
_discretization_ of $I_{\mathrm{dch}}(1_{0};e_{1}e_{0}^{k_{1}-1}\cdots
e_{1}e_{0}^{k_{r}-1};(-1)_{1})$ In this paper, we consider a discretization
$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))$ for more general iterated integrals
and associated notions. The basic three formulas for iterated integrals, which
are known as the _path composition formula_ , _reversal formula_ and _shuffle
product formula_ respectively, are discretized:

######  Theorem 1.3.

Let $\gamma$ and $\gamma^{\prime}$ be composable discrete paths. Then, for
$a_{1},\ldots,a_{k}\in D$ and a $\gamma$-shuffleable pair
$(w,w^{\prime})\in(\mathfrak{H}_{D})^{2}$, we have

  1. (1)

(Path composition formula, Corollary 3.11)

$\Delta_{N,\gamma\gamma^{\prime}}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k}};t(\gamma^{\prime}))=\sum_{i=0}^{k}\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{i}};t(\gamma))\Delta_{N,\gamma^{\prime}}(s(\gamma^{\prime});e_{a_{i+1}}\cdots
e_{a_{k}};t(\gamma^{\prime})).$

  2. (2)

(Reversal formula, Theorem 3.12)

$\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k}};t(\gamma))=(-1)^{k}\Delta_{N,\gamma^{-1}}(t(\gamma);e_{a_{k}}\cdots
e_{a_{1}};s(\gamma)).$

  3. (3)

(Asymptotic shuffle product formula, Theorem 3.17)

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))\Delta_{N,\gamma}(s(\gamma);w^{\prime};t(\gamma))=\Delta_{N,\gamma}(s(\gamma);w\mathbin{\mathcyr{sh}}w^{\prime};t(\gamma))+O(N^{-1}\log^{\bullet}N),\qquad\text{as
}N\to\infty.$

In the second main theorem, we evaluate a cyclic sum on discrete iterated
integrals along a straight path.

######  Theorem 1.4 ($=$ Theorem 5.3).

Let $w=e_{a_{1}}\cdots e_{a_{k}}$ ($k\geq 2$) be an element of
$\mathfrak{H}_{D}$ and $\gamma=(x,y)$ be a positive straight piece. Then we
have

$\sum_{\begin{subarray}{c}1\leq i\leq k\\\ a_{i}\neq
x\end{subarray}}\Delta_{N,\gamma}((x,+);e_{a_{i}}\cdots
e_{a_{k}}e_{a_{1}}\cdots
e_{a_{i-1}}e_{x};(y,-))+\sum_{\begin{subarray}{c}1\leq i\leq k\\\ a_{i}\neq
y\end{subarray}}\Delta_{N,\gamma}((x,+);e_{y}e_{a_{i+1}}\cdots
e_{a_{k}}e_{a_{1}}\cdots e_{a_{i-1}}e_{x})\\\ =\sum_{\begin{subarray}{c}1\leq
i\leq k\\\
a_{i}\in\\{x,y\\}\end{subarray}}\Delta_{N,\gamma}((x,+);e_{a_{i}}\cdots
e_{a_{k}}e_{a_{1}}\cdots
e_{a_{i}};(y,-))-\sum_{i=1}^{k}\Delta_{N,\gamma}((x,+);e_{a_{1},\ldots,a_{i},a_{i},\ldots,a_{k}};(y,-)).$
(1.2)

As its consequence, we show that there is a generalization of the cyclic sum
formula for multiple zeta values ([10]) for multiple polylogarithms (see
Corollary 5.4).
This paper is organized as follows: we give a brief review about usual
iterated integrals in Section 2. After introducing discrete iterated integrals
and proving basic properties in Section 3, we see how discrete iterated
integrals converge to usual ones in Section 4. Finally, in Section 5, we prove
the cyclic sum formula for discrete iterated integrals along a straight path.

## Acknowledgements

The author gratefully thanks to Prof. Minoru Hirose, Prof. Toshiki Matsusaka
and Prof. Shin-ichiro Seki for kindly letting him know about their result [5].

##  2\. Review on usual iterated integrals

In what follows, we fix a finite subset $D$ of $\mathbb{C}$ and put
$M\coloneqq\mathbb{C}\setminus D$ unless noted otherwise.

###  2.1. Tangential base points and paths

For $p\in D$ and a non-zero tangential vector $v\in
T_{p}(\mathbb{C})\simeq\mathbb{C}$, we call a tuple $v_{p}\coloneqq(p,v)$ a
_tangential base point_ , where the isomorphism is induced by the canonical
coordinate of $\mathbb{C}$. Let $\mathrm{pr}(v_{p})\coloneqq p$ for a
tangential base point $v_{p}$ and $\mathrm{pr}(p)\coloneqq p$ for a usual base
point ($=$ an element of $\mathbb{C}\setminus D$) $p$.

######  Definition 2.1 (Path).

Let $x$ and $y$ be (tangential or not) base points. We call $\gamma$ a path if
$\gamma\colon[0,1]\to\mathbb{C}$ is a piecewise smooth path satisfying
$\gamma(0)=\mathrm{pr}(x)$, $\gamma(1)=\mathrm{pr}(y)$, $\gamma((0,1))\in M$,

$\lim_{t\to+0}\frac{\gamma(t)-\mathrm{pr}(x)}{t}=v\qquad\text{if
}x=v_{\mathrm{pr}(x)}\text{ is tangential}$

and

$\lim_{t\to 1-0}\frac{\gamma(t)-\mathrm{pr}(y)}{t-1}=-w\qquad\text{if
}y=w_{\mathrm{pr}(y)}\text{ is tangential}.$

The point $x$ (resp. $y$) is called the _initial point_ (resp. the _final
point_) of $\gamma$.

The set $\pi_{1}(M;y,x)$ of homotopy classes for such a path $\gamma$ (with
the initial point $x$ and the final point $y$) is defined and they admit the
composition

$\pi_{1}(M;y,x)\times\pi_{1}(M;z,y)\to\pi_{1}(M;z,x);~{}(\gamma,\gamma^{\prime})\mapsto\gamma\gamma^{\prime}$

and inversion $\pi_{1}(M;y,x)\to\pi_{1}(M;x,y);~{}\gamma\mapsto\gamma^{-1}$
(cf. [1, §3.8.3]). For a path $\gamma$ and real numbers $a,b$ satisfying
$0\leq a<b\leq 1$, we put $\gamma_{a,b}(t)\coloneqq\gamma((1-t)a+tb)$. Note
that, for a path $\gamma$, choosing suffciently small positive numbers $z$ and
$z^{\prime}$, we get the path $\gamma_{z,1-z^{\prime}}$ whose initial and
final point are not tangential. When $(0,1)\cap D=\varnothing$, the path
defined by $t\mapsto t$ is usually denoted by $\mathrm{dch}$.

###  2.2. Algebraic setup

Define $\mathfrak{H}_{D}$ be the non-commutative free $\mathbb{Q}$-algebra
generated by formal symbols $e_{z}$, where $z$ runs over $D$. For a path
$\gamma$ with the initial point $x$ and the final point $y$, the subalgebra
$\mathfrak{H}_{D}^{\gamma}$ of $\mathfrak{H}_{D}$ is defined by

$\mathfrak{H}_{D}^{\gamma}\coloneqq\mathbb{Q}\oplus\bigoplus_{a\in
D\setminus\\{\mathrm{pr}(x),\mathrm{pr}(y)\\}}\mathbb{Q}e_{a}\oplus\bigoplus_{\begin{subarray}{c}a\in
D\setminus\\{\mathrm{pr}(x)\\}\\\ b\in
D\setminus\\{\mathrm{pr}(y)\\}\end{subarray}}e_{a}\mathfrak{H}_{D}e_{b}.$

######  Definition 2.2 (Shuffle product).

Define a $\mathbb{Q}$-bilinear product $\mathbin{\mathcyr{sh}}$ on
$\mathfrak{H}_{D}$ by $1\mathbin{\mathcyr{sh}}w=w\mathbin{\mathcyr{sh}}1=w$
for all $w\in\mathfrak{H}_{D}$ and the recursive rule

$e_{a}w\mathbin{\mathcyr{sh}}e_{b}w^{\prime}=e_{a}(w\mathbin{\mathcyr{sh}}e_{b}w^{\prime})+e_{b}(e_{a}w\mathbin{\mathcyr{sh}}w^{\prime})\qquad(a,b\in
D,~{}w,w^{\prime}\in\mathfrak{H}_{D}).$

This product induces an associative and commutative $\mathbb{Q}$-algebra
structure on $\mathfrak{H}_{D}$ and $\mathfrak{H}_{D}^{\gamma}$ for any path
$\gamma$.

######  Proposition 2.3.

With the following structures, $\mathfrak{H}_{D}$ is a Hopf
$\mathbb{Q}$-algebra.

  1. (1)

The product $\mathfrak{H}_{D}\otimes\mathfrak{H}_{D}\to\mathfrak{H}_{D}$ is
the shuffle product.

  2. (2)

The unit $\mathbb{Q}\to\mathfrak{H}_{D}$ is determined by $1\mapsto 1$.

  3. (3)

The coproduct $\mathfrak{H}_{D}\to\mathfrak{H}_{D}\otimes\mathfrak{H}_{D}$ is
determined by $\displaystyle\Delta^{\mathbin{\mathcyr{sh}}}(e_{a_{1}}\cdots
e_{a_{k}})\coloneqq\sum_{i=0}^{k}e_{a_{1}}\cdots e_{a_{i}}\otimes
e_{a_{i+1}}\cdots e_{a_{k}}$ ($a_{j}\in D$).

  4. (4)

The counit $\mathfrak{H}_{D}\to\mathbb{Q}$ is determined by $e_{a}\mapsto 0$
($a\in D$).

  5. (5)

The antipode $\mathfrak{H}_{D}\to\mathfrak{H}_{D}$ is determined by
$S^{\mathbin{\mathcyr{sh}}}(e_{a_{1}}\cdots
e_{a_{k}})\coloneqq(-1)^{k}e_{a_{k}}\cdots e_{a_{1}}$.

###  2.3. Iterated integrals and regularization

######  Definition 2.4 (Convergent iterated integrals).

Let $\gamma$ be a path with the initial point $x$ and the final point $y$. For
$w=e_{a_{1}}\cdots e_{a_{k}}\in\mathfrak{H}_{D}$, we put

$I_{\gamma}(x;w;y)\coloneqq\int_{0<t_{1}<\cdots<t_{k}<1}\prod_{i=1}^{k}\omega_{a_{i}}(\gamma(t_{i})),$

where $\omega_{a}(t)$ is the $1$-form $dt/(t-a)$. Extending it
$\mathbb{Q}$-linearly, we define the $\mathbb{Q}$-linear map
$I_{\gamma}(x;-;y)\colon\mathfrak{H}_{D}^{\gamma}\to\mathbb{C}$. We call this
map (or its image) the (convergent) _iterated integral_.

######  Definition 2.5 (Regularized iterated integrals).

Take a sufficiently small number $0<z<1$. Then, for any
$w\in\mathfrak{H}_{D}$, there uniquely exists $P(x)\in\mathbb{C}[x]$ such that

$I_{\gamma_{z,1-z}}(\gamma(z);w;\gamma(1-z))=P(\log
z)+O(z\log^{J}z)\qquad(z\to+0)$

for some $J>0$ (cf. [1, Lemma 3.351]). From these data, we define the
_regularized iterated integral_ by $I_{\gamma}(x;w;y)\coloneqq P(0)$.

######  Remark 2.6.

On the above setting, if $w$ is an element of $\mathfrak{H}_{D}^{\gamma}$ the
regularized iterated integral $I_{\gamma}(x;w;y)$ coincides with the iterated
integral in the sense of Definition 2.4.

The following three formulas are basic properties of iterated integrals.

