2.1 Multiple Eisenstein series

In this subsection, we define the multiple Eisenstein series and compute its Fourier expansion.

Recall the computation of the Fourier expansion of $G_{n_{1}}(\unicode[STIX]{x1D70F})$ , which is well-known (see also [Reference Gangl, Kaneko and Zagier7, Section 7]):

$$\begin{eqnarray}\displaystyle G_{n_{1}}(\unicode[STIX]{x1D70F}) & = & \displaystyle \mathop{\sum }_{0\prec l\unicode[STIX]{x1D70F}+m}\frac{1}{(l\unicode[STIX]{x1D70F}+m)^{n_{1}}}=\mathop{\sum }_{m>0}\frac{1}{m^{n_{1}}}+\mathop{\sum }_{l>0}\mathop{\sum }_{m\in \mathbb{Z}}\frac{1}{(l\unicode[STIX]{x1D70F}+m)^{n_{1}}}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D701}(n_{1})+\frac{(-2\unicode[STIX]{x1D70B}\sqrt{-1})^{n_{1}}}{(n_{1}-1)!}\mathop{\sum }_{n>0}\unicode[STIX]{x1D70E}_{n_{1}-1}(n)q^{n},\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{k}(n)=\sum _{d\mid n}d^{k}$ is the divisor function and $q=e^{2\unicode[STIX]{x1D70B}\sqrt{-1}\unicode[STIX]{x1D70F}}$ . Here for the last equality we have used the Lipschitz formula

(4) $$\begin{eqnarray}\mathop{\sum }_{m\in \mathbb{Z}}\frac{1}{(\unicode[STIX]{x1D70F}+m)^{n_{1}}}=\frac{(-2\unicode[STIX]{x1D70B}\sqrt{-1})^{n_{1}}}{(n_{1}-1)!}\mathop{\sum }_{0<c_{1}}c_{1}^{n_{1}-1}q^{c_{1}}\quad (n_{1}\geqslant 2).\end{eqnarray}$$

When $n_{1}=2$ , the above computation (the second equality) can be justified by using a limit argument which in general is treated in Definition 2.1 below. We remark that the function $G_{n_{1}}(\unicode[STIX]{x1D70F})$ is a modular form of weight $n_{1}$ for $\operatorname{SL}_{2}(\mathbb{Z})$ when $n_{1}$ is even $({>}2)$ ( $G_{2}(\unicode[STIX]{x1D70F})$ is called the quasimodular form) and a nontrivial holomorphic function even if $n_{1}$ is odd.

The following definition enables us to compute the Fourier expansion of $G_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F})$ for integers $n_{1},\ldots ,n_{r}\geqslant 2$ and coincides with the iterated multiple sum (1) when the series defining (1) converges absolutely, that is, $n_{1},\ldots ,n_{r-1}\geqslant 2$ and $n_{r}\geqslant 3$ .

Definition 2.1. For integers $n_{1},\ldots ,n_{r}\geqslant 2$ , we define the holomorphic function $G_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F})$ on the upper half-plane called the multiple Eisenstein series by

$$\begin{eqnarray}\displaystyle & & \displaystyle G_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F}):=\lim _{L\rightarrow \infty }\lim _{M\rightarrow \infty }\mathop{\sum }_{\substack{ 0\prec \unicode[STIX]{x1D706}_{1}\prec \cdots \prec \unicode[STIX]{x1D706}_{r} \\ \unicode[STIX]{x1D706}_{i}\in \mathbb{Z}_{L}\unicode[STIX]{x1D70F}+\mathbb{Z}_{M}}}\frac{1}{\unicode[STIX]{x1D706}_{1}^{n_{1}}\cdots \unicode[STIX]{x1D706}_{r}^{n_{r}}}\nonumber\\ \displaystyle & & \displaystyle \quad =\lim _{L\rightarrow \infty }\lim _{M\rightarrow \infty }\mathop{\sum }_{\substack{ 0\prec (l_{1}\unicode[STIX]{x1D70F}+m_{1})\prec \cdots \prec (l_{r}\unicode[STIX]{x1D70F}+m_{r}) \\ -L\leqslant l_{1},\ldots ,l_{r}\leqslant L \\ -M\leqslant m_{1},\ldots ,m_{r}\leqslant M}}\frac{1}{(l_{1}\unicode[STIX]{x1D70F}+m_{1})^{n_{1}}\cdots (l_{r}\unicode[STIX]{x1D70F}+m_{r})^{n_{r}}},\nonumber\end{eqnarray}$$

where we set $\mathbb{Z}_{M}=\{-M,-M+1,\ldots ,-1,0,1,\ldots ,M-1,M\}$ for an integer $M>0$ .

The Fourier expansion of $G_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F})$ for integers $n_{1},\ldots ,n_{r}\geqslant 2$ is obtained by splitting up the sum into $2^{r}$ terms, which was first done in [Reference Gangl, Kaneko and Zagier7] for the case $r=2$ and in [Reference Bachmann1] for the general case (they use the opposite convention, so that the $\unicode[STIX]{x1D706}_{i}$ ’s are ordered by $\unicode[STIX]{x1D706}_{1}\succ \cdots \succ \unicode[STIX]{x1D706}_{r}\succ 0$ ). To describe each term we introduce the holomorphic function $G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r};\unicode[STIX]{x1D70F})$ on the upper half-plane below. For convenience, we express the set $P$ of positive elements in the lattice $\mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}$ as the disjoint union of two sets

$$\begin{eqnarray}\displaystyle P_{\mathbf{x}} & := & \displaystyle \left\{l\unicode[STIX]{x1D70F}+m\in \mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}\mid l=0\wedge m>0\right\},\nonumber\\ \displaystyle P_{\mathbf{y}} & := & \displaystyle \left\{l\unicode[STIX]{x1D70F}+m\in \mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}\mid l>0\right\},\nonumber\end{eqnarray}$$

that is, $P_{\mathbf{x}}$ are the lattice points on the positive real axis, $P_{\mathbf{y}}$ are the lattice points in the upper half-plane and $P=P_{\mathbf{x}}\cup P_{\mathbf{y}}$ . We notice that $\unicode[STIX]{x1D706}_{1}\prec \unicode[STIX]{x1D706}_{2}$ is equivalent to $\unicode[STIX]{x1D706}_{2}-\unicode[STIX]{x1D706}_{1}\in P$ . Let us denote by $\{\mathbf{x},\mathbf{y}\}^{\ast }$ the set of all words consisting of letters $\mathbf{x}$ and $\mathbf{y}$ . For integers $n_{1},\ldots ,n_{r}\geqslant 2$ and a word $w_{1}\cdots w_{r}\in \{\mathbf{x},\mathbf{y}\}^{\ast }$ ( $w_{i}\in \{\mathbf{x},\mathbf{y}\}$ ) we define

$$\begin{eqnarray}\displaystyle G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r}) & = & \displaystyle G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r};\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & := & \displaystyle \lim _{L\rightarrow \infty }\lim _{M\rightarrow \infty }\mathop{\sum }_{\substack{ \unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D706}_{0}\in P_{w_{1}} \\ \vdots \\ \unicode[STIX]{x1D706}_{r}-\unicode[STIX]{x1D706}_{r-1}\in P_{w_{r}} \\ \unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{r}\in \mathbb{Z}_{L}\unicode[STIX]{x1D70F}+\mathbb{Z}_{M}}}\frac{1}{\unicode[STIX]{x1D706}_{1}^{n_{1}}\cdots \unicode[STIX]{x1D706}_{r}^{n_{r}}},\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{0}:=0$ . Note that in the above sum, adjoining elements $\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{i-1}=(l_{i}-l_{i-1})\unicode[STIX]{x1D70F}+(m_{i}-m_{i-1}),\ldots ,\unicode[STIX]{x1D706}_{j}-\unicode[STIX]{x1D706}_{j-1}=(l_{j}-l_{j-1})\unicode[STIX]{x1D70F}+(m_{j}-m_{j-1})$ are in $P_{\mathbf{x}}$ (i.e., $w_{i}=\cdots =w_{j}=\mathbf{x}$ with $i\leqslant j$ ) if and only if they satisfy $m_{i-1}<m_{i}<\cdots <m_{j}$ with $l_{i-1}=l_{i}=\cdots =l_{j}$ (since $(l-l^{\prime })\unicode[STIX]{x1D70F}+(m-m^{\prime })\in P_{\mathbf{x}}$ if and only if $l=l^{\prime }$ and $m>m^{\prime }$ ), and hence the function $G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})$ is expressible in terms of the following function:

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F})=\mathop{\sum }_{-\infty <m_{1}<\cdots <m_{r}<\infty }\frac{1}{(\unicode[STIX]{x1D70F}+m_{1})^{n_{1}}\cdots (\unicode[STIX]{x1D70F}+m_{r})^{n_{r}}},\end{eqnarray}$$

which was studied thoroughly in [Reference Bouillot3]. In fact, as is easily seen that the series defining $\unicode[STIX]{x1D6F9}_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F})$ converges absolutely when $n_{1},\ldots ,n_{r}\geqslant 2$ , we obtain the following expression:

(5) $$\begin{eqnarray}\displaystyle & & \displaystyle G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D701}(n_{1},\ldots ,n_{t_{1}-1})\mathop{\sum }_{0<l_{1}<\cdots <l_{h}}\unicode[STIX]{x1D6F9}_{n_{t_{1}},\ldots ,n_{t_{2}-1}}(l_{1}\unicode[STIX]{x1D70F})\cdots \unicode[STIX]{x1D6F9}_{n_{t_{h}},\ldots ,n_{r}}(l_{h}\unicode[STIX]{x1D70F}),\end{eqnarray}$$

where $0<t_{1}<\cdots <t_{h}<r+1$ describe the positions of $\mathbf{y}$ ’s in the word $w_{1}\cdots w_{r}$ , that is,

$$\begin{eqnarray}w_{1}\cdots w_{r}=\underbrace{\mathbf{x}\cdots \mathbf{x}}_{t_{1}-1}\mathbf{y}\underbrace{\mathbf{x}\cdots \mathbf{x}}_{t_{2}-t_{1}-1}\mathbf{y}\mathbf{x}\cdots \mathbf{y}\underbrace{\mathbf{x}\cdots \mathbf{x}}_{t_{h}-t_{h-1}-1}\mathbf{y}\underbrace{\mathbf{x}\cdots \mathbf{x}}_{r-t_{h}},\end{eqnarray}$$

and $\unicode[STIX]{x1D701}(n_{1},\ldots ,n_{t_{1}-1})=1$ when $t_{1}=1$ .

We remark that the above expression of words gives a one-to-one correspondence between words of length $r$ in $\{\mathbf{x},\mathbf{y}\}^{\ast }$ and the ordered subsets of $\{1,2,\ldots ,r\}$ . As we use later, this correspondence is written as the map

(6) $$\begin{eqnarray}\begin{array}{@{}rcl@{}}\unicode[STIX]{x1D70C}:\{\mathbf{x},\mathbf{y}\}_{r}^{\ast }\ & \longrightarrow \ & 2^{\{1,2,\ldots ,r\}}\\ w_{1}\cdots w_{r}\ & \longmapsto \ & \{t_{1},\ldots ,t_{h}\},\end{array}\end{eqnarray}$$

where $\{\mathbf{x},\mathbf{y}\}_{r}^{\ast }$ is the set of words of length $r$ and $2^{\{1,2,\ldots ,r\}}$ is the set of all subsets of $\{1,2,\ldots ,r\}$ . For instance, we have $\unicode[STIX]{x1D70C}(\mathbf{x}\mathbf{y}\mathbf{x}\mathbf{y}\mathbf{x}^{r-4})=\{2,4\}$ and $\unicode[STIX]{x1D70C}(\mathbf{x}^{r})=\{\emptyset \}$ .

Proposition 2.2. For integers $n_{1},\ldots ,n_{r}\geqslant 2$ , we have

$$\begin{eqnarray}G_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F})=\mathop{\sum }_{w_{1},\ldots ,w_{r}\in \{\mathbf{x},\mathbf{y}\}}G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r}).\end{eqnarray}$$

Example 2.3. In the case of $r=2$ , one has for $n_{1}\geqslant 2,n_{2}\geqslant 3$

$$\begin{eqnarray}\displaystyle G_{n_{1},n_{2}}(\unicode[STIX]{x1D70F}) & = & \displaystyle \mathop{\sum }_{\substack{ 0\prec \unicode[STIX]{x1D706}_{1}\prec \unicode[STIX]{x1D706}_{2} \\ \unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}\in \mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}}}\unicode[STIX]{x1D706}_{1}^{-n_{1}}\unicode[STIX]{x1D706}_{2}^{-n_{2}}=\mathop{\sum }_{\substack{ \unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D706}_{0}\in P \\ \unicode[STIX]{x1D706}_{2}-\unicode[STIX]{x1D706}_{1}\in P \\ \unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}\in \mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}}}\unicode[STIX]{x1D706}_{1}^{-n_{1}}\unicode[STIX]{x1D706}_{2}^{-n_{2}}\nonumber\\ \displaystyle & = & \displaystyle \bigg(\!\mathop{\sum }_{\substack{ \unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D706}_{0}\in P_{\mathbf{x}} \\ \unicode[STIX]{x1D706}_{2}-\unicode[STIX]{x1D706}_{1}\in P_{\mathbf{x}} \\ \unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}\in \mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}}}+\mathop{\sum }_{\substack{ \unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D706}_{0}\in P_{\mathbf{x}} \\ \unicode[STIX]{x1D706}_{2}-\unicode[STIX]{x1D706}_{1}\in P_{\mathbf{y}} \\ \unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}\in \mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}}}+\mathop{\sum }_{\substack{ \unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D706}_{0}\in P_{\mathbf{y}} \\ \unicode[STIX]{x1D706}_{2}-\unicode[STIX]{x1D706}_{1}\in P_{\mathbf{x}} \\ \unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}\in \mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}}}+\mathop{\sum }_{\substack{ \unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D706}_{0}\in P_{\mathbf{y}} \\ \unicode[STIX]{x1D706}_{2}-\unicode[STIX]{x1D706}_{1}\in P_{\mathbf{y}} \\ \unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}\in \mathbb{Z}\unicode[STIX]{x1D70F}+\mathbb{Z}}}\bigg)\unicode[STIX]{x1D706}_{1}^{-n_{1}}\unicode[STIX]{x1D706}_{2}^{-n_{2}}\nonumber\\ \displaystyle & = & \displaystyle G_{n_{1},n_{2}}(\mathbf{x}\mathbf{x})+G_{n_{1},n_{2}}(\mathbf{x}\mathbf{y})+G_{n_{1},n_{2}}(\mathbf{y}\mathbf{x})+G_{n_{1},n_{2}}(\mathbf{y}\mathbf{y}).\nonumber\end{eqnarray}$$

2.2 Computing the Fourier expansion

In this subsection, we give a Fourier expansion of $G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})$ .

