Shuffle regularization for multiple Eisenstein series of level $\displaystyle N$

Hayato Kanno

Abstract

In [BT], Bachmann and Tasaka discovered a relationship between multiple Eisenstein series (MES) of level $\displaystyle 1$ and formal iterated integrals corresponding to multiple zeta value (MZV). They also constructed shuffle regularized MES of level $\displaystyle 1$ , which satisfies the shuffle relation that is same as multiple zeta values. In this paper, we expand their results for arbitrary level and give some linear relations among MES of level $\displaystyle N$ .

1 Introduction

Multiple zeta values are generalizations of the Riemann zeta values which are defined for integers $\displaystyle n_{1},\dots,n_{r-1}\geq 1$ and $\displaystyle n_{r}\geq 2$ by

\displaystyle\zeta(n_{1},\dots,n_{r})=\sum_{0<k_{1}<\cdots<k_{r}}\frac{1}{k_{1}^{n_{1}}\cdots k_{r}^{n_{r}}}.

MZVs have the iterated integral representation. The two different representations give us one of the $\displaystyle\mathbb{Q}$ -linear relations among MZVs, which is called double shuffle relation. For example, the product of $\displaystyle\zeta(2)$ and $\displaystyle\zeta(3)$ can be expanded in two ways as

	$\displaystyle\displaystyle\zeta(2)\zeta(3)$	$\displaystyle\displaystyle=\zeta(2,3)+\zeta(3,2)+\zeta(5)$		(1)
		$\displaystyle\displaystyle=3\zeta(2,3)+\zeta(3,2)+6\zeta(1,4),$		(2)

from which we deduce the linear relation

\displaystyle\zeta(5)=2\zeta(2,3)+6\zeta(1,4).

It is known that there are numerous $\displaystyle\mathbb{Q}$ -linear relations among MZVs. Zagier ([Zag92], [Zag93]) initially noted that there exists some relations among double zeta values whose coefficients are originated from modular forms on $\displaystyle\mathrm{SL}_{2}(\mathbb{Z})$ . He also conjectured the dimension of the space spanned by double zeta values, which implies that the number of all $\displaystyle\mathbb{Q}$ -linear relations among double zeta values of weight $\displaystyle k$ is equal to $\displaystyle\mathrm{dim}_{\mathbb{C}}\mathcal{M}_{k}(\mathrm{SL}_{2}(\mathbb{Z}))$ , the dimension of the space of modular forms of weight $\displaystyle k$ on $\displaystyle\mathrm{SL}_{2}(\mathbb{Z})$ .

A formulation of modular relations for double zeta values was first established by Gangl, Kaneko and Zagier [GKZ]. It uses the space of even period polynomials, which by the theory of Eichler-Shimura is isomorphic to the space of modular forms on $\displaystyle\mathrm{SL}_{2}(\mathbb{Z})$ . They also introduced double Eisenstein series $\displaystyle G_{r,s}(\tau)$ defined for integers $\displaystyle r,s\geq 2$ as a holomorphic function on the upper half plane $\displaystyle\mathbb{H}$ by

\displaystyle G_{r,s}(\tau)\coloneq\sum_{\begin{subarray}{c}0\prec\lambda_{1}\prec\lambda_{2}\\ \lambda_{1},\lambda_{2}\in\mathbb{Z}\tau+\mathbb{Z}\end{subarray}}\frac{1}{\lambda_{1}^{r}\lambda_{2}^{s}},

where the order of lattice points $\displaystyle l_{1}\tau+m_{1}\succ l_{2}\tau+m_{2}$ means $\displaystyle l_{1}>l_{2}$ or $\displaystyle l_{1}=l_{2},m_{1}>m_{2}$ . They constructed regularized double Eisenstein series $\displaystyle G_{r,s}(q)$ defined for integers $\displaystyle r,s\geq 1$ as a $\displaystyle q$ -series, which satisfies extended double shuffle relation. Double Eisenstein series are not modular forms in general, but they are expected to play an important role for modular relations among MZVs. Tasaka ([Tasaka20]) gave the explicit formula for decomposing a Hecke eigenform into double Eisenstein series. The example of the decomposition is the following:

\displaystyle 22680G^{\frac{1}{2}}_{9,3}(q)-35364G^{\frac{1}{2}}_{7,5}(q)-29145G^{\frac{1}{2}}_{5,7}(q)+13006G^{\frac{1}{2}}_{3,9}(q)+22680G^{\frac{1}{2}}_{1,11}(q)=\frac{1}{680}\Delta(q).

Here, we write $\displaystyle G^{\frac{1}{2}}_{r,s}(q)=G_{r,s}(q)+\frac{1}{2}G_{r+s}(q)$ and $\displaystyle\Delta(q)=q\prod_{i>0}(1-q^{i})^{24}$ .

In [BT], Bachmann and Tasaka studied multiple Eisenstein series (MES) $\displaystyle G_{n_{1},\dots,n_{r}}(\tau)$ for general depth $\displaystyle r>0$ , which is defined for $\displaystyle n_{1},\dots,n_{r}\geq 2$ as a generalization of double Eisenstein series. They revealed the relationship between the Fourier expansion of MES and the Goncharov coproduct on Hopf algebra of iterated integrals. They also constructed shuffle regularized MES $\displaystyle G^{\,\hbox{\sevency X}\,}_{n_{1},\dots,n_{r}}(q)$ as a $\displaystyle q$ -series, which satisfies restricted double shuffle relations. For example, we have

\displaystyle G_{5}(q)=2G_{2,3}(q)+6G^{\,\hbox{\sevency X}\,}_{1,4}(q).

In this paper, we expand their results for general level. Since MES can be viewed as a multivariate generalization of the classical Eisenstein series, we can consider MES with levels as a generalization of classical Eisenstein series for the congruence subgroup. Kaneko and Tasaka ([KTa]) considered double Eisenstein series of level 2. They provided the relationship between double zeta values of level 2 and modular forms of level 2, and obtained the analogous results of Gangl–Kaneko–Zagier [GKZ] and Kaneko [Kaneko04]. Recently, Kina ([Kina24]) considered double Eisenstein series of level 4, whose constant term of its Fourier expansion is double $\displaystyle\widetilde{T}$ -value introduced in Kaneko–Tsumura [KTs], and obtained the analogous results of Gangl–Kaneko–Zagier [GKZ] and Kaneko [Kaneko04]. For general level, Yuan and Zhao ([YZ15]) studied double Eisenstein series of level $\displaystyle N$ and gave the regularization such that it satisfies extended double shuffle relation.

Let $\displaystyle N\in\mathbb{Z}_{>0}$ and $\displaystyle\eta=\exp(2\pi\sqrt{-1}/N)$ . Let $\displaystyle\mathfrak{H}^{1}$ , $\displaystyle\widetilde{\mathfrak{H}^{0}}$ and $\displaystyle\mathfrak{H}^{2}$ be non-commutative polynomial rings of double indices defined by

$\displaystyle\displaystyle\mathfrak{H}^{1}$	$\displaystyle\displaystyle\coloneq\mathbb{Q}\left\langle\binom{n}{a}\mathrel{}\middle\|\mathrel{}n\in\mathbb{Z}_{\geq 1},a\in\mathbb{Z}/N\mathbb{Z}\right\rangle,$	(3)
$\displaystyle\displaystyle\widetilde{\mathfrak{H}^{0}}$	$\displaystyle\displaystyle\coloneq\mathbb{Q}+\sum_{n\geq 2,a\in\mathbb{Z}/N\mathbb{Z}}\mathfrak{H}^{1}\binom{n}{a},$	(4)
$\displaystyle\displaystyle\mathfrak{H}^{2}$	$\displaystyle\displaystyle\coloneq\mathbb{Q}\left\langle\binom{n}{a}\mathrel{}\middle\|\mathrel{}n\in\mathbb{Z}_{\geq 2},a\in\mathbb{Z}/N\mathbb{Z}\right\rangle.$	(5)

Note that $\displaystyle\widetilde{\mathfrak{H}^{0}}$ , $\displaystyle\mathfrak{H}^{2}$ are the space of admissible indices for MZV and MES of level $\displaystyle N$ , respectively. We can equip $\displaystyle\mathfrak{H}^{1}$ the commutative and associative algebra structure with the shuffle product $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ . For $\displaystyle r\geq 2$ , $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , let $\displaystyle\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$ denote the concatenation $\displaystyle\binom{n_{1}}{a_{1}}\cdots\binom{n_{r}}{a_{r}}$ . We define MES of level $\displaystyle N$ as a holomorphic function on upper half plane $\displaystyle\mathbb{H}$ , as an image of the $\displaystyle\mathbb{Q}$ -linear map $\displaystyle G:\mathfrak{H}^{2}\to\mathcal{O}(\mathbb{H})$ , by

\displaystyle G\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}\coloneq\lim_{L\to\infty}\lim_{M\to\infty}\sum_{\begin{subarray}{c}0\prec l_{1}N\tau+m_{1}\prec\dots\prec l_{r}N\tau+m_{r}\\ l_{i}\in\mathbb{Z}_{L},m_{i}\in\mathbb{Z}_{M},m_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{(l_{1}N\tau+m_{1})^{n_{1}}\dots(l_{r}N\tau+m_{r})^{n_{r}}}

for integers $\displaystyle n_{1},\dots,n_{r}\geq 2$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , and $\displaystyle G(\emptyset;\tau)=1$ . Here, $\displaystyle\mathbb{Z}_{L}=\{l\in\mathbb{Z}\mid\left|l\right|\leq L\}$ and $\displaystyle\mathbb{Z}_{M}=\{m\in\mathbb{Z}\mid\left|m\right|\leq M\}$ . The order of lattice points is defined by

\displaystyle l_{1}\tau+m_{1}\succ l_{2}\tau+m_{2}\xLeftrightarrow{\mathrm{def}}l_{1}>l_{2}\quad\text{or}\quad\begin{cases}l_{1}=l_{2}\\ m_{1}>m_{2}\end{cases}.

MES of arbitrary levels are studied in Yuan–Zhao [YZ15] in the case of depth 2. The Fourier expansion can be expressed with MZVs and multiple divisor function of level $\displaystyle N$ . For example, we have

\displaystyle\displaystyle G\bigg{(}\begin{matrix}2,3\\ 1,1\end{matrix};\tau\bigg{)}=\zeta\binom{2,3}{1,1}+\zeta\binom{2}{0}g\bigg{(}\begin{matrix}3\\ 1\end{matrix};q\bigg{)}+\zeta\binom{2}{1}g\bigg{(}\begin{matrix}3\\ 1\end{matrix};q\bigg{)}+3\zeta\binom{3}{0}g\bigg{(}\begin{matrix}2\\ 1\end{matrix};q\bigg{)}+g\bigg{(}\begin{matrix}2,3\\ 1,1\end{matrix};q\bigg{)}.

(6)

where $\displaystyle q=e^{2\pi\sqrt{-1}\tau}$ . Here, MZVs and multiple divisor functions of level $\displaystyle N$ are defined as the image of $\displaystyle\mathbb{Q}$ -linear maps $\displaystyle\zeta:\widetilde{\mathfrak{H}^{0}}\to\mathbb{R}$ and $\displaystyle g:\mathfrak{H}^{1}\to\mathbb{C}\llbracket q\rrbracket$ by

\displaystyle\zeta\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\sum_{\begin{subarray}{c}0<k_{1}<\dots<k_{r}\\ {}^{\forall}i,k_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{k_{1}^{n_{1}}\cdots k_{r}^{n_{r}}}

for integers $\displaystyle n_{1},\dots,n_{r-1}\geq 1$ , $\displaystyle n_{r}\geq 2$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , and

\displaystyle g\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}\coloneq\left(\frac{-2\pi\sqrt{-1}}{N}\right)^{n}\sum_{\begin{subarray}{c}0<d_{1}<\cdots<d_{r}\\ c_{1},\dots,c_{r}>0\end{subarray}}\prod_{i=1}^{r}\frac{\eta^{a_{i}c_{i}}c_{i}^{n_{i}-1}}{(n_{i}-1)!}q^{c_{i}d_{i}}

for integers $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , where $\displaystyle n=n_{1}+\cdots+n_{r}$ . Goncharov considered the algebra generated by formal iterated integrals $\displaystyle\mathrm{I}(a_{0};a_{1},\dots,a_{m};a_{m+1})$ , which is corresponding to iterated integrals

\displaystyle\int_{a_{0}}^{a_{m+1}}\frac{dt}{t-a_{1}}\cdots\frac{dt}{t-a_{m}}.

He proved the algebra has Hopf algebra structure with Goncharov coproduct $\displaystyle\Delta$ . Let $\displaystyle\widetilde{\mathcal{I}^{1}}$ be a $\displaystyle\mathbb{Q}$ -algebra spanned by formal iterated integrals corresponding to MZVs of level $\displaystyle N$ . $\displaystyle\widetilde{\mathcal{I}^{1}}$ is also Hopf algebra with a coproduct $\displaystyle\Delta_{\mu}$ induced from Goncharov coproduct $\displaystyle\Delta$ . Since $\displaystyle\widetilde{\mathcal{I}^{1}}$ is isomorphic to $\displaystyle(\mathfrak{H}^{1},\widetilde{{\,\hbox{\sevency X}\,}})$ as $\displaystyle\mathbb{Q}$ -algebras, we can equip $\displaystyle\mathfrak{H}^{1}$ with Hopf algebra structure. The coproduct is explicitly given, for example we have

\displaystyle\displaystyle\Delta_{\mu}\left(\binom{2,3}{1,1}\right)=\binom{2,3}{1,1}\otimes 1+\binom{2}{0}\otimes\binom{3}{1}+\binom{2}{1}\otimes\binom{3}{1}+3\binom{3}{0}\otimes\binom{2}{1}+1\otimes\binom{2,3}{1,1}.

(7)

Now, comparing equations (6) and (7), we can see the relationships between Fourier expansion of MES of level $\displaystyle N$ and Goncharov coproduct.

Theorem 1.1.

For any $\displaystyle w\in\mathfrak{H}^{2}$ , we have

\displaystyle G(w;\tau)=(\zeta\star g)(w;q)\qquad(q=e^{2\pi\sqrt{-1}\tau}),

where $\displaystyle\zeta\star g=m\circ(\zeta\otimes g)\circ\Delta_{\mu}$ , $\displaystyle m$ is multiplication on $\displaystyle\mathbb{C}\llbracket q\rrbracket$ .

By using this, we can construct shuffle regularized MES of level $\displaystyle N$ . Kitada ([Kitada23]) constructed shuffle regularized multiple divisor function of level $\displaystyle N$ , $\displaystyle g^{\widetilde{{\,\hbox{\sevency X}\,}}}:\mathfrak{H}^{1}\to\mathbb{C}\llbracket q\rrbracket$ . Then, we define shuffle regularized MES of level $\displaystyle N$ as the image of $\displaystyle G^{\widetilde{{\,\hbox{\sevency X}\,}}}:\mathfrak{H}^{1}\to\mathcal{O}(\mathbb{H})$ by

\displaystyle G^{\widetilde{{\,\hbox{\sevency X}\,}}}(w;\tau)\coloneq(\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}\star g^{\widetilde{{\,\hbox{\sevency X}\,}}})(w;q)\quad(q=e^{2\pi\sqrt{-1}\tau})

for $\displaystyle w\in\mathfrak{H}^{1}$ , where $\displaystyle\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}:\mathfrak{H}^{1}\to\mathbb{C}$ is the shuffle regularized MZVs of level $\displaystyle N$ . The regularized MES of level $\displaystyle N$ satisfy restricted double shuffle relation.

Theorem 1.2 (Restricted double shuffle relation).

For any words $\displaystyle w_{1},w_{2}\in\mathfrak{H}^{2}$ , we have

\displaystyle G(w_{1}\widetilde{*}w_{2};\tau)=G^{\widetilde{{\,\hbox{\sevency X}\,}}}(w_{1}\widetilde{{\,\hbox{\sevency X}\,}}w_{2};\tau).

Kitada ([Kitada23]) obtained distribution relation and sum and weighted sum formulas for double Eisenstein series. We give distribution relation for MES of level $\displaystyle N$ .

The organization of this paper is as follows. In Section 2, we recall multiple $\displaystyle L$ -values (MLVs) introduced in Arakawa–Kaneko [AK] and MZVs of level $\displaystyle N$ introduced in Yuan–Zhao [YZ16]. Note that their shuffle (harmonic) products are different in the algebra of indices $\displaystyle\mathfrak{H}$ . We give the relationships between their shuffle (harmonic) products. We also give the antipode relations for MLVs and MZVs of level $\displaystyle N$ . In Section 3, we give an explicit formula of the Fourier expansion of MES of level $\displaystyle N$ in the same way as Bachmann–Tasaka [BT]. In Section 4, we study formal iterated integrals corresponding to MZVs of level $\displaystyle N$ and observe relationship between the Fourier expansion of MES of level $\displaystyle N$ and them. Using this, we construct shuffle regularized MES of level $\displaystyle N$ in Section 5. In Section 6, we give some linear relations among regularized MES of level $\displaystyle N$ .

Acknowledgment

The author would like to express his deepest gratitude to his supervisor, Professor Yasuo Ohno of Tohoku University. He taught the author attitude to study mathematics and gave a great deal of advice. The author also would like to express his deepest gratitude to Professor Koji Tasaka of Kindai University for comments, corrections, and ideas on this paper. In particular, he gave the author very helpful advice for the argument in Section 4.2, and introduced the author thesis [Kitada23] of his former student Shui Kitada. The author also would like to thank the members of Ohno Laboratory for their kindness on both academically and as friends.