######  Theorem 2.7 ($=$ Theorem 1.1).

Let $\gamma$ and $\gamma^{\prime}$ be paths. Denote by $x_{\circ}$ (resp.
$y_{\circ}$) the initial (resp. final) point of
$\circ\in\\{\gamma,\gamma^{\prime}\\}$. Assume that $y_{\gamma}$ is equal to
$x_{\gamma^{\prime}}$. Then, for $a_{1},\ldots,a_{k}\in D$ and
$w,w^{\prime}\in\mathfrak{H}_{D}$, we have

  1. (1)

(Path composition formula) $\displaystyle
I_{\gamma\gamma^{\prime}}(x_{\gamma};e_{a_{1}}\cdots
e_{a_{k}};y_{\gamma^{\prime}})=\sum_{i=0}^{k}I_{\gamma}(x_{\gamma};e_{a_{1}}\cdots
e_{a_{i}};y_{\gamma})I_{\gamma^{\prime}}(x_{\gamma^{\prime}};e_{a_{i+1}}\cdots
e_{a_{k}};y_{\gamma^{\prime}})$.

  2. (2)

(Reversal formula) $I_{\gamma}(x_{\gamma};e_{a_{1}}\cdots
e_{a_{k}};y_{\gamma})=(-1)^{k}I_{\gamma^{-1}}(y_{\gamma};e_{a_{k}}\cdots
e_{a_{1}};x_{\gamma})$.

  3. (3)

(Shuffle product formula)
$I_{\gamma}(x_{\gamma};w\mathbin{\mathcyr{sh}}w^{\prime};y_{\gamma})=I_{\gamma}(x_{\gamma};w;y_{\gamma})I_{\gamma}(x_{\gamma};w^{\prime};y_{\gamma})$.

##  3\. Discrete iterated integrals

In what follows, we fix a large positive even integer $N$ and put
$\mathbb{Z}_{/N}\coloneqq\\{n/N\mid n\in\mathbb{Z}\\}\subseteq\mathbb{Q}$. The
symbol $D$ stands for a finite subset of $\mathbb{C}$.

###  3.1. Discrete paths

In this subsection, we discretize the theory of continuous paths connecting
(tangential or not) base points.

######  Definition 3.1 (Discrete paths).

  1. (1)

The set of _endpoints_ is $B\coloneqq(\mathbb{Z}_{/N}\setminus
D)\sqcup((\mathbb{Z}_{/N}\cap D)\times\\{1,\ldots,N\\})$. If $a\in D$, we also
write $(a,+)\coloneqq(a,N/2)$ and $(a,-)\coloneqq(a,0)\coloneqq(a,N)$.

  2. (2)

Define a map $B\to\mathbb{Z}_{/N}$; $x\mapsto\hat{x}$ as $\hat{x}\coloneqq x$
if $x\in\mathbb{Z}_{/N}\setminus D$ and $\hat{x}\coloneqq a$ if
$x=(a,i)\in(\mathbb{Z}_{/N}\cap D)\times\\{1,\ldots,N\\}$.

  3. (3)

We define the following three types of _pieces_ :

    1. (a)

A _positive straight piece_ is an element $(x,y)\in(\mathbb{Z}_{/N})^{2}$ such
that $x<y$ and the open interval $(x,y)$ does not intersect with $D$. Its
_initial point_ is $(x,+)$ if $x\in D$ and is $x$ if $x\notin D$. Its _final
point_ is $(y,-)$ if $y\in D$ and is $y$ if $y\notin D$.

    2. (b)

A _negative straight piece_ is an element $(x,y)\in(\mathbb{Z}_{/N})^{2}$ such
that $y<x$ and the open interval $(-x,-y)$ does not intersect with $D$. Its
initial point is $(x,-)$ if $x\in D$ and is $x$ if $x\notin D$. Its final
point is $(y,+)$ if $y\in D$ and is $y$ if $y\notin D$.

    3. (c)

A _positive singular piece_ is an element $(x,a,b,+)$ of
$(D\cap\mathbb{Z}_{/N})\times\\{0,\ldots,N\\}^{2}\times\\{+,-\\}$ such that
$a<b$. Its initial point is $(x,a)$ and final point is $(x,b)$.

    4. (d)

A _negative singular piece_ is an element $(x,a,b,-)$ of
$(D\cap\mathbb{Z}_{/N})\times\\{0,\ldots,N\\}^{2}\times\\{+,-\\}$ such that
$a<b$. Its initial point is $(x,N-b)$ and final point is $(x,N-a)$.

    5. (e)

A _trivial piece_ is an element $x$ of copy $B^{\prime}$ of $B$. Its initial
and final points are $x$. If $p\in D$, we use the notation
$(p,a,a,+)\coloneqq(p,a,a,-)\coloneqq(p,a)\in B^{\prime}$ for every
$a\in\\{1,\ldots,N\\}$ and regard it as one of singular pieces.

  4. (4)

Define maps $s$ (resp. $t$) from the set of pieces to $B$ to be what assigns a
piece to its initial (resp. final) point.

  5. (5)

We say that a finite tuple $\gamma=(\gamma_{1},\ldots,\gamma_{r})$ ($r\geq 1$)
is a _discrete path_ (on $\mathbb{Z}_{/N}\setminus D$) if each $\gamma_{i}$ is
one of pieces above such that $t(\gamma_{i})=s(\gamma_{i+1})$ for every
$i\in\\{1,\ldots,k-1\\}$. Define $s(\gamma)\coloneqq s(\gamma_{1})$ and
$t(\gamma)\coloneqq t(\gamma_{r})$.

Note that the above definition of discrete paths depends on $D$. However,
since we fix $D$, we omit the information of $D$ and simply say as “a discrete
path $\gamma$”.

######  Definition 3.2 (Operations on discrete paths).

  1. (1)

For a discrete path $\gamma_{i}=(\gamma_{i,1},\ldots,\gamma_{i,r_{i}})$
($i\in\\{1,2\\}$) satisfying $t(\gamma_{1})=s(\gamma_{2})$, the _composition_
$\gamma_{1}\gamma_{2}$ is the concatenation
$(\gamma_{1,1},\ldots,\gamma_{1,r_{1}},\gamma_{2,1},\ldots,\gamma_{2,r_{2}})$
of them as tuples of pieces.

  2. (2)

For a discrete path $\gamma=(\gamma_{1},\ldots,\gamma_{r})$, the _inverse_ is
a discrete path
$\gamma^{-1}\coloneqq(\gamma_{r}^{-1},\ldots,\gamma_{1}^{-1})$, where the
inverse of pieces are defined as:

    1. (a)

If $\gamma_{i}=(x,y)$ is a (positive or negative) straight piece, put
$\gamma_{i}^{-1}\coloneqq(y,x)$.

    2. (b)

If $\gamma_{i}=(x,a,b,\sigma)$ is a (positive or negative) singular piece, put
$\gamma_{i}^{-1}\coloneqq(x,N-b,N-a,-\sigma)$.

    3. (c)

If $\gamma_{i}$ is a trivial piece, its inverse is itself.

######  Remark 3.3.

From the definition, it is clear that a discrete path
$\gamma=(\gamma_{1},\ldots,\gamma_{r})$ is composition of its pieces
$\gamma_{1}\cdots\gamma_{r}$ by identifying the piece $\gamma_{i}$ with the
discrete path $(\gamma_{i})$. Hereafter we sometimes use this identification.

######  Definition 3.4 (Provision of discrete paths).

Let $\gamma=(\gamma_{1},\ldots,\gamma_{r})$ be a discrete path. We define the
_provision_ of $\gamma$ as the set
$S_{\gamma}\coloneqq\bigsqcup_{i=1}^{r}(S_{\gamma_{i}}\times\\{i\\})$, where
$S_{\gamma_{i}}$ is determined as follows:

  1. (1)

If $\gamma_{i}=(a,b)$ is a positive straight piece,
$S_{\gamma_{i}}\coloneqq\\{aN+1,\ldots,bN-1\\}\subseteq\mathbb{Z}$.

  2. (2)

If $\gamma_{i}=(a,b)$ is a negative straight piece,
$S_{\gamma_{i}}\coloneqq\\{-aN+1,\ldots,-bN,-1\\}\subseteq\mathbb{Z}$.

  3. (3)

If $\gamma_{i}=(x,a,b,\sigma)$ is a (positive or negative) singular piece,
$S_{\gamma_{i}}$ is a set
$\\{\theta_{a+1}^{\sigma},\ldots,\theta_{b}^{\sigma}\\}$, which consists of
formal symbols $\theta_{j}^{\pm}$ (the sign represents whether the piece is
positive or negative, which does not mean the inverse).

  4. (4)

If $\gamma_{i}$ is trivial, set $S_{\gamma_{i}}\coloneqq\varnothing$.

For an element $n\in S_{\gamma}$, denote by $\tilde{n}$ the first component of
$n$ and define
$n+a\coloneqq(\tilde{n}+a,i)\in\mathbb{Z}\times\\{1,\ldots,r\\}$ for
$a\in\mathbb{Z}$ and $n\in S_{\gamma_{i}}\subseteq S_{\gamma}$ if $\gamma_{i}$
is straight. The projection on the second component
$S_{\gamma}\to\\{1,\ldots,r\\}$ is denoted by $J$.

Moreover, we introduce the order $<_{\gamma}$ for the provision of a path
$\gamma=(\gamma_{1},\ldots,\gamma_{r})$ in the following way.

  1. (1)

$(m,i)<_{\gamma}(n,i+1)$ for every $i\in\\{1,\ldots,r-1\\}$, $(m,i)\in
S_{\gamma_{i}}$ and $(n,i+1)\in S_{\gamma_{i+1}}$.

  2. (2)

$(m,i)<_{\gamma}(n,i)$ if $\gamma_{i}$ is a positive or negative piece and
$m<n$.

  3. (3)

$(\theta_{j}^{\pm},i)<_{\gamma}(\theta_{j+1}^{\pm},i)$ for
$j\in\\{1,\ldots,N-1\\}$ if $\gamma_{i}$ is a (positive or negative) singular
piece.

The condition that $A<_{\gamma}B$ or $A=B$ holds is denoted by
$A\leq_{\gamma}B$ as usual.

######  Definition 3.5 (Elimination of discrete paths).

Let $\gamma=(\gamma_{1},\ldots,\gamma_{r})$ be a discrete path and $m,n\in
S_{\gamma}$ satisfying $m\leq_{\gamma}n$. Write $m=(\tilde{m},i)$ and
$n=(\tilde{n},j)$. When $i\neq j$, we define a new discrete path
$\gamma_{m,n}$ by
$(\gamma^{\prime}_{i},\gamma_{i+1},\ldots,\gamma_{j-1},\gamma^{\prime\prime}_{j})$
where

$\gamma^{\prime}_{i}\coloneqq\begin{cases}((\tilde{m}-1)/N,\widehat{t(\gamma_{i})})&\text{if
}\gamma_{i}\text{ is a positive straight path},\\\
(-(\tilde{m}-1)/N,\widehat{t(\gamma_{i})})&\text{if }\gamma_{i}\text{ is a
negative straight path},\\\ (x,h,b,\pm)&\text{if }\gamma_{i}=(x,a,b,\pm)\text{
is a (positive or negative) singular path and
}\tilde{m}=\theta_{h}^{\pm},\end{cases}$

and

$\gamma^{\prime\prime}_{j}\coloneqq\begin{cases}(\widehat{s(\gamma_{j})},(\tilde{n}+1)/N)&\text{if
}\gamma_{j}\text{ is a positive straight path},\\\
(\widehat{s(\gamma_{j})},-(\tilde{n}+1)/N)&\text{if }\gamma_{j}\text{ is a
negative straight path},\\\ (x,a,h,\pm)&\text{if }\gamma_{j}=(x,a,b,\pm)\text{
is a (positive or negative) singular path and
}\tilde{n}=\theta_{h}^{\pm}.\end{cases}$

If $i=j$, we define

$\gamma_{m,n}\coloneqq\begin{cases}((\tilde{m}-1)/N,(\tilde{n}+1)/N)&\text{if
}\gamma_{i}\text{ is a positive straight path},\\\
(-(\tilde{m}-1)/N,-(\tilde{n}+1)/N)&\text{if }\gamma_{i}\text{ is a negative
straight path},\\\ (x,j,j^{\prime},\pm)&\text{if }\gamma_{i}=(x,a,b,\pm)\text{
is a (positive or negative) singular path, }\tilde{m}=\theta_{j}^{\pm}\text{
and }\tilde{n}=\theta_{j^{\prime}}^{\pm}.\end{cases}$

If $m=n$, the path $\gamma_{m,n}$ is defined to be a trivial piece $\tilde{m}$
if $\gamma_{i}$ ($=\gamma_{j}$) is straight and to be a trivial piece $(x,j)$
if $\gamma_{i}=\gamma_{j}=(x,a,b,\pm)$ is singular, where $j$ is the subscript
of $\tilde{m}=\theta_{j}^{\pm}$.