Let us define the $q$ -series $g_{n_{1},\ldots ,n_{r}}(q)$ for integers $n_{1},\ldots ,n_{r}\geqslant 1$ by

(7) $$\begin{eqnarray}g_{n_{1},\ldots ,n_{r}}(q)=\frac{(-2\unicode[STIX]{x1D70B}\sqrt{-1})^{n_{1}+\cdots +n_{r}}}{(n_{1}-1)!\cdots (n_{r}-1)!}\mathop{\sum }_{\substack{ 0<d_{1}<\cdots <d_{r} \\ c_{1},\ldots ,c_{r}\in \mathbb{N} \\ d_{1},\ldots ,d_{r}\in \mathbb{N}}}c_{1}^{n_{1}-1}\cdots c_{r}^{n_{r}-1}q^{c_{1}d_{1}+\cdots +c_{r}d_{r}},\end{eqnarray}$$

which divided by $(-2\unicode[STIX]{x1D70B}\sqrt{-1})^{n_{1}+\cdots +n_{r}}$ was studied in [Reference Bachmann and Kühn2]. We remark that since $g_{n_{1}}(q)$ is the generating series of the divisor function $\unicode[STIX]{x1D70E}_{n_{1}-1}(n)$ up to a scalar factor, the coefficient of $q^{n}$ in the $q$ -series $g_{n_{1},\ldots ,n_{r}}(q)$ is called the multiple divisor sum in [Reference Bachmann and Kühn2] with the opposite convention:

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{n_{1},\ldots ,n_{r}}(n)=\mathop{\sum }_{\substack{ c_{1}d_{1}+\cdots +c_{r}d_{r}=n \\ 0<d_{1}<\cdots <d_{r} \\ c_{1},\ldots ,c_{r}\in \mathbb{N} \\ d_{1},\ldots ,d_{r}\in \mathbb{N}}}c_{1}^{n_{1}-1}\cdots c_{r}^{n_{r}-1},\end{eqnarray}$$

which is regarded as a multiple version of the divisor sum (we do not discuss their properties in this paper). We investigate an algebraic structure related to the $q$ -series $g_{n_{1},\ldots ,n_{r}}(q)$ in a subsequent paper.

To give the Fourier expansion of $G_{n_{1},\ldots ,n_{r}}(w_{1},\ldots ,w_{r})$ , we need the following lemma.

Lemma 2.4. For integers $n_{1},\ldots ,n_{r}\geqslant 2$ , we have

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{q=1}^{r}\mathop{\sum }_{\substack{ k_{1}+\cdots +k_{r}=n_{1}+\cdots +n_{r} \\ k_{i}\geqslant n_{i},k_{q}=1}}\bigg((-1)^{n_{q}+k_{q+1}+\cdots +k_{r}}\mathop{\prod }_{\substack{ j=1 \\ j\neq q}}^{r}\binom{k_{j}-1}{n_{j}-1}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\unicode[STIX]{x1D701}(k_{q-1},k_{q-2},\ldots ,k_{1})\unicode[STIX]{x1D701}(k_{q+1},k_{q+2},\ldots ,k_{r})\bigg)=0,\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D701}(n_{1},\ldots ,n_{r})=1$ when $r=0$ .

Proposition 2.5. For integers $n_{1},\ldots ,n_{r}\geqslant 2$ and a word $w_{1}\cdots w_{r}\in \{\mathbf{x},\mathbf{y}\}^{\ast }$ , we set $N_{t_{m}}=n_{t_{m}}+\cdots +n_{t_{m+1}-1}$ for $m\in \{1,\ldots ,h\}$ where $\{t_{1},\ldots ,t_{h}\}=\unicode[STIX]{x1D70C}(w_{1}\cdots w_{r})$ given by (6) and $t_{h+1}=r+1$ . Then the function $G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r};\unicode[STIX]{x1D70F})$ has the following Fourier expansion:

$$\begin{eqnarray}\displaystyle & & \displaystyle G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})=\unicode[STIX]{x1D701}(n_{1},\ldots ,n_{t_{1}-1})\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\mathop{\sum }_{\substack{ t_{1}\leqslant q_{1}\leqslant t_{2}-1 \\ t_{2}\leqslant q_{2}\leqslant t_{3}-1 \\ \vdots \\ t_{h}\leqslant q_{h}\leqslant r}}\mathop{\sum }_{\substack{ k_{t_{1}}+\cdots +k_{t_{2}-1}=N_{t_{1}} \\ k_{t_{2}}+\cdots +k_{t_{3}-1}=N_{t_{2}} \\ \vdots \\ k_{t_{h}}+\cdots +k_{r}=N_{t_{h}} \\ k_{t_{1}},k_{t_{1}+1},\ldots ,k_{r}\geqslant 2}}\bigg\{(-1)^{\mathop{\sum }_{m=1}^{h}(N_{t_{m}}+n_{q_{m}}+k_{q_{m}+1}+k_{q_{m}+2}+\cdots +k_{q_{m+1}-1})}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\bigg(\mathop{\prod }_{\substack{ j=t_{1} \\ j\neq q_{1},\ldots ,q_{h}}}^{r}\binom{k_{j}-1}{n_{j}-1}\bigg)\bigg(\mathop{\prod }_{m=1}^{h}\unicode[STIX]{x1D701}(\underbrace{k_{q_{m}-1},\ldots ,k_{t_{m}}}_{q_{m}-t_{m}})\unicode[STIX]{x1D701}(\underbrace{k_{q_{m}+1},\ldots ,k_{t_{m+1}-1}}_{t_{m+1}-q_{m}-1})\bigg)\nonumber\\ \displaystyle & & \displaystyle \quad \times \,g_{k_{q_{1}},\ldots ,k_{q_{h}}}(q)\bigg\},\nonumber\end{eqnarray}$$

where $q=e^{2\unicode[STIX]{x1D70B}\sqrt{-1}\unicode[STIX]{x1D70F}}$ , $\unicode[STIX]{x1D701}(n_{1},\ldots ,n_{r})=g_{n_{1},\ldots ,n_{r}}(q)=1$ whenever $r=0$ and $\prod _{\substack{ j=t_{1} \\ j\neq q_{1},\ldots ,q_{h}}}^{r}\binom{k_{j}-1}{n_{j}-1}=1$ when the product is empty, that is, when $\{t_{1},t_{1}+1,\ldots ,r\}=\{q_{1},\ldots ,q_{h}\}$ .

Proof. Put $N=n_{1}+\cdots +n_{r}$ . Using the partial fraction decomposition

$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{1}{(\unicode[STIX]{x1D70F}+m_{1})^{n_{1}}\cdots (\unicode[STIX]{x1D70F}+m_{r})^{n_{r}}}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{q=1}^{r}\mathop{\sum }_{\substack{ k_{1}+\cdots +k_{r}=N \\ k_{1},\ldots ,k_{r}\geqslant 1}}\left(\mathop{\prod }_{j=1}^{q-1}\frac{\binom{k_{j}-1}{n_{j}-1}}{(m_{q}-m_{j})^{k_{j}}}\right)\frac{(-1)^{N+n_{q}}}{(\unicode[STIX]{x1D70F}+m_{q})^{k_{q}}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \left(\mathop{\prod }_{j=q+1}^{r}\frac{(-1)^{k_{j}}\binom{k_{j}-1}{n_{j}-1}}{(m_{j}-m_{q})^{k_{j}}}\right),\nonumber\end{eqnarray}$$

we obtain

(8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F}) & = & \displaystyle \mathop{\sum }_{q=1}^{r}\mathop{\sum }_{\substack{ k_{1}+\cdots +k_{r}=N \\ k_{1},\ldots ,k_{r}\geqslant 1}}\bigg((-1)^{N+n_{q}+k_{q+1}+\cdots +k_{r}}\mathop{\prod }_{\substack{ j=1 \\ j\neq q}}^{r}\binom{k_{j}-1}{n_{j}-1}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\unicode[STIX]{x1D701}(\underbrace{k_{q-1},k_{q-2},\ldots ,k_{1}}_{q-1})\unicode[STIX]{x1D6F9}_{k_{q}}(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D701}(\underbrace{k_{q+1},k_{q+2},\ldots ,k_{r}}_{r-q})\bigg),\end{eqnarray}$$

where the implied interchange of order of summation is justified because the binomial coefficient $\binom{k_{i}-1}{n_{i}-1}$ vanishes if $k_{1}=1$ or $k_{r}=1$ and by Lemma 2.4 the coefficient of $\unicode[STIX]{x1D6F9}_{1}(\unicode[STIX]{x1D70F})$ is zero. Using the Lipschitz formula (4) we easily find that

(9) $$\begin{eqnarray}g_{n_{1},\ldots ,n_{r}}(q)=\mathop{\sum }_{0<d_{1}<\cdots <d_{r}}\unicode[STIX]{x1D6F9}_{n_{1}}(d_{1}\unicode[STIX]{x1D70F})\ldots \unicode[STIX]{x1D6F9}_{n_{r}}(d_{r}\unicode[STIX]{x1D70F})\end{eqnarray}$$

for integers $n_{1},\ldots ,n_{r}\geqslant 2$ . Thus, combining the above formulas (8) and (9) with (5), we have the desired formula.◻

We remark that the formula (8), which in the case of $r=2$ was done in [Reference Gangl, Kaneko and Zagier7, Proof of Theorem 6], is found in [Reference Bouillot3, Theorem 3] and holds when $n_{1},\ldots ,n_{r}\geqslant 1$ with $n_{1},n_{r}\geqslant 2$ , but we use only the formula (8) for $n_{1},\ldots ,n_{r}\geqslant 2$ in this paper.

Let us illustrate a few examples.

Example 2.6. When $r=1$ , we have for $n_{1}>1$

$$\begin{eqnarray}G_{n_{1}}(\mathbf{x};\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D701}(n_{1})\qquad \text{and}\qquad G_{n_{1}}(\mathbf{y};\unicode[STIX]{x1D70F})=\mathop{\sum }_{0<l}\unicode[STIX]{x1D6F9}_{n_{1}}(l\unicode[STIX]{x1D70F})=g_{n_{1}}(q),\end{eqnarray}$$

and hence

$$\begin{eqnarray}G_{n_{1}}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D701}(n_{1})+g_{n_{1}}(q).\end{eqnarray}$$

Example 2.7. We compute the case $r=2$ , which was carried out in [Reference Gangl, Kaneko and Zagier7]. From (5) and (9), it follows

$$\begin{eqnarray}\displaystyle G_{n_{1},n_{2}}(\mathbf{x}\mathbf{x}) & = & \displaystyle \unicode[STIX]{x1D701}(n_{1},n_{2}),\nonumber\\ \displaystyle G_{n_{1},n_{2}}(\mathbf{x}\mathbf{y}) & = & \displaystyle \unicode[STIX]{x1D701}(n_{1})\mathop{\sum }_{0<l}\unicode[STIX]{x1D6F9}_{n_{2}}(l\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D701}(n_{1})g_{n_{2}}(q),\nonumber\\ \displaystyle G_{n_{1},n_{2}}(\mathbf{y}\mathbf{y}) & = & \displaystyle \mathop{\sum }_{0<l_{1}<l_{2}}\unicode[STIX]{x1D6F9}_{n_{1}}(l_{1}\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6F9}_{n_{2}}(l_{2}\unicode[STIX]{x1D70F})=g_{n_{1},n_{2}}(q),\nonumber\end{eqnarray}$$

and using (8), we have

$$\begin{eqnarray}G_{n_{1},n_{2}}(\mathbf{y}\mathbf{x})=\mathop{\sum }_{0<l}\unicode[STIX]{x1D6F9}_{n_{1},n_{2}}(l\unicode[STIX]{x1D70F})=\mathop{\sum }_{\substack{ k_{1}+k_{2}=n_{1}+n_{2} \\ k_{1},k_{2}\geqslant 2}}\mathtt{b}_{n_{1},n_{2}}^{k_{1}}\unicode[STIX]{x1D701}(k_{1})g_{k_{2}}(q),\end{eqnarray}$$

where for integers $n,n^{\prime },k>0$ we set

(10) $$\begin{eqnarray}\mathtt{b}_{n,n^{\prime }}^{k}=(-1)^{n}\binom{k-1}{n-1}+(-1)^{k-n^{\prime }}\binom{k-1}{n^{\prime }-1}.\end{eqnarray}$$

Thus the Fourier expansion of $G_{n_{1},n_{2}}(\unicode[STIX]{x1D70F})$ is given by

(11) $$\begin{eqnarray}G_{n_{1},n_{2}}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D701}(n_{1},n_{2})+\mathop{\sum }_{\substack{ k_{1}+k_{2}=n_{1}+n_{2} \\ k_{1},k_{2}\geqslant 2}}\big(\unicode[STIX]{x1D6FF}_{n_{1},k_{1}}+\mathtt{b}_{n_{1},n_{2}}^{k_{1}}\big)\unicode[STIX]{x1D701}(k_{1})g_{k_{2}}(q)+g_{n_{1},n_{2}}(q),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{n,k}$ is the Kronecker delta.

Example 2.8. For the future literature, we present the Fourier expansion of $G_{n_{1},n_{2},n_{3}}(\unicode[STIX]{x1D70F})$ with $n_{1},n_{2},n_{3}\geqslant 2$ :

$$\begin{eqnarray}\displaystyle & & \displaystyle G_{n_{1},n_{2},n_{3}}(\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D701}(n_{1},n_{2},n_{3})+\unicode[STIX]{x1D701}(n_{1},n_{2})g_{n_{3}}(q)+\unicode[STIX]{x1D701}(n_{1})g_{n_{2},n_{3}}(q)+g_{n_{1},n_{2},n_{3}}(q)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\mathop{\sum }_{\substack{ k_{1}+k_{2}+k_{3}=n_{1}+n_{2}+n_{3} \\ k_{1},k_{2},k_{3}\geqslant 2}}\bigg\{\!(\unicode[STIX]{x1D6FF}_{n_{3},k_{3}}\mathtt{b}_{n_{1},n_{2}}^{k_{1}}+\unicode[STIX]{x1D6FF}_{n_{1},k_{2}}\mathtt{b}_{n_{2},n_{3}}^{k_{1}})\unicode[STIX]{x1D701}(k_{1})g_{k_{2},k_{3}}(q)\nonumber\\ \displaystyle & & \displaystyle \qquad +\left((-1)^{n_{1}+k_{3}}\binom{k_{2}-1}{n_{3}-1}+(-1)^{n_{1}+n_{2}}\binom{k_{2}-1}{n_{1}-1}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\binom{k_{1}-1}{n_{2}-1}\unicode[STIX]{x1D701}(k_{1},k_{2})g_{k_{3}}(q)\nonumber\\ \displaystyle & & \displaystyle \qquad +\left((-1)^{n_{1}+n_{3}+k_{2}}\binom{k_{1}-1}{n_{1}-1}\binom{k_{2}-1}{n_{3}-1}\right.\nonumber\\ \displaystyle & & \displaystyle \left.\qquad +\,\unicode[STIX]{x1D6FF}_{k_{1},n_{1}}\mathtt{b}_{n_{2},n_{3}}^{k_{2}}\right)\unicode[STIX]{x1D701}(k_{1})\unicode[STIX]{x1D701}(k_{2})g_{k_{3}}(q)\bigg\}.\nonumber\end{eqnarray}$$

3.1 Regularized multiple zeta values

In this subsection, we recall the regularized multiple zeta value with respect to the shuffle product defined in [Reference Ihara, Kaneko and Zagier10]. We first review an iterated integral expression of the multiple zeta value due to Kontsevich and Drinfel’d, and then recall the algebraic setup of multiple zeta values given by Hoffman.

