New functional equations of finite multiple polylogarithms

TL;DR

This paper establishes new finite analogues of classical multiple polylogarithm formulas, revealing novel functional equations and error term phenomena specific to finite multiple polylogarithms of Ono-Yamamoto type.

Contribution

It introduces finite analogues of known polylogarithm formulas using shuffle relations, and derives new functional equations unique to the finite setting.

Findings

Finite analogue of the formula for multiple polylogarithms with error terms.

New functional equations of the form 't ↔ 1 - t' for finite multiple polylogarithms.

Identification of error terms in finite polylogarithm formulas.

Abstract

We give a finite analogue of the well-known formula $\mathrm{Li}_{\underbrace{1, \ldots, 1}_n}(t) = \frac{1}{n!}\mathrm{Li}_1(t)^n$ of multiple polylogarithms for any positive integer n by using the shuffle relation of finite multiple polylogarithms of Ono-Yamamoto type. Unlike the usual case, the terms regarded as error terms appear in this formula. As a corollary, we obtain $"t \leftrightarrow 1 - t"$ type new functional equations of finite multiple polylogarithms of Ono-Yamamoto type and Sakugawa-Seki type.

Equations

$${\pounds }_{\mathcal{A},\mathbf{k}}^{\mathrm{OY}}(t):=\text{sideset}{}^{\prime }{\sum }_{0<l_1,\dots ,l_r<p}\frac{t^{l_1+\cdots +l_r}}{l_1^{k_1}(l_1+l_2{)}^{k_2}\cdots (l_1+\cdots +l_r{)}^{k_r}}\mathrm{mod}p$$

$${\zeta }_{\mathcal{A}}^{(i)}(\mathbf{k}):=\text{sideset}{}^{\prime }{\sum }_{\begin{array}{c}0<l_1,\dots ,l_r<p\\ (i-1)p<l_1+\cdots +l_r<ip\end{array}}\frac{1}{l_1^{k_1}(l_1+l_2{)}^{k_2}\cdots (l_1+\cdots +l_r{)}^{k_r}}\mathrm{mod}p.$$

$${\zeta }_{\mathcal{A}}(\mathbf{k})=\sum _{0<n_1<\cdots <n_r<p}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}\mathrm{mod}p$$

$$f_n(t):=\sum _{k=0}^{n-2}\left(\sum _{i=1}^{n-k-1}{\zeta }_{\mathcal{A}}^{(i)}(\{1{\}}^{n-k-2},2)t^{i\mathbf{p}}\right){\pounds }_{\mathcal{A},\{1{\}}^k}^{\mathrm{OY}}(t)$$

$$g_n(t):=\sum _{k=0}^{n-2}\left(\sum _{i=1}^{n-k-2}{\zeta }_{\mathcal{A}}^{(i)}(\{1{\}}^{n-k-2})t^{i\mathbf{p}}\right){\pounds }_{\mathcal{A},(2,\{1{\}}^k)}^{\mathrm{OY}}(t)$$

$${\pounds }_{\mathcal{A},\{1{\}}^n}^{\mathrm{OY}}(t)=\frac{1}{n!}{\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^n+\frac{1}{n!}\sum _{k=1}^n(k-1)!(f_k(t)+g_k(t)){\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^{n-k}.$$

$${\mathcal{L}}_{\mathcal{A},n}(t):={\pounds }_{\mathcal{A},\{1{\}}^n}^{\mathrm{OY}}(t)-\frac{1}{n!}\sum _{k=1}^n(k-1)!(f_k(t)+g_k(t)){\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^{n-k}.$$

$${\mathcal{L}}_{\mathcal{A},n}(t)={\mathcal{L}}_{\mathcal{A},n}(1-t).$$

$${\pounds }_{\mathcal{A},\{1{\}}^{n-1}}^{\mathrm{OY}}(t){\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t)=n{\pounds }_{\mathcal{A},\{1{\}}^n}^{\mathrm{OY}}(t)-f_n(t)-g_n(t).$$

$$F_k(t):={\pounds }_{\mathcal{A}}^{\mathrm{OY}}(\{1{\}}^{n-k-1},(1),\{1{\}}^k;t).$$