Note that, for a discrete path $\gamma$ and $a,b,m,n\in S_{\gamma}$ satisfying
$a\leq_{\gamma}m\leq_{\gamma}n\leq_{\gamma}b$, there is an inclusion
$S_{\gamma_{m,n}}\subseteq S_{\gamma_{a,b}}\subseteq S_{\gamma}$ and the
iterated elimination $(\gamma_{a,b})_{m,n}$ is equal to $\gamma_{m,n}$.
Moreover we see that $S_{\gamma_{a,b}}=\\{n\in S_{\gamma}\mid
a<_{\gamma}n<_{\gamma}b\\}$ holds.

###  3.2. Definition of discrete iterated integrals

######  Definition 3.6 (Discrete differential forms).

Let $\gamma=(\gamma_{1},\ldots,\gamma_{r})$ be a discrete path,
$n=(\tilde{n},i)\in S_{\gamma}$ and $a\in D$. Then we define
$\omega_{a}(n)\in\mathbb{C}[\theta]$ by

$\omega_{a}^{N}(n)\coloneqq\begin{cases}\displaystyle\frac{1}{\tilde{n}-aN}&\text{if
}\gamma_{i}\text{ is a positive straight piece},\\\
\displaystyle\frac{1}{\tilde{n}+aN}&\text{if }\gamma_{i}\text{ is a negative
straight piece},\\\ \displaystyle\frac{\theta}{N}&\text{if }\gamma_{i}\text{
is a positive singular piece whose first entry is }a,\\\
-\displaystyle\frac{\theta}{N}&\text{if }\gamma_{i}\text{ is a negative
singular piece whose first entry is }a,\\\ 0&\text{otherwise.}\end{cases}$

Furthermore, for elements $m=(\tilde{m},i)$ and $n=(\tilde{n},j)$ of
$S_{\gamma}$, we define

$\omega^{N}(m,n)\coloneqq\begin{cases}\displaystyle\frac{1}{\tilde{n}-\tilde{m}}&\text{if
}\gamma_{i}\text{ and }\gamma_{j}\text{ are both positive or negative straight
pieces and }\tilde{n}\neq\tilde{m},\\\ 0&\text{otherwise.}\end{cases}$

######  Definition 3.7 (Discrete iterated integrals).

Let $\gamma$ be a discrete path and $w=e_{a_{1}}\cdots
e_{a_{k}}\in\mathfrak{H}_{D}$. Then we define the _discrete iterated integral_
by

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))\coloneqq\sum_{\begin{subarray}{c}n_{1},\ldots,n_{k}\in
S_{\gamma}\\\
n_{1}\leq_{\gamma}\cdots\leq_{\gamma}n_{k}\end{subarray}}\prod_{i=1}^{k}\omega_{a_{i}}^{N}(n_{i}).$

We put $\Delta_{N,\gamma}(s(\gamma);1;t(\gamma))\coloneqq 1$. Moreover, by the
$\mathbb{Q}$-linear extension, we define
$\Delta_{N,\gamma}(s(\gamma);-;t(\gamma))\colon\mathfrak{H}_{D}^{\gamma}\to\mathbb{C}[\theta]$.

######  Example 3.8.

Let $D=\\{0,1\\}$, $w=e_{1}e_{0}^{2}$ and $\gamma=(\gamma_{1})$ be a discrete
path which consists of only the positive straight piece $\gamma_{1}=(0,1)$.
Then $S_{\gamma}=\\{(1,1),(2,1)\ldots,(N-1,1)\\}$ and
$(i,1)\leq_{\gamma}(j,1)$ if and only if $i\leq j$. Thus we have

$\Delta_{N,\gamma}((0,+);e_{1}e_{0}^{2};(1,-))=\sum_{0<n_{1}\leq n_{2}\leq
n_{3}<N}\frac{1}{(n_{1}-N)n_{2}n_{3}}.$

Taking the limit $N\to\infty$ and using [15, Theorem 1.3], we obtain
$\lim_{N\to\infty}\Delta_{N,\gamma}((0,+);e_{1}e_{0}^{2};(1,-))=I_{\mathrm{dch}}(1_{0};e_{1}e_{0}^{2};(-1)_{1})$.
More general examples explaining such a limit which tends to the usual
iterated integral appears in §4.

###  3.3. Basic formulas for discrete iterated integrals

######  Proposition 3.9.

Let $\gamma=(x,a,b,\pm)$ is a singular piece and identify it with a discrete
path $(\gamma)$. Then we have

$\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k}};t(\gamma))=\begin{cases}\displaystyle\frac{(b-a)_{k}}{k!}\left(\frac{\pm\theta}{N}\right)^{k}&\text{if
}a_{1}=\cdots=a_{k}=x,\\\ 0&\text{otherwise}\end{cases}$

for $a_{1},\ldots,a_{k}\in D$. Here $(X)_{k}\coloneqq X(X+1)\cdots(X+k-1)$
stands for the Pochhammer symbol.

###### Proof.

We proceed the proof by induction on $k$. The case where $k=0$ is obvious. If
the case for some $k$ holds, we have

$\displaystyle\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k+1}};t(\gamma))$ $\displaystyle=\sum_{a<n_{1}\leq\cdots\leq n_{k+1}\leq
b}\prod_{i=1}^{k+1}\omega_{a_{i}}^{N}(\theta_{n_{i}}^{\pm})$ (3.1)
$\displaystyle=\sum_{a\leq n\leq
b}\Delta_{N,(x,a,n,\pm)}((x,a);e_{a_{1}}\cdots
e_{a_{k}};(x,n))\omega_{a_{k+1}}(\theta_{n}^{\pm}).$ (3.2)

If $\\{a_{1},\ldots,a_{k+1}\\}\neq\\{x\\}$ we obtain the desired result.
Otherwise from this expression we have

$\displaystyle\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k+1}};t(\gamma))$ $\displaystyle=\sum_{a<n\leq
b}\frac{(n-a)_{k}}{k!}\left(\frac{\pm\theta}{N}\right)^{k}\cdot\frac{\pm\theta}{N}$
(3.3) $\displaystyle=\left(\frac{\pm\theta}{N}\right)^{k+1}\sum_{a<n\leq
b}\frac{(n-a)_{k}}{k!}$ (3.4)
$\displaystyle=\left(\frac{\pm\theta}{N}\right)^{k+1}\sum_{a<n\leq
b}\left(\frac{(n-a)_{k+1}}{(k+1)!}-\frac{(n-a-1)_{k+1}}{(k+1)!}\right)$ (3.5)
$\displaystyle=\frac{(b-a)_{k+1}}{(k+1)!}\left(\frac{\pm\theta}{N}\right)^{k+1}.$
(3.6)

∎

The following two lemmas are obvious from the definition.

######  Lemma 3.10.

Let $\gamma=(\gamma_{1},\ldots,\gamma_{r})$ be a discrete path and
$w=e_{a_{1}}\cdots e_{a_{k}}\in\mathfrak{H}_{D}$. Then we have

$\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k}};t(\gamma))=\sum_{0=i_{0}\leq i_{1}\leq\cdots\leq i_{r-1}\leq
i_{r}=k}\prod_{j=1}^{r}\sum_{\begin{subarray}{c}n_{i_{j-1}+1},\ldots,n_{i_{j}}\in
S_{\gamma_{j}}\\\
n_{i_{j-1}+1}\leq_{\gamma_{j}}\cdots\leq_{\gamma_{j}}n_{i_{j}}\end{subarray}}\prod_{h=i_{j-1}+1}^{i_{j}}\omega_{a_{h}}^{N}(n_{h}).$

######  Corollary 3.11 (Path composition formula).

Let $\gamma$ and $\gamma^{\prime}$ be composable discrete paths (i.e.,
$t(\gamma)=s(\gamma^{\prime})$). Then, for $w=e_{a_{1}}\cdots
e_{a_{k}}\in\mathfrak{H}_{D}$, we have

$\Delta_{N,\gamma\gamma^{\prime}}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k}};t(\gamma^{\prime}))=\sum_{i=0}^{k}\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{i}};t(\gamma))\Delta_{N,\gamma^{\prime}}(s(\gamma^{\prime});e_{a_{i+1}}\cdots
e_{a_{k}};t(\gamma^{\prime})).$

######  Theorem 3.12 (Reversal formula).

Let $\gamma$ be a discrete path. Then, for $a_{1},\ldots,a_{k}\in D$, we have

$\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k}};t(\gamma))=(-1)^{k}\Delta_{N,\gamma^{-1}}(t(\gamma);e_{a_{k}}\cdots
e_{a_{1}};s(\gamma)).$

###### Proof.

Let $\gamma=(\gamma_{1},\ldots,\gamma_{r})$. Applying Corollary 3.11
repeatedly, we have

$\Delta_{N,\gamma}(s(\gamma);e_{a_{1}}\cdots
e_{a_{k}};t(\gamma))=\sum_{0=i_{0}\leq i_{1}\leq\cdots\leq i_{r-1}\leq
i_{r}=k}\prod_{j=1}^{r}\Delta_{N,\gamma_{j}}(s(\gamma_{j});e_{a_{i_{j-1}+1}}\cdots
e_{a_{i_{j}}};t(\gamma_{j}))$

and

$\Delta_{N,\gamma^{-1}}(t(\gamma);e_{a_{k}}\cdots
e_{a_{1}};s(\gamma))=\sum_{0=i^{\prime}_{0}\leq i^{\prime}_{1}\leq\cdots\leq
i^{\prime}_{r-1}\leq
i^{\prime}_{r}=k}\prod_{j=1}^{r}\Delta_{N,\gamma_{r+1-j}^{-1}}(s(\gamma_{r+1-j});e_{a_{i^{\prime}_{r+1-j}}}\cdots
e_{a_{i^{\prime}_{r-j}+1}};s(\gamma_{r+1-j})).$

Since we can give a one-to-one correspondence
$(i_{1},\ldots,i_{r-1})\mapsto(i^{\prime}_{1},\ldots,i^{\prime}_{r})\coloneqq(k-i_{r-1},\ldots,k-i_{1})$
between the ranges of each sum above, it is sufficient to prove

$\Delta_{N,\gamma_{j}}(s(\gamma_{j});e_{a_{i_{j-1}+1}}\cdots
e_{a_{i_{j}}};t(\gamma_{j}))=(-1)^{i_{j}-i_{j-1}}\Delta_{N,\gamma_{j}^{-1}}(t(\gamma_{j});e_{a_{i_{j}}}\cdots
e_{a_{i_{j-1}+1}};s(\gamma_{j}))$

for $(i_{1},\ldots,i_{r-1})$ satisfying $0=i_{0}\leq i_{1}\leq\cdots\leq
i_{r-1}\leq i_{r}=k$ and $j\in\\{1,\ldots,r\\}$. If $\gamma_{j}$ is a
(positive or negative) straight path, we obtain this identity by substitution
$n\mapsto-n$. When $\gamma_{j}=(x,a,b,\pm)$ is a singular path, from
Proposition 3.9 we have