2 Multiple zeta values of level $\displaystyle N$

2.1 Multiple $\displaystyle L$ -values of level $\displaystyle N$

Let $\displaystyle\mathfrak{H}$ denote the non-commutative polynomial ring generated by letters $\displaystyle x,y_{a}$ $\displaystyle(a\in\mathbb{Z}/N\mathbb{Z})$ and let $\displaystyle\mathfrak{H}^{0}$ be a $\displaystyle\mathbb{Q}$ -subalgebra of $\displaystyle\mathfrak{H}$ defined by

	$\displaystyle\displaystyle\mathfrak{H}$	$\displaystyle\displaystyle\coloneq\mathbb{Q}\left\langle x,y_{a}\mathrel{}\middle\|\mathrel{}a\in\mathbb{Z}/N\mathbb{Z}\right\rangle,$		(8)
	$\displaystyle\displaystyle\mathfrak{H}^{0}$	$\displaystyle\displaystyle\coloneq\mathbb{Q}+\sum_{a\in\mathbb{Z}/N\mathbb{Z}}y_{a}\mathfrak{H}x+\sum_{\begin{subarray}{c}a,b\in\mathbb{Z}/N\mathbb{Z}\\ b\neq 0\end{subarray}}y_{a}\mathfrak{H}y_{b}.$		(9)

We can identify $\displaystyle\mathfrak{H}^{0}$ with the algebra of double indices by sending $\displaystyle z_{n,a}=y_{a}x^{n-1}$ to the double index $\displaystyle\binom{n}{a}$ :

\displaystyle\mathfrak{H}^{0}\simeq\mathbb{Q}+\sum_{\begin{subarray}{c}n\geq 1,a\in\mathbb{Z}/N\mathbb{Z}\\ (n,a)\neq(1,0)\end{subarray}}\mathfrak{H}^{1}\binom{n}{a}.

We also apply the identification to $\displaystyle\mathfrak{H}^{1}$ , $\displaystyle\widetilde{\mathfrak{H}^{0}}$ and $\displaystyle\mathfrak{H}^{2}$ .

Definition 2.1 (cf. Arakawa–Kaneko [AK]).

We define the multiple $\displaystyle L$ -value (MLV) of shuffle and harmonic type $\displaystyle L_{\,\hbox{\sevency X}\,},L_{\ast}:\mathfrak{H}^{0}\to\mathbb{C}$ by

	$\displaystyle\displaystyle L_{{\,\hbox{\sevency X}\,}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\int_{0}^{1}\frac{\eta^{a_{1}}dt}{1-\eta^{a_{1}}t}\left(\frac{dt}{t}\right)^{n_{1}-1}\cdots\frac{\eta^{a_{r}}dt}{1-\eta^{a_{r}}t}\left(\frac{dt}{t}\right)^{n_{r}-1},$		(10)
	$\displaystyle\displaystyle L_{\ast}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\sum_{0<k_{1}<\cdots<k_{r}}\frac{\eta^{a_{1}k_{1}+\cdots+a_{r}k_{r}}}{k_{1}^{n_{1}}\cdots k_{r}^{n_{r}}}$		(11)

for any word $\displaystyle\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\in\mathfrak{H}^{0}$ , and $\displaystyle L_{\,\hbox{\sevency X}\,}(\emptyset)=L_{\ast}(\emptyset)=1$ , together with $\displaystyle\mathbb{Q}$ -linearly.

MLVs of shuffle type also have a series expression:

\displaystyle L_{{\,\hbox{\sevency X}\,}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}=\sum_{0<k_{1}<\cdots<k_{r}}\frac{\eta^{(a_{1}-a_{2})k_{1}+\cdots(a_{r-1}-a_{r})k_{r-1}+a_{r}k_{r}}}{k_{1}^{n_{1}}\cdots k_{r}^{n_{r}}}.

We define shuffle product X on $\displaystyle\mathfrak{H}$ recursively by

(S1)

$\displaystyle w{\,\hbox{\sevency X}\,}1=1{\,\hbox{\sevency X}\,}w=w,$
(S2)

$\displaystyle u_{1}w_{1}{\,\hbox{\sevency X}\,}u_{2}w_{2}=u_{1}(w_{1}{\,\hbox{\sevency X}\,}u_{2}w_{2})+u_{2}(u_{1}w_{1}{\,\hbox{\sevency X}\,}w_{2})$

for any $\displaystyle u_{1},u_{2}\in\{x,y_{a}\mid a\in\mathbb{Z}/N\mathbb{Z}\}$ and any words $\displaystyle w,w_{1},w_{2}\in\mathfrak{H}$ , together with $\displaystyle\mathbb{Q}$ -bilinearly. Note that $\displaystyle\mathfrak{H}^{1}$ is generated by $\displaystyle z_{n,a}=y_{a}x^{n-1}$ $\displaystyle(n>0,a\in\mathbb{Z}/N\mathbb{Z})$ . We also define the harmonic product $\displaystyle\ast$ on $\displaystyle\mathfrak{H}^{1}$ recursively by

(H1)

$\displaystyle w\ast 1=1\ast w=w,$
(H2)

$\displaystyle z_{n_{1},a_{1}}w_{1}\ast z_{n_{2},a_{2}}w_{2}=z_{n_{1},a_{1}}(w_{1}\ast z_{n_{2},a_{2}}w_{2})+z_{n_{2},a_{2}}(z_{n_{1},a_{1}}w_{1}\ast w_{2})+z_{n_{1}+n_{2},a_{1}+a_{2}}(w_{1}\ast w_{2}),$

for any $\displaystyle z_{n_{1},a_{1}},z_{n_{2},a_{2}}$ and any words $\displaystyle w_{1},w_{2}\in\mathfrak{H}^{1}$ , together with $\displaystyle\mathbb{Q}$ -bilinearly. It is known that $\displaystyle L_{\,\hbox{\sevency X}\,}$ and $\displaystyle L_{\ast}$ are algebra homomorphisms i.e.

\displaystyle L_{\#}(w_{1}\#w_{2})=L_{\#}(w_{1})L_{\#}(w_{2})\quad(w_{1},w_{2}\in\mathfrak{H}^{0},\#\in\{{\,\hbox{\sevency X}\,},\ast\}).

Remark 2.2.

There exist regularized MLVs, which are defined as polynomials, $\displaystyle L_{\#}^{\mathrm{reg}}:(\mathfrak{H}^{1},\#)\to\mathbb{C}[T]$ and they are algebra homomorphisms (see Arakawa–Kaneko [AK]).

2.2 Multiple zeta values of level N

Definition 2.3 (cf. Yuan–Zhao [YZ16]).

We define the multiple zeta value (MZV) of level $\displaystyle N$ as the image of $\displaystyle\mathbb{Q}$ -linear map $\displaystyle\zeta:\widetilde{\mathfrak{H}^{0}}\to\mathbb{R}$ by

\displaystyle\zeta\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\sum_{\begin{subarray}{c}0<k_{1}<\dots<k_{r}\\ {}^{\forall}i,k_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{k_{1}^{n_{1}}\cdots k_{r}^{n_{r}}}

for integers $\displaystyle n_{1},\dots,n_{r-1}\geq 1$ , $\displaystyle n_{r}\geq 2$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , together with $\displaystyle\mathbb{Q}$ -linearly and $\displaystyle\zeta(\emptyset)=1$ .

MZVs of level $\displaystyle N$ have an iterated integral representation:

\displaystyle\zeta\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}=\int_{0}^{1}\frac{t^{a_{1}-1}dt}{1-t^{N}}\left(\frac{dt}{t}\right)^{n_{1}-1}\frac{t^{a_{2}-a_{1}-1}dt}{1-t^{N}}\left(\frac{dt}{t}\right)^{n_{2}-1}\cdots\frac{t^{a_{r}-a_{r-1}-1}dt}{1-t^{N}}\left(\frac{dt}{t}\right)^{n_{r}-1}.

We define a $\displaystyle\mathbb{Q}$ -linear bijection $\displaystyle\rho:\mathfrak{H}^{1}\to\mathfrak{H}^{1}$ and a $\displaystyle\mathbb{Q}(\eta)$ -linear bijection $\displaystyle\pi:\mathfrak{H}^{1}\otimes_{\mathbb{Q}}\mathbb{Q}(\eta)\to\mathfrak{H}^{1}\otimes_{\mathbb{Q}}\mathbb{Q}(\eta)$ by

	$\displaystyle\displaystyle\rho(z_{n_{1},a_{1}}\cdots z_{n_{r},a_{r}})=z_{n_{1},a_{1}}z_{n_{2},a_{2}-a_{1}}\cdots z_{n_{r},a_{r}-a_{r-1}},$		(12)
	$\displaystyle\displaystyle\pi(z_{n,a})=N^{-1}\sum_{b\in\mathbb{Z}/N\mathbb{Z}}\eta^{-ab}z_{n,b},$		(13)

We define two products $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}},\widetilde{*}:\mathfrak{H}^{1}\times\mathfrak{H}^{1}\to\mathfrak{H}^{1}$ by

	$\displaystyle\displaystyle w_{1}\widetilde{{\,\hbox{\sevency X}\,}}w_{2}\coloneq\rho^{-1}\left(\rho(w_{1}){\,\hbox{\sevency X}\,}\rho(w_{2})\right),$		(14)
	$\displaystyle\displaystyle w_{1}\widetilde{*}w_{2}\coloneq\pi^{-1}\left(\pi(w_{1})\ast\pi(w_{2})\right).$		(15)

MZVs of level $\displaystyle N$ can be written as $\displaystyle\mathbb{Q}(\eta)$ -linear combination of MLVs by using the linear maps $\displaystyle\rho$ and $\displaystyle\pi$ .

Proposition 2.4 (Yuan–Zhao [YZ16]).

For $\displaystyle w\in\widetilde{\mathfrak{H}^{0}}$ , we have

\displaystyle\zeta(w)=(L_{\,\hbox{\sevency X}\,}\circ\pi\circ\rho)(w)=(L_{\ast}\circ\pi)(w).

MZVs of level $\displaystyle N$ satisfy $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ and $\displaystyle\widetilde{*}$ -products.

Proposition 2.5.

$\displaystyle\zeta:(\widetilde{\mathfrak{H}^{0}},\widetilde{{\,\hbox{\sevency X}\,}})\to\mathbb{R}$ is an algebra homomorphism.

Proof.

Let $\displaystyle y_{a}=\frac{t^{a-1}dt}{1-t^{N}}$ and $\displaystyle x=\frac{dt}{t}$ . By the integral representation of MZV, it holds

\displaystyle\zeta(w)=\int_{0}^{1}\rho(w)

for any word $\displaystyle w\in\widetilde{\mathfrak{H}}^{0}$ . For any words $\displaystyle w_{1},w_{2}\in\widetilde{\mathfrak{H}}^{0}$ , we have

	$\displaystyle\displaystyle\zeta(w_{1}\widetilde{{\,\hbox{\sevency X}\,}}w_{2})$	$\displaystyle\displaystyle=\int_{0}^{1}\rho(w_{1}\widetilde{{\,\hbox{\sevency X}\,}}w_{2})=\int_{0}^{1}\rho(w_{1}){\,\hbox{\sevency X}\,}\rho(w_{2})$		(16)
		$\displaystyle\displaystyle=\left(\int_{0}^{1}\rho(w_{1})\right)\left(\int_{0}^{1}\rho(w_{2})\right)=\zeta(w_{1})\zeta(w_{2}).$		(17)

∎

Proposition 2.6.

The $\displaystyle\widetilde{*}$ -product is well-defined and $\displaystyle\zeta:(\widetilde{\mathfrak{H}^{0}},\widetilde{*})\to\mathbb{R}$ is an algebra homomorphism.

Proof.

It suffices to show the $\displaystyle\widetilde{*}$ -product is determined recursively by

(T1)

$\displaystyle w\widetilde{*}1=1\widetilde{*}w=w,$
(T2)

$\displaystyle z_{n_{1},a_{1}}w_{1}\widetilde{*}z_{n_{2},a_{2}}w_{2}=z_{n_{1},a_{1}}(w_{1}\widetilde{*}z_{n_{2},a_{2}}w_{2})+z_{n_{2},a_{2}}(z_{n_{1},a_{1}}w_{1}\widetilde{*}w_{2})+\delta_{a_{1},a_{2}}z_{n_{1}+n_{2},a_{1}+a_{2}}(w_{1}\widetilde{*}w_{2})$

for any $\displaystyle z_{n_{1},a_{1}},z_{n_{2},a_{2}}$ and any words $\displaystyle w_{1},w_{2}$ . (T1) is clear by definition. We prove (T2) by induction on $\displaystyle l(w_{1})+l(w_{2})$ . Here, $\displaystyle l(w)$ denote the length of the word $\displaystyle w$ . When $\displaystyle l(w_{1})+l(w_{2})=0$ , we have

	$\displaystyle\displaystyle z_{n_{1},a_{1}}\widetilde{*}z_{n_{2},a_{2}}$	$\displaystyle\displaystyle=\pi^{-1}\bigg{(}N^{-2}\sum_{b_{1},b_{2}\in\mathbb{Z}/N\mathbb{Z}}\eta^{-a_{1}b_{1}-a_{2}b_{2}}(z_{n_{1},b_{1}}z_{n_{2},b_{2}}+z_{n_{2},b_{2}}z_{n_{1},b_{1}}+z_{n_{1}+n_{2},b_{1}+b_{2}})\bigg{)}$		(18)
		$\displaystyle\displaystyle=z_{n_{1},a_{1}}z_{n_{2},a_{2}}+z_{n_{2},a_{2}}z_{n_{1},a_{1}}+\pi^{-1}\bigg{(}N^{-2}\sum_{b_{1},b_{2}\in\mathbb{Z}/N\mathbb{Z}}\eta^{-a_{1}b_{1}-a_{2}b_{2}}z_{n_{1}+n_{2},b_{1}+b_{2}}\bigg{)}.$		(19)

In the sum of the third term, replacing $\displaystyle b_{1}^{\prime}=b_{1}+b_{2}$ , we have

	$\displaystyle\displaystyle\pi^{-1}\bigg{(}N^{-2}\sum_{b_{1},b_{2}\in\mathbb{Z}/N\mathbb{Z}}\eta^{-a_{1}b_{1}-a_{2}b_{2}}z_{n_{1}+n_{2},b_{1}+b_{2}}\bigg{)}$	$\displaystyle\displaystyle=\pi^{-1}\bigg{(}N^{-2}\sum_{b_{2}\in\mathbb{Z}/N\mathbb{Z}}\eta^{(a_{1}-a_{2})b_{2}}\sum_{b_{1}^{\prime}\in\mathbb{Z}/N\mathbb{Z}}\eta^{-a_{1}b_{1}^{\prime}}z_{n_{1}+n_{2},b_{1}^{\prime}}\bigg{)}$		(20)
		$\displaystyle\displaystyle=\pi^{-1}\bigg{(}\delta_{a_{1},a_{2}}N^{-1}\sum_{b_{1}^{\prime}\in\mathbb{Z}/N\mathbb{Z}}\eta^{-a_{1}b_{1}^{\prime}}z_{n_{1}+n_{2},b_{1}^{\prime}}\bigg{)}=\delta_{a_{1},a_{2}}z_{n_{1}+n_{2},a_{1}}.$		(21)

When $\displaystyle l(w_{1})+l(w_{2})>0$ , we put $\displaystyle w_{1}=z_{n_{2},a_{2}}\cdots z_{n_{r},a_{r}}$ and $\displaystyle w_{2}=z_{n_{r+2},a_{r+2}}\cdots z_{n_{r+s},a_{r+s}}$ . Using inductive hypothesis, we have

$\displaystyle\displaystyle z_{n_{1},a_{1}}w_{1}\widetilde{*}z_{n_{r+1},a_{r+1}}w_{2}$	$\displaystyle\displaystyle=\pi^{-1}\bigg{(}N^{-(r+s)}\sum_{b_{1},\dots,b_{r+s}}\eta^{-\bm{a}\cdot\bm{b}}z_{n_{1},b_{1}}\cdots z_{n_{r},b_{r}}\ast z_{n_{r+1},b_{r+1}}\cdots z_{n_{r+s},b_{r+s}}\bigg{)}$	(22)
	$\displaystyle\displaystyle=z_{n_{1},a_{1}}(w_{1}\widetilde{}z_{n_{r+1},a_{r+1}}w_{2})+z_{n_{r+1},a_{r+1}}(z_{n_{1},a_{1}}w_{1}\widetilde{}w_{2})$	(23)
	$\displaystyle\displaystyle+\pi^{-1}\bigg{(}N^{-2}\sum_{b_{1},b_{r+1}\in\mathbb{Z}/N\mathbb{Z}}\eta^{-a_{1}b_{1}-a_{r+1}b_{r+1}}z_{n_{1}+n_{r+1},b_{1}+b_{r+1}}\bigg{)}w_{1}\widetilde{*}w_{2},$	(24)

where $\displaystyle\bm{a}=(a_{1},\dots,a_{r})$ and $\displaystyle\bm{b}={}^{t}(b_{1},\dots,b_{r})$ . As mentioned above, the third term is $\displaystyle\delta_{a_{1},a_{2}}w_{1}\widetilde{*}w_{2}$ . ∎

We define regularized MZVs of level $\displaystyle N$ by using regularized MLVs and 2.4.

Definition 2.7.

We define the regularized multiple zeta values of level $\displaystyle N$ as the images of $\displaystyle\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}:\mathfrak{H}^{1}\to\mathbb{C}$ and $\displaystyle\zeta^{\widetilde{*}}:\mathfrak{H}^{1}\to\mathbb{C}$ , by

	$\displaystyle\displaystyle\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}(w)\coloneq(L^{\mathrm{reg}}_{\,\hbox{\sevency X}\,}\circ\pi\circ\rho)(w)\|_{T=0},$		(25)
	$\displaystyle\displaystyle\zeta^{\widetilde{*}}(w)\coloneq(L^{\mathrm{reg}}_{\ast}\circ\pi)(w)\|_{T=0}.$		(26)

By definition, regularized MZV of level $\displaystyle N$ , $\displaystyle\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}$ and $\displaystyle\zeta^{\widetilde{*}}$ satisfy $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ and $\displaystyle\widetilde{*}$ -product, respectively.

Now, we give the antipode relation for MLVs and MZVs of level $\displaystyle N$ . We can extend $\displaystyle L_{\,\hbox{\sevency X}\,}$ on $\displaystyle\mathfrak{H}$ with X -homomorphy by putting $\displaystyle L_{\,\hbox{\sevency X}\,}(x)=L_{\,\hbox{\sevency X}\,}(y_{0})=0$ . Hoffman ([Hoff00]) provided Hopf algebra structure on the quasi-shuffle algebra.