$${\pounds }_{\mathcal{A}}^{\mathrm{OY}}(\mathbf{\lambda },\mathbf{\mu },\mathbf{\nu };t):=\text{sideset}{}^{\prime }{\sum }_{\begin{array}{c}0<l_1,\dots ,l_a<p\\ 0<m_1,\dots ,m_b<p\\ 0<n_1,\dots ,n_c<p\end{array}}\frac{t^{L_a+M_b+N_c}}{\prod _{x=1}^a{L}_x^{{\lambda }_x}\prod _{y=1}^b{M}_y^{{\mu }_y}\prod _{z=1}^c{\left(L_a+M_b+N_z\right)}^{{\nu }_z}}\in {\mathcal{A}}_{\mathbb{Z}[t]},$$

$$\begin{gathered} F_k(t)=F_{k+1}(t)+{\pounds }_{\mathcal{A},\{1{\}}^n}^{\mathrm{OY}}(t)-\left(\sum _{i=1}^{n-k-1}{\zeta }_{\mathcal{A}}^{(i)}(\{1{\}}^{n-k-2},2)t^{i\mathbf{p}}\right){\pounds }_{\mathcal{A},\{1{\}}^k}^{\mathrm{OY}}(t) \\ -\left(\sum _{i=1}^{n-k-2}{\zeta }_{\mathcal{A}}^{(i)}(\{1{\}}^{n-k-2})t^{i\mathbf{p}}\right){\pounds }_{\mathcal{A},(2,\{1{\}}^k)}^{\mathrm{OY}}(t). \end{gathered}$$

$${\pounds }_{\mathcal{A},\{1{\}}^{n-1}}^{\mathrm{OY}}(t)=\frac{1}{(n-1)!}{\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^{n-1}+\frac{1}{(n-1)!}\sum _{k=1}^{n-1}(k-1)!(f_k(t)+g_k(t)){\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^{n-k-1}.$$

$${\pounds }_{\mathcal{A},\{1{\}}^{n-1}}^{\mathrm{OY}}(t){\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t)=n{\pounds }_{\mathcal{A},\{1{\}}^n}^{\mathrm{OY}}(t)-f_n(t)-g_n(t).$$

$$\begin{gathered} \frac{1}{(n-1)!}{\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^n+\frac{1}{(n-1)!}\sum _{k=1}^{n-1}(k-1)!(f_k(t)+g_k(t)){\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^{n-k} \\ =n\left(\frac{1}{n!}{\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^n+\frac{1}{n!}\sum _{k=1}^{n-1}(k-1)!(f_k(t)+g_k(t)){\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^{n-k}\right). \end{gathered}$$

$$f_3(t)={\zeta }_{\mathcal{A}}^{(1)}(1,2)t^{\mathbf{p}}+{\zeta }_{\mathcal{A}}^{(2)}(1,2)t^{2\mathbf{p}}={\zeta }_{\mathcal{A}}(1,2)t^{\mathbf{p}}(1-t^{\mathbf{p}})={\zeta }_{\mathcal{A}}(1,2)t^p(1-t{)}^{\mathbf{p}}.$$

$${\pounds }_{\mathcal{A},\{1{\}}^3}^{\mathrm{OY}}(t)=\frac{1}{3!}{\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^3+\frac{1}{3}{\zeta }_{\mathcal{A}}(1,2)t^{\mathbf{p}}(1-t{)}^{\mathbf{p}}.$$

$$f_4(t)=$$

$$+({\zeta }_{\mathcal{A}}(1,2)t^{\mathbf{p}}+{\zeta }_{\mathcal{A}}^{(2)}(1,2)t^{2\mathbf{p}}){\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t)+{\zeta }_{\mathcal{A}}(2)t^{\mathbf{p}}{\pounds }_{\mathcal{A},(1,1)}^{\mathrm{OY}}(t)$$

$$=$$

$$g_4(t)=({\zeta }_{\mathcal{A}}(1,1)t^{\mathbf{p}}+{\zeta }_{\mathcal{A}}^{(2)}(1,1)t^{2\mathbf{p}}){\pounds }_{\mathcal{A},2}^{\mathrm{OY}}(t)+{\zeta }_{\mathcal{A}}(1)t^{\mathbf{p}}{\pounds }_{\mathcal{A},(2,1)}^{\mathrm{OY}}(t)=0.$$