$\displaystyle\Delta_{N,\gamma_{j}}(s(\gamma_{j});e_{a_{i_{j-1}+1}}\cdots
e_{a_{i_{j}}};t(\gamma_{j}))$ (3.7)
$\displaystyle=\begin{cases}\displaystyle\frac{(b-a)_{i_{j}-i_{j-1}}}{(i_{j}-i_{j-1})!}\left(\frac{\pm\theta}{N}\right)^{i_{j}-i_{j-1}}&\text{if
}a_{i_{j-1}+1}=\cdots=a_{i_{j}}=x,\\\ 0&\text{otherwise},\end{cases}$ (3.8)
$\displaystyle=(-1)^{i_{j}-i_{j-1}}\begin{cases}\displaystyle\frac{((N-a)-(N-b))_{i_{j}-i_{j-1}}}{(i_{j}-i_{j-1})!}\left(\frac{\mp\theta}{N}\right)^{i_{j}-i_{j-1}}&\text{if
}a_{i_{j-1}+1}=\cdots=a_{i_{j}}=x,\\\ 0&\text{otherwise},\end{cases}$ (3.9)
$\displaystyle=(-1)^{i_{j}-i_{j-1}}\Delta_{N,\gamma_{j}^{-1}}(t(\gamma_{j});e_{a_{i_{j}}}\cdots
e_{a_{i_{j-1}+1}};s(\gamma_{j})).$ (3.10)

∎

######  Lemma 3.13.

Let $\gamma=(x,y)$ be a positive straight piece (regarded as a discrete path)
and $a\in D$. Then we have

$\sum_{xN\leq n\leq yN}\frac{1}{|n-aN|}=O(\log N),\qquad\text{as }N\to\infty.$

###### Proof.

For the case where $a\notin\\{x,y\\}$, put

$\rho\coloneqq\begin{cases}\mathrm{Re}(a)-y&\text{if }\mathrm{Re}(a)>y,\\\
x-\mathrm{Re}(a)&\text{if }x>\mathrm{Re}(a),\\\
\mathrm{Im}(a)&\text{otherwise.}\end{cases}$

Then this quantity does not depend on $N$ and $|n-aN|\geq\rho N>0$ always
holds for an integer $xN<n<yN$. Thus we have

$\sum_{xN<n<yN}\frac{1}{|n-aN|}\leq\frac{(y-x)N-1}{\rho N}.$

If $a=x$, we obtain

$\displaystyle\sum_{xN<n<yN}\frac{1}{|n-xN|}$
$\displaystyle=\sum_{0<n<(y-x)N}\frac{1}{n}$ (3.11) $\displaystyle=\log
N+O(1),\qquad\text{as }N\to\infty.$ (3.12)

The estimation for the $a=y$ case is also obtained by a similar computation. ∎

For treat more general convergences, we define an auxiliary algebra
$\mathcal{P}_{D}\coloneqq\mathbb{Q}\langle e_{a_{1},\ldots,a_{m}}\mid a_{i}\in
D,~{}m\geq 1\rangle$ and auxiliary sums

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))\coloneqq\sum_{\begin{subarray}{c}n_{1},\ldots,n_{k}\in
S_{\gamma}\\\
n_{1}\leq_{\gamma}\cdots\leq_{\gamma}n_{k}\end{subarray}}\prod_{i=1}^{k}\prod_{j=1}^{m_{i}}\omega_{a_{i,j}}(n_{i}),$

for a straight piece $\gamma$ and $w=e_{a_{1,1},\ldots,a_{1,m_{1}}}\cdots
e_{a_{k,1},\ldots,a_{k,m_{k}}}\in\mathcal{P}_{D}$. Under the same setting, we
also define its variant as

$|\Delta|_{N,\gamma}(s(\gamma);w;t(\gamma))\coloneqq\sum_{\begin{subarray}{c}n_{1},\ldots,n_{k}\in
S_{\gamma}\\\
n_{1}\leq_{\gamma}\cdots\leq_{\gamma}n_{k}\end{subarray}}\prod_{i=1}^{k}\prod_{j=1}^{m_{i}}|\omega_{a_{i,j}}(n_{i})|.$

######  Lemma 3.14.

Let $\gamma$ be a straight piece and $w=e_{a_{1,1},\ldots,a_{1,m_{1}}}\cdots
e_{a_{k,1}}\cdots e_{a_{k,m_{k}}}\in\mathcal{P}_{D}$. Put
$x\coloneqq\widehat{s(\gamma)}$ and $y\coloneqq\widehat{t(\gamma)}$. Then we
have the following:

  1. (1)

As $N\to\infty$, the sums $\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))$ and
$|\Delta|_{N,\gamma}(s(\gamma);w;t(\gamma))$ are $O(\log^{\bullet}N)$.

  2. (2)

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))=O(N^{-1}\log^{\bullet}N)$ as
$N\to\infty$ if there exists $1\leq i\leq k$ and $1\leq j<j^{\prime}\leq
m_{i}$ such that $m_{i}\geq 2$ and $a_{i,j}\neq a_{i,j^{\prime}}$.

  3. (3)

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))=O(N^{-1}\log^{\bullet}N)$ as
$N\to\infty$ if there exists $1\leq i\leq k$ and $1\leq j<j^{\prime}\leq
m_{i}$ such that $m_{i}\geq 2$ and $a_{i,j}=a_{i,j^{\prime}}\in
D\setminus\\{x,y\\}$.

  4. (4)

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))=O(N^{-1}\log^{\bullet}N)$ as
$N\to\infty$ if there exists $1\leq i<i^{\prime}\leq k$, $1\leq j\leq m_{i}$
and $1\leq j^{\prime}<j^{\prime\prime}\leq m_{i^{\prime}}$ such that
$m_{i^{\prime}}\geq 2$,
$a_{i^{\prime},j^{\prime}}=a_{i^{\prime},j^{\prime\prime}}=x$ and $a_{i,j}\neq
x$.

  5. (5)

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))=O(N^{-1}\log^{\bullet}N)$ as
$N\to\infty$ if there exists $1\leq i^{\prime}<i\leq k$, $1\leq j\leq m_{i}$
and $1\leq j^{\prime}<j^{\prime\prime}\leq m_{i^{\prime}}$ such that
$m_{i^{\prime}}\geq 2$,
$a_{i^{\prime},j^{\prime}}=a_{i^{\prime},j^{\prime\prime}}=y$ and $a_{i,j}\neq
y$.

###### Proof.

Arguments similar to the below proofs for the positive straight case also
works for the negative straight case. Thus we deal with only the positive
straight case. For $1\leq i\leq k$ and $1\leq j\leq m_{i}$ define

$\mathbf{a}_{i}\coloneqq(a_{i,1},\ldots,a_{i,m_{i}})$

and

$\mathbf{a}_{i}\langle
j\rangle\coloneqq(a_{i,1},\ldots,a_{i,j-1},a_{i,j+1},\ldots,a_{i,m_{i}})$

  1. (1)

For the case where $\gamma$ is positive straight, it is sufficient that the
latter case is proven by the triangle inequality. From Lemma 3.13 we have

$|\Delta|_{N,\gamma}(s(\gamma);w;t(\gamma))\leq\sum_{\begin{subarray}{c}xN<n_{i,j}<yN\\\
1\leq i\leq k\\\ 1\leq j\leq
m_{i}\end{subarray}}\prod_{j=1}^{m}\frac{1}{|n_{i,j}-a_{i,j}N|}=O(\log^{mk}N),\qquad\text{as
}N\to\infty.$

  2. (2)

By the partial fraction decomposition

$\frac{1}{n_{i}-a_{i,j}N}\frac{1}{n_{i}-a_{i,j^{\prime}}N}=\frac{1}{N(a_{i,j}-a_{i,j^{\prime}})}\left(\frac{1}{n_{i}-a_{i,j}N}-\frac{1}{n_{i}-a_{i,j^{\prime}}N}\right),$

we obtain

$\displaystyle\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))$
$\displaystyle=\sum_{xN<n_{1}\leq\cdots\leq
n_{k}<yN}\prod_{h=1}^{k}\prod_{h^{\prime}=1}^{m_{i}}\frac{1}{n_{h}-a_{h,h^{\prime}}N}$
(3.13)
$\displaystyle=\begin{multlined}\frac{1}{N(a_{i,j}-a_{i,j^{\prime}})}\left(\Delta_{N,\gamma}(s(\gamma);e_{\mathbf{a}_{1}}\cdots
e_{\mathbf{a}_{i-1}}e_{\mathbf{a}_{i}\langle
j\rangle}e_{\mathbf{a}_{i+1}}\cdots e_{\mathbf{a}_{k}};t(\gamma))\right.\\\
\left.-\Delta_{N,\gamma}(s(\gamma);e_{\mathbf{a}_{1}}\cdots
e_{\mathbf{a}_{i-1}}e_{\mathbf{a}_{i}\langle
j^{\prime}\rangle}e_{\mathbf{a}_{i+1}}\cdots
e_{\mathbf{a}_{k}};t(\gamma))\right).\end{multlined}\frac{1}{N(a_{i,j}-a_{i,j^{\prime}})}\left(\Delta_{N,\gamma}(s(\gamma);e_{\mathbf{a}_{1}}\cdots
e_{\mathbf{a}_{i-1}}e_{\mathbf{a}_{i}\langle
j\rangle}e_{\mathbf{a}_{i+1}}\cdots e_{\mathbf{a}_{k}};t(\gamma))\right.\\\
\left.-\Delta_{N,\gamma}(s(\gamma);e_{\mathbf{a}_{1}}\cdots
e_{\mathbf{a}_{i-1}}e_{\mathbf{a}_{i}\langle
j^{\prime}\rangle}e_{\mathbf{a}_{i+1}}\cdots
e_{\mathbf{a}_{k}};t(\gamma))\right).$ (3.16)

Then the desired result follows from this equality and (1). If $\gamma$ is
singular,

  3. (3)

Put

$\rho\coloneqq\begin{cases}\mathrm{Re}(a_{i,j})-y&\text{if
}\mathrm{Re}(a_{i,j})>y,\\\ x-\mathrm{Re}(a_{i,j})&\text{if
}x>\mathrm{Re}(a_{i,j}),\\\ \mathrm{Im}(a_{i,j})&\text{otherwise.}\end{cases}$

Then $\rho$ is independent of $N$ and the inequality $|n_{i}-a_{i,j}N|\geq\rho
N>0$ always holds for an integer $xN<n_{i,j}<yN$. Thus we have

$\displaystyle|\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))|$
$\displaystyle\leq|\Delta|_{N,\gamma}(s(\gamma);w;t(\gamma))$ (3.17)
$\displaystyle\leq\frac{1}{\rho
N}|\Delta|_{N,\gamma}(s(\gamma);e_{\mathbf{a}_{1}}\cdots
e_{\mathbf{a}_{i-1}}e_{\mathbf{a}_{i}\langle
j\rangle}e_{\mathbf{a}_{i+1}}\cdots e_{\mathbf{a}_{k}};t(\gamma))$ (3.18)
$\displaystyle\stackrel{{\scriptstyle(1)}}{{=}}O(N^{-1}\log^{\bullet}N)\qquad\text{as
}N\to\infty.$ (3.19)

  4. (4)

We have

$\displaystyle|\Delta|_{N,\gamma}(s(\gamma);w;t(\gamma))$
$\displaystyle=\sum_{xN<n_{1}\leq\cdots\leq
n_{k}<yN}\prod_{\begin{subarray}{c}1\leq h\leq k\\\ 1\leq h^{\prime}\leq
m_{h}\end{subarray}}\frac{1}{|n_{h}-a_{h,h^{\prime}}N|}$ (3.20)
$\displaystyle\leq\sum_{xN<n_{1}\leq\cdots\leq
n_{k}<yN}\frac{1}{|n_{i}-xN|}\prod_{\begin{subarray}{c}1\leq h\leq k\\\ 1\leq
h^{\prime}\leq m_{h}\\\
(h,h^{\prime})\neq(i^{\prime},j^{\prime})\end{subarray}}\frac{1}{|n_{h}-a_{h,h^{\prime}}N|}$
(3.21) $\displaystyle=|\Delta|_{N,\gamma}(s(\gamma);e_{\mathbf{a}_{1}}\cdots
e_{\mathbf{a}_{i-1}}e_{\mathbf{a}_{i},x}e_{\mathbf{a}_{i+1}}\cdots
e_{\mathbf{a}_{i^{\prime}-1}}e_{\mathbf{a}_{i^{\prime}}\langle
j^{\prime}\rangle}e_{\mathbf{a}_{i^{\prime}+1}}\cdots
e_{\mathbf{a}_{k}};t(\gamma))$ (3.22)

since $|n_{i^{\prime}}-xN|\geq|n_{i}-xN|$. Then we can use (2) for the most
right-hand side above.