We denote by $\unicode[STIX]{x1D714}_{0}(t)=\frac{dt}{t}$ and $\unicode[STIX]{x1D714}_{1}(t)=\frac{dt}{1-t}$ holomorphic 1-forms on the smooth manifold $\mathbb{P}_{\mathbb{C}}^{1}\backslash \{0,1,\infty \}$ . For integers $n_{1},\ldots ,n_{r-1}\geqslant 1$ and $n_{r}\geqslant 2$ with $N=n_{1}+\cdots +n_{r}$ , the multiple zeta value $\unicode[STIX]{x1D701}(n_{1},\ldots ,n_{r})$ is expressible as an iterated integral on the smooth manifold $\mathbb{P}_{\mathbb{C}}^{1}\backslash \{0,1,\infty \}$ :

(12) $$\begin{eqnarray}\unicode[STIX]{x1D701}(n_{1},\ldots ,n_{r})=\int \cdots\int _{0<t_{1}<t_{2}<\cdots <t_{N}<1}\unicode[STIX]{x1D714}_{a_{1}}(t_{1})\wedge \unicode[STIX]{x1D714}_{a_{2}}(t_{2})\wedge \cdots \wedge \unicode[STIX]{x1D714}_{a_{N}}(t_{N}),\end{eqnarray}$$

where $a_{i}=1$ if $i\in \{1,n_{1}+1,n_{1}+n_{2}+1,\ldots ,n_{1}+\cdots +n_{r-1}+1\}$ and $a_{i}=0$ otherwise.

Let $\mathfrak{H}=\mathbb{Q}\langle e_{0},e_{1}\rangle$ be the noncommutative polynomial algebra in two indeterminates $e_{0}$ and $e_{1}$ , and $\mathfrak{H}^{1}:=\mathbb{Q}+e_{1}\mathfrak{H}$ and $\mathfrak{H}^{0}:=\mathbb{Q}+e_{1}\mathfrak{H}e_{0}$ its subalgebras. Set

$$\begin{eqnarray}y_{n}:=e_{1}e_{0}^{n-1}=e_{1}\underbrace{e_{0}\cdots e_{0}}_{n-1}\end{eqnarray}$$

for each positive integer $n>0$ . It is easily seen that the subalgebra $\mathfrak{H}^{1}$ is freely generated by $y_{n}$ ’s $(n\geqslant 1)$ as a noncommutative polynomial algebra:

$$\begin{eqnarray}\mathfrak{H}^{1}=\mathbb{Q}\langle y_{1},y_{2},y_{3},\ldots \rangle .\end{eqnarray}$$

We define the shuffle product, a $\mathbb{Q}$ -bilinear product on $\mathfrak{H}$ , inductively by

$$\begin{eqnarray}\displaystyle uw~\unicode[STIX]{x0428}~vw^{\prime }=u(w~\unicode[STIX]{x0428}~vw^{\prime })+v(uw~\unicode[STIX]{x0428}~w^{\prime }), & & \displaystyle \nonumber\end{eqnarray}$$

with the initial condition $w~\unicode[STIX]{x0428}~1=1~\unicode[STIX]{x0428}~w=w$ $(1\in \mathbb{Q})$ , where $w,w^{\prime }\in \mathfrak{H}$ and $u,v\in \{e_{0},e_{1}\}$ . This provides the structures of commutative $\mathbb{Q}$ -algebras for spaces $\mathfrak{H},\mathfrak{H}^{1}$ and $\mathfrak{H}^{0}$ (see [Reference Reutenauer14]), which we denote by $\mathfrak{H}_{\unicode[STIX]{x0428}},\mathfrak{H}_{\unicode[STIX]{x0428}}^{1}$ and $\mathfrak{H}_{\unicode[STIX]{x0428}}^{0}$ , respectively. By taking the iterated integral (12), with the identification $w_{i}(t)\leftrightarrow e_{i}~(i\in \{0,1\})$ , one can define an algebra homomorphism

$$\begin{eqnarray}\displaystyle Z:\mathfrak{H}_{\unicode[STIX]{x0428}}^{0} & \longrightarrow & \displaystyle \mathbb{R}\nonumber\\ \displaystyle y_{n_{1}}\cdots y_{n_{r}} & \longmapsto & \displaystyle \unicode[STIX]{x1D701}(n_{1},\ldots ,n_{r})\quad (n_{r}>1)\nonumber\end{eqnarray}$$

with $Z(1)=1$ , since it is shown by Chen [Reference Chen5] that the iterated integral (12) satisfies the shuffle product formulas. By [Reference Ihara, Kaneko and Zagier10, Proposition 1], there is a $\mathbb{Q}$ -algebra homomorphism

$$\begin{eqnarray}Z^{\unicode[STIX]{x0428}}:\mathfrak{H}_{\unicode[STIX]{x0428}}^{1}\rightarrow \mathbb{R}[T]\end{eqnarray}$$

which is uniquely determined by the properties that $Z^{\unicode[STIX]{x0428}}\big|_{\mathfrak{H}_{\unicode[STIX]{x0428}}^{0}}=Z$ and $Z^{\unicode[STIX]{x0428}}(e_{1})=T$ . We note that the image of the word $y_{n_{1}}\cdots y_{n_{r}}$ in $\mathfrak{H}_{\unicode[STIX]{x0428}}^{1}$ under the map $Z^{\unicode[STIX]{x0428}}$ is a polynomial in $T$ whose coefficients are expressed as $\mathbb{Q}$ -linear combinations of multiple zeta values.

Definition 3.1. The regularized multiple zeta value, denoted by $\unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(n_{1},\ldots ,n_{r})$ , is defined as the constant term of $Z^{\unicode[STIX]{x0428}}(y_{n_{1}}\cdots y_{n_{r}})$ in $T$ :

$$\begin{eqnarray}\unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(n_{1},\ldots ,n_{r}):=Z^{\unicode[STIX]{x0428}}(y_{n_{1}}\cdots y_{n_{r}})\big|_{T=0}.\end{eqnarray}$$

For example, we have $\unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(2,1)=-2\unicode[STIX]{x1D701}(1,2)$ and

(13) $$\begin{eqnarray}\unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(n_{1},\ldots ,n_{r})=\unicode[STIX]{x1D701}(n_{1},\ldots ,n_{r})\quad (n_{r}\geqslant 2).\end{eqnarray}$$

3.2 Hopf algebras of iterated integrals

In this subsection, we recall Hopf algebras of formal iterated integrals introduced by Goncharov.

In his paper [Reference Goncharov8, Section 2], Goncharov considered a formal version of the iterated integrals

(14) $$\begin{eqnarray}\int _{a_{0}}^{a_{N+1}}\frac{dt_{1}}{t_{1}-a_{1}}\cdots \int _{t_{N-2}}^{a_{N+1}}\frac{dt_{N-1}}{t_{N-1}-a_{N-1}}\int _{t_{N-1}}^{a_{N+1}}\frac{dt_{N}}{t_{N}-a_{N}}\quad (a_{i}\in \mathbb{C}).\end{eqnarray}$$

He proved that the space ${\mathcal{I}}_{\bullet }(S)$ generated by such formal iterated integrals carries a Hopf algebra structure. Let us recall the definition of the space ${\mathcal{I}}_{\bullet }(S)$ .

Definition 3.2. Let $S$ be a set. Let us denote by ${\mathcal{I}}_{\bullet }(S)$ the commutative graded algebra over $\mathbb{Q}$ generated by the elements

$$\begin{eqnarray}\mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+1}),\quad N\geqslant 0,~a_{i}\in S\end{eqnarray}$$

with degree $N$ which are subject to the following relations.

1. (I1) For any $a,b\in S$ , the unit is given by $\mathbb{I}(a;b):=\mathbb{I}(a;\emptyset ;b)=1$ .

2. (I2) The product is given by the shuffle product: for all integers $N,N^{\prime }\geqslant 0$ and $a_{i}\in S$ , one has

   $$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+N^{\prime }+1})\mathbb{I}(a_{0};a_{N+1},\ldots ,a_{N+N^{\prime }};a_{N+N^{\prime }+1})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}(N,N^{\prime })}\mathbb{I}(a_{0};a_{\unicode[STIX]{x1D70E}^{-1}(1)},\ldots ,a_{\unicode[STIX]{x1D70E}^{-1}(N+N^{\prime })};a_{N+N^{\prime }+1}),\nonumber\end{eqnarray}$$
   where $\unicode[STIX]{x1D6F4}(N,N^{\prime })$ is the set of $\unicode[STIX]{x1D70E}$ in the symmetric group $\mathfrak{S}_{N+N^{\prime }}$ such that $\unicode[STIX]{x1D70E}(1)<\cdots <\unicode[STIX]{x1D70E}(N)$ and $\unicode[STIX]{x1D70E}(N+1)<\cdots <\unicode[STIX]{x1D70E}(N+N^{\prime })$ .

3. (I3) The path composition formula holds: for any $N\geqslant 0$ and $a_{i},x\in S$ , one has

   $$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+1})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{k=0}^{N}\mathbb{I}(a_{0};a_{1},\ldots ,a_{k};x)\mathbb{I}(x;a_{k+1},\ldots ,a_{N};a_{N+1}).\nonumber\end{eqnarray}$$

4. (I4) For $N\geqslant 1$ and $a_{i},a\in S$ , $\mathbb{I}(a;a_{1},\ldots ,a_{N};a)=0$ .

We remark that the element $\mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+1})$ is an analogue of the iterated integral (14), since by Chen [Reference Chen5] iterated integrals satisfy (I1) to (I4) when the integral converges.

The Goncharov coproduct $\unicode[STIX]{x1D6E5}:{\mathcal{I}}_{\bullet }(S)\rightarrow {\mathcal{I}}_{\bullet }(S)\otimes {\mathcal{I}}_{\bullet }(S)$ is defined by

(15) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6E5}\big(\mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+1})\big)\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ 0\leqslant k\leqslant N \\ i_{0}=0<i_{1}<\cdots <i_{k}<i_{k+1}=N+1}}\mathop{\prod }_{p=0}^{k}\mathbb{I}(a_{i_{p}};a_{i_{p}+1},\ldots ,a_{i_{p+1}-1};a_{i_{p+1}})\nonumber\\ \displaystyle & & \displaystyle \qquad \otimes \,\mathbb{I}(a_{0};a_{i_{1}},\ldots ,a_{i_{k}};a_{N+1}),\end{eqnarray}$$

for any $N\geqslant 0$ and $a_{i}\in S$ , and then extending by $\mathbb{Q}$ -linearity. The formula (15) can be found in [Reference Brown4] (see Equation (2.18)), originally given by Goncharov (see [Reference Goncharov8, Equation (27)]) with the factors interchanged.

We remark that the antipode ${\mathcal{S}}$ of the above Hopf algebra is uniquely and inductively determined by the definition (see [Reference Goncharov8, Lemma A.1]). For example, since $\unicode[STIX]{x1D6E5}(\mathbb{I}(a_{0};a_{1};a_{2}))=\mathbb{I}(a_{0};a_{1};a_{2})\otimes 1+1\otimes \mathbb{I}(a_{0};a_{1};a_{2})$ for any $a_{0},a_{1},a_{2}\in S$ , we have

$$\begin{eqnarray}{\mathcal{S}}(\mathbb{I}(a_{0};a_{1};a_{2}))+\mathbb{I}(a_{0};a_{1};a_{2})=0=u\circ \unicode[STIX]{x1D700}(\mathbb{I}(a_{0};a_{1};a_{2})),\end{eqnarray}$$

where $u:\mathbb{Q}\rightarrow {\mathcal{I}}_{\bullet }(S)$ is the unit. In general, the formula is obtained from the fact that

$$\begin{eqnarray}\displaystyle & & \displaystyle {\mathcal{S}}(\mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+1}))+\mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+1})\nonumber\\ \displaystyle & & \displaystyle \quad =\text{a}~\mathbb{Z}\text{-linear combination of products of }\mathbb{I}\text{'s of degree }<N,\nonumber\end{eqnarray}$$

which we do not develop the precise formula in this paper.

3.3 Formal iterated integrals and regularized multiple zeta values

In this subsection, we define the map $\mathfrak{z}^{\unicode[STIX]{x0428}}$ described in the introduction. Hereafter, we restrict only the Hopf algebra ${\mathcal{I}}_{\bullet }:={\mathcal{I}}_{\bullet }(S)$ to the case with $S=\{0,1\}$ .

Consider the quotient algebra

$$\begin{eqnarray}{\mathcal{I}}_{\bullet }^{1}:={\mathcal{I}}_{\bullet }/\mathbb{I}(0;0;1){\mathcal{I}}_{\bullet }.\end{eqnarray}$$

It is easy to verify that $\mathbb{I}(0;0;1)$ is primitive, that is, $\unicode[STIX]{x1D6E5}(\mathbb{I}(0;0;1))=1\otimes \mathbb{I}(0;0;1)+\mathbb{I}(0;0;1)\otimes 1$ . Thus the ideal $\mathbb{I}(0;0;1){\mathcal{I}}_{\bullet }$ generated by $\mathbb{I}(0;0;1)$ in the $\mathbb{Q}$ -algebra ${\mathcal{I}}_{\bullet }$ becomes a Hopf ideal, and hence the quotient map ${\mathcal{I}}_{\bullet }\rightarrow {\mathcal{I}}_{\bullet }^{1}$ induces a Hopf algebra structure on the quotient algebra ${\mathcal{I}}_{\bullet }^{1}$ . Let us denote by

$$\begin{eqnarray}I(a_{0};a_{1},\ldots ,a_{N};a_{N+1})\in {\mathcal{I}}_{\bullet }^{1}\end{eqnarray}$$

an image of $\mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+1})$ in ${\mathcal{I}}_{\bullet }^{1}$ and by the same symbol $\unicode[STIX]{x1D6E5}$ the induced coproduct on ${\mathcal{I}}_{\bullet }^{1}$ given by the same formula as (15) replacing $\mathbb{I}$ with $I$ . As a result, we have the following proposition which we use later.