$${\pounds }_{\mathcal{A},\{1{\}}^4}^{\mathrm{OY}}(t)=\frac{1}{24}{\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t{)}^4+\frac{1}{3}{\zeta }_{\mathcal{A}}(1,2)t^{\mathbf{p}}(1-t{)}^{\mathbf{p}}{\pounds }_{\mathcal{A},1}^{\mathrm{OY}}(t).$$

$$g_5(t)=$$

$$+({\zeta }_{\mathcal{A}}(1,1)t^{\mathbf{p}}+{\zeta }_{\mathcal{A}}^{(2)}(1,1)t^{2\mathbf{p}}){\pounds }_{\mathcal{A},(2,1)}^{\mathrm{OY}}(t)+{\zeta }_{\mathcal{A}}(1)t^{\mathbf{p}}{\pounds }_{\mathcal{A},(2,1,1)}^{\mathrm{OY}}(t)=0.$$

$${\pounds }_{\mathcal{A},\{1{\}}^5}^{\mathrm{OY}}(t)=$$

$$=$$

$${\pounds }_{\mathcal{A},\{1{\}}^5}^{\mathrm{OY}}(t)-\frac{1}{5}{f}_5(t)={\pounds }_{\mathcal{A},\{1{\}}^5}^{\mathrm{OY}}(1-t)-\frac{1}{5}{f}_5(1-t).$$

$$f_5(t)=$$

$$\displaystyle+\biggl{(}\zeta^{(1)}_{\mathcal{A}}(1,1,2)t^{\boldsymbol{p}}+\zeta^{(2)}_{\mathcal{A}}(1,1,2)t^{2\boldsymbol{p}}+\zeta^{(3)}_{\mathcal{A}}(1,1,2)t^{3\boldsymbol{p}}\biggr{)}\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)$$

$$=$$

$${\pounds }_{\mathcal{A},\{1{\}}^5}^{\mathrm{OY}}(t)-{\pounds }_{\mathcal{A},\{1{\}}^5}^{\mathrm{OY}}(1-t)=$$

$$=$$

$${\pounds }_{\mathcal{A},\{1{\}}^5}^{\mathrm{OY}}(t)-{\pounds }_{\mathcal{A},\{1{\}}^5}^{\mathrm{OY}}(1-t)=\frac{B_{\mathbf{p}-5}}{5}{t}^{\mathbf{p}}(1-t^{\mathbf{p}})(2t^{\mathbf{p}}-1).$$

Full text

This research was supported in part by KAKENHI 26247004, as well as the JSPS Core-to-Core program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry” and the KiPAS program 2013–2018 of the Faculty of Science and Technology at Keio University.

1. 1 Introduction
2. 2 Proof of the main theorem
3. 3 Functional equations of finite multiple polylogarithms of Ono-Yamamoto type
4. 4 Functional equations of finite multiple polylogarithms of Sakugawa-Seki type

1. Introduction

In this article, we give a finite analogue of the well-known formula $\mathrm{Li}_{\underbrace{1,\ldots,1}_{n}}(t)=\frac{1}{n!}\mathrm{Li}_{1}(t)^{n}$ of multiple polylogarithms for any positive integer $n$ by using the shuffle relation of finite multiple polylogarithm (FMP) of Ono–Yamamoto type (OY-type). As a corollary, we obtain $``t\leftrightarrow 1-t"$ type new functional equations of FMPs of OY-type. In addition, by using the known relation between FMPs of OY-type and Sakugawa–Seki type (SS-type), we also obtain new functional equations of FMPs of SS-type, which seem to be difficult obtained only by using Sakugawa–Seki’s results [SS].