  5. (5)

We have

$\displaystyle|\Delta|_{N,\gamma}(s(\gamma);w;t(\gamma))$
$\displaystyle=\sum_{xN<n_{1}\leq\cdots\leq
n_{k}<yN}\prod_{\begin{subarray}{c}1\leq h\leq k\\\ 1\leq h^{\prime}\leq
m_{h}\end{subarray}}\frac{1}{|n_{h}-a_{h,h^{\prime}}N|}$ (3.23)
$\displaystyle\leq\sum_{xN<n_{1}\leq\cdots\leq
n_{k}<yN}\frac{1}{|n_{i}-yN|}\prod_{\begin{subarray}{c}1\leq h\leq k\\\ 1\leq
h^{\prime}\leq m_{h}\\\
(h,h^{\prime})\neq(i^{\prime},j^{\prime})\end{subarray}}\frac{1}{|n_{h}-a_{h,h^{\prime}}N|}$
(3.24) $\displaystyle=|\Delta|_{N,\gamma}(s(\gamma);e_{\mathbf{a}_{1}}\cdots
e_{\mathbf{a}_{i^{\prime}-1}}e_{\mathbf{a}_{i^{\prime}}\langle
j^{\prime}\rangle}e_{\mathbf{a}_{i^{\prime}+1}}\cdots
e_{\mathbf{a}_{i-1}}e_{\mathbf{a}_{i},y}e_{\mathbf{a}_{i+1}}\cdots
e_{\mathbf{a}_{k}};t(\gamma))$ (3.25)

since $|n_{i^{\prime}}-yN|\geq|n_{i}-yN|$. Then we can use (2) for the most
right-hand side above.

∎

From here we use the $O$-notation for also elements of $\mathbb{C}[\theta]$,
that is, we write as

$\sum_{i=0}^{k}a_{i,N}\theta^{i}=O(b_{N}),\qquad\text{as }N\to\infty,$

for $k\in\mathbb{Z}_{\geq 0}$ and sequences $a_{i,-}\colon\mathbb{Z}_{\geq
1}\to\mathbb{C}$ ($0\leq i\leq k$) satisfying $a_{i,N}=O(b_{N})$ as
$N\to\infty$ for every $0\leq i\leq k$.

######  Remark 3.15.

The estimations (2)–(5) appearing in Lemma 3.14 hold for also the case where
$\gamma$ is singular. Indeed, for $\gamma=(x,a,b,\pm)$ we easily obtain

$\Delta_{N,\gamma}((s,\gamma);w;(t,\gamma))=\begin{cases}\displaystyle\frac{(b-a)_{k}}{k!}\left(\frac{\pm\theta}{N}\right)^{mk}&\text{if
every }a_{i,j}\text{ is }x,\\\ 0&\text{otherwise.}\end{cases}$

######  Definition 3.16.

Let $\gamma=(\gamma_{1},\ldots,\gamma_{r})$ be a discrete path and
$w=e_{a_{1}}\cdots e_{a_{k}}$ and $w^{\prime}=e_{b_{1}}\cdots e_{b_{l}}$
elements of $\mathfrak{H}_{D}$. Then we say that a pair
$(w,w^{\prime})\in(\mathfrak{H}_{D})^{2}$ is _$\gamma$ -shuffleable_ if, for
every increasing sequences $0=i_{0}\leq i_{1}\leq\cdots\leq i_{r}=k$ and
$0=j_{0}\leq j_{1}\leq\cdots\leq j_{r}=l$ and $1\leq c\leq r$, either
$a_{i_{c-1}+1}$ or $b_{j_{c-1}+1}$ (resp. $a_{i_{c}}$ or $b_{j_{c}}$) does
_not_ coincide with $\widehat{s(\gamma_{c})}$ (resp.
$\widehat{t(\gamma_{c})}$).

We remark that, if $\gamma$ is a piece, the above condition asserts that
$\\{a_{1},b_{1}\\}\neq\\{\widehat{s(\gamma)}\\}$ and
$\\{a_{k},b_{l}\\}\neq\\{\widehat{t(\gamma)}\\}$.

######  Theorem 3.17 (Asymptotic shuffle product formula).

Let $\gamma$ be a discrete path and $w=e_{a_{1}}\cdots e_{a_{k}}$ and
$w^{\prime}=e_{b_{1}}\cdots e_{b_{l}}$ elements of $\mathfrak{H}_{D}$. Assume
that $(w,w^{\prime})$ is $\gamma$-shuffleable. Then we have

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))\Delta_{N,\gamma}(s(\gamma);w^{\prime};t(\gamma))=\Delta_{N,\gamma}(s(\gamma);w\mathbin{\mathcyr{sh}}w^{\prime};t(\gamma))+O(N^{-1}\log^{\bullet}N),\qquad\text{as
}N\to\infty.$ (3.26)

###### Proof.

The proof proceeds in a way similar to [16, Proposition 2.4]. First we
consider the case where $\gamma$ is a piece. By definition we have

$\displaystyle\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))\Delta_{N,\gamma}(s(\gamma);w^{\prime};t(\gamma))$
(3.27)
$\displaystyle=\sum_{\begin{subarray}{c}m_{1},\ldots,m_{k},n_{1},\ldots,n_{l}\in
S_{\gamma}\\\ m_{1}\leq_{\gamma}\cdots\leq_{\gamma}m_{k}\\\
n_{1}\leq_{\gamma}\cdots\leq_{\gamma}n_{l}\end{subarray}}\left(\prod_{i=1}^{k}\omega_{a_{i}}^{N}(m_{i})\right)\left(\prod_{j=1}^{l}\omega_{b_{j}}^{N}(n_{j})\right)$
(3.28)
$\displaystyle=\Delta_{N,\gamma}((s,\gamma);w\mathbin{\mathcyr{sh}}w^{\prime};(t,\gamma))+\sum_{\begin{subarray}{c}m_{1},\ldots,m_{k},n_{1},\ldots,n_{l}\in
S_{\gamma}\\\ m_{1}\leq_{\gamma}\cdots\leq_{\gamma}m_{k}\\\
n_{1}\leq_{\gamma}\cdots\leq_{\gamma}n_{l}\\\
\exists(i,j),~{}m_{i}=n_{j}\end{subarray}}\left(\prod_{i=1}^{k}\omega_{a_{i}}^{N}(m_{i})\right)\left(\prod_{j=1}^{l}\omega_{b_{j}}^{N}(n_{j})\right).$
(3.29)

Focusing on the second term on the right-hand side, we see that
$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))\Delta_{N,\gamma}(s(\gamma);w^{\prime};t(\gamma))-\Delta_{N,\gamma}((s,\gamma);w\mathbin{\mathcyr{sh}}w^{\prime};(t,\gamma))$
can be decomposed to a finite sum of $\Delta_{N,\gamma}$ values satisfying one
of the conditions in (2)–(5) of Lemma 3.14. Indeed, we can use (2) if
$a_{i}\neq b_{j}$, (3) if
$a_{i}=b_{j}\notin\\{\widehat{s(\gamma)},\widehat{t(\gamma)}\\}$, (4) and the
assumption $a_{1}\neq\widehat{s(\gamma)}$ if $a_{i}=b_{j}=\widehat{s(\gamma)}$
or (5) and the assumption $b_{l}\neq\widehat{t(\gamma)}$ if
$a_{i}=b_{j}=\widehat{t(\gamma)}$. Thus (3.26) holds if $\gamma$ is a piece.
Next we deal with the general case. We use the temporary notation
$f_{i}(X)\coloneqq\Delta_{N,\gamma_{i}}(s(\gamma_{i});X;t(\gamma_{i}))$ for
$X\in\mathfrak{H}_{D}$. Using Corollary 3.11, we have

$\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))\Delta_{N,\gamma}(s(\gamma);w^{\prime};t(\gamma))=\sum_{\begin{subarray}{c}0=i_{0}\leq
i_{1}\leq\cdots\leq i_{r}=k\\\ 0=j_{0}\leq j_{1}\leq\cdots\leq
j_{r}=l\end{subarray}}\prod_{c=1}^{r}\left(f_{c}(e_{a_{i_{c-1}+1}}\cdots
e_{a_{i_{c}}})f_{c}(e_{b_{j_{c-1}+1}}\cdots e_{b_{j_{c}}})\right).$

From the assumption that $(w,w^{\prime})$ is $\gamma$-shuffleable and the
piecewise result, we have

$f_{c}(e_{a_{i_{c-1}+1}}\cdots e_{a_{i_{c}}})f_{c}(e_{b_{j_{c-1}+1}}\cdots
e_{b_{j_{c}}})=f_{c}(e_{a_{i_{c-1}+1}}\cdots
e_{a_{i_{c}}}\mathbin{\mathcyr{sh}}e_{b_{j_{c-1}+1}}\cdots
e_{b_{j_{c}}})+O(N^{-1}\log^{\bullet}N),\qquad\text{as }N\to\infty.$

By this fact and Proposition 2.3, we obtain

$\displaystyle(\mu_{\mathbb{C}}\circ(f_{1}\otimes\cdots\otimes
f_{r})\circ\underbrace{\Delta^{\mathbin{\mathcyr{sh}}}\circ\cdots\circ\Delta^{\mathbin{\mathcyr{sh}}}}_{r-1})(w\mathbin{\mathcyr{sh}}w^{\prime})$
(3.30) $\displaystyle=\sum_{\begin{subarray}{c}w_{1}\cdots w_{r}=w\\\
w^{\prime}_{1}\cdots
w^{\prime}_{r}=w^{\prime}\end{subarray}}(\mu_{\mathbb{C}}\circ(f_{1}\otimes\cdots\otimes
f_{r}))((w_{1}\otimes\cdots\otimes
w_{r})\mathbin{\mathcyr{sh}}(w^{\prime}_{1}\otimes\cdots\otimes
w^{\prime}_{r}))$ (3.31) $\displaystyle=\sum_{\begin{subarray}{c}w_{1}\cdots
w_{r}=w\\\ w^{\prime}_{1}\cdots
w^{\prime}_{r}=w^{\prime}\end{subarray}}(\mu_{\mathbb{C}}\circ(f_{1}\otimes\cdots\otimes
f_{r}))((w_{1}\mathbin{\mathcyr{sh}}w^{\prime}_{1})\otimes\cdots\otimes(w_{r}\mathbin{\mathcyr{sh}}w^{\prime}_{r}))$
(3.32) $\displaystyle=\sum_{\begin{subarray}{c}0=i_{0}\leq i_{1}\leq\cdots\leq
i_{r}=k\\\ 0=j_{0}\leq j_{1}\leq\cdots\leq
j_{r}=l\end{subarray}}\prod_{c=1}^{r}\left(f_{c}(e_{a_{i_{c-1}+1}}\cdots
e_{a_{i_{c}}}\mathbin{\mathcyr{sh}}e_{b_{j_{c-1}+1}}\cdots
e_{b_{j_{c}}})\right)$ (3.33)
$\displaystyle=\Delta_{N,\gamma}(s(\gamma);w;t(\gamma))\Delta_{N,\gamma}(s(\gamma);w^{\prime};t(\gamma))+O(N^{-1}\log^{\bullet}N),\qquad\text{as
}N\to\infty,$ (3.34)

where we use the symbol $\mathbin{\mathcyr{sh}}$ for the product on
$(\mathfrak{H}_{D})^{\otimes r}$ induced from $\mathbin{\mathcyr{sh}}$ by
abuse of notation and $\mu_{\mathbb{C}}$ for the product for
$\mathbb{C}[\theta]$. The leftmost side is
$\Delta_{N,\gamma}(s(\gamma);w\mathbin{\mathcyr{sh}}w^{\prime};t(\gamma))$ by
the path composition formula. This completes the proof. ∎

##  4\. Recovering formulas for usual iterated integrals

In this section, we deal with the following types of paths:

######  Definition 4.1.

  1. (1)

Define a continuous path $\beta$ to be
$\mathrm{dch}\cdot\alpha\cdot\mathrm{dch}^{-1}$, where $\alpha$ is the path
which starts from $(-1)_{1}$, circles $1$ counterclockwise and ends on
$(-1)_{1}$.