Theorem 2.8 (Hoffman [Hoff00] Theorem 3.2).

$\displaystyle(\mathfrak{H}_{\,\hbox{\sevency X}\,},\Delta_{H},\varepsilon_{H},\mathcal{S})$ is a Hopf algebra with

\displaystyle\displaystyle\Delta_{H}(w)=\sum_{uv=w}u\otimes v,\quad\varepsilon_{H}(w)=\begin{cases}1&w=1\\ 0&w\neq 1\end{cases},\quad\mathcal{S}(w)=(-1)^{\mathrm{wt}(w)}\overset{\leftarrow}{w},

(27)

where $\displaystyle\overset{\leftarrow}{w}=a_{n}\cdots a_{1}$ for $\displaystyle w=a_{1}\cdots a_{n}$ .

Lemma 2.9.

For any word $\displaystyle a_{1}\cdots a_{n}\in\mathfrak{H}$ $\displaystyle(a_{1},\dots,a_{n}\in\{x,y_{a}\mid a\in\mathbb{Z}/N\mathbb{Z}\})$ , we have

\displaystyle\sum_{i=0}^{n}(-1)^{i}a_{i}a_{i-1}\cdots a_{1}{\,\hbox{\sevency X}\,}a_{i+1}a_{i+2}\cdots a_{n}=0.

Proof.

Let $\displaystyle m:\mathfrak{H}_{\,\hbox{\sevency X}\,}\otimes\mathfrak{H}_{\,\hbox{\sevency X}\,}\to\mathfrak{H}_{\,\hbox{\sevency X}\,}$ and $\displaystyle u:\mathbb{Q}\to\mathfrak{H}_{\,\hbox{\sevency X}\,}$ denote the product and the unit of the shuffle algebra $\displaystyle\mathfrak{H}_{\,\hbox{\sevency X}\,}$ , respectively. Since $\displaystyle\mathcal{S}$ is an antipode, we have

\displaystyle(\mathcal{S}\star 1)(w)=(u\circ\varepsilon_{H})(w)=\begin{cases}1&w=1,\\ 0&w\neq 1.\end{cases}

Meanwhile, by definition of convolution, it holds

\displaystyle(\mathcal{S}\star 1)(w)=m\circ(\mathcal{S}\otimes 1)\circ\Delta_{H}(w)=\sum_{uv=w}\mathcal{S}(u){\,\hbox{\sevency X}\,}v=\sum_{uv=w}(-1)^{\mathrm{wt}(u)}\overset{\leftarrow}{u}{\,\hbox{\sevency X}\,}v.

Putting $\displaystyle w=a_{1}\cdots a_{n}$ , we obtain the claim. ∎

Using this, we obtain antipode relation for MLVs of shuffle type.

Proposition 2.10.

For $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r-1}\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle\sum_{q=1}^{r}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{r}=n\\ k_{q}=1\end{subarray}}(-1)^{m_{q}}\prod_{\begin{subarray}{c}i=1\\ i\neq q\end{subarray}}^{r}\binom{k_{i}-1}{n_{i}-1}L_{\,\hbox{\sevency X}\,}\binom{k_{q-1},\dots,k_{1}}{a_{q-1},\dots,a_{1}}L_{\,\hbox{\sevency X}\,}\binom{k_{q+1},\dots,k_{r}}{a_{q},\dots,a_{r-1}}=0

where $\displaystyle m_{q}=k_{1}+\cdots+k_{q-1}+n_{q}$ and $\displaystyle n=n_{1}+\cdots+n_{r}$ .

Proof.

By applying 2.9 with $\displaystyle w=x^{n_{1}-1}y_{a_{1}}x^{n_{2}-1}y_{a_{2}}\cdots x^{n_{r-1}-1}y_{a_{r-1}}x^{n_{r}-1}$ , we have

	$\displaystyle\displaystyle\sum_{q=1}^{r}\sum_{l_{q}=0}^{n_{q}-1}(-1)^{n_{1}+\cdots+n_{q-1}+l_{q}}$	$\displaystyle\displaystyle(x^{l_{q}}y_{a_{q-1}}x^{n_{q-1}-1}\cdots y_{a_{1}}x^{n_{1}-1}$		(28)
		$\displaystyle\displaystyle{\,\hbox{\sevency X}\,}x^{n_{q}-l_{q}-1}y_{a_{q}}x^{n_{q+1}-1}\cdots y_{a_{r-1}}x^{n_{r}-1})=0.$		(29)

Sending both sides by the map $\displaystyle L_{\,\hbox{\sevency X}\,}$ , we obtain the claim. ∎

Since MZV of level $\displaystyle N$ can be written via MLV, we obtain antipode relation for MZV of level $\displaystyle N$ .

Corollary 2.11 (Antipode relation for MZV of level $\displaystyle N$ ).

For $\displaystyle n_{1},\dots,n_{r}\geq 2$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle\sum_{q=1}^{r}\sum_{\begin{subarray}{c}k_{1}+\dots+k_{r}=n\\ k_{q}=1\end{subarray}}(-1)^{m_{q}}\prod_{\begin{subarray}{c}i=1\\ i\neq q\end{subarray}}^{r}\binom{k_{i}-1}{n_{i}-1}\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q-1}&,\dots,&k_{1}\\ a_{q}-a_{q-1}&,\dots,&a_{q}-a_{1}\end{array}\bigg{)}\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q+1}&,\dots,&k_{r}\\ a_{q+1}-a_{q}&,\dots,&a_{r}-a_{q}\end{array}\bigg{)}=0,

where $\displaystyle m_{q}=k_{1}+\cdots+k_{q-1}+n_{q}$ and $\displaystyle n=n_{1}+\cdots+n_{r}$ .

Proof.

Using 2.4 and 2.10, we have

	$\displaystyle\displaystyle(\text{L.H.S})=$	$\displaystyle\displaystyle\sum_{b_{1},\dots,b_{r-1}\in\mathbb{Z}/N\mathbb{Z}}\eta^{(\bm{a},\bm{b})}\sum_{q=1}^{r}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{r}=n\\ k_{q}=1\end{subarray}}(-1)^{k_{1}+\cdots+k_{q-1}+n_{q}}$		(30)
		$\displaystyle\displaystyle\times\prod_{\begin{subarray}{c}i=1\\ i\neq q\end{subarray}}^{r}\binom{k_{i}-1}{n_{i}-1}L_{\,\hbox{\sevency X}\,}\binom{k_{q-1},\dots,k_{1}}{b_{q-1},\dots,b_{1}}L_{\,\hbox{\sevency X}\,}\binom{k_{q+1},\dots,k_{r}}{b_{q},\dots,b_{r-1}}=0,$		(31)

where $\displaystyle(\bm{a},\bm{b})=-\sum_{i=1}^{r-1}(a_{i+1}-a_{i})b_{i}$ . ∎

3 Fourier expansion for multiple Eisenstein series of level $\displaystyle N$

In this section, we give the Fourier expansion of MES of level $\displaystyle N$ explicitly. By the definition of the order of lattice points, we can split the sum $\displaystyle\sum_{0\prec\lambda_{1}\prec\cdots\prec\lambda_{r}}$ into $\displaystyle 2^{r}$ many terms. In this section, we consider the each term and give its Fourier expansion. Let $\displaystyle\{\mathrm{x},\mathrm{y}\}^{\ast}$ be the set of all words generated by letters $\displaystyle\mathrm{x}$ and $\displaystyle\mathrm{y}$ .

Definition 3.1.

For $\displaystyle n_{1},\dots,n_{r}\geq 2$ , $\displaystyle a_{1},\dots,a_{r}\in{\mathbb{Z}/N\mathbb{Z}}$ and $\displaystyle w_{1}\cdots w_{r}\in\{\mathrm{x},\mathrm{y}\}^{\ast}$ , we define

\displaystyle\displaystyle G_{w_{1}\cdots w_{r}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}\coloneq\lim_{L\to\infty}\lim_{M\to\infty}\sum_{\begin{subarray}{c}\lambda_{i}-\lambda_{i-1}\in P_{w_{i}}\\ \lambda_{i}\in N\mathbb{Z}_{L}\tau+\mathbb{Z}_{M},\lambda_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{\lambda_{1}^{n_{1}}\dots\lambda_{r}^{n_{r}}},

(32)

where $\displaystyle P_{\mathrm{x}}=\{l\tau+m\in\mathbb{Z}\tau+\mathbb{Z}\mid l=0,m>0\}$ , $\displaystyle P_{\mathrm{y}}=\{l\tau+m\in\mathbb{Z}\tau+\mathbb{Z}\mid l>0\}$ and $\displaystyle l\tau+m\equiv a$ means $\displaystyle m\equiv a$ .

Note that $\displaystyle\lambda\in P_{\mathrm{x}}\sqcup P_{\mathrm{y}}\Leftrightarrow\lambda\succ 0$ .

Lemma 3.2.

For $\displaystyle n_{1},\dots,n_{r}\geq 2$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle\displaystyle G\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}=\sum_{w_{1},\dots,w_{r}\in\{\mathrm{x},\mathrm{y}\}}G_{w_{1}\cdots w_{r}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}.

(33)

Proof.

By definition of MES of level $\displaystyle N$ , we have

$\displaystyle\displaystyle(\text{L.H.S})$	$\displaystyle\displaystyle=\lim_{L\to\infty}\lim_{M\to\infty}\sum_{\begin{subarray}{c}\lambda_{i}-\lambda_{i-1}\in P_{\mathrm{x}}\cup P_{\mathrm{y}}\\ \lambda_{i}\in\mathbb{Z}_{L}\tau+\mathbb{Z}_{M},\lambda_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{\lambda_{1}^{n_{1}}\dots\lambda_{r}^{n_{r}}}$	(34)
	$\displaystyle\displaystyle=\sum_{w_{1},\dots,w_{r}\in\{\mathrm{x},\mathrm{y}\}}\lim_{L\to\infty}\lim_{M\to\infty}\sum_{\begin{subarray}{c}\lambda_{i}-\lambda_{i-1}\in P_{w_{i}}\\ \lambda_{i}\in\mathbb{Z}_{L}\tau+\mathbb{Z}_{M},\lambda_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{\lambda_{1}^{n_{1}}\dots\lambda_{r}^{n_{r}}}$	(35)
	$\displaystyle\displaystyle=\sum_{w_{1},\dots,w_{r}\in\{\mathrm{x},\mathrm{y}\}}G_{w_{1}\cdots w_{r}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}=(\text{R.H.S}).$	(36)

∎

3.1 Multitangent function of level $\displaystyle N$

Multitangent function is defined by Bouillot [Bouillot] in the case of level 1, and he studied the algebraic structure of multitangent functions. In this subsection, we define multitangent function of level $\displaystyle N$ and give its Fourier expansion.

Definition 3.3.

We define the multitangent function of level $\displaystyle N$ as the image of $\displaystyle\mathbb{Q}$ -linear map $\displaystyle\Psi:\mathfrak{H}^{2}\to\mathcal{O}(\mathbb{H})$ by

\displaystyle\Psi\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}\coloneq\sum_{\begin{subarray}{c}-\infty<m_{1}<\dots<m_{r}<+\infty\\ m_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{(\tau+m_{1})^{n_{1}}\dots(\tau+m_{r})^{n_{r}}}

for $\displaystyle n_{1},\dots,n_{r}\geq 2$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , and $\displaystyle\Psi(\emptyset;\tau)=1$ , together with $\displaystyle\mathbb{Q}$ -linearly. We define

\displaystyle\Psi\bigg{(}\begin{matrix}1\\ a\end{matrix}\>;\tau\bigg{)}\coloneq\lim_{M\to\infty}\sum_{\begin{subarray}{c}|m|<M\\ m\equiv a\pmod{N}\end{subarray}}\frac{1}{\tau+m}.

for $\displaystyle a\in\mathbb{Z}/N\mathbb{Z}$ and $\displaystyle\tau\in\mathbb{H}$ .

Bouillot ([Bouillot]) proved that any multitangent fuction can be written as a $\displaystyle\mathbb{Q}$ -linear sum of products of MZVs and monotangent functions. In the case of level $\displaystyle N$ , multitangent function can be reduced into monotangent function.

Lemma 3.4.

For $\displaystyle n_{1},\dots,n_{r}\geq 2$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we have

	$\displaystyle\displaystyle\Psi\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}$	$\displaystyle\displaystyle=\sum_{q=1}^{r}\sum_{\begin{subarray}{c}k_{1}+\dots+k_{r}=n\\ k_{q}\geq 2\end{subarray}}(-1)^{n+n_{q}+k_{q+1}+\dots+k_{r}}\prod_{\begin{subarray}{c}p=1\\ p\neq q\end{subarray}}^{r}\binom{k_{p}-1}{n_{p}-1}$		(37)
		$\displaystyle\displaystyle\times\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q-1}&,\dots,&k_{1}\\ a_{q}-a_{q-1}&,\dots,&a_{q}-a_{1}\end{array}\bigg{)}\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q+1}&,\dots,&k_{r}\\ a_{q+1}-a_{q}&,\dots,&a_{r}-a_{q}\end{array}\bigg{)}\Psi\bigg{(}\begin{matrix}k_{q}\\ a_{q}\end{matrix}\>;\tau\bigg{)},$		(42)

where $\displaystyle n=n_{1}+\cdots+n_{r}$ .

Proof.

Using partial fraction decomposition,

\displaystyle\frac{1}{(\tau+m_{1})^{n_{1}}\cdots(\tau+m_{r})^{n_{r}}}=\sum_{q=1}^{r}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{r}=n\\ k_{i}\geq 1\end{subarray}}\left(\prod_{p=1}^{q-1}\frac{\binom{k_{p}-1}{n_{p}-1}}{(m_{q}-m_{p})^{k_{j}}}\right)\frac{(-1)^{n+n_{q}}}{(\tau+m_{q})^{k_{q}}}\left(\prod_{p=q+1}^{r}\frac{(-1)^{k_{p}}\binom{k_{p}-1}{n_{p}-1}}{(m_{p}-m_{q})^{k_{p}}}\right)

for $\displaystyle m_{1},\dots,m_{r}\in\mathbb{Z}$ , we have

	$\displaystyle\displaystyle\Psi\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}$		(43)
	$\displaystyle\displaystyle=\sum_{\begin{subarray}{c}-\infty<m_{1}<\cdots<m_{r}<+\infty\\ m_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{(\tau+m_{1})^{n_{1}}\cdots(\tau+m_{r})^{n_{r}}}$		(44)
	$\displaystyle\displaystyle=\sum_{q=1}^{r}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{r}=n\\ k_{i}\geq 1\end{subarray}}(-1)^{n+n_{q}+k_{q+1}+\cdots+k_{r}}\prod_{\begin{subarray}{c}p=1\\ p\neq q\end{subarray}}^{r}\binom{k_{p}-1}{n_{p}-1}\sum_{\begin{subarray}{c}m_{q}\in\mathbb{Z}\\ m_{q}\equiv a_{q}\pmod{N}\end{subarray}}\frac{1}{(\tau+m_{q})^{k_{q}}}$		(45)
	$\displaystyle\displaystyle\times\sum_{\begin{subarray}{c}-\infty<m_{1}<\cdots<m_{q-1}<m_{q}\\ m_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{(m_{q}-m_{1})^{k_{1}}\cdots(m_{q}-m_{q-1})^{k_{q-1}}}$		(46)
	$\displaystyle\displaystyle\times\sum_{\begin{subarray}{c}m_{q}<m_{q+1}<\cdots<m_{r}<+\infty\\ m_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{(m_{q+1}-m_{q})^{k_{q}}\cdots(m_{r}-m_{q})^{k_{r}}}$		(47)
	$\displaystyle\displaystyle=\sum_{q=1}^{r}\sum_{\begin{subarray}{c}k_{1}+\dots+k_{r}=n\\ k_{q}\geq 1\end{subarray}}(-1)^{n+n_{q}+k_{q+1}+\dots+k_{r}}\prod_{\begin{subarray}{c}p=1\\ p\neq q\end{subarray}}^{r}\binom{k_{p}-1}{n_{p}-1}$		(48)
	$\displaystyle\displaystyle\times\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q-1}&,\dots,&k_{1}\\ a_{q}-a_{q-1}&,\dots,&a_{q}-a_{1}\end{array}\bigg{)}\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q+1}&,\dots,&k_{r}\\ a_{q+1}-a_{q}&,\dots,&a_{r}-a_{q}\end{array}\bigg{)}\sum_{\begin{subarray}{c}m_{q}\in\mathbb{Z}\\ m_{q}\equiv a_{q}\pmod{N}\end{subarray}}\frac{1}{(\tau+m_{q})^{k_{q}}}.$		(53)

∎

The following lemma gives us the Fourier expansion for multitangent functions of level $\displaystyle N$ .

Lemma 3.5 (Yuan–Zhao [YZ15], Lemma 4.1).

For an integer $\displaystyle n\geq 1$ and $\displaystyle a\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle\Psi\bigg{(}\begin{matrix}n\\ a\end{matrix}\>;N\tau\bigg{)}=\left(\frac{-2\pi\sqrt{-1}}{N}\right)^{n}\sum_{c>0}\frac{c^{n-1}\eta^{ac}}{(n-1)!}q^{c}-\delta_{n,1}\frac{\pi\sqrt{-1}}{N},

where $\displaystyle q=e^{2\pi\sqrt{-1}\tau}$ .

3.2 Multiple divisor function of level $\displaystyle N$

Multiple divisor function is initially studied by Bachmann and Kühn ([BK]). Yuan and Zhao ([YZ16]) generalized it to arbitrary level and studied the relationship with MZV of level $\displaystyle N$ .

Definition 3.6 ([YZ16]).

For $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r}\in{\mathbb{Z}/N\mathbb{Z}}$ , we define multiple divisor function of level $\displaystyle N$ as the image of $\displaystyle\mathbb{Q}$ -linear map $\displaystyle g:\mathfrak{H}^{1}\to\mathbb{C}\llbracket q\rrbracket$ by

\displaystyle g\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}\coloneq\left(\frac{-2\pi\sqrt{-1}}{N}\right)^{n}\sum_{\begin{subarray}{c}0<d_{1}<\cdots<d_{r}\\ c_{1},\dots,c_{r}>0\end{subarray}}\prod_{i=1}^{r}\frac{\eta^{a_{i}c_{i}}c_{i}^{n_{i}-1}}{(n_{i}-1)!}q^{c_{i}d_{i}}

for $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , and $\displaystyle g(\emptyset;q)=1$ , together with $\displaystyle\mathbb{Q}$ -linearly, where $\displaystyle n=n_{1}+\cdots+n_{r}$ .