First, we review of the history of FMPs. Recently, the author and Yamamoto [OY] introduced finite multiple polylogarithms (FMPs) $\pounds^{\mathrm{OY}}_{\mathcal{A},\mathbf{k}}(t)$ as an element of the $\mathbb{Q}$-algebra \mathcal{A}_{\mathbb{Z}[t]}:=\left.\bigl{(}\prod_{p}(\mathbb{Z}/p\mathbb{Z})[t]\bigr{)}\right/\bigl{(}\bigoplus_{p}(\mathbb{Z}/p\mathbb{Z})[t]\bigr{)}, where $p$ runs through all the rational primes. Thus, an element of $\mathcal{A}_{\mathbb{Z}[t]}$ is represented by a family $(f_{p})_{p}$ of polynomials $f_{p}\in(\mathbb{Z}/p\mathbb{Z})[t]$, and two families $(f_{p})_{p}$ and $(g_{p})_{p}$ represent the same element of $\mathcal{A}_{\mathbb{Z}[t]}$ if and only if $f_{p}=g_{p}$ for all but finitely many primes $p$. We denote such an element of $\mathcal{A}_{\mathbb{Z}[t]}$ simply by $f_{p}$ omitting $(\ )_{p}$ if there is no fear of confusion. For example, we denote an element $(t^{p}\bmod{p})_{p}$ of $\mathcal{A}_{\mathbb{Z}[t]}$ by $t^{\boldsymbol{p}}$.

Note that the idea considering several objects depending on a fixed prime $p$ in some adelic rings is due to Kaneko–Zagier’s theory of finite multiple zeta values [KZ]. Also note that the $\mathbb{Q}$-algebra $\mathcal{A}_{\mathbb{Z}[t]}$ is denoted by $\mathcal{B}$ in [OY]. The symbol $\mathcal{A}_{\mathbb{Z}[t]}$ is Sakugawa–Seki’s notation in [SS].

Definition 1.1** ([OY, Definition 1.2]).**

For a positive integer $r$ and an index $\mathbf{k}=(k_{1},\ldots,k_{r})\in(\mathbb{Z}_{\geq 1})^{r}$, we define a finite multiple polylogarithm of Ono–Yamamoto type (OY-type for short) by

[TABLE]

as an element of $\mathcal{A}_{\mathbb{Z}[t]}$. Here, $\sideset{}{{}^{\prime}}{\sum}$ denotes the sum of fractions whose denominators are prime to $p$.

We respectively call integers $r$ and $k_{1}+\cdots+k_{r}$ the depth and weight of $\mathbf{k}=(k_{1},\ldots,k_{r})$ and we denote the weight of index of $\mathbf{k}$ by $\mathrm{wt}(\mathbf{k})$.

One of the reasons why we introduces FMPs of OY-type was to establish a finite analogue of the shuffle relation for multiple polylogarithms.

On the other hand, Sakugawa and Seki introduced another type of FMPs in [SS]. We call their FMPs SS-type in this article. One of their motivations of introducing their FMPs was to establish functional equations of their FMPs.

More recently, Seki put a question about functional equations of FMPs of OY-type with the index $\{1\}^{n}:=(\underbrace{1,\ldots,1}_{n})$ for a positive integer $n$. The results of Kontsevich [K, (A)] and Elbaz-Vincent and Gangl [EG, PROPOSITION 5.9 (1)] say that $\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)=\pounds^{\mathrm{OY}}_{\mathcal{A},1}(1-t)$ holds. Futhermore, by using functional equations of FMPs of SS-type and the fact that FMPs of OY-type can be written in terms of FMPs of SS-type [SS, Proposition 3.26], Seki [Se, Theorem 14.6] proved the equality $\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{2}}(t)=\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{2}}(1-t)$.

In this article, we give an answer to Seki’s question, that is, we give functional equations between $\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{n}}(t)$ and $\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{n}}(1-t)$ for a positive integer $n$, which contain Seki’s result as the case of $n=2$. In order to state our main theorem, we recall the definition of a variant $\zeta^{(i)}_{\mathcal{A}}(\mathbf{k})$ of finite multiple zeta values (FMZVs) $\zeta_{\mathcal{A}}(\mathbf{k})$.

Definition 1.2** ([OY, Definition 2.1]).**

For an index $\mathbf{k}=(k_{1},\ldots,k_{r})$ and $1\leq i\leq r$, we define a variant of FMZVs as an element of \mathcal{A}:=\left.\bigl{(}\prod_{p}\mathbb{Z}/p\mathbb{Z}\bigr{)}\right/\bigl{(}\bigoplus_{p}\mathbb{Z}/p\mathbb{Z}\bigr{)} by

[TABLE]

Note that $\zeta^{(1)}_{\mathcal{A}}(\mathbf{k})$ coincides with the usual FMZV $\zeta_{\mathcal{A}}(\mathbf{k})$ defined by

[TABLE]

and we see that $\zeta^{(r)}_{\mathcal{A}}(\mathbf{k})=(-1)^{\mathrm{wt}(\mathbf{k})}\zeta_{\mathcal{A}}(\mathbf{k})$.