  2. (2)

Define a discrete path $\beta^{\Delta}=(\gamma_{1},\gamma_{2},\gamma_{3})$,
where $\gamma_{1}=(0,1)$, $\gamma_{2}=(1,0,N,+)$ and $\gamma_{3}=(1,0)$.

Moreover, we also write the straight piece $(0,1)$ as $\mathrm{dch}^{\Delta}$.

Hereafter we assume that $D$ contains $0$ and $1$ and that
$D\setminus\\{0\\}\subseteq M^{\prime}\coloneqq\\{z\mid|z|\geq
1\\}\subseteq\mathbb{C}$. Put

$\mathcal{I}^{\prime}\coloneqq\left\\{\begin{pmatrix}z_{1},\ldots,z_{r}\\\
k_{1},\ldots,k_{r}\end{pmatrix}\left|\begin{array}[]{c}r\geq 1,\\\
k_{1},\ldots,k_{r}\in\mathbb{Z}_{\geq 1},\\\ z_{1},\ldots,z_{r}\in D,\\\
(k_{r},z_{r})\neq(1,1)\end{array}\right.\right\\}.$

###  4.1. Preliminaries: harmonic product

Let $\mathfrak{H}^{1}\coloneqq\mathbb{Q}\oplus e_{1}\mathbb{Q}\langle
e_{0},e_{1}\rangle$ and $\mathfrak{H}^{0}\coloneqq\mathbb{Q}\oplus
e_{1}\mathbb{Q}\langle e_{0},e_{1}\rangle e_{0}$. Endow $\mathfrak{H}^{1}$
with a product $\ast$ by the following rules: $1\ast w=w\ast 1=w$ for every
$w\in\mathfrak{H}^{1}$ and

$e_{a}w\ast e_{b}w^{\prime}\coloneqq e_{ab}(w\ast e_{b}w^{\prime}+e_{a}w\ast
w^{\prime}-e_{0}(w\ast w^{\prime}))$

for $a,b\in\\{0,1\\}$ and $w,w^{\prime}\in\mathfrak{H}^{1}$. This provides a
$\mathbb{Q}$-algebra structure on $\mathfrak{H}^{1}$ and its subalgebra
structure on $\mathfrak{H}^{0}$.

######  Proposition 4.2 (Hoffman [9]).

Every $w\in\mathfrak{H}^{1}$ has the expression

$w=\sum_{i=0}^{n}w_{i}\ast e_{1}^{i},$

by the unique non-negative integer $n$ and $w_{i}\in\mathfrak{H}^{0}$.

The constant term $w_{0}$ on the above proposition is denoted by
$\operatorname{reg}_{\ast}(w)$. Moreover we introduce a variant of harmonic
regularization in [12].

######  Definition 4.3.

Assume that a sequence $a\colon\mathbb{Z}_{\geq 1}\to\mathbb{C}$ has the
asymptotic expansion of type

$a_{N}=P(\log N+\gamma)+O(N^{-1}\log^{\bullet}N),\qquad\text{as }N\to\infty,$

where $P(X)\in\mathbb{C}[X]$ and $\gamma$ denotes the Euler constant. Note
that such an expansion is unique if its exists. Then we define

$\operatorname*{Reg}_{N\to\infty}^{*}a_{N}\coloneqq P(0).$

Moreover, for $f\in(\mathbb{C}^{\mathbb{Z}_{\geq 1}})[\theta]$, we extend the
definition as

$\operatorname*{Reg}_{N\to\infty}^{\ast}f(N,\theta)\coloneqq\sum_{i=0}^{n}\left(\operatorname*{Reg}_{N\to\infty}^{\ast}a_{N,i}\right)\theta^{i},$

where we write the coefficients as
$f(N,\theta)=\sum_{i=0}^{n}a_{N,i}\theta^{i}$ and note that the degree $n$
does not depend on $N$.

######  Remark 4.4.

The operation $\operatorname*{Reg}_{N\to\infty}^{\ast}$ is compatible with the
addition and product. Furthermore, when $\\{a_{n}\\}_{n}$ converges,
$\operatorname*{Reg}_{N\to\infty}^{*}a_{N}=\lim_{N\to\infty}a_{N}$ holds.

######  Definition 4.5.

For $w\in\mathfrak{H}_{D}$ and $\gamma\in\\{\mathrm{dch},\beta\\}$, we define

$I^{\Delta}_{\gamma}(w)\coloneqq\operatorname*{Reg}_{N\to\infty}^{\ast}\Delta_{N,\gamma^{\Delta}}(s(\gamma);w;t(\gamma)).$

######  Lemma 4.6.

Let $k_{1},\ldots,k_{r}\geq 1$. Then we have

$I^{\Delta}_{\mathrm{dch}}(e_{1}e_{0}^{k_{1}-1}\cdots
e_{1}e_{0}^{k_{r}-1})=(-1)^{r}\zeta^{\ast}(k_{1},\ldots,k_{r}),$ (4.1)

where $\zeta^{\ast}$ stands for the harmonic regularization of multiple zeta
values:

$\zeta^{\ast}(k_{1},\ldots,k_{r})\coloneqq\operatorname*{Reg}_{N\to\infty}^{*}\sum_{0<n_{1}<\cdots<n_{r}<N}\frac{1}{n_{1}^{k_{1}}\cdots
n_{r}^{k_{r}}}.$

###### Proof.

Combining [15, Theorem 1.3] and Lemma 3.14, we have

$\Delta_{N,(0,1)}((0,+);u;(1,-))=Z_{<N}(u)+O(N^{-1}\log^{\bullet}N),$

for $u\in\mathfrak{H}^{1}$, where $Z_{<N}\colon\mathfrak{H}^{1}\to\mathbb{Q}$
is the $\mathbb{Q}$-linear map determined by $1\mapsto 1$ and

$e_{1}e_{0}^{l_{1}-1}\cdots
e_{1}e_{0}^{l_{s}-1}\mapsto(-1)^{s}\sum_{0<n_{1}<\cdots<n_{s}<N}\frac{1}{n_{1}^{l_{1}}\cdots
n_{s}^{l_{s}}},\qquad(l_{1},\ldots,l_{s}\geq 1).$

Therefore we obtain

$\displaystyle I^{\Delta}_{\mathrm{dch}}(e_{1}e_{0}^{k_{1}-1}\cdots
e_{1}e_{0}^{k_{r}-1})$
$\displaystyle=\operatorname*{Reg}_{N\to\infty}^{\ast}\Delta_{N,(0,1)}((0,+);w;(1,-))$
(4.2) $\displaystyle=\operatorname*{Reg}_{N\to\infty}^{\ast}Z_{<N}(w)$ (4.3)
$\displaystyle=(-1)^{r}\zeta^{\ast}(k_{1},\ldots,k_{r}),$ (4.4)

where $w=e_{1}e_{0}^{k_{1}-1}\cdots e_{1}e_{0}^{k_{r}-1}$. ∎

###  4.2. The straight case

######  Theorem 4.7 (Hirose–Matsusaka–Seki [5]).

Let $\iota=\begin{pmatrix}z_{1},\ldots,z_{r}\\\
k_{1},\ldots,k_{r}\end{pmatrix}\in\mathcal{I}^{\prime}$. Then we have

$\lim_{N\to\infty}\Delta_{N,(0,1)}((0,+);W(\iota);(1,-))=I_{\mathrm{dch}_{0,1}}(1_{0};W(\iota);(-1)_{1}).$

###### Proof.

Combine [5, Theorem 1.2] and [5, Proposition 2.5]. ∎

######  Definition 4.8.

The _multiple polylogarithm_ (of shuffle-type) is defined by

$\mathrm{Li}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}z_{1},\ldots,z_{r}\\\
k_{1},\ldots,k_{r}\end{pmatrix}\coloneqq\sum_{0<n_{1}<\cdots<n_{r}}\frac{z_{1}^{n_{1}}z_{2}^{n_{2}-n_{1}}\cdots
z_{r}^{n_{r}-n_{r-1}}}{n_{1}^{k_{1}}\cdots n_{r}^{k_{r}}}$

for complex numbers $z_{1},\ldots,z_{r}$ and
$k_{1},\ldots,k_{r}\in\mathbb{Z}_{\geq 1}$ satisfying
$\begin{pmatrix}z_{1}^{-1},\ldots,z_{r}^{-1}\\\
k_{1},\ldots,k_{r}\end{pmatrix}\in\mathcal{I}^{\prime}$.

For $\iota=\begin{pmatrix}z_{1},\ldots,z_{r}\\\
k_{1},\ldots,k_{r}\end{pmatrix}\in\mathcal{I}^{\prime}$, put

$W(\iota)\coloneqq e_{z_{1}}e_{0}^{k_{1}-1}\cdots e_{z_{r}}e_{0}^{k_{r}-1}.$

Then it holds that

$I_{\mathrm{dch}_{0,1}}(1_{0};W(\iota);(-1)_{1})=(-1)^{r}\mathrm{Li}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}1/z_{1},\ldots,1/z_{r}\\\
k_{1},\ldots,k_{r}\end{pmatrix}.$

Therefore, combining this and Theorem 4.7 we obtain

$\lim_{N\to\infty}\Delta_{N,(0,1)}((0,+);W(\iota);(1,-))=(-1)^{r}\mathrm{Li}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}1/z_{1},\ldots,1/z_{r}\\\
k_{1},\ldots,k_{r}\end{pmatrix}.$ (4.5)

For the next subsection, we introduce the extension

$\mathrm{L}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}z_{1},\ldots,z_{r}\\\
k_{1},\ldots,k_{r}\end{pmatrix}\coloneqq
I_{\mathrm{dch}}^{\Delta}(e_{1/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{1/z_{r}}e_{0}^{k_{r}-1}),$

for $k_{1},\ldots,k_{r}\in\mathbb{Z}_{\geq 1}$ and
$z_{1},\ldots,z_{r}\in\mathbb{C}$ satisfying $z_{i}^{-1}\in D$.

###  4.3. Variants of symmetric multiple polylogarithms

In this subsection, imitating the construction of $\zeta^{RS}$ in [4], we show
that the symmetric multiple polylogarithms and symmetric multiple zeta values
are recovered from discrete iterated integrals. For a polynomial
$P(\theta)=a_{0}+\cdots+a_{n}\theta^{n}\in\mathbb{C}[\theta]$, we put
$P(\theta)[\theta]\coloneqq a_{1}$.