As a holomorphic function on $\displaystyle\mathbb{H}$ , multiple divisor function $\displaystyle g(q)$ can be written as sum of products of monotangent function. The following lemma follows from 3.5.

Lemma 3.7.

For any $\displaystyle n_{1},\dots,n_{r}\in\mathbb{Z}_{\geq 2}$ , $\displaystyle a_{1},\dots,a_{r}\in{\mathbb{Z}/N\mathbb{Z}}$ and $\displaystyle\tau\in\mathbb{H}$ , we have

\displaystyle\displaystyle g\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}=\sum_{0<d_{1}<\dots<d_{r}}\Psi\bigg{(}\begin{matrix}n_{1}\\ a_{1}\end{matrix}\>;d_{1}N\tau\bigg{)}\cdots\Psi\bigg{(}\begin{matrix}n_{r}\\ a_{r}\end{matrix}\>;d_{r}N\tau\bigg{)},

(54)

where $\displaystyle q=e^{2\pi\sqrt{-1}\tau}$ .

3.3 The Fourier expansion of MES of level $\displaystyle N$

The Fourier expansion of MES can be written with MZVs and multiple divisor functions.

Proposition 3.8.

For any $\displaystyle n_{1},\dots,n_{r}\geq 2$ , $\displaystyle a_{1},\dots,a_{r}\in{\mathbb{Z}/N\mathbb{Z}}$ and $\displaystyle w_{1}\cdots w_{r}\in\{\mathrm{x},\mathrm{y}\}^{*}$ , we put $\displaystyle w_{1}\cdots w_{r}=\mathrm{x}^{t_{1}-1}\mathrm{y}\mathrm{x}^{t_{2}-t_{1}-1}\mathrm{y}\cdots\mathrm{x}^{t_{h}-t_{h-1}-1}\mathrm{y}\mathrm{x}^{r-t_{h}}$ . Then we have

	$\displaystyle\displaystyle G_{w_{1}\cdots w_{r}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}$		(55)
	$\displaystyle\displaystyle=\zeta\binom{n_{1},\dots,n_{t_{1}-1}}{a_{1},\dots,a_{t_{1}-1}}\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}t_{1}\leq q_{1}\leq t_{2}-1\\[-5.0pt] \vdots\\ t_{h}\leq q_{h}\leq r\end{subarray}}$}}\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}k_{t_{j}}+\cdots+k_{t_{j+1}-1}\\ =n_{t_{j}}+\cdots+n_{t_{j+1}-1}\\ (1\leq j\leq h),k_{i}\geq 1\end{subarray}}$}}\prod_{j=1}^{h}\Bigg{\{}(-1)^{l_{j}}\Bigg{(}\prod_{\begin{subarray}{c}p=t_{j}\\ p\neq q_{j}\end{subarray}}^{t_{j+1}-1}\binom{k_{p}-1}{n_{p}-1}$
	$\displaystyle\displaystyle\times\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}-1}&,\dots,&k_{t_{j}}\\ a_{q_{j}}-a_{q_{j}-1}&,\dots,&a_{q_{j}}-a_{t_{j}}\end{array}\bigg{)}\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}+1}&,\dots,&k_{t_{j+1}-1}\\ a_{q_{j}+1}-a_{q_{j}}&,\dots,&a_{t_{j+1}-1}-a_{q_{j}}\end{array}\bigg{)}\Bigg{\}}g\bigg{(}\begin{matrix}k_{q_{1}},\dots,k_{q_{h}}\\ a_{q_{1}},\dots,a_{q_{h}}\end{matrix};q\bigg{)},$		(60)

where $\displaystyle l_{j}=n_{t_{j}}+\cdots+n_{t_{j+1}-1}+n_{q_{j}}+k_{q_{j}+1}+\cdots+k_{q_{j+1}-1}$ , $\displaystyle q=e^{2\pi\sqrt{-1}\tau}$ , $\displaystyle t_{h+1}=r+1$ and $\displaystyle a_{r+1}=0$ .

Proof.

By definition of the portion $\displaystyle G_{w_{1}\cdots w_{r}}$ , we have

	$\displaystyle\displaystyle G_{w_{1}\cdots w_{r}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}$		(61)
	$\displaystyle\displaystyle=\sum_{\begin{subarray}{c}0<l_{1}<\cdots<l_{h}\\ 0\leq{}^{\forall}j\leq h,m_{t_{j}}<\cdots<m_{t_{j+1}-1}\\ {}^{\forall}i>0,m_{i}>0,m_{i}\equiv a_{i}\pmod{N}\end{subarray}}\frac{1}{m_{1}^{n_{1}}\cdots m_{t_{1}-1}^{n_{t_{1}-1}}}\prod_{j=1}^{h}\frac{1}{(l_{j}N\tau+m_{t_{j}})^{n_{t_{j}}}\cdots(l_{j}N\tau+m_{t_{j+1}-1})^{n_{t_{j+1}-1}}}$		(62)
	$\displaystyle\displaystyle=\zeta\binom{n_{1},\dots,n_{t_{1}-1}}{a_{1},\dots,a_{t_{1}-1}}\sum_{0<l_{1}<\cdots<l_{h}}\prod_{j=1}^{h}\Psi\bigg{(}\begin{matrix}n_{t_{j}},\dots,n_{t_{j+1}-1}\\ a_{t_{j}},\dots,a_{t_{j+1}-1}\end{matrix};l_{j}N\tau\bigg{)}.$		(63)

By 3.4, we have

	$\displaystyle\displaystyle\sum_{0<l_{1}<\cdots<l_{h}}\prod_{j=1}^{h}\Psi\bigg{(}\begin{matrix}n_{t_{j}},\dots,n_{t_{j+1}-1}\\ a_{t_{j}},\dots,a_{t_{j+1}-1}\end{matrix};l_{j}N\tau\bigg{)}$		(64)
	$\displaystyle\displaystyle=\lim_{L\to\infty}\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}t_{1}\leq q_{1}\leq t_{2}-1\\[-5.0pt] \vdots\\ t_{h}\leq q_{h}\leq r\end{subarray}}$}}\,\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}k_{t_{j}}+\cdots+k_{t_{j+1}-1}\\ =n_{t_{j}}+\cdots+n_{t_{j+1}-1}\\ (1\leq j\leq h),k_{i}\geq 1\end{subarray}}$}}\prod_{j=1}^{h}\Bigg{\{}(-1)^{l_{j}}\prod_{\begin{subarray}{c}p=t_{j}\\ p\neq q_{j}\end{subarray}}^{t_{j+1}-1}$
	$\displaystyle\displaystyle\times\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}-1}&,\dots,&k_{t_{j}}\\ a_{q_{j}}-a_{q_{j}-1}&,\dots,&a_{q_{j}}-a_{t_{j}}\end{array}\bigg{)}\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}+1}&,\dots,&k_{t_{j+1}-1}\\ a_{q_{j}+1}-a_{q_{j}}&,\dots,&a_{t_{j+1}-1}-a_{q_{j}}\end{array}\bigg{)}\Bigg{\}}$		(69)
	$\displaystyle\displaystyle\times\sum_{0<l_{1}<\dots<l_{h}<L}\Psi\bigg{(}\begin{matrix}k_{q_{1}}\\ a_{q_{1}}\end{matrix}\>;l_{1}N\tau\bigg{)}\cdots\Psi\bigg{(}\begin{matrix}k_{q_{h}}\\ a_{q_{h}}\end{matrix}\>;l_{h}N\tau\bigg{)}.$		(70)

It follows from 3.7 since the constant term in the right hand side vanish if $\displaystyle k_{q_{i}}=1$ for some $\displaystyle i=1,\dots,h$ by 2.11.

∎

4 Formal iterated integrals corresponding to MZV of level $\displaystyle N$

In this section, we consider the algebra generated by formal iterated integrals, which is introduced by Goncharov [Gon05] and calculate the coproduct for the formal iterated integrals corresponding to MZV of level $\displaystyle N$ .

4.1 Hopf algebra of formal iterated integrals

Goncharov([Gon05]) considered the formal version of iterated integrals

\displaystyle\displaystyle\int_{a_{0}}^{a_{m+1}}\frac{dt}{t-a_{1}}\cdots\frac{dt}{t-a_{m}}\quad(a_{0},\dots,a_{m+1}\in\mathbb{C}),

(71)

and proved the algebra generated by formal iterated integrals has Hopf algebra structure.

Definition 4.1 (Goncharov [Gon05]).

Let $\displaystyle S$ be a set. We define a commutative graded $\displaystyle\mathbb{Q}$ -algebra $\displaystyle\mathcal{I}_{\bullet}(S)$ by

\displaystyle\mathcal{I}_{\bullet}(S)\coloneq\mathbb{Q}\left[\mathrm{I}(a_{0};a_{1},\dots,a_{m};a_{m+1})\mid m\geq 0,a_{i}\in S\right]/_{(\mathrm{i})\sim(\mathrm{iv})},

where $\displaystyle\mathrm{deg}(\mathrm{I}(a_{0};a_{1},\dots,a_{m};a_{m+1}))=m$ and the relations $\displaystyle(\mathrm{i})\sim(\mathrm{iv})$ are the following:

(i)

$\displaystyle\mathrm{I}(a;b)=1,\quad(a,b\in S).$

(ii)

(Shuffle product formula) For $\displaystyle a,b,a_{1},\dots,a_{n+m}\in S$ , it holds

\displaystyle\mathrm{I}(a;a_{1},\dots,a_{n};b)\mathrm{I}(a;a_{n+1},\dots,a_{n+m};b)=\sum_{\sigma\in Sh^{(n+m)}_{n}}\mathrm{I}(a;a_{\sigma^{-1}(1)},\dots,a_{\sigma^{-1}(n+m)};b),

where

\displaystyle Sh^{(n+m)}_{n}\coloneq\left\{\sigma\in\mathfrak{S}_{n+m}\mid\sigma(1)<\cdots<\sigma(n),\sigma(n+1)<\cdots<\sigma(n+m)\right\}.

(iii)

(Path composition formula) For $\displaystyle x,a_{0},\dots,a_{m+1}\in S$ , it holds

\displaystyle\mathrm{I}(a_{0};a_{1},\dots,a_{m};a_{m+1})=\sum_{i=0}^{m}\mathrm{I}(a_{0};a_{1},\dots,a_{i};x)\mathrm{I}(x;a_{i+1},\dots,a_{m};a_{m+1}).

(iv)

$\displaystyle\mathrm{I}(a;a_{1},\dots,a_{m};a)=0,\quad(a,a_{1},\dots,a_{m}\in S,m\geq 1).$

Theorem 4.2 (Goncharov [Gon05]).

$\displaystyle\mathcal{I}_{\bullet}(S)$ is a graded Hopf algebra with the coproduct $\displaystyle\Delta:\mathcal{I}_{\bullet}(S)\rightarrow\mathcal{I}_{\bullet}(S)\otimes_{\mathbb{Q}}\mathcal{I}_{\bullet}(S)$ defined by

		$\displaystyle\displaystyle\Delta(\mathrm{I}(a_{0};a_{1},\dots,a_{m};a_{m+1}))$		(72)
	$\displaystyle\displaystyle\coloneq$	$\displaystyle\displaystyle\sum_{0=i_{0}<i_{1}<\cdots<i_{k}<i_{k+1}=m+1}\prod_{p=0}^{k}\mathrm{I}(a_{i_{p}};a_{i_{p+1}},\dots,a_{i_{p+1}-1};a_{i_{p+1}})\otimes\mathrm{I}(a_{0};a_{i_{1}},\dots,a_{i_{k}};a_{m+1}).$		(73)

Remark 4.3.

The counit $\displaystyle\varepsilon_{G}:\mathcal{I}_{\bullet}(S)\to\mathbb{Q}$ is defined by

\displaystyle\varepsilon_{G}(u)=\begin{cases}1&\mathrm{deg}(u)=0\\ 0&\mathrm{deg}(u)>0\end{cases}.

The antipode is determined inductively on the degree.

4.2 Formal iterated integrals corresponding MLV and MZV of level $\displaystyle N$

Hereinafter, we consider the case $\displaystyle S=\{\eta,\eta^{2},\dots,\eta^{N},0\}$ and denote $\displaystyle\mathcal{I}_{\bullet}=\mathcal{I}_{\bullet}(S)$ . Let $\displaystyle\mathfrak{a}$ be an ideal of $\displaystyle\mathcal{I}$ generated by $\displaystyle\{\mathrm{I}(0;0;a)\mid a\in S\setminus\{0\}\}$ , and let $\displaystyle\mathcal{I}^{0}$ be the quotient $\displaystyle\mathcal{I}^{0}=\mathcal{I}/\mathfrak{a}$ .

Proposition 4.4 (Bachmann–Tasaka [BT]).

$\displaystyle(\mathcal{I}^{0},\Delta)$ is a Hopf algebra.

Proof.

For any $\displaystyle a\in S\setminus\{0\}$ , $\displaystyle\mathrm{I}(0;0;a)$ is a primitive element since

\displaystyle\Delta(\mathrm{I}(0;0;a))=\mathrm{I}(0;0;a)\otimes 1+1\otimes\mathrm{I}(0;0;a).

Therefore, $\displaystyle\mathfrak{a}=\{\mathrm{I}(0;0;a)\mid a\in S\setminus\{0\}\}\mathcal{I}$ is a Hopf ideal and the quotient is a Hopf algebra. ∎

We give some important properties of formal iterated integrals.

Lemma 4.5 (Goncharov [Gon05]).

For any $\displaystyle a_{0},\dots,a_{m+1}\in S$ , we have

\displaystyle\mathrm{I}(a_{0};a_{1},\dots,a_{m};a_{m+1})=(-1)^{m}\mathrm{I}(a_{m+1};a_{m},\dots,a_{1};a_{0}).

For $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a,a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , denote

	$\displaystyle\displaystyle\mathrm{I}_{a}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\mathrm{I}(0;\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a}),$		(74)
	$\displaystyle\displaystyle\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\mathrm{I}_{0}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}.$		(75)

2.1 implies $\displaystyle\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$ corresponds to MLV of shuffle type

\displaystyle(-1)^{r}L_{{\,\hbox{\sevency X}\,}}\binom{n_{1},\dots,n_{r}}{-a_{1},\dots,-a_{r}}=\int_{0}^{1}\frac{dt}{t-\eta^{a_{1}}}\left(\frac{dt}{t}\right)^{n_{1}-1}\cdots\frac{dt}{t-\eta^{a_{r}}}\left(\frac{dt}{t}\right)^{n_{r}-1}.

Lemma 4.6.

For any $\displaystyle n,n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a,a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we have

	$\displaystyle\displaystyle\mathrm{I}(0;\{0\}^{n},\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a})$		(76)
	$\displaystyle\displaystyle=(-1)^{n}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{r}=n+n_{1}+\cdots+n_{r}\\ k_{i}\geq n_{i}\end{subarray}}\prod_{p=1}^{r}\binom{k_{p}-1}{n_{p}-1}\mathrm{I}_{a}\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}.$		(77)

Proof.

We prove by induction on $\displaystyle n\geq 1$ . When $\displaystyle n=1$ , using (ii) and (v), we have

$\displaystyle\displaystyle 0$	$\displaystyle\displaystyle=\mathrm{I}(0;0;\eta^{a})\mathrm{I}(0;\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a})$	(78)
	$\displaystyle\displaystyle=\mathrm{I}(0;0,\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a})$	(79)
	$\displaystyle\displaystyle+\sum_{q=1}^{r}n_{q}\mathrm{I}(0;\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{q}},\{0\}^{n_{q}},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a}).$	(80)

When $\displaystyle n>1$ , using (ii) and (v), we have

$\displaystyle\displaystyle 0$	$\displaystyle\displaystyle=\mathrm{I}(0;0;\eta^{a})\mathrm{I}(0;\{0\}^{n-1},\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a})$	(81)
	$\displaystyle\displaystyle=n\mathrm{I}(0;\{0\}^{n},\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a})$	(82)
	$\displaystyle\displaystyle+\sum_{q=1}^{r}n_{q}\mathrm{I}(0;\{0\}^{n-1},\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{q}},\{0\}^{n_{q}},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a}).$	(83)

By the inductive hypothesis, we have

	$\displaystyle\displaystyle\mathrm{I}(0;\{0\}^{n},\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1};\eta^{a})$		(84)
	$\displaystyle\displaystyle=(-1)^{n}\sum_{q=1}^{r}\frac{n_{q}}{n}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{r}=n+n_{1}+\cdots+n_{r}\\ k_{i}\geq n_{i}\end{subarray}}\prod_{\begin{subarray}{c}p=1\\ p\neq q\end{subarray}}^{r}\binom{k_{p}-1}{n_{p}-1}\binom{k_{q}-1}{n_{q}}\mathrm{I}_{a}\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}$		(85)
	$\displaystyle\displaystyle=(-1)^{n}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{r}=n+n_{1}+\cdots+n_{r}\\ k_{i}\geq n_{i}\end{subarray}}\prod_{p=1}^{r}\binom{k_{p}-1}{n_{p}-1}\mathrm{I}_{a}\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}.$		(86)

∎

Using these properties, we know that any elements of $\displaystyle\mathcal{I}^{0}$ can be written as a polynomial of $\displaystyle\mathrm{I}_{a}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$

Proposition 4.7.

It holds

\displaystyle\mathcal{I}^{0}=\mathbb{Q}\left[\mathrm{I}_{a}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\mathrel{}\middle|\mathrel{}r\geq 0,n_{i}\geq 1,a,a_{i}\in\mathbb{Z}/N\mathbb{Z}\right].

Proof.