The main theorem of this article is the following.

Theorem 1.3**.**

For a positive integer $n$, we define two elements

[TABLE]

and

[TABLE]

of $\mathcal{A}_{\mathbb{Z}[t]}$. Here, we understand that these elements are equal to 0 if the sums are empty. Then, we have

[TABLE]

We remark that this equality can be regarded as an finite analogue of the well-known formula $\mathrm{Li}_{\{1\}^{n}}(t)=\frac{1}{n!}\mathrm{Li}_{1}(t)^{n}$, where $\mathrm{Li}_{\mathbf{k}}(t):=\sum_{0<n_{1}<\cdots<n_{r}}\frac{t^{n_{r}}}{n^{k_{1}}_{1}\cdots n^{k_{r}}_{r}}$ is the (one variable) multiple polylogarithm.

As a corollary of our main theorem and the equality $\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)=\pounds^{\mathrm{OY}}_{\mathcal{A},1}(1-t)$, we obtain $``t\leftrightarrow 1-t"$ type functional equations of FMPs of OY-type.

Corollary 1.4**.**

For a positive integer $n$, set

[TABLE]

Then we obtain

[TABLE]

Proof.

By Theorem 1.3, we have $\mathcal{L}_{\mathcal{A},n}(t)=\frac{1}{n!}\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)^{n}$. Therefore, the assertion holds by the result of Elbaz-Vincent and Gangl [EG, PROPOSITION 5.9 (1)]. ∎

The contents of this article is as follows. In Section 2, we prove Theorem 1.3 by using the shuffle relation of FMPs of OY-type. In Section 3, by using our main theorem, we give examples of functional equations of FMPs of OY-type. In the final section, by using the relation between FMPs of OY-type and SS-type, we also give functional equations of FMPs of SS-type, which seem to be difficult to obtain only using the results of [SS].

2. Proof of the main theorem

In this section, we prove the main theorem by using the shuffle relation of FMPs of OY-type, which was proved by the author and S. Yamamoto in [OY].

First, we explicitly calculate the shuffle relation of $\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{n-1}}(t)$ and $\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)$ for a positive integer $n$.

Lemma 2.1**.**

For a positive integer $n$, we have

[TABLE]

Recall the definitions of $f_{n}(t)$ and $g_{n}(t)$. See (1) and (2).

Proof.

For $0\leq k\leq n-1$, set

[TABLE]

Here, for indices $\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{a}),\boldsymbol{\mu}=(\mu_{1},\ldots,\mu_{b})$ and $\boldsymbol{\nu}=(\nu_{1},\ldots,\nu_{c})$ ($a,b,c\in\mathbb{Z}_{\geq 0}$), $\pounds^{\mathrm{OY}}_{\mathcal{A}}(\boldsymbol{\lambda},\boldsymbol{\mu},\boldsymbol{\nu};t)$ is the FMP of type $(\boldsymbol{\lambda},\boldsymbol{\mu},\boldsymbol{\nu})$ [OY, Definition 3.1] defined by

[TABLE]

where $L_{x}:=l_{1}+\cdots+l_{x},M_{y}:=m_{1}+\cdots+m_{y}$ and $N_{z}:=n_{1}+\cdots+n_{z}$. By [OY, Remark 3.2], we have $F_{0}(t)=\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{n-1}}(t)\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)$ and $F_{n-1}(t)=\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{n}}(t)$. By using [OY, Proposition 3.7] in the case $\boldsymbol{\lambda}:=\{1\}^{n-k-1},\boldsymbol{\mu}:=(1)$ and $\boldsymbol{\nu}:=\{1\}^{k}$ $(0\leq k\leq n-2)$, we obtain

[TABLE]

Therefore, the statement holds from taking the telescoping sum of (4). ∎

Proof of Theorem 1.3.