######  Theorem 4.9.

Let $\alpha,k_{1},\ldots,k_{r}$ be positive integers and
$z_{1},\ldots,z_{r+1}\in D$. Assume that $u/z_{i}\in D$ for every $1\leq i\leq
r+1$ and $u\in D$. Then we have

$\sum_{u\in D}u^{\alpha}I^{\Delta}_{\beta}(e_{u/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{u/z_{r}}e_{0}^{k_{r}-1}e_{u/z_{r+1}})[\theta]\\\
=\sum_{i=0}^{r}(-1)^{k_{i+1}+\cdots+k_{r}}z_{i+1}^{\alpha}\mathrm{L}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}z_{1}/z_{i+1},\ldots,z_{i}/z_{i+1}\\\
k_{1},\ldots,k_{i}\end{pmatrix}\mathrm{L}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}z_{r+1}/z_{i+1},\ldots,z_{i+2}/z_{i+1}\\\
k_{r},\ldots,k_{i+1}\end{pmatrix}.$ (4.6)

###### Proof.

Set $k\coloneqq k_{1}+\cdots+k_{r}$ and

$w_{u}\coloneqq e_{a_{1}}\cdots e_{a_{k+1}}\coloneqq
e_{u/z_{1}}e_{0}^{k_{1}-1}\cdots e_{u/z_{r}}e_{0}^{k_{r}-1}e_{u/z_{r+1}}$

for $u\in D$. By the (discrete) path composition formula (Corollary 3.11), for
each $u\in D$ we have

$\Delta_{N,\beta^{\Delta}}((0,+);w_{u};(+,0))\\\ =\sum_{0\leq i\leq j\leq
k+1}\Delta_{N,\mathrm{dch}^{\Delta}}((0,+);e_{a_{1}}\cdots
e_{a_{i}};(1,-))\Delta_{N,\alpha^{\Delta}}((1,-);e_{a_{i+1}}\cdots
e_{a_{j}};(1,-))\Delta_{N,(\mathrm{dch}^{\Delta})^{-1}}((1,-);e_{a_{j+1}}\cdots
e_{a_{k}};(0,+)).$ (4.7)

Since Proposition 3.9 guarantees that
$\Delta_{N,\alpha^{\Delta}}((1,-);e_{a_{i+1}}\cdots e_{a_{j}};(1,-))$ only if
$a_{i+1}=\cdots=a_{j}=1$, we obtain

$\Delta_{N,\beta^{\Delta}}((0,+);w_{u};(+,0))\\\
=\sum_{\begin{subarray}{c}0\leq i\leq j\leq r\\\ k_{i+1}=\cdots=k_{j}=1\\\
z_{i+1}=\cdots=z_{j+1}=u\end{subarray}}\Delta_{N,\mathrm{dch}^{\Delta}}((0,+);e_{u/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{u/z_{i}}e_{0}^{k_{i}-1};(1,-))\frac{(N)_{j-i+1}}{(j-i+1)!}\left(\frac{\theta}{N}\right)^{j-i+1}\\\
\cdot\Delta_{N,(\mathrm{dch}^{\Delta})^{-1}}((1,-);e_{0}^{k_{j+1}-1}e_{u/z_{j+2}}\cdots
e_{0}^{k_{r}-1}e_{u/z_{r+1}};(0,+)).$ (4.8)

Applying $\operatorname*{Reg}_{N\to\infty}^{\ast}\sum_{u\in D}$ after taking
the coefficients at $\theta$ on both sides, we have

$\displaystyle\sum_{u\in D}u^{\alpha}I^{\Delta}_{\beta}(w_{u})[\theta]$
$\displaystyle=\begin{multlined}\operatorname*{Reg}_{N\to\infty}^{\ast}\sum_{u\in
D}u^{\alpha}\sum_{\begin{subarray}{c}0\leq i=j\leq r\\\
k_{i+1}=\cdots=k_{j}=1\\\
z_{i+1}=\cdots=z_{j+1}=u\end{subarray}}\Delta_{N,\mathrm{dch}^{\Delta}}((0,+);e_{u/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{u/z_{i}}e_{0}^{k_{i}-1};(1,-))\\\
\cdot\Delta_{N,(\mathrm{dch}^{\Delta})^{-1}}((1,-);e_{0}^{k_{j+1}-1}e_{u/z_{j+2}}\cdots
e_{0}^{k_{r}-1}e_{u/z_{r+1}};(0,+))\end{multlined}\operatorname*{Reg}_{N\to\infty}^{\ast}\sum_{u\in
D}u^{\alpha}\sum_{\begin{subarray}{c}0\leq i=j\leq r\\\
k_{i+1}=\cdots=k_{j}=1\\\
z_{i+1}=\cdots=z_{j+1}=u\end{subarray}}\Delta_{N,\mathrm{dch}^{\Delta}}((0,+);e_{u/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{u/z_{i}}e_{0}^{k_{i}-1};(1,-))\\\
\cdot\Delta_{N,(\mathrm{dch}^{\Delta})^{-1}}((1,-);e_{0}^{k_{j+1}-1}e_{u/z_{j+2}}\cdots
e_{0}^{k_{r}-1}e_{u/z_{r+1}};(0,+))$ (4.11)
$\displaystyle=\begin{multlined}\sum_{0\leq i\leq
r}z_{i+1}^{\alpha}\operatorname*{Reg}_{N\to\infty}^{\ast}\left(\Delta_{N,\mathrm{dch}^{\Delta}}((0,+);e_{z_{i+1}/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{z_{i+1}/z_{i}}e_{0}^{k_{i}-1};(1,-))\right.\\\
\cdot\left.\Delta_{N,(\mathrm{dch}^{\Delta})^{-1}}((1,-);e_{0}^{k_{j+1}-1}e_{z_{i+1}/z_{i+2}}\cdots
e_{0}^{k_{r}-1}e_{z_{i+1}/z_{r+1}};(0,+))\right)\end{multlined}\sum_{0\leq
i\leq
r}z_{i+1}^{\alpha}\operatorname*{Reg}_{N\to\infty}^{\ast}\left(\Delta_{N,\mathrm{dch}^{\Delta}}((0,+);e_{z_{i+1}/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{z_{i+1}/z_{i}}e_{0}^{k_{i}-1};(1,-))\right.\\\
\cdot\left.\Delta_{N,(\mathrm{dch}^{\Delta})^{-1}}((1,-);e_{0}^{k_{j+1}-1}e_{z_{i+1}/z_{i+2}}\cdots
e_{0}^{k_{r}-1}e_{z_{i+1}/z_{r+1}};(0,+))\right)$ (4.14)
$\displaystyle=\begin{multlined}\sum_{0\leq i\leq
r}(-1)^{k_{i+1}+\cdots+k_{r}}z_{i+1}^{\alpha}\operatorname*{Reg}_{N\to\infty}^{\ast}\left(((0,+);e_{z_{i+1}/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{z_{i+1}/z_{i}}e_{0}^{k_{i}-1};(1,-))\right.\\\
\cdot\left.\Delta_{N,\mathrm{dch}^{\Delta}}((0,+);e_{z_{i+1}/z_{r+1}}e_{0}^{k_{r}-1}\cdots
e_{z_{i+1}/z_{i+2}}e_{0}^{k_{i+1}-1};(1,-))\right)\end{multlined}\sum_{0\leq
i\leq
r}(-1)^{k_{i+1}+\cdots+k_{r}}z_{i+1}^{\alpha}\operatorname*{Reg}_{N\to\infty}^{\ast}\left(((0,+);e_{z_{i+1}/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{z_{i+1}/z_{i}}e_{0}^{k_{i}-1};(1,-))\right.\\\
\cdot\left.\Delta_{N,\mathrm{dch}^{\Delta}}((0,+);e_{z_{i+1}/z_{r+1}}e_{0}^{k_{r}-1}\cdots
e_{z_{i+1}/z_{i+2}}e_{0}^{k_{i+1}-1};(1,-))\right)$ (4.17)
$\displaystyle=\sum_{0\leq i\leq
r}(-1)^{k_{i+1}+\cdots+k_{r}}z_{i+1}^{\alpha}\mathrm{L}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}z_{1}/z_{i+1},\ldots,z_{i}/z_{i+1}\\\
k_{1},\ldots,k_{i}\end{pmatrix}\mathrm{L}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}z_{r+1}/z_{i+1},\ldots,z_{i+2}/z_{i+1}\\\
k_{r},\ldots,k_{i+1}\end{pmatrix}.$ (4.18)

∎

######  Remark 4.10.

In addition to the assumption in Theorem 4.9, we assume that $z_{i}\neq
z_{i+1}$ for $1\leq i\leq r$ satisfying $k_{i}=1$. Then we see that the right-
hand side of Theorem 4.9 coincides with the symmetric multiple polylogarithm
$\pounds_{\mathcal{S},\alpha}\begin{pmatrix}z_{1},\ldots,z_{r}\\\
k_{1},\ldots,k_{r}\end{pmatrix}$ (for definition, take modulo $t$ in [13,
Definition 4.15]. See also [2, Definition B.1.1]).

######  Corollary 4.11.

Let $k_{1},\ldots,k_{r}$ be positive integers and $D=\\{0,1\\}$. Then we have

$I_{\beta}^{\Delta}(e_{1}e_{0}^{k_{1}-1}\cdots
e_{1}e_{0}^{k_{r}-1}e_{1})[\theta]=\zeta_{\mathcal{S}}^{\ast}(k_{1},\ldots,k_{r})\coloneqq\sum_{i=0}^{r}(-1)^{k_{i+1}+\cdots+k_{r}}\zeta^{\ast}(k_{1},\ldots,k_{i})\zeta^{\ast}(k_{r},\ldots,k_{i+1}).$

###### Proof.

It follows from Theorem 4.9 and Lemma 4.6. ∎

##  5\. Cyclic sum formulas for discrete iterated integrals

In this section, we prove the cyclic sum formulas for discrete iterated
integrals along a straight piece. Throughout this section, the symbol $\gamma$
always stands for a positive straight piece $(x,y)$ such that $x,y\in D$.

######  Definition 5.1 (Hoffman–Ohno type sum).

Let $w=e_{a_{1}}\cdots e_{a_{k}}$ be an element of $\mathfrak{H}_{D}$. We
define

$\Delta_{N,\gamma}^{O}((x,+);w;(y,-))\coloneqq\sum_{\begin{subarray}{c}n_{1},\ldots,n_{k+1}\in
S_{\gamma}\\\
n_{1}\leq_{\gamma}\cdots\leq_{\gamma}n_{k+1}\end{subarray}}\left(\prod_{i=1}^{k}\omega_{a_{i}}^{N}(n_{i})\right)\left(\omega^{N}(n_{1},n_{k+1})-\omega^{N}_{x}(n_{k+1})\right)$

and

$\Delta_{N,\gamma}^{O^{\prime}}((x,+);w;(y,-))\coloneqq\sum_{\begin{subarray}{c}n_{0},\ldots,n_{k}\in
S_{\gamma}\\\
n_{0}\leq_{\gamma}\cdots\leq_{\gamma}n_{k}\end{subarray}}\left(\prod_{i=1}^{k}\omega_{a_{i}}^{N}(n_{i})\right)\left(\omega_{y}^{N}(n_{0})-\omega^{N}(n_{k},n_{0})\right)$

######  Proposition 5.2 (Transport relations).

Let $a_{1},\ldots,a_{k},z$ ($k\geq 1$) be elements of $D$ and put
$w=e_{a_{1}}\cdots e_{a_{k}}$. Then we have

  1. (1)

$\displaystyle\Delta_{N,\gamma}^{O}((x,+);we_{z};(y,-))=\Delta_{N,\gamma}^{O^{\prime}}((x,+);we_{z};(y,-)),$

  2. (2)

$\Delta_{N,\gamma}^{O^{\prime}}((x,+);we_{z};(y,-))=\Delta_{N,\gamma}^{O}((x,+);e_{z}w;(y,-))+(1-\delta_{x,z})\Delta_{N,\gamma}((x,+);e_{z}we_{x};(y,-))\\\
+(1-\delta_{y,z})\Delta_{N,\gamma}((x,+);e_{y}we_{z};(y,-))-(1-\delta_{x,z})(1-\delta_{y,z})\Delta_{N,\gamma}((x,+);e_{z}we_{z};(y,-))+\Delta_{N,\gamma}((x,+);e_{z,a_{1},\ldots,a_{k},z};(y,-)).$
(5.1)

###### Proof.

  1. (1)