We remark that dividing ${\mathcal{I}}_{\bullet }$ by $\mathbb{I}(0;0;1){\mathcal{I}}_{\bullet }$ can be viewed as a regularization for “ $\int _{0}^{1}dt/t=-\log (0)$ ” which plays a role as $\mathbb{I}(0;0;1)$ in the evaluation of iterated integrals. For example, one can write $I(0;0,1,0;1)=-2I(0;1,0,0;1)$ in ${\mathcal{I}}_{\bullet }^{1}$ since it follows $\mathbb{I}(0;0,1,0;1)=\mathbb{I}(0;0;1)\mathbb{I}(0;1,0;1)-2\mathbb{I}(0;1,0,0;1)$ , and this computation corresponds to taking the constant term of $\int _{\unicode[STIX]{x1D700}}^{1}\frac{dt_{1}}{t_{1}}\int _{\unicode[STIX]{x1D700}}^{t_{1}}\frac{dt_{2}}{1-t_{2}}\int _{\unicode[STIX]{x1D700}}^{t_{2}}\frac{dt_{3}}{t_{3}}$ as a polynomial of $\log (\unicode[STIX]{x1D700})$ and letting $\unicode[STIX]{x1D700}\rightarrow 0$ .

By the standard calculation about the shuffle product formulas, we obtain more identities in the space ${\mathcal{I}}_{\bullet }^{1}$ (see [Reference Brown4, p. 955]).

1. (1) For $n\geqslant 1$ and $a,b\in \{0,1\}$ , we have

   (16) $$\begin{eqnarray}I(a;\underbrace{0,\ldots ,0}_{n};b)=0.\end{eqnarray}$$

2. (2) For integers $n\geqslant 0,n_{1},\ldots ,n_{r}\geqslant 1$ , we have

   (17) $$\begin{eqnarray}\displaystyle & & \displaystyle I(0;\underbrace{0,\ldots ,0}_{n},\underbrace{1,0,\ldots ,0}_{n_{1}},\ldots ,\underbrace{1,0,\ldots ,0}_{n_{r}};1)\nonumber\\ \displaystyle & & \displaystyle \quad =(-1)^{n}\mathop{\sum }_{\substack{ k_{1}+\cdots +k_{r}=n_{1}+\cdots +n_{r}+n \\ k_{1},\ldots ,k_{r}\geqslant 1}}\bigg(\mathop{\prod }_{j=1}^{r}\binom{k_{j}-1}{n_{j}-1}\bigg)I(k_{1},\ldots ,k_{r}),\end{eqnarray}$$
   where we set
   $$\begin{eqnarray}I(n_{1},\ldots ,n_{r}):=I(0;\underbrace{1,0,\ldots ,0}_{n_{1}},\ldots ,\underbrace{1,0,\ldots ,0}_{n_{r}};1).\end{eqnarray}$$

In order to define the map $\mathfrak{z}^{\unicode[STIX]{x0428}}$ as a $\mathbb{Q}$ -linear map, we give a linear basis of the space ${\mathcal{I}}_{\bullet }^{1}$ first.

Proof. Recall the result of Goncharov [Reference Goncharov8, Proposition 2.1]: for each integer $N\geqslant 0$ and $a_{0},\ldots ,a_{N+1}\in \{0,1\}$ one has

(18) $$\begin{eqnarray}\mathbb{I}(a_{0};a_{1},\ldots ,a_{N};a_{N+1})=(-1)^{N}\mathbb{I}(a_{N+1};a_{N},\ldots ,a_{1};a_{0}),\end{eqnarray}$$

which essentially follows from (I3) and (I4). Then, we find that the collection

$$\begin{eqnarray}\{\mathbb{I}(0;a_{1},\ldots ,a_{N};1)\mid N\geqslant 0,a_{i}\in \{0,1\}\}\end{eqnarray}$$

forms a linear basis of the linear space ${\mathcal{I}}_{\bullet }$ , since none of the relations (I1) to (I4) yield $\mathbb{Q}$ -linear relations among them. Combining this with (17), we obtain the desired basis.◻

Definition 3.6. Let $\mathfrak{z}^{\unicode[STIX]{x0428}}:{\mathcal{I}}_{\bullet }^{1}\rightarrow \mathbb{R}$ be the $\mathbb{Q}$ -linear map given by

$$\begin{eqnarray}\displaystyle \mathfrak{z}^{\unicode[STIX]{x0428}}:{\mathcal{I}}_{\bullet }^{1} & \longrightarrow & \displaystyle \mathbb{R}\nonumber\\ \displaystyle I(n_{1},\ldots ,n_{r}) & \longmapsto & \displaystyle \unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(n_{1},\ldots ,n_{r})\nonumber\end{eqnarray}$$

and $\mathfrak{z}^{\unicode[STIX]{x0428}}(1)=1$ .

3.4 Computing the Goncharov coproduct

In this subsection, we rewrite the Goncharov coproduct $\unicode[STIX]{x1D6E5}$ for $I(n_{1},\ldots ,n_{r})$ in terms of $I(k_{1},\ldots ,k_{i})$ ’s. Although one can compute the formula by using Propositions 3.8 and 3.12, we do not give explicit formulas for $\unicode[STIX]{x1D6E5}(I(n_{1},\ldots ,n_{r}))$ in general. We present an explicit formula for only $\unicode[STIX]{x1D6E5}(I(n_{1},n_{2},n_{3}))$ in the end of this subsection.

To describe the formula, it is convenient to use the algebraic setup. Let $\mathfrak{H}^{\prime }:=\langle e_{0},e_{1},e_{0}^{\prime },e_{1}^{\prime }\rangle$ be the noncommutative polynomial algebra in four indeterminates $e_{0},e_{1},e_{0}^{\prime }$ and $e_{1}^{\prime }$ . For integers $0<i_{1}<i_{2}<\cdots <i_{k}<N+1$   $(0\leqslant k\leqslant N)$ , we define

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbf{e}_{i_{1},\ldots ,i_{k}}(a_{1},\ldots ,a_{N})\nonumber\\ \displaystyle & & \displaystyle \quad :=e_{a_{1}}\cdots e_{a_{i_{1}-1}}\left(\mathop{\prod }_{p=1}^{k-1}e_{a_{i_{p}}}^{\prime }e_{a_{i_{p}+1}}\cdots e_{a_{i_{p+1}-1}}\right)e_{a_{i_{k}}}^{\prime }e_{a_{i_{k}+1}}\cdots e_{a_{N}}\nonumber\end{eqnarray}$$

with each $a_{j}=0$ or $1$ , where the product $\prod _{p=1}^{k-1}$ means the concatenation product. Let

$$\begin{eqnarray}\unicode[STIX]{x1D711}:\mathfrak{H}^{\prime }\rightarrow {\mathcal{I}}_{\bullet }^{1}\otimes {\mathcal{I}}_{\bullet }^{1}\end{eqnarray}$$

be the $\mathbb{Q}$ -linear map that assigns to each word $\mathbf{e}_{i_{1},\ldots ,i_{k}}(a_{1},\ldots ,a_{N})$ the factor of the right-hand side of the equation (15) with $a_{0}=0$ and $a_{N+1}=1$ :

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D711}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(a_{1},\ldots ,a_{N}))\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\prod }_{p=0}^{k}I(a_{i_{p}};a_{i_{p}+1},\ldots ,a_{i_{p+1}-1};a_{i_{p+1}})\otimes I(0;a_{i_{1}},\ldots ,a_{i_{k}};1),\nonumber\end{eqnarray}$$

where we set $a_{i_{0}}=0$ and $a_{i_{k+1}}=1$ . For example, we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(\mathbf{e}_{2,3}(a_{1},\ldots ,a_{4})) & = & \displaystyle \unicode[STIX]{x1D711}(e_{a_{1}}e_{a_{2}}^{\prime }e_{a_{3}}^{\prime }e_{a_{4}})\nonumber\\ \displaystyle & = & \displaystyle I(0;a_{1};a_{2})I(a_{2};a_{3})I(a_{3};a_{4};1)\otimes I(0;a_{2},a_{3};1).\nonumber\end{eqnarray}$$

In the rest of this subsection, for integers $n_{1},\ldots ,n_{r}\geqslant 1$ with $N=n_{1}+\cdots +n_{r}$ , we set

$$\begin{eqnarray}\{a_{1},\ldots ,a_{N}\}=\{1,\underbrace{0,\ldots ,0}_{n_{1}-1},1,\underbrace{0,\ldots ,0}_{n_{2}-1},\ldots ,1,\underbrace{0,\ldots ,0}_{n_{r}-1}\},\end{eqnarray}$$

and write $\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r}):=\mathbf{e}_{i_{1},\ldots ,i_{k}}(a_{1},\ldots ,a_{N})$ . We note that $a_{j}=1$ if $j$ lies in the set

$$\begin{eqnarray}P_{n_{1},\ldots ,n_{r}}:=\{1,n_{1}+1,\ldots ,n_{1}+\cdots +n_{r-1}+1\},\end{eqnarray}$$

and $a_{j}=0$ otherwise.

Using the above notations, one has

(19) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}(I(n_{1},\ldots ,n_{r}))=\mathop{\sum }_{k=0}^{N}\mathop{\sum }_{0<i_{1}<\cdots <i_{k}<N+1}\unicode[STIX]{x1D711}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r})).\end{eqnarray}$$

To compute (19), we split the right-hand side of (19) into $2^{r}$ sums of $\unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})$ defined below.

Define the map

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{n_{1},\ldots ,n_{r}}:2^{\{1,2,\ldots ,r\}}\longrightarrow 2^{P_{n_{1},\ldots ,n_{r}}}\end{eqnarray}$$

given by $1\mapsto 1$ and $i\mapsto n_{1}+\cdots +n_{i-1}+1~(2\leqslant i\leqslant r)$ , where for a set $X$ the set of all subsets of $X$ is denoted by $2^{X}$ . It follows that the map $\unicode[STIX]{x1D704}_{n_{1},\ldots ,n_{r}}$ is bijective. Let

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D70C}}_{n_{1},\ldots ,n_{r}}:=\unicode[STIX]{x1D704}_{n_{1},\ldots ,n_{r}}\circ \unicode[STIX]{x1D70C}:\{\mathbf{x},\mathbf{y}\}_{r}^{\ast }\longrightarrow 2^{P_{n_{1},\ldots ,n_{r}}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is defined in (6). The map $\overline{\unicode[STIX]{x1D70C}}$ is also bijective. For instance, we have $\overline{\unicode[STIX]{x1D70C}}_{n_{1},\ldots ,n_{r}}(\mathbf{y}\mathbf{x}\mathbf{y}\mathbf{x}^{r-3})=\{1,n_{1}+n_{2}+1\}$ and $\overline{\unicode[STIX]{x1D70C}}_{n_{1},\ldots ,n_{r}}(\mathbf{x}^{r})=\{\emptyset \}$ . With the map $\overline{\unicode[STIX]{x1D70C}}_{n_{1},\ldots ,n_{r}}$ , for integers $n_{1},\ldots ,n_{r}\geqslant 1$ and a word $w_{1}\cdots w_{r}$ ( $w_{i}\in \{\mathbf{x},\mathbf{y}\}$ ) we define

(20) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})\nonumber\\ \displaystyle & & \displaystyle \quad :=\mathop{\sum }_{k=h}^{N}\mathop{\sum }_{\substack{ 0<i_{1}<\cdots <i_{k}<N+1 \\ \{i_{1},\ldots ,i_{k}\}\cap P_{n_{1},\ldots ,n_{r}}=\overline{\unicode[STIX]{x1D70C}}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})}}\unicode[STIX]{x1D711}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r})),\end{eqnarray}$$

where $h$ is the number of $\mathbf{y}$ ’s in the word $w_{1}\cdots w_{r}$ (i.e., $h=\deg _{\mathbf{y}}(w_{1}\cdots w_{r})$ ).

Proposition 3.8. For integers $n_{1},\ldots ,n_{r}\geqslant 1$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}\big(I(n_{1},\ldots ,n_{r})\big)=\mathop{\sum }_{w_{1},\ldots ,w_{r}\in \{\mathbf{x},\mathbf{y}\}}\unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r}).\end{eqnarray}$$

Proof. For the word $\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r})$ , we denote by $h$ the number of $e_{1}^{\prime }$ ’s in the prime symbols $e_{a_{i_{1}}}^{\prime },\ldots ,e_{a_{i_{k}}}^{\prime }$ , that is,

$$\begin{eqnarray}h=\deg _{e_{1}^{\prime }}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r})).\end{eqnarray}$$

Since $a_{j}=1$ if and only if $j\in P_{n_{1},\ldots ,n_{r}}$ , we have

$$\begin{eqnarray}h=\sharp (\{i_{1},\ldots ,i_{k}\}\cap P_{n_{1},\ldots ,n_{r}}).\end{eqnarray}$$

We notice that $h$ can be chosen from $\{0,1,\ldots ,\min \{r,k\}\!\}$ for each $k$ . With this, the formula (19) can be written in the form

$$\begin{eqnarray}\displaystyle (19) & = & \displaystyle \mathop{\sum }_{k=0}^{N}\mathop{\sum }_{h=0}^{\min \{r,k\}}\mathop{\sum }_{\substack{ 0<i_{1}<\cdots <i_{k}<N+1 \\ \sharp (\{i_{1},\ldots ,i_{k}\}\cap P_{n_{1},\ldots ,n_{r}})=h}}\unicode[STIX]{x1D711}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r}))\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{h=0}^{r}\mathop{\sum }_{k=h}^{N}\mathop{\sum }_{\substack{ 0<i_{1}<\cdots <i_{k}<N+1 \\ \sharp (\{i_{1},\ldots ,i_{k}\}\cap P_{n_{1},\ldots ,n_{r}})=h}}\unicode[STIX]{x1D711}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r})).\nonumber\end{eqnarray}$$