Any elements $\displaystyle\mathrm{I}(a_{0};a_{1},\dots,a_{m};a_{m+1})\in\mathcal{I}^{0}$ can be expressed as a sum of products for some $\displaystyle\mathrm{I}(0;\dots;a)$ $\displaystyle(a\in S)$ since (iii) and 4.5. Using (ii) and 4.6, it can be written as a sum of products for some $\displaystyle\mathrm{I}_{a}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$ . Now, the products can be expressed as a $\displaystyle\mathbb{Q}$ -linear combination of them by shuffle product. ∎

Let $\displaystyle\mathcal{I}^{1}$ be a subalgebra of $\displaystyle\mathcal{I}^{0}$ defined by

\displaystyle\mathcal{I}^{1}\coloneq\left\langle\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\in\mathcal{I}^{0}\mathrel{}\middle|\mathrel{}r\geq 0,n_{i}>0,a_{i}\in\mathbb{Z}/N\mathbb{Z}\right\rangle_{\mathbb{Q}}

and let $\displaystyle\mu:\mathcal{I}^{0}\twoheadrightarrow\mathcal{I}^{1}$ be a surjective algebra homomorphism defined by

\displaystyle\mu\left(\mathrm{I}_{a}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\right)=\mathrm{I}\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}c}n_{1}&,\dots,&n_{r}\\ a_{1}-a&,\dots,&a_{r}-a\end{array}\bigg{)}.

Remark 4.8.

$\displaystyle\mathcal{I}^{0}$ has no information about homothety property. $\displaystyle\mathcal{I}^{1}$ is an algebra spanned by formal iterated integrals that satisfy such homothety. $\displaystyle\mu$ is an operator corresponding to variable changing of integrals.

Let $\displaystyle\Delta_{\mu}:\mathcal{I}^{1}\to\mathcal{I}^{1}\otimes\mathcal{I}^{1}$ be an algebra homomorphism defined by

\displaystyle\Delta_{\mu}\coloneq(\mu\otimes\mu)\circ\Delta|_{\mathcal{I}^{1}}.

Proposition 4.9.

$\displaystyle(\mathcal{I}^{1},\Delta_{\mu},\varepsilon_{G})$ is a Hopf algebra.

Proof.

It is clear that $\displaystyle\varepsilon_{G}$ and $\displaystyle\Delta_{\mu}$ satisfy the counitary property. Let us check the coassociativity. If it holds $\displaystyle\Delta_{\mu}\circ\mu=\Delta_{\mu}$ on $\displaystyle\mathcal{I}^{0}$ , we have

\displaystyle(\Delta_{\mu}\circ\mathrm{id})\circ\Delta_{\mu}(u)=(\Delta_{\mu}\otimes\mathrm{id})\sum\mu(u_{1})\otimes\mu(u_{2})=\sum\mu(u_{1})\otimes\mu(u_{2})\otimes\mu(u_{3})=(\mathrm{id}\otimes\Delta_{\mu})\circ\Delta_{\mu}(u).

So it suffices to show that

\displaystyle\Delta_{\mu}\circ\mu\left(\mathrm{I}_{a}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\right)=\Delta_{\mu}\left(\mathrm{I}_{a}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\right)

for any $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a,a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ . This statement follows from the calculation of $\displaystyle\Delta$ in the next subsection. The antipode is determined inductively since the product and coproduct preserve the degree. ∎

4.3 Computing Goncharov coproduct

In this subsection, we give the explicit formula for the Goncharov coproduct of $\displaystyle\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$ , which is corresponding to MLVs of shuffle type. Then, we consider formal iterated integrals corresponding MZVs of level $\displaystyle N$ and give its coproduct.

For positive integers $\displaystyle 0=i_{0}<i_{1}<\cdots<i_{k}<i_{k+1}=n+1$ $\displaystyle(0\leq k\leq n)$ and $\displaystyle\varepsilon_{1},\dots,\varepsilon_{n}\in S$ , we define $\displaystyle\varphi_{i_{1},\dots,i_{k}}(\varepsilon_{1},\dots,\varepsilon_{n})\in\mathcal{I}^{0}\otimes\mathcal{I}^{0}$ by

\displaystyle\varphi_{i_{1},\dots,i_{k}}(\varepsilon_{1},\dots,\varepsilon_{n})\coloneq\prod_{p=0}^{k}\mathrm{I}(\varepsilon_{i_{p}};\varepsilon_{i_{p+1}},\dots,\varepsilon_{i_{p+1}-1};\varepsilon_{i_{p+1}})\otimes\mathrm{I}(0;\varepsilon_{i_{1}},\dots,\varepsilon_{i_{k}};1),

where $\displaystyle\varepsilon_{0}=0$ , $\displaystyle\varepsilon_{n+1}=1$ , and denote

\displaystyle\varphi_{i_{1},\dots,i_{k}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\varphi_{i_{1},\dots,i_{k}}(\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{r}},\{0\}^{n_{r}-1}).

Further, we put

\displaystyle\iota_{n_{1},\dots,n_{r}}(w_{1}\cdots w_{r})\coloneq\{n_{1}+\cdots+n_{t_{1}-1}+1,\dots,n_{1}+\cdots+n_{t_{h}-1}+1\}

for $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle w_{1}\cdots w_{r}=\mathrm{x}^{t_{1}-1}\mathrm{y}\mathrm{x}^{t_{2}-t_{1}-1}\mathrm{y}\cdots\mathrm{x}^{t_{h}-t_{h-1}-1}\mathrm{y}\mathrm{x}^{r-t_{h}}\in\{\mathrm{x},\mathrm{y}\}^{*}$ $\displaystyle(0<t_{1}<\cdots<t_{n}<r+1)$ .

Definition 4.10.

For any word $\displaystyle w_{1}\cdots w_{r}=\mathrm{x}^{t_{1}-1}\mathrm{y}\mathrm{x}^{t_{2}-t_{1}-1}\mathrm{y}\cdots\mathrm{x}^{t_{h}-t_{h-1}-1}\mathrm{y}\mathrm{x}^{r-t_{h}}\in\{\mathrm{x},\mathrm{y}\}^{*}$ , $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we define

\displaystyle\displaystyle\Phi_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\sum_{k=h}^{n}\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{k}\leq n\\ \{i_{1},\dots,i_{k}\}\cap\{1,n_{1}+1,\dots,n_{1}+\cdots+n_{r-1}+1\}\\ =\iota_{n_{1},\dots,n_{r}}(w_{1}\cdots w_{r})\end{subarray}}\varphi_{i_{1},\dots,i_{k}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}},

(89)

where $\displaystyle n=n_{1}+\cdots+n_{r}$ .

Remark 4.11.

Roughly speaking, $\displaystyle\Phi_{\bm{w}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$ is a portion of $\displaystyle\Delta\left(\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\right)$ such that it doesn’t mark the point of $\displaystyle\eta^{a_{i}}$ corresponding to $\displaystyle\mathrm{x}$ and does the point of $\displaystyle\eta^{a_{i}}$ corresponding to $\displaystyle\mathrm{y}$ .

Example 4.12.

\displaystyle\Phi_{\mathrm{x}\mathrm{y}\mathrm{x}}\binom{n_{1},n_{2},n_{3}}{a_{1},a_{2},a_{3}}=\sum_{k=1}^{n}\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{k}\leq n\\ {}^{\exists}l,i_{l}=n_{1}+1\\ {}^{\forall}l,i_{l}\neq 1,n_{1}+n_{2}+1\end{subarray}}\varphi_{i_{1},\dots,i_{k}}\binom{n_{1},n_{2},n_{3}}{a_{1},a_{2},a_{3}}.

In the right-hand side, the term for $\displaystyle k=1$ is only $\displaystyle\varphi_{n_{1}+1}$ . In general, the term for $\displaystyle k=p$ is sum of $\displaystyle\varphi$ such that it marks $\displaystyle p-1$ white points.

Lemma 4.13.

For any $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle\Delta\left(\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\right)=\sum_{w_{1},\dots,w_{r}\in\{\mathrm{x},\mathrm{y}\}}\Phi_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}.

Proof.

By definition of the coproduct $\displaystyle\Delta$ , we have

\displaystyle\Delta\left(\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\right)=\sum_{k=0}^{n}\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}\varphi_{i_{1},\dots,i_{k}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}.

Meanwhile, it holds

$\displaystyle\displaystyle\sum_{w_{1},\dots,w_{r}\in\{\mathrm{x},\mathrm{y}\}}\Phi_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$	$\displaystyle\displaystyle=\sum_{h=0}^{r}\sum_{\begin{subarray}{c}w_{1},\dots,w_{r}\in\{\mathrm{x},\mathrm{y}\}\\ \deg_{\mathrm{y}}(w_{1}\cdots w_{r})=h\end{subarray}}\Phi_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$	(92)
	$\displaystyle\displaystyle=\sum_{h=0}^{r}\sum_{k=h}^{r}\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{k}\leq n\\ \#\{i_{1},\dots,i_{k}\}\cap P_{n_{1},\dots,n_{r}}=h\end{subarray}}\varphi_{i_{1},\dots,i_{k}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$	(93)
	$\displaystyle\displaystyle=\sum_{k=0}^{n}\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}\varphi_{i_{1},\dots,i_{k}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}},$	(94)

where $\displaystyle P_{n_{1},\dots,n_{r}}=\{1,n_{1}+1,n_{1}+n_{2}+1,\dots,n_{1}+\cdots+n_{r-1}+1\}$ . ∎

The following lemma gives us the explicit formula for the Goncharov coproduct of $\displaystyle\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$ .

Lemma 4.14.

For any $\displaystyle n_{1},\dots,n_{r}\geq 1$ , $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ and any word $\displaystyle w_{1}\cdots w_{r}=\mathrm{x}^{t_{1}-1}\mathrm{y}\mathrm{x}^{t_{2}-t_{1}-1}\mathrm{y}$ $\displaystyle\cdots\mathrm{x}^{t_{h}-t_{h-1}-1}\mathrm{y}\mathrm{x}^{r-t_{h}}\in\{\mathrm{x},\mathrm{y}\}^{*}$ , we have

	$\displaystyle\displaystyle\Phi_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}=\left(\mathrm{I}_{a_{t_{1}}}\binom{n_{1},\dots,n_{t_{1}-1}}{a_{1},\dots,a_{t_{1}-1}}\otimes 1\right)$		(95)
	$\displaystyle\displaystyle\times\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}t_{1}\leq q_{1}\leq t_{2}-1\\[-5.0pt] \vdots\\ t_{h}\leq q_{h}\leq r\end{subarray}}$}}\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}k_{t_{j}}+\cdots+k_{t_{j+1}-1}\\ =n_{t_{j}}+\cdots+n_{t_{j+1}-1}\\ (1\leq j\leq h),k_{i}\geq 1\end{subarray}}$}}\prod_{j=1}^{h}\Bigg{\{}(-1)^{l_{j}}\Bigg{(}\prod_{\begin{subarray}{c}p=t_{j}\\ p\neq q_{j}\end{subarray}}^{t_{j+1}-1}\binom{k_{p}-1}{n_{q}-1}$
	$\displaystyle\displaystyle\times\mathrm{I}_{a_{t_{j}}}\binom{k_{q_{j}-1},\dots,k_{t_{j}}}{a_{q_{j}},\dots,a_{t_{j}+1}}\mathrm{I}_{a_{t_{j+1}}}\binom{k_{q_{j}+1},\dots,k_{t_{j+1}-1}}{a_{q_{j}+1},\dots,a_{t_{j+1}-1}}\Bigg{\}}\otimes\mathrm{I}\binom{k_{q_{1}},\dots,k_{q_{h}}}{a_{t_{1}},\dots,a_{t_{h}}},$		(96)

where $\displaystyle l_{j}=n_{t_{j}}+\cdots+n_{t_{j+1}-1}+n_{q_{j}}+k_{q_{j}+1}+\cdots+k_{q_{j+1}-1}$ , $\displaystyle t_{h+1}=r+1$ and $\displaystyle a_{r+1}=0$ .

Proof.

The left-hand side is a sum of all terms of $\displaystyle\Delta\left(\mathrm{I}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\right)$ such that its edge is $\displaystyle 0,\eta^{a_{t_{1}}},\dots,\eta^{a_{t_{h}}}$ . Using path composition formula

	$\displaystyle\displaystyle\mathrm{I}(\eta^{a_{t_{j}}};\{0\}^{n_{t_{j}-1}},\eta^{a_{t_{j}+1}},\dots,\{0\}^{n_{t_{j+1}-1}-1};\eta^{a_{t_{j+1}}})$		(97)
	$\displaystyle\displaystyle=\sum_{t_{j}\leq q_{j}\leq t_{j+1}-1}\sum_{0\leq l_{q_{j}}\leq n_{q_{j}}-1}\mathrm{I}(\eta^{a_{t_{j}}};\{0\}^{n_{t_{j}}-1},\eta^{a_{t_{j}+1}},\dots,\eta^{a_{q_{j}}},\{0\}^{l_{q_{j}}};0)$		(98)
	$\displaystyle\displaystyle\times\mathrm{I}(0;\{0\}^{n_{q_{j}}-l_{q_{j}}-1},\eta^{a_{q_{j}+1}},\dots,\{0\}^{n_{t_{j+1}-1}-1};\eta^{a_{t_{j+1}}}),$		(99)

we have

	$\displaystyle\displaystyle\Phi_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$		(100)
	$\displaystyle\displaystyle=(\mathrm{I}(0;\eta^{a_{1}},\{0\}^{n_{1}-1},\dots,\eta^{a_{t_{1}-1}},\{0\}^{n_{t_{1}-1}};\eta^{a_{t_{1}}})\otimes 1)$		(101)
	$\displaystyle\displaystyle\times\bigg{(}\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}t_{1}\leq q_{1}\leq t_{2}-1\\[-5.0pt] \vdots\\ t_{h}\leq q_{h}\leq r\end{subarray}}\sum_{\begin{subarray}{c}0<l_{q_{1}}+k_{q_{1}}\leq n_{q_{1}}\\[-7.0pt] \vdots\\[1.0pt] 0<l_{q_{h}}+k_{q_{h}}\leq n_{q_{h}}\\ l_{i}\geq 0,k_{i}\geq 1\end{subarray}}$}}\prod_{j=1}^{h}\mathrm{I}(\eta^{a_{t_{j}}};\{0\}^{n_{t_{j}}-1},\eta^{a_{t_{j}+1}},\{0\}^{n_{t_{j}+1}-1},\dots,\eta^{a_{q_{j}}},\{0\}^{l_{q_{j}}};0$
	$\displaystyle\displaystyle\times(\mathrm{I}(0;0))^{k_{q_{j}}-2}\mathrm{I}(0;\{0\}^{n_{q_{j}}-l_{q_{j}}-k_{q_{j}}},\eta^{a_{q_{j}+1}},\{0\}^{n_{q_{j}+1}-1},\dots,\eta^{a_{t_{j+1}-1}},\{0\}^{n_{t_{j+1}-1}-1};\eta^{a_{t_{j+1}}})$		(102)
	$\displaystyle\displaystyle\otimes\mathrm{I}(0;\eta^{a_{t_{1}}},\{0\}^{k_{q_{1}}-1},\dots,\eta^{a_{t_{h}}},\{0\}^{k_{q_{h}}-1};1)\bigg{)}.$		(103)

Here, when $\displaystyle k_{q_{j}}=1$ , we understand $\displaystyle(I(0;0))^{k_{q_{j}}-2}=1$ . By 4.5,4.6, we have

	$\displaystyle\displaystyle\mathrm{I}(\eta^{a_{t_{j}}};\{0\}^{n_{t_{j}}-1},\eta^{a_{t_{j}+1}},\{0\}^{n_{t_{j}+1}-1},\dots,\eta^{a_{q_{j}}},\{0\}^{l_{q_{j}}};0)$		(104)
	$\displaystyle\displaystyle=(-1)^{n_{t_{j}}+\cdots+n_{q_{j}-1}+l_{q_{j}}}\mathrm{I}(0;\{0\}^{l_{q_{j}}},\eta^{a_{q_{j}}},\dots,\{0\}^{n_{t_{j}+1}-1},\eta^{a_{t_{j}+1}},\{0\}^{n_{t_{j}}-1};\eta^{a_{t_{j}}})$		(105)
	$\displaystyle\displaystyle=(-1)^{n_{t_{j}}+\cdots+n_{q_{j}-1}}\sum_{\begin{subarray}{c}k_{t_{j}}+\cdots+k_{q_{j}-1}\\ =n_{t_{j}}+\cdots+n_{q_{j}-1}+l_{q_{j}}\end{subarray}}\prod_{p=t_{j}}^{q_{j}-1}\binom{k_{p}-1}{n_{p}-1}\mathrm{I}_{a_{t_{j}}}\binom{k_{q_{j}-1},\dots,k_{t_{j}}}{a_{q_{j}},\dots,a_{t_{j+1}}},$		(106)

and

	$\displaystyle\displaystyle\mathrm{I}(0;\{0\}^{n_{q_{j}}-l_{q_{j}}-k_{q_{j}}},\eta^{a_{q_{j}+1}},\{0\}^{n_{q_{j}+1}-1},\dots,\eta^{a_{t_{j+1}-1}},\{0\}^{n_{t_{j+1}-1}-1};\eta^{a_{t_{j+1}}})$		(107)
	$\displaystyle\displaystyle=(-1)^{n_{q_{j}}-l_{q_{j}}-k_{q_{j}}}\sum_{\begin{subarray}{c}k_{q_{j}+1}+\cdots+k_{t_{j+1}-1}\\ =n_{q_{j}}-l_{q_{j}}-k_{q_{j}}+n_{q_{j}+1}+\cdots+n_{t_{j+1}-1}\end{subarray}}\prod_{p=q_{j}+1}^{t_{j+1}-1}\binom{k_{p}-1}{n_{p}-1}\mathrm{I}_{a_{t_{j+1}}}\binom{k_{q_{j}+1},\dots,k_{t_{j+1}-1}}{a_{q_{j}+1},\dots,a_{t_{j+1}-1}}.$		(108)

Therefore, we have

	$\displaystyle\displaystyle\Phi_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}=\left(\mathrm{I}_{a_{t_{1}}}\binom{n_{1},\dots,n_{t_{1}-1}}{a_{1},\dots,a_{t_{1}-1}}\otimes 1\right)$		(109)
	$\displaystyle\displaystyle\times\sum_{\begin{subarray}{c}t_{1}\leq q_{1}\leq t_{2}-1\\[-5.0pt] \vdots\\ t_{h}\leq q_{h}\leq r\end{subarray}}\sum_{(k_{q_{1}},l_{q_{1}},\dots,k_{q_{h}},l_{q_{h}})}\prod_{j=1}^{h}\Bigg{\{}(-1)^{n_{t_{j}}+\cdots+n_{q_{j}}-l_{q_{j}}-k_{q_{j}}}\Bigg{(}\prod_{\begin{subarray}{c}p=t_{j}\\ p\neq q_{j}\end{subarray}}^{t_{j+1}-1}\binom{k_{p}-1}{n_{q}-1}\Bigg{)}$		(110)
	$\displaystyle\displaystyle\times\mathrm{I}_{a_{t_{j}}}\binom{k_{q_{j}-1},\dots,k_{t_{j}}}{a_{q_{j}},\dots,a_{t_{j}+1}}\mathrm{I}_{a_{t_{j+1}}}\binom{k_{q_{j}+1},\dots,k_{t_{j+1}-1}}{a_{q_{j}+1},\dots,a_{t_{j+1}-1}}\Bigg{\}}\otimes\mathrm{I}\binom{k_{q_{1}},\dots,k_{q_{h}}}{a_{t_{1}},\dots,a_{t_{h}}}.$		(111)

Here, the second sum runs over

\displaystyle\displaystyle\left\{(k_{q_{1}},l_{q_{1}},\dots,k_{q_{h}},l_{q_{h}})\mathrel{}\middle|\mathrel{}\begin{array}[]{l}l_{i}\geq 0,k_{i}\geq 1,0<l_{q_{j}}+k_{q_{j}}\leq n_{q_{j}},\\ k_{t_{j}}+\cdots+k_{q_{j}-1}=n_{t_{j}}+\cdots+n_{q_{j}-1}+l_{q_{j}},\\ k_{q_{j}+1}+\cdots+k_{t_{j+1}-1}=n_{q_{j}+1}+\cdots+n_{t_{j+1}-1}+n_{q_{j}}-l_{q_{j}}-k_{q_{j}}.\end{array}\right\}.