We prove the statement by the induction on $n\geq 1$. Note that the statement for $n=1$ holds by [K, (A)] or [EG, PROPOSITION 5.9]. For $n\geq 2$, assume that the statement holds for $n-1$:

[TABLE]

By Lemma 2.1, the product of the left hand side of (5) and $\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)$ coincides with

[TABLE]

On the other hand, the product of the right hand side of (5) and $\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)$ coincides with

[TABLE]

Therefore, by (6) and (7), we see that the statement holds for $n$. ∎

3. Functional equations of finite multiple polylogarithms of Ono-Yamamoto type

Next, we give examples of our main theorem for $n=2,3$ and $4$. We use the following lemmas for calculating our examples.

Lemma 3.1** ([H, Theorem 4.3]).**

For any positive integer $k$ and $r$, we have $\zeta_{\mathcal{A}}(\underbrace{k,\ldots,k}_{r})=0$. In particular, we have $\zeta_{\mathcal{A}}(k)=0$ for any positive integer $k$.

Lemma 3.2** ([Sa, Table 2], for example).**

For any index $\mathbf{k}$ of weight 4, we have $\zeta_{\mathcal{A}}(\mathbf{k})=0$.

Example 3.3**.**

1. (i)

First, we consider the case $n=2$. By the definition of $\zeta^{(i)}_{\mathcal{A}}$ and Lemma 3.1, we obtain $f_{2}(t)=\zeta_{\mathcal{A}}(2)t^{\boldsymbol{p}}=0$. By the definition of $g_{2}(t)$, we have $g_{2}(t)=0$. Thus Theorem 1.3 says the equality $\pounds^{\mathrm{OY}}_{\mathcal{A},(1,1)}(t)=\frac{1}{2}\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)^{2}$. Moreover, by Corollary 1.4, we obtain Seki’s result $\pounds^{\mathrm{OY}}_{\mathcal{A},(1,1)}(t)=\pounds^{\mathrm{OY}}_{\mathcal{A},(1,1)}(1-t)$. 2. (ii)

Next, we consider the case of $n=3$. By an easy calculation and $\zeta^{(2)}_{\mathcal{A}}(1,2)=-\zeta_{\mathcal{A}}(1,2)$ [OY, Remark 2.2], $f_{3}(t)$ can be calculated as follows.

[TABLE]

The last equality holds since $1-t^{p}=(1-t)^{p}$ holds in $(\mathbb{Z}/p\mathbb{Z})[t]$ for all primes $p$. On the other hand, we have $g_{3}(t)=\zeta_{\mathcal{A}}(1)t^{\boldsymbol{p}}\pounds^{\mathrm{OY}}_{\mathcal{A},2}(t)=0$ by Lemma 3.1. Therefore, Theorem 1.3 says the equality

[TABLE]

Moreover, since $f_{3}(t)=f_{3}(1-t)$ by (8), we see that Corollary 1.4 says the equality $\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{3}}(t)=\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{3}}(1-t)$. 3. (iii)

Furthermore, we consider the case of $n=4$. By an easy calculation, we obtain

[TABLE]

By [OY, Example 2.6, (ii)], $\zeta^{(2)}_{\mathcal{A}}(1,1,2)$ is a sum of FMZVs of weight 4, we see that $f_{4}(t)=f_{3}(t)\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)$ by Lemma 3.2. On the other hand, since $\zeta_{\mathcal{A}}(1,1)=\zeta^{(2)}_{\mathcal{A}}(1,1)=0$ by Lemma 3.1, we have

[TABLE]

Therefore, we obtain

[TABLE]

Moreover, since $f_{4}(t)=f_{4}(1-t)$ by (3), we have $\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{4}}(t)=\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{4}}(1-t)$. 4. (iv)

Finally, consider the case of $n=5$. In this case, it is difficult to expect the $"t\leftrightarrow 1-t"$ type relation of $\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{n}}(t)$ for $n\geq 5$.