By specifying the subset of $P_{n_{1},\ldots ,n_{r}}$ with length $h$ , the above third sum can be split into the following sums:

$$\begin{eqnarray}\displaystyle (19) & = & \displaystyle \mathop{\sum }_{h=0}^{r}\mathop{\sum }_{k=h}^{N}\mathop{\sum }_{\substack{ \{j_{1},\ldots ,j_{h}\}\subset P_{n_{1},\ldots ,n_{r}} \\ j_{1}<\cdots <j_{h}}}\mathop{\sum }_{\substack{ 0<i_{1}<\cdots <i_{k}<N+1 \\ \{i_{1},\ldots ,i_{k}\}\cap P_{n_{1},\ldots ,n_{r}}=\{j_{1},\ldots ,j_{h}\}}}\!\!\!\!\!\!\!\!\!\unicode[STIX]{x1D711}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r}))\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{h=0}^{r}\mathop{\sum }_{\substack{ \{j_{1},\ldots ,j_{h}\}\subset P_{n_{1},\ldots ,n_{r}} \\ j_{1}<\cdots <j_{h}}}\mathop{\sum }_{k=h}^{N}\mathop{\sum }_{\substack{ 0<i_{1}<\cdots <i_{k}<N+1 \\ \{i_{1},\ldots ,i_{k}\}\cap P_{n_{1},\ldots ,n_{r}}=\{j_{1},\ldots ,j_{h}\}}}\!\!\!\!\!\!\!\!\!\unicode[STIX]{x1D711}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r}))\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{h=0}^{r}\mathop{\sum }_{\substack{ w_{1},\ldots ,w_{r}\in \{\mathbf{x},\mathbf{y}\} \\ \deg _{\mathbf{y}}w_{1}\cdots w_{r}=h}}\unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{w_{1},\ldots ,w_{r}\in \{\mathbf{x},\mathbf{y}\}}\unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r}),\nonumber\end{eqnarray}$$

which completes the proof. ◻

We express (20) as algebraic combinations of $I(k_{1},\ldots ,k_{i})$ ’s. To do this, we extract possible nonzero terms from the right-hand side of (20) by using (I4). For a positive integer $n$ , we define $\unicode[STIX]{x1D702}_{0}(n)$ as the sum of all words of degree $n-1$ consisting of $e_{0}$ and a consecutive $e_{0}^{\prime }$ :

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{0}(n)=\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}+k+\unicode[STIX]{x1D6FD}=n \\ \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\geqslant 0 \\ k\geqslant 1}}e_{0}^{\unicode[STIX]{x1D6FC}}(e_{0}^{\prime })^{k-1}e_{0}^{\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$

Proposition 3.9. For integers $n_{1},\ldots ,n_{r}\geqslant 1$ and a word $w_{1}\cdots w_{r}$ of length $r$ in $\{\mathbf{x},\mathbf{y}\}^{\ast }$ , we have

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ t_{1}\leqslant q_{1}\leqslant t_{2}-1 \\ t_{2}\leqslant q_{2}\leqslant t_{3}-1 \\ \vdots \\ t_{h}\leqslant q_{h}\leqslant r}}\unicode[STIX]{x1D711}\biggl(\!y_{n_{1}}\cdots y_{n_{t_{1}-1}}\mathop{\prod }_{m=1}^{h}\big(\!e_{1}^{\prime }\underbrace{e_{0}^{n_{t_{m}}-1}y_{n_{t_{m}+1}}\cdots y_{n_{q_{m-1}}}e_{1}}_{\text{degree in}~e_{1}=q_{m}-t_{m}}\unicode[STIX]{x1D702}_{0}(n_{q_{m}})y_{n_{q_{m}+1}}\cdots y_{n_{t_{m+1}-1}}\!\big)\!\biggr)\!,\nonumber\end{eqnarray}$$

where $\{t_{1},\ldots ,t_{h}\}=\unicode[STIX]{x1D70C}(w_{1}\cdots w_{r})$ given by (6),  $t_{h+1}=r+1$ and the product $\prod _{m=1}^{h}$ means the concatenation product of words.

Proof. It follows $\unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(\mathbf{x}^{r})=\unicode[STIX]{x1D711}(y_{n_{1}}\cdots y_{n_{r}})$ , so we consider the case $h>0$ which means the number of $\mathbf{y}$ ’s in $w_{1}\cdots w_{r}$ is greater than 0. We note that the sum defining (20) runs over all words $c_{a_{1}}\cdots c_{a_{N}}$ ( $c_{a_{i}}\in \{e_{a_{i}},e_{a_{i}}^{\prime }\}$ ) with $k$ ( $h\leqslant k\leqslant N$ ) prime symbols whose positions of $e_{1}^{\prime }$ ’s are placed on the set $\{j_{1},\ldots ,j_{h}\}=\overline{\unicode[STIX]{x1D70C}}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})$ :

$$\begin{eqnarray}(20)=\mathop{\sum }_{k=h}^{N}\mathop{\sum }_{\substack{ w_{0},w_{1},\ldots ,w_{h}\in \{e_{0},e_{1},e_{0}^{\prime }\}^{\ast } \\ \deg _{e_{0}^{\prime }}(w_{0}w_{1}\cdots w_{h})=k-h \\ \deg (w_{0})=j_{1}-1 \\ \deg (e_{1}^{\prime }w_{m})=j_{m+1}-j_{m}~(1\leqslant m\leqslant h) \\ j_{h+1}=N+1}}\unicode[STIX]{x1D711}\Big(w_{0}\mathop{\prod }_{m=1}^{h}\big(e_{1}^{\prime }w_{m}\big)\Big).\end{eqnarray}$$

We find by (I4) that $\unicode[STIX]{x1D711}(\mathbf{e}_{i_{1},\ldots ,i_{k}}(n_{1},\ldots ,n_{r}))$ is 0 whenever $a_{i_{1}}=0$ (notice that $a_{0}=0$ and $a_{1}=1$ ). This implies that if the degree of the above $w_{0}$ in the letter $e_{0}^{\prime }$ is greater than 0, then $\unicode[STIX]{x1D711}\big(w_{0}\prod _{m=1}^{h}(e_{1}^{\prime }w_{m})\big)=0$ . For a word $w\in \mathfrak{H}^{\prime }$ , we also find $\unicode[STIX]{x1D711}(w)=0$ if $w$ contains a subword of the form $e_{0}^{\prime }ve_{0}^{\prime }$ with $v\in \mathfrak{H}~(v\neq \emptyset )$ , that is, $w=w_{1}e_{0}^{\prime }ve_{0}^{\prime }w_{2}$ for some $w_{1},w_{2}\in \mathfrak{H}^{\prime }$ , because the factor of the left-hand side of $\unicode[STIX]{x1D711}(w)$ involves $I(0;v;0)$ which by (I4) is 0. This implies that the above second sum regarding $w_{m}$ ( $1\leqslant m\leqslant h$ ) of the form $w_{m}=w_{1}e_{0}^{\prime }ve_{0}^{\prime }w_{2}$ with $v\in \{e_{0},e_{1}\}^{\ast }~(v\neq \emptyset )$ and $w_{1},w_{2}\in \{e_{0},e_{1},e_{0}^{\prime }\}^{\ast }$ can be excluded. Thus, the possible nonzero terms in (20), sieved out by (I4), occur if $w_{0}=y_{n_{1}}\cdots y_{n_{t_{1}-1}}$ and $w_{m}$ is written in the form

$$\begin{eqnarray}\underbrace{e_{0}^{n_{t_{m}}-1}y_{n_{t_{m}+1}}\cdots y_{n_{q_{m-1}}}e_{1}}_{\text{degree in}~e_{1}=q_{m}-t_{m}}e_{0}^{\unicode[STIX]{x1D6FC}}(e_{0}^{\prime })^{k}e_{0}^{\unicode[STIX]{x1D6FD}}y_{n_{q_{m}+1}}\cdots y_{n_{t_{m+1}-1}},\end{eqnarray}$$

where $q_{m}\in \{t_{m},t_{m}+1,\ldots ,t_{m+1}-1\}$ , $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},k\in \mathbb{Z}_{{\geqslant}0}$ with $\unicode[STIX]{x1D6FC}+k+\unicode[STIX]{x1D6FD}=n_{q_{m}}-1$ and $\{t_{1},\ldots ,t_{h}\}=\unicode[STIX]{x1D70C}(w_{1}\cdots w_{r})$ . This completes the proof.◻

Before giving an explicit formula for $\unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})$ , we illustrate examples for $r=1$ and $2$ .

Example 3.10. By definition, for $n_{1}\geqslant 1$ one has $\unicode[STIX]{x1D713}_{n_{1}}(\mathbf{x})=\unicode[STIX]{x1D711}(e_{1}e_{0}^{n_{1}-1})=I(n_{1})\otimes 1$ . By (16), we obtain

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{n_{1}}(\mathbf{y})=\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}+k+\unicode[STIX]{x1D6FD}=n_{1} \\ \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\geqslant 0 \\ k\geqslant 1}}\unicode[STIX]{x1D711}(e_{1}^{\prime }e_{0}^{\unicode[STIX]{x1D6FC}}(e_{0}^{\prime })^{k-1}e_{0}^{\unicode[STIX]{x1D6FD}})=1\otimes I(n_{1}).\end{eqnarray}$$

Thus, we get

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(I(n_{1}))=I(n_{1})\otimes 1+1\otimes I(n_{1}).\end{eqnarray}$$

Example 3.11. It follows $\unicode[STIX]{x1D713}_{n_{1},n_{2}}(\mathbf{x}\mathbf{x})=\unicode[STIX]{x1D711}(e_{1}e_{0}^{n_{1}-1}e_{1}e_{0}^{n_{2}-1})=I(n_{1},n_{2})\otimes 1$ . By (16) one can compute

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{n_{1},n_{2}}(\mathbf{x}\mathbf{y}) & = & \displaystyle \mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}+k+\unicode[STIX]{x1D6FD}=n_{2} \\ \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\geqslant 0 \\ k\geqslant 1}}\unicode[STIX]{x1D711}(e_{1}e_{0}^{n_{1}-1}e_{1}^{\prime }e_{0}^{\unicode[STIX]{x1D6FC}}(e_{0}^{\prime })^{k-1}e_{0}^{\unicode[STIX]{x1D6FD}})=I(n_{1})\otimes I(n_{2}),\nonumber\\ \displaystyle \unicode[STIX]{x1D713}_{n_{1},n_{2}}(\mathbf{y}\mathbf{y}) & = & \displaystyle \mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}_{1}+k_{1}+\unicode[STIX]{x1D6FD}_{1}=n_{1} \\ \unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FD}_{1}\geqslant 0 \\ k_{1}\geqslant 1}}\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}_{2}+k_{2}+\unicode[STIX]{x1D6FD}_{2}=n_{2} \\ \unicode[STIX]{x1D6FC}_{2},\unicode[STIX]{x1D6FD}_{2}\geqslant 0 \\ k_{2}\geqslant 1}}\unicode[STIX]{x1D711}(e_{1}^{\prime }e_{0}^{\unicode[STIX]{x1D6FC}_{1}}(e_{0}^{\prime })^{k_{1}-1}e_{0}^{\unicode[STIX]{x1D6FD}_{1}}e_{1}^{\prime }e_{0}^{\unicode[STIX]{x1D6FC}_{2}}(e_{0}^{\prime })^{k_{2}-1}e_{0}^{\unicode[STIX]{x1D6FD}_{2}})\nonumber\\ \displaystyle & = & \displaystyle 1\otimes I(n_{1},n_{2}),\nonumber\end{eqnarray}$$

and using (17) and (18) we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{n_{1},n_{2}}(\mathbf{y}\mathbf{x}) & = & \displaystyle \mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}+k+\unicode[STIX]{x1D6FD}=n_{1} \\ \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\geqslant 0 \\ k\geqslant 1}}\unicode[STIX]{x1D711}(e_{1}^{\prime }e_{0}^{\unicode[STIX]{x1D6FC}}(e_{0}^{\prime })^{k-1}e_{0}^{\unicode[STIX]{x1D6FD}}e_{1}e_{0}^{n_{2}-1})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}+k+\unicode[STIX]{x1D6FD}=n_{2} \\ \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\geqslant 0 \\ k\geqslant 1}}\unicode[STIX]{x1D711}(e_{1}^{\prime }e_{0}^{n_{1}-1}e_{1}e_{0}^{\unicode[STIX]{x1D6FC}}(e_{0}^{\prime })^{k-1}e_{0}^{\unicode[STIX]{x1D6FD}})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\substack{ k_{1}+k_{2}=n_{1}+n_{2} \\ k_{1},k_{2}\geqslant 1}}\mathtt{b}_{n_{1},n_{2}}^{k_{1}}I(k_{1})\otimes I(k_{2}),\nonumber\end{eqnarray}$$

where $\mathtt{b}_{n,n^{\prime }}^{k}$ is defined in (10). Therefore, by Proposition 3.8 we have

(21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}(I(n_{1},n_{2})) & = & \displaystyle I(n_{1},n_{2})\otimes 1\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{\substack{ k_{1}+k_{2}=n_{1}+n_{2} \\ k_{1},k_{2}\geqslant 1}}\!\!\big(\unicode[STIX]{x1D6FF}_{n_{1},k_{1}}+\mathtt{b}_{n_{1},n_{2}}^{k_{1}}\big)I(k_{1})\otimes I(k_{2})+1\otimes I(n_{1},n_{2}).\end{eqnarray}$$

Proposition 3.12. For integers $n_{1},\ldots ,n_{r}\geqslant 2$ and a word $w_{1}\cdots w_{r}\in \{\mathbf{x},\mathbf{y}\}^{\ast }$ , we set

$$\begin{eqnarray}N_{t_{m}}=n_{t_{m}}+n_{t_{m}+1}+\cdots +n_{t_{m+1}-1}\end{eqnarray}$$

for each $m\in \{1,2,\ldots ,h\}$ , where $\{t_{1},\ldots ,t_{h}\}=\unicode[STIX]{x1D70C}(w_{1}\cdots w_{r})$ and $t_{h+1}=r+1$ . Then we have

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})=(I(n_{1},\ldots ,n_{t_{1}-1})\otimes 1)\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\mathop{\sum }_{\substack{ t_{1}\leqslant q_{1}\leqslant t_{2}-1 \\ t_{2}\leqslant q_{2}\leqslant t_{3}-1 \\ \vdots \\ t_{h}\leqslant q_{h}\leqslant r}}\mathop{\sum }_{\substack{ k_{t_{1}}+\cdots +k_{t_{2}-1}=N_{t_{1}} \\ k_{t_{2}}+\cdots +k_{t_{3}-1}=N_{t_{2}} \\ \vdots \\ k_{t_{h}}+\cdots +k_{r}=N_{t_{h}} \\ k_{i}\geqslant 1}}\bigg\{(-1)^{\mathop{\sum }_{m=1}^{h}(N_{t_{m}}+n_{q_{m}}+k_{q_{m}+1}+k_{q_{m}+2}+\cdots +k_{q_{m+1}-1})}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\bigg(\mathop{\prod }_{\substack{ j=t_{1} \\ j\neq q_{1},\ldots ,q_{h}}}^{r}\binom{k_{j}-1}{n_{j}-1}\bigg)\bigg(\mathop{\prod }_{m=1}^{h}I(\underbrace{k_{q_{m}-1},\ldots ,k_{t_{m}}}_{q_{m}-t_{m}})I(\underbrace{k_{q_{m}+1},\ldots ,k_{t_{m+1}-1}}_{t_{m+1}-q_{m}-1})\bigg)\nonumber\\ \displaystyle & & \displaystyle \qquad \otimes I(k_{q_{1}},\ldots ,k_{q_{h}})\bigg\},\nonumber\end{eqnarray}$$

where $\prod _{\substack{ j=t_{1} \\ j\neq q_{1},\ldots ,q_{h}}}^{r}\binom{k_{j}-1}{n_{j}-1}=1$ when $\{t_{1},t_{1}+1,\ldots ,r\}=\{q_{1},\ldots ,q_{h}\}$ .