(115)

This is exactly the right-hand side of the claim. ∎

Let $\displaystyle\widetilde{\mathrm{I}}$ be a formal iterated integral corresponding to MZV of level $\displaystyle N$ defined by

\displaystyle\widetilde{\mathrm{I}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq\frac{(-1)^{r}}{N^{r}}\sum_{b_{1},\dots,b_{r}\in\mathbb{Z}/N\mathbb{Z}}\eta^{\rho(\bm{a})\cdot\bm{b}}\mathrm{I}\binom{n_{1},\dots,n_{r}}{b_{1},\dots,b_{r}}\in\mathcal{I}^{1}\otimes_{\mathbb{Q}}\mathbb{Q}(\eta)

for $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , where $\displaystyle\rho(\bm{a})=(a_{1},a_{2}-a_{1},\dots,a_{r}-a_{r-1})$ . 2.4 implies that $\displaystyle\widetilde{\mathrm{I}}$ corresponds to $\displaystyle\zeta$ . Let

\displaystyle\widetilde{\Phi}_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\coloneq(\mu\otimes\mu)\left(\frac{(-1)^{r}}{N^{r}}\sum_{b_{1},\dots,b_{r}\in\mathbb{Z}/N\mathbb{Z}}\eta^{\rho(\bm{a})\cdot\bm{b}}\Phi_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{b_{1},\dots,b_{r}}\right).

By 4.13, we have

\displaystyle\Delta_{\mu}\left(\widetilde{\mathrm{I}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\right)=\sum_{w_{1},\dots,w_{r}\in\{\mathrm{x},\mathrm{y}\}}\widetilde{\Phi}_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}.

The following proposition gives us the explicit formula for the Goncharov coproduct of formal iterated integrals corresponding to MZVs of level $\displaystyle N$ .

Proposition 4.15.

For any $\displaystyle n_{1},\dots,n_{r}\geq 1$ , $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ and $\displaystyle w_{1},\dots,w_{r}\in\{\mathrm{x},\mathrm{y}\}$ , we have

	$\displaystyle\displaystyle\widetilde{\Phi}_{w_{1}\cdots w_{r}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}=\left(\widetilde{\mathrm{I}}\binom{n_{1},\dots,n_{t_{1}-1}}{a_{1},\dots,a_{t_{1}-1}}\otimes 1\right)$		(116)
	$\displaystyle\displaystyle\times\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}t_{1}\leq q_{1}\leq t_{2}-1\\[-5.0pt] \vdots\\ t_{h}\leq q_{h}\leq r\end{subarray}}$}}\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}k_{t_{j}}+\cdots+k_{t_{j+1}-1}\\ =n_{t_{j}}+\cdots+n_{t_{j+1}-1}\\ (1\leq j\leq h),k_{i}\geq 1\end{subarray}}$}}\prod_{j=1}^{h}\Bigg{\{}(-1)^{l_{j}}\Bigg{(}\prod_{\begin{subarray}{c}p=t_{j}\\ p\neq q_{j}\end{subarray}}^{t_{j+1}-1}\binom{k_{p}-1}{n_{p}-1}$
	$\displaystyle\displaystyle\times\widetilde{\mathrm{I}}\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}-1}&,\dots,&k_{t_{j}}\\ a_{q_{j}}-a_{q_{j}-1}&,\dots,&a_{q_{j}}-a_{t_{j}}\end{array}\bigg{)}\widetilde{\mathrm{I}}\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}+1}&,\dots,&k_{t_{j+1}-1}\\ a_{q_{j}+1}-a_{q_{j}}&,\dots,&a_{t_{j+1}-1}-a_{q_{j}}\end{array}\bigg{)}\Bigg{\}}\otimes\widetilde{\mathrm{I}}\binom{k_{q_{1}},\dots,k_{q_{h}}}{a_{q_{1}},\dots,a_{q_{h}}},$		(121)

where $\displaystyle l_{j}=n_{t_{j}}+\cdots+n_{t_{j+1}-1}+n_{q_{j}}+k_{q_{j}+1}+\cdots+k_{q_{j+1}-1}$ , $\displaystyle t_{h+1}=r+1$ and $\displaystyle a_{r+1}=0$ .

Proof.

By 4.14, we have

	$\displaystyle\displaystyle(\text{L.H.S})=\frac{(-1)^{r}}{N^{r}}\sum_{\begin{subarray}{c}t_{1}\leq q_{1}\leq t_{2}-1\\[-4.0pt] \vdots\\ t_{h}\leq q_{h}\leq r\end{subarray}}\sum_{\begin{subarray}{c}k_{t_{1}}+\cdots+k_{t_{2}-1}=n_{t_{1}}+\cdots+n_{t_{2}-1}\\[-7.0pt] \vdots\\ k_{t_{h}}+\cdots+k_{r}=n_{t_{h}}+\cdots+n_{r}\\ k_{i}\geq 1\end{subarray}}(-1)^{\sum_{j=1}^{h}l_{j}}\left(\prod_{\begin{subarray}{c}p=t_{1}\\ p\neq q_{1},\dots,q_{h}\end{subarray}}^{r}\binom{k_{p}-1}{n_{p}-1}\right)$		(122)
	$\displaystyle\displaystyle\times\sum_{b_{1},\dots,b_{r}\in\mathbb{Z}/N\mathbb{Z}}\eta^{\rho(\bm{a})\cdot\bm{b}}\left(\mathrm{I}\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}n_{1}&,\dots,&n_{t_{1}-1}\\ b_{1}-b_{t_{1}}&,\dots,&b_{t_{1}-1}-b_{t_{1}}\end{array}\bigg{)}\otimes 1\right)$		(125)
	$\displaystyle\displaystyle\times\prod_{j=1}^{h}\left(\mathrm{I}\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}-1}&,\dots,&k_{t_{j}}\\ b_{q_{j}}-b_{t_{j}}&,\dots,&b_{t_{j}+1}-b_{t_{j}}\end{array}\bigg{)}\mathrm{I}\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}+1}&,\dots,&k_{t_{j+1}-1}\\ b_{q_{j}+1}-b_{t_{j+1}}&,\dots,&b_{t_{j+1}-1}-b_{t_{j+1}}\end{array}\bigg{)}\right)\otimes\mathrm{I}\binom{k_{q_{1}},\dots,k_{q_{h}}}{b_{t_{1}},\dots,b_{t_{h}}}.$		(130)

Replacing a running index as

\displaystyle b_{k}\mapsto\begin{cases}b_{k}+b_{q_{1}}&1\leq k<t_{1},\\ b_{q_{j}}&k=t_{j}\;(j\in\{1,\dots,h\}),\\ b_{k-1}+b_{q_{j}}&t_{j}<k\leq q_{j}\;(j\in\{1,\dots,h\}),\\ b_{k}+b_{q_{j+1}}&q_{j}<k<t_{j+1}\;(j\in\{1,\dots,h\}).\end{cases},

we obtain the claim. ∎

We define a $\displaystyle\mathbb{Q}$ -algebra generated by formal MZVs $\displaystyle\widetilde{\mathcal{I}^{1}}$ by

\displaystyle\widetilde{\mathcal{I}^{1}}\coloneq\left\langle\widetilde{\mathrm{I}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\mathrel{}\middle|\mathrel{}r\geq 0,n_{1},\dots,n_{r}\geq 1,a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}\right\rangle_{\mathbb{Q}}.

$\displaystyle\widetilde{\mathcal{I}^{1}}$ is isomorphic to $\displaystyle(\mathfrak{H}^{1},\widetilde{{\,\hbox{\sevency X}\,}})$ as a $\displaystyle\mathbb{Q}$ -algebra by sending $\displaystyle\widetilde{\mathrm{I}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}$ to $\displaystyle z_{n_{1},a_{1}}\cdots z_{n_{r},a_{r}}$ . In other words, we can equip $\displaystyle(\mathfrak{H}^{1},\widetilde{{\,\hbox{\sevency X}\,}})$ with a Hopf algebra structure by the identification.

1.1.

For any $\displaystyle w\in\mathfrak{H}^{2}$ , we have

\displaystyle G(w;\tau)=(\zeta\star g)(w;q)\qquad(q=e^{2\pi\sqrt{-1}\tau}).

Proof.

It follows from 3.8 and 4.15 ∎

5 Shuffle regularization for MES of level $\displaystyle N$

In this section, we construct shuffle regularized MES of level $\displaystyle N$ by using the $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ -homomorphism $\displaystyle g^{\widetilde{{\,\hbox{\sevency X}\,}}}$ .

5.1 Shuffle regularization for multiple divisor function

Kitada ([Kitada23]) constructed shuffle regularized multiple divisor function. In this subsection, we introduce it. Let $\displaystyle\mathcal{K}$ denote $\displaystyle\mathbb{C}\llbracket q\rrbracket$ .

Definition 5.1 (Kitada [Kitada23]).

For $\displaystyle n_{1},\dots,n_{r}\geq 1$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we define $\displaystyle H,g\in\mathcal{K}\llbracket x_{1},\dots,x_{r}\rrbracket$ by

	$\displaystyle\displaystyle H\begin{pmatrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\\ x_{1},\dots,x_{r}\end{pmatrix}\coloneq\sum_{0<d_{1}<\cdots<d_{r}}\prod_{j=1}^{r}e^{d_{j}x_{j}}\eta^{a_{j}d_{j}}\left(\frac{q^{d_{j}}}{1-q^{d_{j}}}\right)^{n_{j}},$		(131)
	$\displaystyle\displaystyle g\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}\coloneq\sum_{k_{1},\dots,k_{r}>0}\left(\frac{N}{-2\pi\sqrt{-1}}\right)^{k_{1}+\cdots+k_{r}}\!g\bigg{(}\begin{matrix}k_{1},\dots,k_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}x_{1}^{k_{1}-1}\cdots x_{r}^{k_{r}-1}.$		(132)

Lemma 5.2 ([Kitada23]).

For any $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle g\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}=H\left(\begin{array}[]{@{}c@{}c@{}c@{}l@{}}1&,\dots,&1&,1\\ a_{r}-a_{r-1}&,\dots,&a_{2}-a_{1}&,a_{1}\\ x_{r}-x_{r-1}&,\dots,&x_{2}-x_{1}&,x_{1}\end{array}\right).

Proof.

By 3.6, we have

\displaystyle\displaystyle(\text{L.H.S})=\sum_{\begin{subarray}{c}0<d_{1}<\cdots<d_{r}\\ k_{1},\dots,k_{r}>0\\ c_{1},\dots,c_{r}>0\end{subarray}}\prod_{j=1}^{r}\frac{(c_{j}x_{j})^{k_{j}-1}}{(k_{j}-1)!}\eta^{a_{j}c_{j}}q^{c_{j}d_{j}}=\sum_{\begin{subarray}{c}0<d_{1}<\cdots<d_{r}\\ c_{1},\dots,c_{r}>0\end{subarray}}\prod_{j=1}^{r}e^{c_{j}x_{j}}\eta^{a_{j}c_{j}}q^{c_{j}d_{j}}.

(136)

On the other hand, we have

\displaystyle\displaystyle(\text{R.H.S})

\displaystyle\displaystyle=\sum_{\begin{subarray}{c}0<d_{1}<\cdots<d_{r}\\ c_{1},\dots,c_{r}>0\end{subarray}}\prod_{j=1}^{r}e^{(d_{r-j+1}-d_{r-j})x_{j}}\eta^{(d_{r-j+1}-d_{r-j})a_{j}}q^{c_{j}d_{j}}.

(137)

Replacing

\displaystyle d_{j}=c_{r-j+1}^{\prime}+\cdots+c_{r}^{\prime},\quad c_{j}=d_{r-j+1}^{\prime}-d_{r-j}^{\prime}\quad(j\in\{1,\dots,r\}),

we have

\displaystyle\displaystyle\sum_{\begin{subarray}{c}0<d_{1}<\cdots<d_{r}\\ c_{1},\dots,c_{r}>0\end{subarray}}\prod_{j=1}^{r}e^{(d_{r-j+1}-d_{r-j})x_{j}}\eta^{(d_{r-j+1}-d_{r-j})a_{j}}q^{c_{j}d_{j}}=\sum_{\begin{subarray}{c}c_{1}^{\prime},\dots,c_{r}^{\prime}>0\\ 0<d_{1}^{\prime}<\cdots<d_{r}^{\prime}\end{subarray}}\prod_{j=1}^{r}e^{c_{j}^{\prime}x_{j}}\eta^{a_{j}c_{j}^{\prime}}q^{c_{j}^{\prime}d_{j}^{\prime}}.

(138)

∎

Let $\displaystyle\mathcal{U}$ be a non-commutative polynomial ring defined by

\displaystyle\mathcal{U}\coloneq\mathbb{Q}\left\langle\begin{pmatrix}n\\ a\\ z\end{pmatrix}\mathrel{}\middle|\mathrel{}n\in\mathbb{Z}_{>0},a\in\mathbb{Z}/N\mathbb{Z},z\in X\right\rangle,

where the set $\displaystyle X$ is the following

\displaystyle X\coloneq\left\{\sum_{i>0}m_{i}x_{i}\mathrel{}\middle|\mathrel{}m_{i}\in\mathbb{Z}_{\geq 0},m_{i}=0\text{ for almost all $\displaystyle i$}\right\}.

Denote the concatenation of letters as

\displaystyle\begin{pmatrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\\ z_{1},\dots,z_{r}\end{pmatrix}=\begin{pmatrix}n_{1}\\ a_{1}\\ z_{1}\end{pmatrix}\cdots\begin{pmatrix}n_{r}\\ a_{r}\\ z_{r}\end{pmatrix}.

We define two products $\displaystyle\ast,{\,\hbox{\sevency X}\,}$ on $\displaystyle\mathcal{U}$ inductively by

	$\displaystyle\displaystyle w1=1w=w,\quad w{\,\hbox{\sevency X}\,}1=1{\,\hbox{\sevency X}\,}w=w,$		(139)
	$\displaystyle\displaystyle u_{1}w_{1}u_{2}w_{2}=u_{1}\left(w_{1}u_{2}w_{2}\right)+u_{2}\left(u_{1}w_{1}w_{2}\right)+\begin{pmatrix}n_{1}+n_{2}\\ a_{1}+a_{2}\\ z_{1}+z_{2}\end{pmatrix}(w_{1}w_{2}),$		(140)
	$\displaystyle\displaystyle u_{1}w_{1}{\,\hbox{\sevency X}\,}u_{2}w_{2}=u_{1}\left(w_{1}{\,\hbox{\sevency X}\,}u_{2}w_{2}\right)+u_{2}\left(u_{1}w_{1}{\,\hbox{\sevency X}\,}w_{2}\right),$		(141)

for letters $\displaystyle u_{1}=\Bigg{(}\begin{matrix}n_{1}\\[-3.0pt] a_{1}\\[-3.0pt] z_{1}\end{matrix}\Bigg{)},u_{2}=\Bigg{(}\begin{matrix}n_{2}\\[-3.0pt] a_{2}\\[-3.0pt] z_{2}\end{matrix}\Bigg{)}\in\mathcal{U}$ and words $\displaystyle w,w_{1},w_{2}\in\mathcal{U}$ . Note that $\displaystyle H$ satisfies $\displaystyle\ast$ -product. We define exponential map on $\displaystyle\mathcal{U}$ by

\displaystyle\displaystyle\exp\left(\begin{pmatrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\\ z_{1},\dots,z_{r}\end{pmatrix}\right)\coloneq\sum_{\begin{subarray}{c}1\leq m\leq r\\ i_{1}+\cdots+i_{m}=r\\ i_{1},\dots,i_{m}>0\end{subarray}}\frac{1}{i_{1}!\cdots i_{m}!}\begin{pmatrix}n^{\prime}_{i_{1}},\dots,n^{\prime}_{i_{m}}\\ a^{\prime}_{i_{1}},\dots,a^{\prime}_{i_{m}}\\ z^{\prime}_{i_{1}},\dots,z^{\prime}_{i_{m}}\end{pmatrix},

(142)

where $\displaystyle p^{\prime}_{i_{k}}=p_{i_{1}+\cdots+i_{k-1}+1}+\cdots+p_{i_{1}+\cdots+i_{k}}$ for $\displaystyle p\in\{n,a,z\}$ and $\displaystyle k=1,\dots,m$ .