First, by [OY, Example 2.6] and Lemma 3.1, we have \zeta_{\mathcal{A}}(1,1,1)$$=-\zeta^{(3)}_{\mathcal{A}}(1,1,1)=0 and $\zeta^{(2)}_{\mathcal{A}}(1,1,1)=4\zeta_{\mathcal{A}}(1,1,1)+\zeta_{\mathcal{A}}(2,1)+\zeta_{\mathcal{A}}(1,2)=0$. Therefore, we obtain

[TABLE]

Thus, Theorem 1.3 says that

[TABLE]

Since $f_{3}(t)\pounds^{\mathrm{OY}}_{\mathcal{A},1}(t)^{2}=f_{3}(1-t)\pounds^{\mathrm{OY}}_{\mathcal{A},1}(1-t)^{2}$, we have

[TABLE]

Next, we have

[TABLE]

Therefore, we see that

[TABLE]

By [OY, Example 2.6 (2)] and [Sa, Table 2], we see that $\zeta_{\mathcal{A}}(1,1,1,2)=B_{\boldsymbol{p}-5}$ and $\zeta^{(2)}_{\mathcal{A}}(1,1,1,2)=0$. Here, we set $B_{\boldsymbol{p}-5}:=(B_{p-5}\bmod{p})_{p}\in\mathcal{A}$. Therefore, we obtain

[TABLE]

Thus, since it is conjectured that $B_{\boldsymbol{p}-5}$ does not vanish in $\mathcal{A}$ (for example, see [Z, Conjecture 2.1]), we see that $\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{5}}(t)\neq\pounds^{\mathrm{OY}}_{\mathcal{A},\{1\}^{5}}(1-t)$.

4. Functional equations of finite multiple polylogarithms of Sakugawa-Seki type

We end this article with new functional equations of FMPs of SS-type.

Definition 4.1** ([SS, Definition 3.8]).**

Let $r$ be a positive integer and $\mathbf{k}=(k_{1},\ldots,k_{r})$ an index. Then we define finite harmonic multiple polylogarithms and 1-variable finite multiple polylogarithms as follows:

[TABLE]

[TABLE]

Here, for an $r$-tuple of variables $\boldsymbol{t}:=(t_{1},\ldots,t_{r})$, we set \mathcal{A}_{\mathbb{Z}[\boldsymbol{t}]}:=\left.\bigl{(}\prod_{p}(\mathbb{Z}/p\mathbb{Z})[\boldsymbol{t}]\bigr{)}\right/\bigl{(}\bigoplus_{p}(\mathbb{Z}/p\mathbb{Z})[\boldsymbol{t}]\bigr{)}.

Now we prepare the following notation to describe the relation between the FMPs of OY-type and SS-type (cf. [OY, Section 2]). First, for a positive integer $r$, set

[TABLE]

and

[TABLE]

Next, for $\phi\in\Phi_{r,s}$, set $s_{\phi}:=s$. Furthermore, for $\phi\in\Phi_{r}$ and $1\leq i\leq r$, we define an integer $\delta_{\phi}(i)$ by

[TABLE]

Finally, a map $\beta:\Phi_{r}\rightarrow[r]$ is defined by $\beta(\phi):=\delta_{\phi}(r)+1$ and we set

[TABLE]

Proposition 4.2** ([SS, Proposition 3.26]).**

For an index $\mathbf{k}=(k_{1},\ldots,k_{r})$, we have

[TABLE]

By Proposition 4.2, our main theorem gives functional equations of FMPs of SS-type. It seems very difficult to obtain our functional equations of FMPs of SS-type only using Sakugawa-Seki’s theory [SS] and without using the shuffle relation of FMPs of OY-type. We describe only two functional equations of FMPs of SS-type which are obtained from that of FMPs of OY-type with indices $\{1\}^{3}$ and $\{1\}^{4}$.

Corollary 4.3**.**

We have

[TABLE]

Corollary 4.4**.**

We have

[TABLE]

Acknowledgement

The author expresses his sincere gratitude to Dr. Shin-ichiro Seki for introducing me a question concerning the functional equations of FMPs of OY-type. He would like to thank Dr. Kenji Sakugawa and Dr. Shin-ichiro Seki for their valuable comments and helpful discussions at Keio University. He also would like to thank Prof. Kenichi Bannai, Prof. Shuji Yamamoto, Dr. Kenji Sakugawa and Dr. Shin-ichiro Seki for careful reading of the manuscript.