By definition we have

$\displaystyle\Delta_{N,\gamma}^{O}((x,+);we_{z};(y,-))$ (5.2)
$\displaystyle=\sum_{\begin{subarray}{c}n_{1},n_{k+1},n_{k+2}\in S_{\gamma}\\\
n_{1}\leq_{\gamma}n_{k+1}\leq_{\gamma}n_{k+2}\end{subarray}}\omega_{a_{1}}(n_{1})\Delta_{N,\gamma_{n_{1},n_{k}}}(s(\gamma_{n_{1},n_{k}});e_{a_{2}}\cdots
e_{a_{k}};t(\gamma_{n_{1},n_{k}}))\omega_{z}(n_{k})\left(\omega^{N}(n_{1},n_{k+1})-\omega_{x}^{N}(n_{k+1})\right).$
(5.3)

Let us compute the sum about $n_{k+1}$. For $m,n\in S_{\gamma}$ satisfying
$m\leq_{\gamma}n$, define

$A(m,n)\coloneqq\sum_{\begin{subarray}{c}c\in S_{\gamma}\\\
n\leq_{\gamma}c\end{subarray}}\left(\omega^{N}(m,c)-\omega_{x}^{N}(c)\right)$

and

$B(m,n)\coloneqq\sum_{\begin{subarray}{c}d\in S_{\gamma}\\\
d\leq_{\gamma}m\end{subarray}}\left(\omega_{y}^{N}(d)-\omega^{N}(n,d)\right).$

It suffices to prove $A(m,n)=B(m,n)$. If $m\neq n$, by making the sum
telescopic, we have

$\displaystyle A(m,n)$ $\displaystyle=\sum_{\tilde{n}\leq
c<yN}\left(\frac{1}{c-\tilde{m}}-\frac{1}{c-xN}\right)$ (5.4)
$\displaystyle=\sum_{\tilde{n}\leq
c<yN}\sum_{xN<d\leq\tilde{m}}\left(\frac{1}{c-d}-\frac{1}{c-(d-1)}\right)$
(5.5)
$\displaystyle=\sum_{xN<d\leq\tilde{m}}\left(\frac{1}{\tilde{n}-d}-\frac{1}{yN-d}\right)$
(5.6) $\displaystyle=B(m,n).$ (5.7)

The $m=n$ case is also proved by a similar argument.

  2. (2)

In the definition

$\Delta_{N,\gamma}^{O^{\prime}}((x,+);we_{z};(y,-))=\sum_{\begin{subarray}{c}n_{0},n_{k+1}\in
S_{\gamma}\\\
n_{0}\leq_{\gamma}n_{k+1}\end{subarray}}\Delta_{N,\gamma_{n_{0},n_{k+1}}}(s(\gamma_{n_{0},n_{k+1}});w;t(\gamma_{n_{0},n_{k+1}}))\omega_{z}^{N}(n_{k+1})\left(\omega_{y}^{N}(n_{0})-\omega^{N}(n_{k+1},n_{0})\right),$

we can apply the decomposition

$\displaystyle\omega_{z}^{N}(n_{k+1})\left(\omega_{y}^{N}(n_{0})-\omega^{N}(n_{k+1},n_{0})\right)$
(5.8)
$\displaystyle=\frac{1}{\tilde{n_{k+1}}-zN}\left(\frac{1}{\tilde{n_{0}}-yN}-\frac{1}{\tilde{n_{0}}-\tilde{n_{k+1}}}\right)$
(5.9)
$\displaystyle=\frac{1}{\tilde{n_{0}}-zN}\left(\frac{1}{\tilde{n_{k+1}}-\tilde{n_{0}}}-\frac{1}{\tilde{n_{k+1}}-xN}\right)+\frac{1-\delta_{x,z}}{\tilde{n_{0}}-zN}\frac{1}{\tilde{n_{k+1}}-xN}+\frac{1-\delta_{y,z}}{\tilde{n_{0}}-yN}\frac{1}{\tilde{n_{k+1}}-zN}-\frac{1-\delta_{x,z}}{\tilde{n_{0}}-zN}\frac{1-\delta_{y,z}}{\tilde{n_{k+1}}-zN}$
(5.10)
$\displaystyle=\begin{multlined}\omega_{z}^{N}(n_{0})(\omega^{N}(n_{0},n_{k+1})-\omega^{N}_{x}(n_{k+1}))+(1-\delta_{x,z})\omega^{N}_{z}(n_{0})\omega^{N}_{x}(n_{k+1})\\\
+(1-\delta_{y,z})\omega^{N}_{y}(n_{0})\omega^{N}_{z}(n_{k+1})-(1-\delta_{x,z})(1-\delta_{y,z})\omega^{N}_{z}(n_{0})\omega^{N}_{z}(n_{k+1}),\end{multlined}\omega_{z}^{N}(n_{0})(\omega^{N}(n_{0},n_{k+1})-\omega^{N}_{x}(n_{k+1}))+(1-\delta_{x,z})\omega^{N}_{z}(n_{0})\omega^{N}_{x}(n_{k+1})\\\
+(1-\delta_{y,z})\omega^{N}_{y}(n_{0})\omega^{N}_{z}(n_{k+1})-(1-\delta_{x,z})(1-\delta_{y,z})\omega^{N}_{z}(n_{0})\omega^{N}_{z}(n_{k+1}),$
(5.13)

when $n_{0}\neq n_{k+1}$. If $n_{0}=n_{k+1}$, such a composition becomes the
form

$\displaystyle\omega_{z}^{N}(n_{k+1})\left(\omega_{y}^{N}(n_{0})-\omega^{N}(n_{k+1},n_{0})\right)$
(5.14)
$\displaystyle=\omega_{z}^{N}(n_{0})(\omega^{N}(n_{0},n_{k+1})-\omega^{N}_{x}(n_{k+1}))+(1-\delta_{x,z})\omega^{N}_{z}(n_{0})\omega^{N}_{x}(n_{k+1})$
(5.15)
$\displaystyle+(1-\delta_{y,z})\omega^{N}_{y}(n_{0})\omega^{N}_{z}(n_{k+1}).$
(5.16)

Therefore we obtain the result by summing up them.

∎

######  Theorem 5.3 (Cyclic sum formula).

Let $w=e_{a_{1}}\cdots e_{a_{k}}$ ($k\geq 2$) be an element of
$\mathfrak{H}_{D}$. Then we have

$\sum_{\begin{subarray}{c}1\leq i\leq k\\\ a_{i}\neq
x\end{subarray}}\Delta_{N,\gamma}((x,+);e_{a_{i}}\cdots
e_{a_{k}}e_{a_{1}}\cdots
e_{a_{i-1}}e_{x};(y,-))+\sum_{\begin{subarray}{c}1\leq i\leq k\\\ a_{i}\neq
y\end{subarray}}\Delta_{N,\gamma}((x,+);e_{y}e_{a_{i+1}}\cdots
e_{a_{k}}e_{a_{1}}\cdots e_{a_{i-1}}e_{x})\\\ =\sum_{\begin{subarray}{c}1\leq
i\leq k\\\
a_{i}\in\\{x,y\\}\end{subarray}}\Delta_{N,\gamma}((x,+);e_{a_{i}}\cdots
e_{a_{k}}e_{a_{1}}\cdots
e_{a_{i}};(y,-))-\sum_{i=1}^{k}\Delta_{N,\gamma}((x,+);e_{a_{1},\ldots,a_{i},a_{i},\ldots,a_{k}};(y,-)).$
(5.17)

###### Proof.

Using two assertinons in Proposition 5.2 and putting $w=e_{a_{1}}\cdots
e_{a_{k-1}}$ and $z=a_{k}$, we have

$\Delta_{N,\gamma}^{O}((x,+);e_{a_{1}}\cdots
e_{a_{k}};(y,-))=\Delta_{N,\gamma}^{O}((x,+);e_{a_{k}}e_{a_{1}}\cdots
e_{a_{k-1}};(y,-))+(1-\delta_{x,a_{k}})\Delta_{N,\gamma}((x,+);e_{a_{k}}e_{a_{1}}\cdots
e_{a_{k-1}}e_{x};(y,-))\\\
+(1-\delta_{y,a_{k}})\Delta_{N,\gamma}((x,+);e_{y}e_{a_{1}}\cdots
e_{a_{k}};(y,-))-(1-\delta_{x,a_{k}})(1-\delta_{y,a_{k}})\Delta_{N,\gamma}((x,+);e_{a_{k}}e_{a_{1}}\cdots
e_{a_{k}};(y,-))\\\
+\Delta_{N,\gamma}((x,+);e_{a_{1},\ldots,a_{i},a_{i},\ldots,a_{k}};(y,-)),$
(5.18)

Applying this equation $k$ times derives that the $\Delta_{N,\gamma}^{O}$ term
vanishes and thus obtain the desired formula. ∎

Making the setting suitable for multiple polylogarithms in the above theorem,
we obtain the cyclic sum formulas for them as below. This is a generalization
of that for multiple $L$-values ([14]).

######  Corollary 5.4 (Cyclic sum formula for multiple polylogarithms).

Let $k_{1},\ldots,k_{r}$ ($r\geq 1$) be positive integers and
$z_{1},\ldots,z_{r}$ complex numbers $|z_{i}|\leq 1$ for all $1\leq i\leq r$.
Suppose that $\\{z_{1},\ldots,z_{r}\\}\neq 1$ if $k_{1}=\cdots=k_{r}=1$. Then
we have

$\sum_{i=1}^{r}\mathrm{Li}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}z_{i+1},\ldots,z_{r},z_{1},\ldots,z_{i}\\\
k_{i+1},\ldots,k_{r},k_{1},\ldots,k_{i-1},k_{i}+1\end{pmatrix}=\sum_{i=1}^{r}\sum_{j=0}^{k_{i}-2}\mathrm{Li}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}1,z_{i+1},\ldots,z_{r},z_{1},\ldots,z_{i}\\\
j+1,k_{i+1},\ldots,k_{r},k_{1},\ldots,k_{i-1},k_{i}-j\end{pmatrix}\\\
+\sum_{\begin{subarray}{c}1\leq i\leq r\\\ z_{i}\neq
1\end{subarray}}\mathrm{Li}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}1,z_{i+1},\ldots,z_{r},z_{1},\ldots,z_{i}\\\
k_{i+1},\ldots,k_{r},k_{1},\ldots,k_{i},1\end{pmatrix}-\sum_{\begin{subarray}{c}1\leq
i\leq r\\\ z_{i}\neq
1\end{subarray}}\mathrm{Li}^{\mathbin{\mathcyr{sh}}}\begin{pmatrix}z_{i},\ldots,z_{r},z_{1},\ldots,z_{i}\\\
k_{i+1},\ldots,k_{r},k_{1},\ldots,k_{i},1\end{pmatrix}.$ (5.19)

###### Proof.

Put $e_{a_{1}}\cdots e_{a_{k}}=e_{1/z_{1}}e_{0}^{k_{1}-1}\cdots
e_{1/z_{r}}e_{0}^{k_{r}-1}$ in Theorem 5.3. The assumption guarantees that
$\\{a_{1},\ldots,a_{k}\\}\neq 1$ and then we see that

$\lim_{N\to\infty}\Delta_{N,\gamma}((x,+);e_{a_{1},\ldots,a_{i},a_{i},\ldots,a_{k}};(y,-))=0$

by Lemma 3.14. Thus we have the theorem by using (4.5). ∎

##  6\. Expected future works

  1. (1)

Using notion of the discrete pullback (cf. [3, Definition 3.5.2]), it is
expected that iterated integrals along more general paths are well
discretized. The cyclic sum formulas for more general paths (for example, that
for symmetric multiple zeta values [8]) are as well.

  2. (2)

In the definition of discrete iterated integrals, permitting $e_{z}$’s for
$(z,i)\in S_{\gamma}$, one may obtain new (asymptotic) relations. In
particular, we expect that, by extending the proof of Theorem 5.3, we may have
the derivation relation for multiple zeta values ([12]) and its generalization
to multiple polylogarithms.

  3. (3)

There are many formulas other than what treats cyclic sums, which is valid for
general iterated integrals or hyperlogarithms, such as the confluence relation
([6, Proposition 6.9]), block shuffle identity ([7, Theorem 42]), etc. Do them
have any good discretization?

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