Proposition 5.3 (Hoffman [Hoff00]).

$\displaystyle\mathcal{U}_{{\,\hbox{\sevency X}\,}}$ and $\displaystyle\mathcal{U}_{\ast}$ are commutative $\displaystyle\mathbb{Q}$ -algebras, and the exponential map gives the isomorphism between them:

\displaystyle\exp:\mathcal{U}_{{\,\hbox{\sevency X}\,}}\xlongrightarrow{\sim}\mathcal{U}_{\ast}.

Definition 5.4 ([Kitada23]).

For $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we define $\displaystyle h\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}\in\mathcal{K}\llbracket x_{1},\dots,x_{r}\rrbracket$ by

	$\displaystyle\displaystyle h\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}$	$\displaystyle\displaystyle\coloneq H\circ\exp\left(\begin{pmatrix}1,\dots,1\\ a_{1},\dots,a_{r}\\ x_{1},\dots,x_{r}\end{pmatrix}\right)$		(143)
		$\displaystyle\displaystyle=\sum_{\begin{subarray}{c}1\leq m\leq r\\ i_{1}+\cdots+i_{m}=r\\ i_{1},\dots,i_{m}>0\end{subarray}}\frac{1}{i_{1}!\cdots i_{m}!}H\begin{pmatrix}n^{\prime}_{i_{1}},\dots,n^{\prime}_{i_{m}}\\ a^{\prime}_{i_{1}},\dots,a^{\prime}_{i_{m}}\\ z^{\prime}_{i_{1}},\dots,z^{\prime}_{i_{m}}\end{pmatrix}.$		(144)

Lemma 5.5 ([Kitada23]).

For any $\displaystyle a_{1},\dots,a_{r+s}\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle h\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}h\binom{a_{r+1},\dots,a_{r+s}}{x_{r+1},\dots,x_{r+s}}=\left.h\binom{a_{1},\dots,a_{r+s}}{x_{1},\dots,x_{r+s}}\mathrel{}\middle|\mathrel{}\mathrm{sh}^{(r+s)}_{r}\right.,

where

\displaystyle\mathrm{sh}^{(r+s)}_{r}\coloneq\sum_{\sigma\in\mathrm{Sh}^{(r+s)}_{r}}\sigma\in\mathbb{Z}[\mathfrak{S}_{r+s}],

and the action extends to an action of the group ring $\displaystyle\mathbb{Z}[\mathfrak{S}_{r+s}]$ by linearity.

Proof.

Since $\displaystyle H$ satisfies harmonic product, the map

\displaystyle H:\mathcal{U}_{*}\to\varinjlim_{r}\mathcal{K}\llbracket x_{1},\dots,x_{r}\rrbracket:\begin{pmatrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\\ z_{1},\dots,z_{r}\end{pmatrix}\mapsto H\begin{pmatrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\\ z_{1},\dots,z_{r}\end{pmatrix}

is homomorphism. Clearly, we have

\displaystyle h\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}=H\circ\exp\left(\begin{pmatrix}1,\dots,1\\ a_{1},\dots,a_{r}\\ x_{1},\dots,x_{r}\end{pmatrix}\right).

Therefore, $\displaystyle h:\mathcal{U}_{\,\hbox{\sevency X}\,}\to\mathcal{K}\llbracket x_{1},\dots,x_{r}\rrbracket$ is a homomorphism. ∎

The following lemma gives us the characterization of the product $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ through generating function.

Lemma 5.6.

Let $\displaystyle F\in\mathcal{K}\llbracket x_{1},\dots,x_{r}\rrbracket$ and $\displaystyle f\in\mathcal{K}$ satisfying

\displaystyle F\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}=\sum_{k_{1},\dots,k_{r}>0}f\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}x_{1}^{k_{1}-1}\cdots x_{r}^{k_{r}-1}.

Then, the following statements are equivalent:

(i)

$\displaystyle f\in\mathcal{K}$ satisfies $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ -product i.e. it holds

\displaystyle f\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}f\binom{k_{r+1},\dots,k_{r+s}}{a_{r+1},\dots,a_{r+s}}=f\left(\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}\widetilde{{\,\hbox{\sevency X}\,}}\binom{k_{r+1},\dots,k_{r+s}}{a_{r+1},\dots,a_{r+s}}\right).

for any $\displaystyle k_{1},\dots,k_{r+s}\in\mathfrak{H}^{1}$ .

(ii)

For any positive integers $\displaystyle r,s>0$ and $\displaystyle a_{1},\dots,a_{r+s}\in\mathbb{Z}/N\mathbb{Z}$ , it holds

\displaystyle F^{\#}\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}F^{\#}\binom{a_{r+1},\dots,a_{r+s}}{x_{r+1},\dots,x_{r+s}}=\left.F^{\#}\binom{a_{1},\dots,a_{r+s}}{x_{1},\dots,x_{r+s}}\mathrel{}\middle|\mathrel{}\mathrm{sh}^{(r+s)}_{r}\right..

Here, $\displaystyle F^{\#}\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}\coloneq F^{\#}\binom{a_{1},a_{1}+a_{2},\dots,a_{1}+\cdots+a_{r}}{x_{1},x_{1}+x_{2},\dots,x_{1}+\cdots+x_{r}}$ .

Proof.

Since

\displaystyle F^{\#}\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}=\sum_{k_{1},\dots,k_{r}>0}f\left(\rho^{-1}\binom{k_{1},\dots,k_{1}}{a_{1},\dots,a_{r}}\right)x_{1}^{k_{1}-1}(x_{1}+x_{2})^{k_{2}-1}\cdots(x_{1}+\cdots+x_{r})^{k_{r}-1},

the statement (ii) is equivalent to

\displaystyle f\left(\rho^{-1}\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}\right)f\left(\rho^{-1}\binom{k_{r+1},\dots,k_{r+s}}{a_{r+1},\dots,a_{r+s}}\right)=f\left(\rho^{-1}\left(\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}{\,\hbox{\sevency X}\,}\binom{k_{r+1},\dots,k_{r+s}}{a_{r+1},\dots,a_{r+s}}\right)\right)

for any $\displaystyle k_{1},\dots,k_{r+s}\geq 1$ and $\displaystyle a_{1},\dots,a_{r+s}\in\mathbb{Z}/N\mathbb{Z}$ (see Ihara–Kaneko–Zagier [IKZ], Section 8). By definition of $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ and bijectiveness of $\displaystyle\rho$ , this statement is equivalent to

\displaystyle f\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}f\binom{k_{r+1},\dots,k_{r+s}}{a_{r+1},\dots,a_{r+s}}=f\left(\binom{k_{1},\dots,k_{r}}{a_{1},\dots,a_{r}}\widetilde{{\,\hbox{\sevency X}\,}}\binom{k_{r+1},\dots,k_{r+s}}{a_{r+1},\dots,a_{r+s}}\right)

for any $\displaystyle k_{1},\dots,k_{r+s}\geq 1$ and $\displaystyle a_{1},\dots,a_{r+s}\in\mathbb{Z}/N\mathbb{Z}$ . ∎

Definition 5.7 ([Kitada23]).

We define $\displaystyle g_{\widetilde{{\,\hbox{\sevency X}\,}}}\in\mathcal{K}\llbracket x_{1},\dots,x_{r}\rrbracket$ by

\displaystyle g_{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}\coloneq h\binom{a_{r}-a_{r-1},\dots,a_{2}-a_{1},a_{1}}{x_{r}-x_{r-1},\dots,x_{2}-x_{1},x_{1}},

and we define shuffle regularization $\displaystyle g^{\widetilde{{\,\hbox{\sevency X}\,}}}:\mathfrak{H}^{1}\to\mathbb{C}\llbracket q\rrbracket$ as the coefficient of the generating function:

\displaystyle\sum_{k_{1},\dots,k_{r}>0}\left(\frac{N}{-2\pi\sqrt{-1}}\right)^{k_{1}+\cdots+k_{r}}g^{\widetilde{{\,\hbox{\sevency X}\,}}}\bigg{(}\begin{matrix}k_{1},\dots,k_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}x_{1}^{k_{1}-1}\cdots x_{r}^{k_{r}-1}\coloneq g_{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}.

Proposition 5.8 ([Kitada23]).

For any $\displaystyle n_{1},\dots,n_{r+s}\geq 1$ and $\displaystyle a_{1},\dots,a_{r+s}\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle g^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}g^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{n_{r+1},\dots,n_{r+s}}{a_{r+1},\dots,a_{r+s}}=g^{\widetilde{{\,\hbox{\sevency X}\,}}}\left(\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}\widetilde{{\,\hbox{\sevency X}\,}}\binom{n_{r+1},\dots,n_{r+s}}{a_{r+1},\dots,a_{r+s}}\right).

Proof.

Let $\displaystyle\rho_{r,s}\in\mathfrak{S}_{r+s}$ , $\displaystyle\tau_{r}\in\mathfrak{S}_{r}$ be

\displaystyle\rho_{r,s}=\binom{1\;\cdots\;r\quad r+1\;\cdots\;r+s}{r\;\cdots\;1\quad r+s\;\cdots\;r+1},\quad\tau_{r}=\binom{1\;\cdots\;r}{r\;\cdots\;1}.

We have

	$\displaystyle\displaystyle g_{\widetilde{{\,\hbox{\sevency X}\,}}}^{\#}\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}g_{\widetilde{{\,\hbox{\sevency X}\,}}}^{\#}\binom{a_{r+1},\dots,a_{r+s}}{x_{r+1},\dots,x_{r+s}}$	$\displaystyle\displaystyle=h\binom{a_{r},\dots,a_{1}}{x_{r},\dots,x_{1}}h\binom{a_{r+s},\dots,a_{r+1}}{x_{r+s},\dots,x_{r+1}}$		(145)
		$\displaystyle\displaystyle=h\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}h\binom{a_{r+1},\dots,a_{r+s}}{x_{r+1},\dots,x_{r+s}}\Big{\|}\rho_{r,s}.$		(146)

Since $\displaystyle h$ satisfies X -product, we have

	$\displaystyle\displaystyle h\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}h\binom{a_{r+1},\dots,a_{r+s}}{x_{r+1},\dots,x_{r+s}}\Big{\|}\rho_{r,s}$	$\displaystyle\displaystyle=h\binom{a_{1},\dots,a_{r+s}}{x_{1},\dots,x_{r+s}}\Big{\|}\mathrm{sh}_{r}^{(r+s)}\Big{\|}\rho_{r,s}$		(147)
		$\displaystyle\displaystyle=g_{\widetilde{{\,\hbox{\sevency X}\,}}}^{\#}\binom{a_{1},\dots,a_{r+s}}{x_{1},\dots,x_{r+s}}\Big{\|}\tau_{r+s}\Big{\|}\mathrm{sh}_{r}^{(r+s)}\Big{\|}\rho_{r,s}.$		(148)

Since $\displaystyle\tau_{r+s}\sigma\rho_{r,s}\in\mathrm{Sh}^{(r+s)}_{r}$ for any $\displaystyle\sigma\in\mathrm{Sh}^{(r+s)}_{r}$ , we have

\displaystyle g_{\widetilde{{\,\hbox{\sevency X}\,}}}^{\#}\binom{a_{1},\dots,a_{r+s}}{x_{1},\dots,x_{r+s}}\Big{|}\tau_{r+s}\Big{|}\mathrm{sh}_{r}^{(r+s)}\Big{|}\rho_{r,s}=g_{\widetilde{{\,\hbox{\sevency X}\,}}}^{\#}\binom{a_{1},\dots,a_{r+s}}{x_{1},\dots,x_{r+s}}\Big{|}\mathrm{sh}_{r}^{(r+s)}.

By 5.6, $\displaystyle g^{\widetilde{{\,\hbox{\sevency X}\,}}}$ satisfies $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ -product. ∎

For an index $\displaystyle w\in\mathfrak{H}^{1}$ such that all upper components are greater than 1, we have $\displaystyle g^{\widetilde{{\,\hbox{\sevency X}\,}}}(w;q)=g(w;q)$ .

Lemma 5.9 ([Kitada23]).

For $\displaystyle n_{1},\dots,n_{r}\geq 2$ and $\displaystyle a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle g^{\widetilde{{\,\hbox{\sevency X}\,}}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}=g\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}.

Proof.

By definition of $\displaystyle g_{\widetilde{{\,\hbox{\sevency X}\,}}},h,H\in\mathcal{K}\llbracket x_{1},\dots,x_{r}\rrbracket$ , we have

\displaystyle g_{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}=\sum_{\begin{subarray}{c}1\leq m\leq r\\ i_{1}+\cdots+i_{m}=r\\ i_{j}>0\end{subarray}}\frac{1}{i_{1}!\cdots i_{m}!}H\left(\begin{matrix}i_{1},\dots,i_{m}\\ a_{i_{1}}^{\prime\prime},\dots,a_{i_{m}}^{\prime\prime}\\ x_{i_{1}}^{\prime\prime},\dots,x_{i_{m}}^{\prime\prime}\end{matrix}\right),

where $\displaystyle p_{i_{j}}^{\prime\prime}\coloneq p_{r-(i_{1}+\cdots+i_{j-1})}-p_{r-(i_{1}+\cdots+i_{j})}$ for $\displaystyle p\in\{a,x\}$ and $\displaystyle j=1,\dots,r$ . Since

\displaystyle\text{The coefficients of $\displaystyle x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}$ in }H\left(\begin{matrix}i_{1},\dots,i_{m}\\ a_{i_{1}}^{\prime\prime},\dots,a_{i_{m}}^{\prime\prime}\\ x_{i_{1}}^{\prime\prime},\dots,x_{i_{m}}^{\prime\prime}\end{matrix}\right)=0

for any $\displaystyle n_{1},\dots,n_{r}\geq 2$ when $\displaystyle 1\leq m<r$ . By 5.2, we have

	$\displaystyle\displaystyle\text{The coefficients of $\displaystyle x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}$ in }g_{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}$		(149)
	$\displaystyle\displaystyle=\text{The coefficients of $\displaystyle x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}$ in }H\left(\begin{array}[]{@{}c@{}c@{}c@{}l@{}}1&,\dots,&1&,1\\ a_{r}-a_{r-1}&,\dots,&a_{2}-a_{1}&,a_{1}\\ x_{r}-x_{r-1}&,\dots,&x_{2}-x_{1}&,x_{1}\end{array}\right)$		(153)
	$\displaystyle\displaystyle=\text{The coefficients of $\displaystyle x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}$ in }g\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}.$		(154)

∎

5.2 Shuffle regularization for MES of level $\displaystyle N$

Definition 5.10.

We define the shuffle regularized MES of level $\displaystyle N$ as the image of $\displaystyle\mathbb{Q}$ -linear map $\displaystyle G:\mathfrak{H}^{1}\to\mathcal{O}(\mathbb{H})$ by

\displaystyle G^{\widetilde{{\,\hbox{\sevency X}\,}}}(w;\tau)\coloneq(\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}\star g^{\widetilde{{\,\hbox{\sevency X}\,}}})(w;q)

for $\displaystyle w\in\mathfrak{H}^{1}$ and $\displaystyle q=e^{2\pi\sqrt{-1}\tau}$ .

This regularization makes sense, in other words, the regularized MES is equal to the original MES for the cases of convergence.

Proposition 5.11.

It holds $\displaystyle G^{\widetilde{{\,\hbox{\sevency X}\,}}}=G$ on $\displaystyle\mathfrak{H}^{2}$ .

Proof.

By the explicit formula of Fourier expansion of MES, we have

	$\displaystyle\displaystyle(G^{\widetilde{{\,\hbox{\sevency X}\,}}}-G)\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}$		(155)
	$\displaystyle\displaystyle=\sum_{\begin{subarray}{c}0\leq h\leq r\\ 0<t_{1}<\cdots<t_{h}<r+1\end{subarray}}\zeta\binom{n_{1},\dots,n_{t_{1}-1}}{a_{1},\dots,a_{t_{1}-1}}\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}t_{1}\leq q_{1}\leq t_{2}-1\\[-5.0pt] \vdots\\ t_{h}\leq q_{h}\leq r\end{subarray}}$}}\raisebox{0.0pt}{{$\displaystyle\sum_{\begin{subarray}{c}k_{t_{j}}+\cdots+k_{t_{j+1}-1}\\ =n_{t_{j}}+\cdots+n_{t_{j+1}-1}\\ (1\leq j\leq h),k_{i}\geq 1\end{subarray}}$}}\prod_{j=1}^{h}\Bigg{\{}(-1)^{l_{j}}\Bigg{(}\prod_{\begin{subarray}{c}p=t_{j}\\ p\neq q_{j}\end{subarray}}^{t_{j+1}-1}\binom{k_{p}-1}{n_{p}-1}$
	$\displaystyle\displaystyle\times\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}-1}&,\dots,&k_{t_{j}}\\ a_{q_{j}}-a_{q_{j}-1}&,\dots,&a_{q_{j}}-a_{t_{j}}\end{array}\bigg{)}\zeta\bigg{(}\begin{array}[]{@{}c@{}c@{}c@{}}k_{q_{j}+1}&,\dots,&k_{t_{j+1}-1}\\ a_{q_{j}+1}-a_{q_{j}}&,\dots,&a_{t_{j+1}-1}-a_{q_{j}}\end{array}\bigg{)}\Bigg{\}}(g^{\widetilde{{\,\hbox{\sevency X}\,}}}-g)\bigg{(}\begin{matrix}k_{q_{1}},\dots,k_{q_{h}}\\ a_{q_{1}},\dots,a_{q_{h}}\end{matrix};q\bigg{)}$		(160)

for $\displaystyle n_{1},\dots,n_{r}\geq 2$ . The terms with $\displaystyle k_{q_{1}},\dots,k_{q_{h}}\geq 2$ vanish by 5.9. When $\displaystyle k_{q_{j_{1}}},\dots,k_{q_{j_{s}}}=1$ for some $\displaystyle 1\leq j_{1}<\cdots<j_{s}\leq h$ , the terms vanish by 2.11 since we can write $\displaystyle(g^{\widetilde{{\,\hbox{\sevency X}\,}}}-g)\bigg{(}\begin{matrix}k_{q_{1}},\dots,k_{q_{h}}\\ a_{q_{1}},\dots,a_{q_{h}}\end{matrix};q\bigg{)}=\sum_{l=1}^{s}f_{l}(q)$ such that $\displaystyle f_{l}$ does not depend on $\displaystyle a_{q_{j_{l}}}$ . ∎

6 Linear relations among regularized MES of level $\displaystyle N$

In this final section, we obtain restricted double shuffle relation, distribution relation for MES, and sum and weighted sum formulas for double Eisenstein series (DES).