Example 3.13. For the future literature, we present an explicit formula for $\unicode[STIX]{x1D6E5}(I(n_{1},n_{2},n_{3}))$ obtained from Propositions 3.8 and 3.12:

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6E5}(I(n_{1},n_{2},n_{3}))\nonumber\\ \displaystyle & & \displaystyle \quad =I(n_{1},n_{2},n_{3})\otimes 1+I(n_{1},n_{2})\otimes I(n_{3})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,I(n_{1})\otimes I(n_{2},n_{3})+1\otimes I(n_{1},n_{2},n_{3})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\mathop{\sum }_{\substack{ k_{1}+k_{2}+k_{3}=n_{1}+n_{2}+n_{3} \\ k_{1},k_{2},k_{3}\geqslant 1}}\bigg\{\!(\unicode[STIX]{x1D6FF}_{n_{3},k_{3}}\mathtt{b}_{n_{1},n_{2}}^{k_{1}}+\unicode[STIX]{x1D6FF}_{n_{1},k_{2}}\mathtt{b}_{n_{2},n_{3}}^{k_{1}})I(k_{1})\otimes I(k_{2},k_{3})\nonumber\\ \displaystyle & & \displaystyle \qquad +\left((-1)^{n_{1}+k_{3}}\binom{k_{2}-1}{n_{3}-1}+(-1)^{n_{1}+n_{2}}\binom{k_{2}-1}{n_{1}-1}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\binom{k_{1}-1}{n_{2}-1}I(k_{1},k_{2})\otimes I(k_{3})\nonumber\\ \displaystyle & & \displaystyle \qquad +\left((-1)^{n_{1}+n_{3}+k_{2}}\binom{k_{1}-1}{n_{1}-1}\binom{k_{2}-1}{n_{3}-1}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D6FF}_{k_{1},n_{1}}\left.\mathtt{b}_{n_{2},n_{3}}^{k_{2}}\right)I(k_{1})I(k_{2})\otimes I(k_{3})\bigg\}.\nonumber\end{eqnarray}$$

3.5 Proof of Theorem 1.1

We now give a proof of Theorem 1.1. Recall the $q$ -series $g_{n_{1},\ldots ,n_{r}}(q)$ defined in (7). Let

$$\begin{eqnarray}\mathfrak{g}:{\mathcal{I}}_{\bullet }^{1}\rightarrow \mathbb{C}[[q]]\end{eqnarray}$$

be the $\mathbb{Q}$ -linear map given by $\mathfrak{g}(I(n_{1},\ldots ,n_{r}))=g_{n_{1},\ldots ,n_{r}}(q)$ and $\mathfrak{g}(1)=1$ .

Proof of Theorem 1.1.

Taking $\mathfrak{z}^{\unicode[STIX]{x0428}}\otimes \mathfrak{g}$ for the explicit formula in Proposition 3.12 and comparing this with Proposition 2.5, we have

$$\begin{eqnarray}\big(\mathfrak{z}^{\unicode[STIX]{x0428}}\otimes \mathfrak{g}\big)\big(\unicode[STIX]{x1D713}_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r})\big)=G_{n_{1},\ldots ,n_{r}}(w_{1}\cdots w_{r}).\end{eqnarray}$$

Here the second sum (relating to $k_{i}$ ) of the formula in Proposition 3.12 differs from that of the formula in Proposition 2.5, but apparently it is the same because binomial coefficient terms allow us to take $k_{i}\geqslant n_{i}$ for $t_{1}\leqslant i\leqslant r$ with $i\neq q_{1},\ldots ,q_{h}$ , and by Lemma 2.4 it turns out that the coefficient of $g_{k_{q_{1}},\ldots ,k_{q_{h}}}(q)$ becomes 0 if $k_{q_{j}}=1$ for some $1\leqslant j\leqslant h$ . With this, the statement follows from Propositions 2.2 and 3.8.◻

4.1 The shuffle algebra consisting of certain $q$ -series

In this subsection, we define the map $\mathfrak{g}^{\unicode[STIX]{x0428}}:{\mathcal{I}}_{\bullet }^{1}\rightarrow \mathbb{C}[[q]]$ described in the introduction. We first introduce the power series $H\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ x_{1},\ldots ,x_{r}\end{smallmatrix}\!\right)$ satisfying the harmonic product formulas. Then, by using Hoffman’s results, the power series $h(x_{1},\ldots ,x_{r})$ (see (22)) satisfying the shuffle relation (24), which is a variant of the shuffle relation (I2) (reformulated in (26)), is constructed. Finally, we introduce the power series $g_{\unicode[STIX]{x0428}}(x_{1},\ldots ,x_{r})$ (see (25)) and prove in Theorem 4.6 that their coefficients, which are $q$ -series, satisfy the shuffle relations given in (I2).

Consider the iterated multiple sum

$$\begin{eqnarray}H\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ x_{1},\ldots ,x_{r}\end{smallmatrix}\!\right):=\mathop{\sum }_{0<d_{1}<\cdots <d_{r}}e^{d_{1}x_{1}}\left(\frac{q^{d_{1}}}{1-q^{d_{1}}}\right)^{n_{1}}\cdots e^{d_{r}x_{r}}\left(\frac{q^{d_{r}}}{1-q^{d_{r}}}\right)^{n_{r}},\end{eqnarray}$$

where $n_{1},\ldots ,n_{r}$ are positive integers and $x_{1},\ldots ,x_{r}$ are commutative variables, that is, these are elements in the power series ring ${\mathcal{K}}[[x_{1},\ldots ,x_{r}]]$ , where ${\mathcal{K}}=\mathbb{Q}[[q]]$ . It is easily seen that the power series $H\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ x_{1},\ldots ,x_{r}\end{smallmatrix}\!\right)$ satisfies the harmonic product: for example, one has

$$\begin{eqnarray}\displaystyle H\left(\!\begin{smallmatrix}1\\ x_{1}\end{smallmatrix}\!\right)H\left(\!\begin{smallmatrix}1\\ x_{2}\end{smallmatrix}\!\right) & = & \displaystyle \mathop{\sum }_{0<d_{1},d_{2}}e^{d_{1}x_{1}}\frac{q^{d_{1}}}{1-q^{d_{1}}}e^{d_{2}x_{2}}\frac{q^{d_{2}}}{1-q^{d_{2}}}\nonumber\\ \displaystyle & = & \displaystyle \bigg(\mathop{\sum }_{0<d_{1}<d_{2}}+\mathop{\sum }_{0<d_{2}<d_{1}}+\mathop{\sum }_{0<d_{1}=d_{2}}\bigg)e^{d_{1}x_{1}}\frac{q^{d_{1}}}{1-q^{d_{1}}}e^{d_{2}x_{2}}\frac{q^{d_{2}}}{1-q^{d_{2}}}\nonumber\\ \displaystyle & = & \displaystyle H\left(\!\begin{smallmatrix}1,1\\ x_{1},x_{2}\end{smallmatrix}\!\right)+H\left(\!\begin{smallmatrix}1,1\\ x_{2},x_{1}\end{smallmatrix}\!\right)+H\left(\!\begin{smallmatrix}2\\ x_{1}+x_{2}\end{smallmatrix}\!\right).\nonumber\end{eqnarray}$$

The power series $H\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ x_{1},\ldots ,x_{r}\end{smallmatrix}\!\right)$ naturally appears in an expression of the generating series of $g_{n_{1},\ldots ,n_{r}}(q)$ .

Lemma 4.1. For any $r>0$ , let

$$\begin{eqnarray}g(x_{1},\ldots ,x_{r}):=\mathop{\sum }_{n_{1},\ldots ,n_{r}\geqslant 1}\frac{g_{n_{1},\ldots ,n_{r}}(q)}{(-2\unicode[STIX]{x1D70B}\sqrt{-1})^{n_{1}+\cdots +n_{r}}}x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}.\end{eqnarray}$$

Then, we have

$$\begin{eqnarray}g(x_{1},\ldots ,x_{r})=H\left(\!\begin{smallmatrix}1,\ldots ,1,1\\ x_{r}-x_{r-1},\ldots ,x_{2}-x_{1},x_{1}\end{smallmatrix}\!\right).\end{eqnarray}$$

For $r>0$ , consider the power series

(22) $$\begin{eqnarray}h(x_{1},\ldots ,x_{r}):=\mathop{\sum }_{m=1}^{r}\mathop{\sum }_{(i_{1},i_{2},\ldots ,i_{m})}\frac{1}{i_{1}!i_{2}!\cdots i_{m}!}H\left(\!\begin{smallmatrix}i_{1},i_{2},\ldots ,i_{m}\\ x_{i_{1}}^{\prime },x_{i_{2}}^{\prime },\ldots ,x_{i_{m}}^{\prime }\end{smallmatrix}\!\right),\end{eqnarray}$$

where the second sum runs over all decompositions of the integer $r$ as a sum of $m$ positive integers and the variables are given by $x_{i_{1}}^{\prime }=x_{1}+\cdots +x_{i_{1}},x_{i_{2}}^{\prime }=x_{i_{1}+1}+\cdots +x_{i_{1}+i_{2}},\ldots ,x_{i_{m}}^{\prime }=x_{i_{1}+\cdots +i_{m-1}+1}+\cdots +x_{r}$ . For example, we have

$$\begin{eqnarray}h(x)=H\left(\!\begin{smallmatrix}1\\ x\end{smallmatrix}\!\right),\qquad h(x_{1},x_{2})=H\left(\!\begin{smallmatrix}1,1\\ x_{1},x_{2}\end{smallmatrix}\!\right)+\frac{1}{2}H\left(\!\begin{smallmatrix}2\\ x_{1}+x_{2}\end{smallmatrix}\!\right).\end{eqnarray}$$

We shall prove that the power series $h(x_{1},\ldots ,x_{r})$ satisfies the shuffle relation (24) below, which in the case of $r=2$ is given by

$$\begin{eqnarray}h(x_{1})h(x_{2})=h(x_{1},x_{2})+h(x_{2},x_{1}).\end{eqnarray}$$

To do this, we reformulate the result of Hoffman [Reference Hoffman9, Theorem 2.5] in accordance with our situation.

Let ${\mathcal{U}}$ be the noncommutative polynomial algebra over $\mathbb{Q}$ generated by noncommutative symbols $\left(\!\begin{smallmatrix}n\\ z\end{smallmatrix}\!\right)$ indexed by $n\in \mathbb{N}$ and

$$\begin{eqnarray}z\in X:=\biggl\{\mathop{\sum }_{i>0}a_{i}x_{i}\mid a_{i}\in \mathbb{Z}_{{\geqslant}0}\text{ is zero for almost all }i\text{'s}\biggr\}.\end{eqnarray}$$

The word consisting of the concatenation products of letters $\left(\!\begin{smallmatrix}n_{1}\\ z_{1}\end{smallmatrix}\!\right),\ldots ,\left(\!\begin{smallmatrix}n_{r}\\ z_{r}\end{smallmatrix}\!\right)$ is denoted by $\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ z_{1},\ldots ,z_{r}\end{smallmatrix}\!\right)\,\big(=\left(\!\begin{smallmatrix}n_{1}\\ z_{1}\end{smallmatrix}\!\right)\cdots \left(\!\begin{smallmatrix}n_{r}\\ z_{r}\end{smallmatrix}\!\right)\big)$ , for short. The empty word is viewed as $1\in \mathbb{Q}$ . As usual, the harmonic product $\ast$ on ${\mathcal{U}}$ is inductively defined for $n,n^{\prime }\in \mathbb{N}$ , $z,z^{\prime }\in X$ and words $w,w^{\prime }$ in ${\mathcal{U}}$ by

$$\begin{eqnarray}\displaystyle & & \displaystyle \big(\!\left(\!\begin{smallmatrix}n\\ z\end{smallmatrix}\!\right)\cdot w\big)\ast \big(\!\left(\!\begin{smallmatrix}n^{\prime }\\ z^{\prime }\end{smallmatrix}\!\right)\cdot w^{\prime }\big)\nonumber\\ \displaystyle & & \displaystyle \quad =\left(\!\begin{smallmatrix}n\\ z\end{smallmatrix}\!\right)\cdot \big(w\ast \big(\!\left(\!\begin{smallmatrix}n^{\prime }\\ z^{\prime }\end{smallmatrix}\!\right)\cdot w^{\prime }\big)\big)+\left(\!\begin{smallmatrix}n^{\prime }\\ z^{\prime }\end{smallmatrix}\!\right)\cdot \big(\big(\!\left(\!\begin{smallmatrix}n\\ z\end{smallmatrix}\!\right)\cdot w\big)\ast w^{\prime }\big)+\left(\!\begin{smallmatrix}n+n^{\prime }\\ z+z^{\prime }\end{smallmatrix}\!\right)\cdot (w\ast w^{\prime }),\nonumber\end{eqnarray}$$

with the initial condition $w\ast 1=1\ast w=w$ . The shuffle product on ${\mathcal{U}}$ is defined in the same way as in $\unicode[STIX]{x0428}$ on $\mathfrak{H}=\mathbb{Q}\langle e_{0},e_{1}\rangle$ , replacing the underlying vector space with ${\mathcal{U}}$ . Let us define the $\mathbb{Q}$ -linear map $\exp :{\mathcal{U}}\rightarrow {\mathcal{U}}$ given for each linear basis $\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ z_{1},\ldots ,z_{r}\end{smallmatrix}\!\right)$ by

(23) $$\begin{eqnarray}\exp \left(\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ z_{1},\ldots ,z_{r}\end{smallmatrix}\!\right)\right)=\mathop{\sum }_{m=1}^{r}\mathop{\sum }_{(i_{1},i_{2},\ldots ,i_{m})}\frac{1}{i_{1}!i_{2}!\cdots i_{m}!}\left(\!\begin{smallmatrix}n_{i_{1}}^{\prime },n_{i_{2}}^{\prime },\ldots ,n_{i_{m}}^{\prime }\\ z_{i_{1}}^{\prime },z_{i_{2}}^{\prime },\ldots ,z_{i_{m}}^{\prime }\end{smallmatrix}\!\right),\end{eqnarray}$$

where the second sum runs over all decompositions of the integer $r$ as a sum of $m$ positive integers and the variables are given by $z_{i_{1}}^{\prime }=z_{1}+\cdots +z_{i_{1}}$ , $z_{i_{2}}^{\prime }=z_{i_{1}+1}+\cdots +z_{i_{1}+i_{2}},\ldots ,z_{i_{m}}^{\prime }=z_{i_{1}+\cdots +i_{m-1}+1}+\cdots +z_{r}$ and $n_{i_{1}}^{\prime }=n_{1}+\cdots +n_{i_{1}}$ , $n_{i_{2}}^{\prime }=n_{i_{1}+1}+\cdots +n_{i_{1}+i_{2}},\ldots ,n_{i_{m}}^{\prime }=n_{i_{1}+\cdots +i_{m-1}+1}+\cdots +n_{r}$ . This map was first considered by Hoffman (see [Reference Hoffman9, p. 52]) and called the exponential map. Then, by [Reference Hoffman9, Theorem 2.5], we have the following proposition.