1.2 (Restricted double shuffle relation).

For any words $\displaystyle w_{1},w_{2}\in X^{0}$ , we have

\displaystyle\displaystyle G(w_{1}\widetilde{*}w_{2};\tau)=G^{\widetilde{{\,\hbox{\sevency X}\,}}}(w_{1}\widetilde{{\,\hbox{\sevency X}\,}}w_{2};\tau).

(161)

Proof.

This equation follows by expanding $\displaystyle G(w_{1};\tau)G(w_{2};\tau)$ in $\displaystyle\widetilde{*}$ -product and $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ -product. ∎

Let $\displaystyle G^{\,\hbox{\sevency X}\,}_{n_{1},\dots,n_{r}}(\tau)$ , $\displaystyle\zeta^{\,\hbox{\sevency X}\,}(n_{1},\dots,n_{r})$ and $\displaystyle g^{\,\hbox{\sevency X}\,}_{n_{1},\dots,n_{r}}(q)$ denote those of level $\displaystyle 1$ . Note that $\displaystyle\widetilde{{\,\hbox{\sevency X}\,}}$ -product is equal to X -products when $\displaystyle N=1$ .

Theorem 6.1 (Distribution relation).

For $\displaystyle n_{1},\dots,n_{r}\geq 1$ , we have

\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}G^{\widetilde{{\,\hbox{\sevency X}\,}}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};\tau\bigg{)}=G^{\,\hbox{\sevency X}\,}_{n_{1},\dots,n_{r}}(N\tau).

Proof.

It suffices to show $\displaystyle\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}$ and $\displaystyle g^{\widetilde{{\,\hbox{\sevency X}\,}}}$ satisfy distribution relation i.e.

	$\displaystyle\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}=\zeta^{\,\hbox{\sevency X}\,}(n_{1},\dots,n_{r}),$		(162)
	$\displaystyle\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}g^{\widetilde{{\,\hbox{\sevency X}\,}}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}=g^{{\,\hbox{\sevency X}\,}}_{n_{1},\dots,n_{r}}(q^{N}).$		(163)

By definition of $\displaystyle\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}$ , we have

\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}=\frac{1}{N^{r}}\sum_{b_{1},\dots,b_{r}\in\mathbb{Z}/N\mathbb{Z}}\left(\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}\eta^{-\rho(\bm{a})\cdot\bm{b}}\right)L_{{\,\hbox{\sevency X}\,}}^{\mathrm{reg}}\binom{n_{1},\dots,n_{r}}{b_{1},\dots,b_{r}}\bigg{|}_{T=0}.

Since

\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}\eta^{-\rho(\bm{a})\cdot\bm{b}}=\prod_{j=1}^{r}\sum_{a_{j}\in\mathbb{Z}/N\mathbb{Z}}\eta^{a_{j}(b_{j}-b_{j+1})}=\begin{cases}N^{r}&b_{1}=\cdots=b_{r}=0,\\ 0&\mathrm{otherwise}.\end{cases},

we have

\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}\zeta^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{n_{1},\dots,n_{r}}{a_{1},\dots,a_{r}}=L_{{\,\hbox{\sevency X}\,}}^{\mathrm{reg}}\binom{n_{1},\dots,n_{r}}{0,\dots,0}\bigg{|}_{T=0}=\zeta^{{\,\hbox{\sevency X}\,}}(n_{1},\dots,n_{r}).

By definition of $\displaystyle a_{i_{j}}^{\prime\prime}$ , we have

$\displaystyle\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}g_{\widetilde{{\,\hbox{\sevency X}\,}}}\bigg{(}\begin{matrix}a_{1},\dots,a_{r}\\ x_{1},\dots,x_{r}\end{matrix};q\bigg{)}$	$\displaystyle\displaystyle=\sum_{\begin{subarray}{c}1\leq m\leq r\\ i_{1}+\cdots+i_{m}=r\\ i_{1},\dots,i_{m}>0\end{subarray}}\frac{1}{i_{1}!\cdots i_{m}!}\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}H\left(\begin{matrix}i_{1},\dots,i_{m}\\ a_{i_{1}}^{\prime\prime},\dots,a_{i_{m}}^{\prime\prime}\\ x_{i_{1}}^{\prime\prime},\dots,x_{i_{m}}^{\prime\prime}\end{matrix};q\right)$	(164)
	$\displaystyle\displaystyle=\sum_{\begin{subarray}{c}1\leq m\leq r\\ i_{1}+\cdots+i_{m}=r\\ i_{1},\dots,i_{m}>0\end{subarray}}\frac{1}{i_{1}!\cdots i_{m}!}\Bigg{(}\sum_{\begin{subarray}{c}a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}\\ \widehat{a_{i_{1}}},\dots,\widehat{a_{i_{m}}}\end{subarray}}1\Bigg{)}$	(165)
	$\displaystyle\displaystyle\times\sum_{a_{i_{1}},\dots,a_{i_{m}}\in\mathbb{Z}/N\mathbb{Z}}H\left(\begin{matrix}i_{1},\dots,i_{m}\\ a_{i_{1}},\dots,a_{i_{m}}\\ x_{i_{1}}^{\prime\prime},\dots,x_{i_{m}}^{\prime\prime}\end{matrix};q\right).$	(166)

Therefore, we have

$\displaystyle\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}H\left(\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\\ x_{1},\dots,x_{r}\end{matrix};q\right)$	$\displaystyle\displaystyle=\sum_{0<d_{1}<\cdots<d_{r}}\prod_{j=1}^{r}\left(\sum_{a_{j}\in\mathbb{Z}/N\mathbb{Z}}\eta^{a_{j}d_{j}}\right)e^{d_{j}x_{j}}\left(\frac{q^{d_{j}}}{1-q^{d_{j}}}\right)^{n_{j}}$	(167)
	$\displaystyle\displaystyle=N^{r}\sum_{\begin{subarray}{c}0<d_{1}<\cdots<d_{r}\\ d_{i}=0\end{subarray}}\prod_{j=1}^{r}e^{d_{j}x_{j}}\left(\frac{q^{d_{j}}}{1-q^{d_{j}}}\right)^{n_{j}}$	(168)
	$\displaystyle\displaystyle=N^{r}H\left(\begin{matrix}n_{1},\dots,n_{r}\\ 0,\dots,0\\ Nx_{1},\dots,Nx_{r}\end{matrix};q^{N}\right).$	(169)

By definition of $\displaystyle g_{\widetilde{{\,\hbox{\sevency X}\,}}}$ , we have

	$\displaystyle\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}g_{\widetilde{{\,\hbox{\sevency X}\,}}}\bigg{(}\begin{matrix}a_{1},\dots,a_{r}\\ x_{1},\dots,x_{r}\end{matrix};q\bigg{)}$	$\displaystyle\displaystyle=\sum_{\begin{subarray}{c}1\leq m\leq r\\ i_{1}+\cdots+i_{m}=r\\ i_{1},\dots,i_{m}>0\end{subarray}}\frac{1}{i_{1}!\cdots i_{m}!}N^{r-m}\cdot N^{m}H\left(\begin{matrix}i_{1},\dots,i_{m}\\ 0,\dots,0\\ Nx_{i_{1}}^{\prime\prime},\dots,Nx_{i_{m}}^{\prime\prime}\end{matrix};q^{N}\right)$		(170)
		$\displaystyle\displaystyle=N^{r}g_{\widetilde{{\,\hbox{\sevency X}\,}}}\bigg{(}\begin{matrix}0,\dots,0\\ Nx_{1},\dots,Nx_{r}\end{matrix};q^{N}\bigg{)}.$		(171)

Comparing both coefficients, we have

\displaystyle\sum_{a_{1},\dots,a_{r}\in\mathbb{Z}/N\mathbb{Z}}g^{\widetilde{{\,\hbox{\sevency X}\,}}}\bigg{(}\begin{matrix}n_{1},\dots,n_{r}\\ a_{1},\dots,a_{r}\end{matrix};q\bigg{)}=g^{{\,\hbox{\sevency X}\,}}_{n_{1},\dots,n_{r}}(q^{N}).

∎

We give sum and weighted sum formula for DES through the generating function. Let $\displaystyle F_{a_{1},a_{2}}(X,Y)$ be the generating function of DES of weight $\displaystyle k$ $\displaystyle(\geq 4)$ and level $\displaystyle N$ ,

\displaystyle F_{a_{1},a_{2}}(X,Y)\coloneq\sum_{\begin{subarray}{c}i+j=k\\ i,j>1\end{subarray}}G\bigg{(}\begin{matrix}i,j\\ a_{1},a_{2}\end{matrix};\tau\bigg{)}X^{i-1}Y^{j-1}\quad(a_{1},a_{2}\in\mathbb{Z}/N\mathbb{Z}).

By restricted double shuffle relation for DES, we have the following equation for the generating function.

Lemma 6.2 ([Kitada23]).

For any integer $\displaystyle k\geq 4$ and $\displaystyle a_{1},a_{2}\in\mathbb{Z}/N\mathbb{Z}$ , we have

	$\displaystyle\displaystyle F_{a_{1},a_{2}}(X,Y)+F_{a_{2},a_{1}}(Y,X)+\delta_{a_{1},a_{2}}G\binom{k}{a_{1}}\left(\frac{X^{k-1}-Y^{k-1}}{X-Y}-(X^{k-2}-Y^{k-2})\right)$		(172)
	$\displaystyle\displaystyle=F_{a_{1},a_{1}+a_{2}}(X,X+Y)+F_{a_{2},a_{1}+a_{2}}(Y,X+Y)$		(173)
	$\displaystyle\displaystyle+\left(G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{1,k-1}{a_{1},a_{1}+a_{2}}+G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{1,k-1}{a_{2},a_{1}+a_{2}}\right)(X+Y)^{k-2}$		(174)
	$\displaystyle\displaystyle-\sum_{\begin{subarray}{c}i+j=k\\ i>0,j>1\end{subarray}}G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{i,j}{a_{1},a_{1}+a_{2}}X^{k-2}-\sum_{\begin{subarray}{c}i+j=k\\ i>0,j>1\end{subarray}}G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{i,j}{a_{2},a_{1}+a_{2}}Y^{k-2}.$		(175)

Proof.

By restricted double shuffle relation, we have

$\displaystyle\displaystyle G\binom{i}{a_{1}}G\binom{j}{a_{2}}$	$\displaystyle\displaystyle=G\binom{i,j}{a_{1},a_{2}}+G\binom{j,i}{a_{2},a_{1}}+\delta_{a_{1},a_{2}}G\binom{i+j}{a_{1}}$	(176)
	$\displaystyle\displaystyle=\sum_{\begin{subarray}{c}m+n=i+j\\ m,n>1\end{subarray}}\left\{\binom{n-1}{j-1}G\binom{m,n}{a_{1},a_{1}+a_{2}}+\binom{n-1}{i-1}G\binom{m,n}{a_{2},a_{1}+a_{2}}\right\}$	(177)
	$\displaystyle\displaystyle+\binom{i+j-2}{i-1}\left(G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{1,i+j-1}{a_{1},a_{1}+a_{2}}+G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{1,i+j-1}{a_{2},a_{1}+a_{2}}\right)$	(178)

for $\displaystyle i,j>1$ . Multiplying $\displaystyle X^{i-1}Y^{j-1}$ and adding up for $\displaystyle i+j=k$ , $\displaystyle i,j>1$ , we obtain the equation. ∎

By using this lemma, we have sum and weighted sum formula for DES.

Theorem 6.3 (Sum formula for DES, Kitada [Kitada23]).

For any even integer $\displaystyle k\geq 4$ and $\displaystyle a\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle 2\sum_{\begin{subarray}{c}i+j=k\\ i,j>1\end{subarray}}\left((-1)^{i-1}G\binom{i,j}{a,a}+G\binom{i,j}{a,2a}\right)+4G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{1,k-1}{a,2a}=G\binom{k}{a}.

Proof.

It follows by letting $\displaystyle k$ be even, $\displaystyle(X,Y)=(1,-1)$ and $\displaystyle a_{1}=a_{2}=a$ in 6.2. ∎

Theorem 6.4 (Weighted sum formula for DES).

For any integer $\displaystyle k\geq 4$ and $\displaystyle a\in\mathbb{Z}/N\mathbb{Z}$ , we have

\displaystyle\sum_{\begin{subarray}{c}i+j=k\\ i,j>0\end{subarray}}\left\{(2^{j-1}-1)G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{i,j}{a,2a}+(1-\delta_{j,1})G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{i,j}{a,a}\right\}=\frac{k-3}{2}G^{\widetilde{{\,\hbox{\sevency X}\,}}}\binom{k}{a}.

Proof.

It follows by letting $\displaystyle(X,Y)=(1,1)$ and $\displaystyle a_{1}=a_{2}=a$ in 6.2. ∎

	$\displaystyle\displaystyle h\binom{a_{1},\dots,a_{r}}{x_{1},\dots,x_{r}}h\binom{a_{r+1},\dots,a_{r+s}}{x_{r+1},\dots,x_{r+s}}\Big{\|}\rho_{r,s}$	$\displaystyle\displaystyle=h\binom{a_{1},\dots,a_{r+s}}{x_{1},\dots,x_{r+s}}\Big{\|}\mathrm{sh}_{r}^{(r+s)}\Big{\|}\rho_{r,s}$		(147)
		$\displaystyle\displaystyle=g_{\widetilde{{\,\hbox{\sevency X}\,}}}^{\#}\binom{a_{1},\dots,a_{r+s}}{x_{1},\dots,x_{r+s}}\Big{\|}\tau_{r+s}\Big{\|}\mathrm{sh}_{r}^{(r+s)}\Big{\|}\rho_{r,s}.$		(148)

Shuffle regularization for multiple Eisenstein series of level N\displaystyle N

Abstract

1 Introduction

Theorem 1.1.

Theorem 1.2 (Restricted double shuffle relation).

Acknowledgment

2 Multiple zeta values of level N\displaystyle N

2.1 Multiple L\displaystyle L-values of level N\displaystyle N

Definition 2.1 (cf. Arakawa–Kaneko [AK]).

Remark 2.2.

2.2 Multiple zeta values of level N

Definition 2.3 (cf. Yuan–Zhao [YZ16]).

Proposition 2.4 (Yuan–Zhao [YZ16]).

Proposition 2.5.

Proof.

Proposition 2.6.

Proof.

Definition 2.7.

Theorem 2.8 (Hoffman [Hoff00] Theorem 3.2).

Lemma 2.9.

Proof.

Proposition 2.10.

Proof.

Corollary 2.11 (Antipode relation for MZV of level N\displaystyle N).

Proof.

3 Fourier expansion for multiple Eisenstein series of level N\displaystyle N

Definition 3.1.

Lemma 3.2.

Proof.

3.1 Multitangent function of level N\displaystyle N

Definition 3.3.

Lemma 3.4.

Proof.

Lemma 3.5 (Yuan–Zhao [YZ15], Lemma 4.1).

3.2 Multiple divisor function of level N\displaystyle N

Definition 3.6 ([YZ16]).

Lemma 3.7.

3.3 The Fourier expansion of MES of level N\displaystyle N

Proposition 3.8.

Proof.

4 Formal iterated integrals corresponding to MZV of level N\displaystyle N

4.1 Hopf algebra of formal iterated integrals

Definition 4.1 (Goncharov [Gon05]).

Theorem 4.2 (Goncharov [Gon05]).

Remark 4.3.

4.2 Formal iterated integrals corresponding MLV and MZV of level N\displaystyle N

Proposition 4.4 (Bachmann–Tasaka [BT]).

Proof.

Lemma 4.5 (Goncharov [Gon05]).

Lemma 4.6.

Proof.

Proposition 4.7.

Proof.

Remark 4.8.

Proposition 4.9.

Proof.

4.3 Computing Goncharov coproduct

Definition 4.10.

Remark 4.11.

Example 4.12.

Lemma 4.13.

Proof.

Lemma 4.14.

Proof.

Proposition 4.15.

Proof.

1.1.

Proof.

5 Shuffle regularization for MES of level N\displaystyle N

5.1 Shuffle regularization for multiple divisor function

Definition 5.1 (Kitada [Kitada23]).

Lemma 5.2 ([Kitada23]).

Proof.

Proposition 5.3 (Hoffman [Hoff00]).

Definition 5.4 ([Kitada23]).

Lemma 5.5 ([Kitada23]).

Proof.

Lemma 5.6.

Proof.

Definition 5.7 ([Kitada23]).

Shuffle regularization for multiple Eisenstein series of level $\displaystyle N$

2 Multiple zeta values of level $\displaystyle N$

2.1 Multiple $\displaystyle L$ -values of level $\displaystyle N$

Corollary 2.11 (Antipode relation for MZV of level $\displaystyle N$ ).

3 Fourier expansion for multiple Eisenstein series of level $\displaystyle N$

3.1 Multitangent function of level $\displaystyle N$

3.2 Multiple divisor function of level $\displaystyle N$

3.3 The Fourier expansion of MES of level $\displaystyle N$

4 Formal iterated integrals corresponding to MZV of level $\displaystyle N$

4.2 Formal iterated integrals corresponding MLV and MZV of level $\displaystyle N$

5 Shuffle regularization for MES of level $\displaystyle N$

5.2 Shuffle regularization for MES of level $\displaystyle N$

6 Linear relations among regularized MES of level $\displaystyle N$