Proposition 4.2. The exponential map gives the isomorphism

$$\begin{eqnarray}\exp :{\mathcal{U}}_{\unicode[STIX]{x0428}}\longrightarrow {\mathcal{U}}_{\ast }\end{eqnarray}$$

as commutative $\mathbb{Q}$ -algebras, where ${\mathcal{U}}_{\unicode[STIX]{x0428}}$ (resp. ${\mathcal{U}}_{\ast }$ ) denotes the commutative algebra ${\mathcal{U}}$ equipped with the product $\unicode[STIX]{x0428}$ (resp. $\ast$ ).

We now prove the shuffle relation for $h(x_{1},\ldots ,x_{r})$ ’s. We denote by $\mathfrak{S}_{r}$ the symmetric group of order $r$ . The group $\mathfrak{S}_{r}$ acts on ${\mathcal{K}}[[x_{1},\ldots ,x_{r}]]$ in the obvious way by $(f\big|\unicode[STIX]{x1D70E})(x_{1},\ldots ,x_{r})=f(x_{\unicode[STIX]{x1D70E}^{-1}(1)},\ldots ,x_{\unicode[STIX]{x1D70E}^{-1}(r)})$ , which defines a right action, that is, $f\big|(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F})=(f\big|\unicode[STIX]{x1D70E})\big|\unicode[STIX]{x1D70F}$ . This action extends to an action of the group ring $\mathbb{Z}[\mathfrak{S}_{r}]$ by linearity.

Lemma 4.3. For any $r,s\geqslant 1$ , we have

(24) $$\begin{eqnarray}h(x_{1},\ldots ,x_{r})h(x_{r+1},\ldots ,x_{r+s})=h(x_{1},\ldots ,x_{r+s})\big|sh_{r}^{(r+s)},\end{eqnarray}$$

where $sh_{r}^{(r+s)}:=\sum _{\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}(r,s)}\unicode[STIX]{x1D70E}$ is in the group ring $\mathbb{Z}[\mathfrak{S}_{r+s}]$ and the set $\unicode[STIX]{x1D6F4}(r,s)$ is defined in the shuffle product formula $\text{(I2)}$ .

Proof. Define the $\mathbb{Q}$ -linear map

$$\begin{eqnarray}\displaystyle H:{\mathcal{U}} & \longrightarrow & \displaystyle {\mathcal{R}}:=\mathop{\varinjlim }\nolimits_{r}{\mathcal{K}}[[x_{1},\ldots ,x_{r}]]\nonumber\\ \displaystyle \left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ z_{1},\ldots ,z_{r}\end{smallmatrix}\!\right) & \longmapsto & \displaystyle H\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ z_{1},\ldots ,z_{r}\end{smallmatrix}\!\right)\nonumber\end{eqnarray}$$

for each linear basis $\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ z_{1},\ldots ,z_{r}\end{smallmatrix}\!\right)$ of ${\mathcal{U}}$ with $H(1)=1$ . By combining the exponential map (23) with the map $H$ , one easily finds that

$$\begin{eqnarray}h(x_{1},\ldots ,x_{r})=H\circ \exp \left(\left(\!\begin{smallmatrix}1,\ldots ,1\\ x_{1},\ldots ,x_{r}\end{smallmatrix}\!\right)\right).\end{eqnarray}$$

Since the power series $H\left(\!\begin{smallmatrix}n_{1},\ldots ,n_{r}\\ z_{1},\ldots ,z_{r}\end{smallmatrix}\!\right)$ satisfies the harmonic product, the algebra homomorphism $H:{\mathcal{U}}_{\ast }\rightarrow {\mathcal{R}}$ is obtained, and hence, by Proposition 4.2, the composition map $H\circ \exp :{\mathcal{U}}_{\unicode[STIX]{x0428}}\rightarrow {\mathcal{R}}$ is an algebra homomorphism. Then, we have

$$\begin{eqnarray}\displaystyle h(x_{1},\ldots ,x_{r})h(x_{r+1},\ldots ,x_{r+s}) & = & \displaystyle H\circ \exp \left(\left(\!\begin{smallmatrix}1,\ldots ,1\\ x_{1},\ldots ,x_{r}\end{smallmatrix}\!\right)~\unicode[STIX]{x0428}~\left(\!\begin{smallmatrix}1,\ldots ,1\\ x_{r+1},\ldots ,x_{r+s}\end{smallmatrix}\!\right)\right)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}(r,s)}H\circ \exp \left(\left(\!\begin{smallmatrix}1,\ldots ,1\\ x_{\unicode[STIX]{x1D70E}^{-1}(1)},\ldots ,x_{\unicode[STIX]{x1D70E}^{-1}(r+s)}\end{smallmatrix}\!\right)\right)\nonumber\\ \displaystyle & = & \displaystyle h(x_{1},\ldots ,x_{r+s})\big|sh_{r}^{(r+s)},\nonumber\end{eqnarray}$$

which completes the proof. ◻

For $r>0$ , consider the power series

(25) $$\begin{eqnarray}g_{\unicode[STIX]{x0428}}(x_{1},\ldots ,x_{r}):=h(x_{r}-x_{r-1},\ldots ,x_{2}-x_{1},x_{1}).\end{eqnarray}$$

We shall prove that the $q$ -series obtained from the coefficient of $x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}$ in the power series $g_{\unicode[STIX]{x0428}}(x_{1},\ldots ,x_{r})$ satisfies the shuffle product formulas (I2). For this, we use the generating series expression of the shuffle product formulas (I2) (this expression is found in [Reference Ihara, Kaneko and Zagier10, Proof of Proposition 7] with the opposite convention).

Proposition 4.4. For any $r>0$ , let

$$\begin{eqnarray}\mathbf{I}(x_{1},\ldots ,x_{r}):=\mathop{\sum }_{n_{1},\ldots ,n_{r}>0}I(n_{1},\ldots ,n_{r})x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}.\end{eqnarray}$$

Then, the shuffle product formulas for $I(n_{1},\ldots ,n_{r})$ ’s obtained in $\text{(I2)}$ is equivalent to

(26) $$\begin{eqnarray}\mathbf{I}^{\sharp }(x_{1},\ldots ,x_{r})\mathbf{I}^{\sharp }(x_{r+1},\ldots ,x_{r+s})=\mathbf{I}^{\sharp }(x_{1},\ldots x_{r+s})\big|sh_{r}^{(r+s)},\end{eqnarray}$$

where the operator $\sharp$ is the change of variables defined by

$$\begin{eqnarray}f^{\sharp }(x_{1},\ldots ,x_{r})=f(x_{1},x_{1}+x_{2},\ldots ,x_{1}+\cdots +x_{r}).\end{eqnarray}$$

Proof. Computing the shuffle product formula (I2), one obtains

(27) $$\begin{eqnarray}I(n_{1})I(n_{2})=\mathop{\sum }_{\substack{ k_{1}+k_{2}=n_{1}+n_{2} \\ k_{1},k_{2}\geqslant 1}}\left(\binom{k_{2}-1}{n_{1}-1}+\binom{k_{2}-1}{n_{2}-1}\right)I(k_{1},k_{2}),\end{eqnarray}$$

and hence we have

$$\begin{eqnarray}\mathbf{I}(x_{1})\mathbf{I}(x_{2})=\mathbf{I}(x_{2},x_{1}+x_{2})+\mathbf{I}(x_{1},x_{1}+x_{2}),\end{eqnarray}$$

which coincides with the identity (26) for $r=s=1$ . The reminder (the case $r+s>2$ ) can be verified by induction.◻

We now define the map $\mathfrak{g}^{\unicode[STIX]{x0428}}:{\mathcal{I}}_{\bullet }^{1}\rightarrow \mathbb{C}[[q]]$ and prove that this is an algebra homomorphism.

Definition 4.5. Let us denote by $g_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)$ the coefficient of $x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}$ in the power series

$$\begin{eqnarray}g_{\unicode[STIX]{x0428}}(-2\unicode[STIX]{x1D70B}\sqrt{-1}x_{1},\ldots ,-2\unicode[STIX]{x1D70B}\sqrt{-1}x_{r})=\mathop{\sum }_{n_{1},\ldots ,n_{r}>0}g_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}.\end{eqnarray}$$

With this, we define the $\mathbb{Q}$ -linear map $\mathfrak{g}^{\unicode[STIX]{x0428}}:{\mathcal{I}}_{\bullet }^{1}\rightarrow \mathbb{C}[[q]]$ for each linear basis $I(n_{1},\ldots ,n_{r})$ of ${\mathcal{I}}_{\bullet }^{1}$ by

$$\begin{eqnarray}\mathfrak{g}^{\unicode[STIX]{x0428}}(I(n_{1},\ldots ,n_{r}))=g_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)\end{eqnarray}$$

and $\mathfrak{g}^{\unicode[STIX]{x0428}}(1)=1$ .

Proof. By Proposition 4.4, it is sufficient to show that for any integers $r,s\geqslant 1$ the power series $g_{\unicode[STIX]{x0428}}(x_{1},\ldots ,x_{r+s})$ satisfies the relation

$$\begin{eqnarray}g_{\unicode[STIX]{x0428}}^{\sharp }(x_{1},\ldots ,x_{r})g_{\unicode[STIX]{x0428}}^{\sharp }(x_{r+1},\ldots ,x_{r+s})=g_{\unicode[STIX]{x0428}}^{\sharp }(x_{1},\ldots x_{r+s})\big|sh_{r}^{(r+s)}.\end{eqnarray}$$

For integers $r,s\geqslant 1$ , let

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{r,s}=\left(\begin{array}{@{}cccccccc@{}}1 & 2 & \cdots \, & r & r+1 & \cdots \, & r+s-1 & r+s\\ r & r-1 & \cdots \, & 1 & r+s & \cdots \, & r+2 & r+1\end{array}\right)\in \mathfrak{S}_{r+s}.\end{eqnarray}$$

Applying the operator $\sharp$ to both sides of (25), we obtain $g_{\unicode[STIX]{x0428}}^{\sharp }(x_{1},\ldots ,x_{r})=h(x_{r},\ldots ,x_{1})$ , and hence by Lemma 4.3 we can compute

$$\begin{eqnarray}\displaystyle g_{\unicode[STIX]{x0428}}^{\sharp }(x_{1},\ldots ,x_{r})g_{\unicode[STIX]{x0428}}^{\sharp }(x_{r+1},\ldots ,x_{r+s}) & = & \displaystyle h(x_{r},\ldots ,x_{1})h(x_{r+s},\ldots ,x_{r+1})\nonumber\\ \displaystyle & = & \displaystyle h(x_{1},\ldots ,x_{r})h(x_{r+1},\ldots ,x_{r+s})\big|\unicode[STIX]{x1D70C}_{r,s}\nonumber\\ \displaystyle & = & \displaystyle h(x_{1},\ldots ,x_{r+s})\big|sh_{r}^{(r+s)}\big|\unicode[STIX]{x1D70C}_{r,s}\nonumber\\ \displaystyle & = & \displaystyle g_{\unicode[STIX]{x0428}}^{\sharp }(x_{1},\ldots ,x_{r+s})\big|\unicode[STIX]{x1D70F}_{r+s}\big|sh_{r}^{(r+s)}\big|\unicode[STIX]{x1D70C}_{r,s},\nonumber\end{eqnarray}$$

where we set $\unicode[STIX]{x1D70F}_{r+s}=\left(\begin{array}{@{}cccc@{}}1 & 2 & \cdots \, & r+s\\ r+s & r+s-1 & \cdots \, & 1\end{array}\right)\in \mathfrak{S}_{r+s}$ . For any $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}(r,s)$ , one easily finds that $\unicode[STIX]{x1D70F}_{r+s}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70C}_{r,s}\in \unicode[STIX]{x1D6F4}(r,s)$ , and hence

$$\begin{eqnarray}g_{\unicode[STIX]{x0428}}^{\sharp }(x_{1},\ldots ,x_{r+s})\big|\unicode[STIX]{x1D70F}_{r+s}\big|sh_{r}^{(r+s)}\big|\unicode[STIX]{x1D70C}_{r,s}=g_{\unicode[STIX]{x0428}}^{\sharp }(x_{1},\ldots ,x_{r+s})\big|sh_{r}^{(r+s)},\end{eqnarray}$$

which completes the proof. ◻

Remark 4.7. From Theorem 4.6, we learn that the $q$ -series $g_{n_{1},\ldots ,n_{r}}(q)$ does not satisfy the shuffle product formula (I2). For example, by Theorem 4.6 one has $g_{\unicode[STIX]{x0428}}(x_{1})g_{\unicode[STIX]{x0428}}(x_{2})=g_{\unicode[STIX]{x0428}}(x_{2},x_{1}+x_{2})+g_{\unicode[STIX]{x0428}}(x_{1},x_{1}+x_{2})$ . Since $g_{\unicode[STIX]{x0428}}(x)=g(x)$ and $g_{\unicode[STIX]{x0428}}(x_{1},x_{2})=h(x_{2}-x_{1},x_{1})=g(x_{1},x_{2})+\frac{1}{2}H\left(\!\begin{smallmatrix}2\\ x_{2}\end{smallmatrix}\!\right)$ , one gets

$$\begin{eqnarray}g(x_{1})g(x_{2})=g(x_{2},x_{1}+x_{2})+g(x_{1},x_{1}+x_{2})+H\left(\!\begin{smallmatrix}2\\ x_{1}+x_{2}\end{smallmatrix}\!\right),\end{eqnarray}$$

which proves that the $q$ -series $g_{n_{1},n_{2}}(q)$ ( $n_{1},n_{2}\geqslant 1$ ) do not satisfy the shuffle product formulas (27), because $H\left(\!\begin{smallmatrix}2\\ x_{1}+x_{2}\end{smallmatrix}\!\right)\neq 0$ .

4.2 Proof of Theorem 1.2

In this subsection, we introduce the regularized multiple Eisenstein series $G_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)$ for integers $n_{1},\ldots ,n_{r}\geqslant 1$ as a $q$ -series. By relating $G^{\unicode[STIX]{x0428}}$ ’s with $G$ ’s (Theorem 4.10), we show that the multiple Eisenstein series $G^{\unicode[STIX]{x0428}}$ ’s satisfy the harmonic product (Theorem 4.11). Using these results, we finally prove the double shuffle relations for regularized multiple Eisenstein series (Theorem 1.2).

The regularized multiple Eisenstein series $G_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)$ is defined as follows.

Definition 4.8. For integers $n_{1},\ldots ,n_{r}\geqslant 1$ we define the $q$ -series $G_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)$ by

$$\begin{eqnarray}G_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)=\big(\mathfrak{z}^{\unicode[STIX]{x0428}}\otimes \mathfrak{g}^{\unicode[STIX]{x0428}}\big)\circ \unicode[STIX]{x1D6E5}(I(n_{1},\ldots ,n_{r})).\end{eqnarray}$$

Example 4.9. For an integer $n\geqslant 1$ , we have

$$\begin{eqnarray}G_{n}^{\unicode[STIX]{x0428}}(q)=\unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(n)+g_{n}^{\unicode[STIX]{x0428}}(q)=\unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(n)+g_{n}(q).\end{eqnarray}$$

For integers $n_{1},n_{2}\geqslant 1$ , by (21), one has

$$\begin{eqnarray}\displaystyle G_{n_{1},n_{2}}^{\unicode[STIX]{x0428}}(q) & = & \displaystyle \unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(n_{1},n_{2})+\mathop{\sum }_{\substack{ k_{1}+k_{2}=n_{1}+n_{2} \\ k_{1},k_{2}\geqslant 1}}\big(\unicode[STIX]{x1D6FF}_{n_{1},k_{1}}+\mathtt{b}_{n_{1},n_{2}}^{k_{1}}\big)\unicode[STIX]{x1D701}^{\unicode[STIX]{x0428}}(k_{1})g_{k_{2}}^{\unicode[STIX]{x0428}}(q)\nonumber\\ \displaystyle & & \displaystyle +\,g_{n_{1},n_{2}}^{\unicode[STIX]{x0428}}(q),\nonumber\end{eqnarray}$$

where $\mathtt{b}_{n_{1},n_{2}}^{k_{1}}$ is defined in (10). We remark that the above double Eisenstein series coincides with Kaneko’s double Eisenstein series developed in [Reference Kaneko, Kim and Taguchi11].

We begin by showing a connection with the multiple Eisenstein series $G_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F})$ , which can be regarded as an analogue of (13).

Theorem 4.10. For integers $n_{1},\ldots ,n_{r}\geqslant 2$ , with $q=e^{2\unicode[STIX]{x1D70B}\sqrt{-1}\unicode[STIX]{x1D70F}}$ we have

$$\begin{eqnarray}G_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)=G_{n_{1},\ldots ,n_{r}}(\unicode[STIX]{x1D70F}).\end{eqnarray}$$

Proof. This equation follows once we obtain the following identity: for integers $n_{1},\ldots ,n_{r}\geqslant 2$

$$\begin{eqnarray}g_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)=g_{n_{1},\ldots ,n_{r}}(q).\end{eqnarray}$$

Combining (25) with (22), we have

$$\begin{eqnarray}g_{\unicode[STIX]{x0428}}(x_{1},\ldots ,x_{r})=\mathop{\sum }_{m=1}^{r}\mathop{\sum }_{(i_{1},i_{2},\ldots ,i_{m})}\frac{1}{i_{1}!i_{2}!\cdots \,,i_{m}!}H\left(\!\begin{smallmatrix}i_{1},i_{2},\ldots ,i_{m}\\ x_{i_{1}}^{\prime \prime },x_{i_{2}}^{\prime \prime },\ldots ,x_{i_{m}}^{\prime \prime }\end{smallmatrix}\!\right),\end{eqnarray}$$

where the second sum runs over all decompositions of the integer $r$ as a sum of $m$ positive integers and $x_{i_{1}}^{\prime \prime }=x_{r}-x_{r-i_{1}},x_{i_{2}}^{\prime \prime }=x_{r-i_{1}}-x_{r-i_{1}-i_{2}},\ldots ,x_{i_{m}}^{\prime \prime }=x_{r-i_{1}-\cdots -i_{m-1}}$ . For any $n_{1},\ldots ,n_{r}\geqslant 2$ , since we have

$$\begin{eqnarray}\text{coefficient of}~x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}~\text{in}~H\left(\!\begin{smallmatrix}i_{1},i_{2},\ldots ,i_{m}\\ x_{i_{1}}^{\prime \prime },x_{i_{2}}^{\prime \prime },\ldots ,x_{i_{m}}^{\prime \prime }\end{smallmatrix}\!\right)=0\end{eqnarray}$$

whenever $i_{j}>1$ for some $j\in \{1,2,\ldots ,m\}$ , we obtain

$$\begin{eqnarray}\displaystyle & & \displaystyle \text{coefficient of }x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}\text{ in }g_{\unicode[STIX]{x0428}}(x_{1},\ldots ,x_{r})\nonumber\\ \displaystyle & & \displaystyle \quad =\text{coefficient of }x_{1}^{n_{1}-1}\cdots x_{r}^{n_{r}-1}\text{ in }H\left(\!\begin{smallmatrix}1,\ldots ,1,1\\ x_{r}-x_{r-1},\ldots ,x_{2}-x_{1},x_{1}\end{smallmatrix}\!\right),\nonumber\end{eqnarray}$$

which by Lemma 4.1 is equal to $g_{n_{1},\ldots ,n_{r}}(q)/(-2\unicode[STIX]{x1D70B}\sqrt{-1})^{n_{1}+\cdots +n_{r}}$ . This completes the proof.◻

Let us prove the harmonic product for $G^{\unicode[STIX]{x0428}}$ ’s. The harmonic product $\ast$ on $\mathfrak{H}^{1}$ is defined inductively by

$$\begin{eqnarray}y_{n_{1}}w\ast y_{n_{2}}w^{\prime }=y_{n_{1}}(w\ast y_{n_{2}}w^{\prime })+y_{n_{2}}(y_{n_{1}}w\ast w^{\prime })+y_{n_{1}+n_{2}}(w\ast w^{\prime }),\end{eqnarray}$$

and $w\ast 1=1\ast w=w$ for letters $y_{n_{1}},y_{n_{2}}\in \mathfrak{H}^{1}$ and words $w,w^{\prime }$ in $\mathfrak{H}^{1}$ , together with $\mathbb{Q}$ -bilinearity. For each word $w\in \mathfrak{H}^{1}$ , the dual element of $w$ is denoted by $c_{w}\in (\mathfrak{H}^{1})^{\vee }=\operatorname{Hom}(\mathfrak{H}^{1},\mathbb{Q})$ such that $c_{w}(v)=\unicode[STIX]{x1D6FF}_{w,v}$ for any word $v\in \mathfrak{H}^{1}$ . If $w$ is the empty word $\emptyset$ , $c_{w}$ kills $\mathfrak{H}_{{>}0}^{1}$ and $c_{w}(1)=1$ . Then, the harmonic product of the $q$ -series $G_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)$ with $n_{1},\ldots ,n_{r}\geqslant 2$ is stated as follows:

Theorem 4.11. For any words $w_{1},w_{2}$ in $\{y_{2},y_{3},y_{4},\ldots \}^{\ast }$ , one has

$$\begin{eqnarray}G_{w_{1}}^{\unicode[STIX]{x0428}}(q)G_{w_{2}}^{\unicode[STIX]{x0428}}(q)=\mathop{\sum }_{w\in \{y_{2},y_{3},\ldots \}^{\ast }}c_{w}(w_{1}\ast w_{2})G_{w}^{\unicode[STIX]{x0428}}(q),\end{eqnarray}$$

where we write $G_{w}^{\unicode[STIX]{x0428}}(q)=G_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)$ for each word $w=y_{n_{1}}\cdots y_{n_{r}}$ .

Proof. Consider the following holomorphic function on the upper half-plane: for integers $L,M>0$

$$\begin{eqnarray}G_{n_{1},\ldots ,n_{r}}^{(L,M)}(\unicode[STIX]{x1D70F})=\mathop{\sum }_{\substack{ 0\prec \unicode[STIX]{x1D706}_{1}\prec \cdots \prec \unicode[STIX]{x1D706}_{r} \\ \unicode[STIX]{x1D706}_{i}\in \mathbb{Z}_{L}\unicode[STIX]{x1D70F}+\mathbb{Z}_{M}}}\frac{1}{\unicode[STIX]{x1D706}_{1}^{n_{1}}\cdots \unicode[STIX]{x1D706}_{r}^{n_{r}}}.\end{eqnarray}$$

By definition, it follows that these functions satisfy the harmonic product: for any words $w_{1},w_{2}\in \mathfrak{H}^{1}$ , one has

(28) $$\begin{eqnarray}G_{w_{1}}^{(L,M)}(\unicode[STIX]{x1D70F})G_{w_{2}}^{(L,M)}(\unicode[STIX]{x1D70F})=\mathop{\sum }_{w\in \{y_{1},y_{2},y_{3},\ldots \}^{\ast }}c_{w}(w_{1}\ast w_{2})G_{w}^{(L,M)}(\unicode[STIX]{x1D70F}).\end{eqnarray}$$

Since the space $\mathfrak{H}^{2}:=\mathbb{Q}\langle y_{2},y_{3},y_{4},\ldots \rangle$ is closed under the harmonic product  $\ast$ , taking $\lim _{L\rightarrow \infty }\lim _{M\rightarrow \infty }$ for both sides of (28), one has for words $w_{1}$ , $w_{2}\in \mathfrak{H}^{2}$

$$\begin{eqnarray}G_{w_{1}}(\unicode[STIX]{x1D70F})G_{w_{2}}(\unicode[STIX]{x1D70F})=\mathop{\sum }_{w\in \{y_{2},y_{3},y_{4},\ldots \}^{\ast }}c_{w}(w_{1}\ast w_{2})G_{w}(\unicode[STIX]{x1D70F}).\end{eqnarray}$$

Then the result follows from Theorem 4.10. ◻

We finally prove Theorem 1.2.

Theorem 4.12. For any $w_{1},w_{2}\in \mathfrak{H}^{2}$ , we obtain

$$\begin{eqnarray}\mathop{\sum }_{w\in \{y_{1},y_{2},y_{3},\ldots \}^{\ast }}c_{w}(w_{1}~\unicode[STIX]{x0428}~w_{2})G_{w}^{\unicode[STIX]{x0428}}(q)=\mathop{\sum }_{w\in \{y_{2},y_{3},\ldots \}^{\ast }}c_{w}(w_{1}\ast w_{2})G_{w}^{\unicode[STIX]{x0428}}(q),\end{eqnarray}$$

which is called the restricted finite double shuffle relation in this paper.

It follows from Propositions 3.4, 3.7 and Theorem 4.6 that $G^{\unicode[STIX]{x0428}}$ ’s satisfies the shuffle product formula (I2). With this, Theorem 4.12 follows from Theorem 4.11.

Example 4.13. The first example of $\mathbb{Q}$ -linear relations among $G^{\unicode[STIX]{x0428}}$ ’s obtained in Theorem 4.12 is

$$\begin{eqnarray}G_{4}^{\unicode[STIX]{x0428}}(q)-4G_{1,3}^{\unicode[STIX]{x0428}}(q)=0,\end{eqnarray}$$

which comes from $y_{2}\ast y_{2}-y_{2}~\unicode[STIX]{x0428}~y_{2}$ .

Remark 4.14. Let us discuss the dimension of the space of $G^{\unicode[STIX]{x0428}}$ ’s. For our convenience, we take the normalization

$$\begin{eqnarray}\widetilde{G}_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q)=\frac{1}{(-2\unicode[STIX]{x1D70B}\sqrt{-1})^{n_{1}+\cdots +n_{r}}}G_{n_{1},\ldots ,n_{r}}^{\unicode[STIX]{x0428}}(q).\end{eqnarray}$$

As usual, we call $n_{1}+\cdots +n_{r}$ the weight and $\unicode[STIX]{x1D6FE}_{n_{1},\ldots ,n_{r}}$ admissible if $n_{r}\geqslant 2$ . Let ${\mathcal{E}}_{N}$ be the $\mathbb{Q}$ -vector space spanned by all admissible $\widetilde{G}^{\unicode[STIX]{x0428}}$ ’s of weight $N$ . Set ${\mathcal{E}}_{0}=\mathbb{Q}$ . Koji Tasaka performed numerical experiments of the dimension of the above vector spaces up to $N=7$ by using Mathematica. The list of the conjectural dimension is given as follows.

Using Theorem 4.12, one can reduce the dimension of the space ${\mathcal{E}}_{N}$ . The following is the table of the number of linearly independent relations provided by Theorem 4.12:

This table together with the dimension table shows that the relations obtained in Theorem 4.12 are not enough to capture all relations among $\widetilde{G}^{\unicode[STIX]{x0428}}$ ’s for $N\geqslant 5$ .

Remark 4.16. It is also worth mentioning that the $\mathbb{Q}$ -algebra ${\mathcal{E}}$ contains the ring of quasimodular forms for $\operatorname{SL}_{2}(\mathbb{Z})$ over $\mathbb{Q}$ :

$$\begin{eqnarray}\mathbb{Q}[\widetilde{G}_{2}^{\unicode[STIX]{x0428}},\widetilde{G}_{4}^{\unicode[STIX]{x0428}},\widetilde{G}_{6}^{\unicode[STIX]{x0428}}]\subset {\mathcal{E}},\end{eqnarray}$$

and that the ring of quasimodular forms is stable under the derivative $\text{d}=qd/dq$ (see [Reference Kaneko and Zagier13]). It might be remarkable to consider whether the $\mathbb{Q}$ -algebra ${\mathcal{E}}$ is stable under the derivative, because by expressing $\text{d}\widetilde{G}^{\unicode[STIX]{x0428}}$ as $\mathbb{Q}$ -linear combinations of $\widetilde{G}^{\unicode[STIX]{x0428}}$ ’s and taking the constant term as an element in $\mathbb{C}[[q]]$ one obtains $\mathbb{Q}$ -linear relations among multiple zeta values. For instance, Kaneko [Reference Kaneko, Kim and Taguchi11] proved the identity

$$\begin{eqnarray}\text{d}\widetilde{G}_{N}^{\unicode[STIX]{x0428}}(q)=2N\biggl(\widetilde{G}_{N+2}^{\unicode[STIX]{x0428}}(q)-\mathop{\sum }_{i=1}^{N}\widetilde{G}_{i,N+2-i}^{\unicode[STIX]{x0428}}(q)\biggr),\end{eqnarray}$$

which provides the well-known formula $\unicode[STIX]{x1D701}(N+2)=\sum _{i=1}^{N}\unicode[STIX]{x1D701}(i,N+2-i)$ . We hope to discuss these problems in a future publication.