$\mathbb{Q}$-linear relations of specific families of multiple zeta values and the linear part of Kawashima's relation

TL;DR

This paper investigates specific families of multiple zeta values related to Kawashima's relation, providing explicit bases and exploring their complex function interpolations, revealing connections to duality and derivation relations.

Contribution

It introduces explicit bases for these families and shows how duality and derivation relations follow from Kawashima's linear relation.

Findings

Explicit bases for specific multiple zeta value families

Connections between Kawashima's relation and duality/derivation relations

Interpolation of families to complex functions

Abstract

In this paper, we study specific families of multiple zeta values which closely relate to the linear part of Kawashima's relation. We obtain an explicit basis of these families, and investigate their interpolations to complex functions. As a corollary of our main results, we also see that the duality formula and the derivation relation are deduced from the linear part of Kawashima's relation.

Equations

$$\zeta (k_1,\dots ,k_r):=\sum _{1\le n_1<\cdots <n_r}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.$$

$$f(A,B):=(Z(AB),Z(A(x+y)B),Z(A(x+y{)}^{2}B),\dots ).$$

$$w\diamond 1=1\diamond w=w,$$

$$x{w}_1\diamond x{w}_2=x(w_1\diamond x{w}_2)-x(y{w}_1\diamond w_2),$$

$$x{w}_1\diamond y{w}_2=x(w_1\diamond y{w}_2)+y(x{w}_1\diamond w_2),$$

$$y{w}_1\diamond x{w}_2=y(w_1\diamond x{w}_2)+x(y{w}_1\diamond w_2),$$

$$y{w}_1\diamond y{w}_2=y(w_1\diamond y{w}_2)-y(x{w}_1\diamond w_2)$$

$$f(y{w}_1,w_{2}x)=f(y,(\tau (w_1)\diamond w_2)x).$$

$$f(y,wx)=0\iff w=0.$$

$$f(A,B)=(L_1(s;A,B)+s{L}_2(s;A,B){)}_{s=0}^{\infty }.$$

$$\{f(y,u_1\cdots u_{n}x)\mid n\in {\mathbb{Z}}_{\ge 0},u_1,\dots ,u_n\in \{x,y\}\}$$

$$\{f(y{w}_1,w_{2}x)\mid w_1,w_2\in \mathfrak{H}\}.$$

$$\sum _{A,B}{n}_{A,B}f(A,B)=0,$$

$$f(A,B)=f(\tau (B),\tau (A)).$$

$$Z({\partial }_l(w))=0(w\in y\mathfrak{H}x).$$

$$f({\partial }_l(A),B)+f(A,{\partial }_l(B))=0$$

$$1\ast w$$

$$x{w}_1\ast w_2$$

$$y{w}_1\ast y{w}_2$$

$$Z(\varphi (w_1\ast w_2)x)=0.$$

$$(x+y)w_1\diamond w_2=w_1\diamond (x+y)w_2=(x+y)(w_1\diamond w_2).$$

$$x{w}_1\diamond (x+y)w_2$$

$$=x(w_1\diamond (x+y)w_2)-x(y{w}_1\diamond w_2)+y(x{w}_1\diamond w_2)$$

$$=x((x+y)w_1\diamond w_2)-x(y{w}_1\diamond w_2)+y(x{w}_1\diamond w_2)$$

$$=(x+y)(x{w}_1\diamond w_2)$$

$$=(x+y)x{w}_1\diamond w_2.$$

$$\varphi (w_1\ast w_2)=\varphi (w_1)\diamond \varphi (w_2).$$

$$\varphi (u_1{w}_1\ast u_2{w}_2)$$

$$=\{\begin{array}{ll}\varphi (x)(\varphi (w_1)\diamond \varphi (u_2{w}_2))& (u_1=x),\\ \varphi (x)(\varphi (y{w}_1)\diamond \varphi (w_2))& (u_1=y,u_2=x),\\ \varphi (y)(\varphi (w_1)\diamond \varphi (y{w}_2)+\varphi (y{w}_1)\diamond \varphi (w_2)+\varphi (x{w}_1)\diamond \varphi (w_2))& (u_1=u_2=y)\end{array}$$

$$=\varphi (u_1{w}_1)\diamond \varphi (u_2{w}_2).\text{qed}$$

$$A_1$$

$$A_2$$

$$A_3$$

$$A_4$$

$$(y{w}_1\diamond y{w}_2)x$$

$$=y(w_1\diamond y{w}_2-x{w}_1\diamond w_2)x$$

$$=-\left(y\tau (w_1)(y{w}_2)x-y(w_1\diamond y{w}_2)x\right)+\left(y\tau (x{w}_1)w_{2}x-y(x{w}_1\diamond w_2)x\right).$$

$$y{w}_1{w}_{2}x-y(\tau (w_1)\diamond w_2)x\in \mathrm{span}\{(u\diamond v)x\mid u,v\in y\mathfrak{H}\}$$

$$w_1:=u_1\cdots u_m(m\in {\mathbb{Z}}_{\ge 0},u_1,\dots ,u_m\in \{y,x+y\})$$

Full text

1. Introduction

For positive integers $k_{1},\ldots,k_{r}$ with $k_{r}\geq 2$, the multiple zeta values (MZVs) are defined by

[TABLE]

There exist a lot of $\mathbb{Q}$-linear relations among MZVs, and Kawashima’s relation (with the shuffle product formula) is a big family of such relations. One can conjecturally obtain every $\mathbb{Q}$-linear relation among MZVs from Kawashima’s relation (if the products of MZVs are expanded by the shuffle product formula). The linear part of Kawashima’s relation is a special case of Kawashima’s relation. It does not exhaust all $\mathbb{Q}$-linear relations among MZVs, but contains several types of well-known relations among MZVs such as the duality formula, the derivation relation, the quasi-derivation relation, the cyclic sum formula, the Ohno relation, and so on (see Tanaka [7, Section 1]). The purpose of this paper is to study specific families of MZVs coming from Hoffman’s algebraic setup, and to clarify their close relationship to the linear part of Kawashima’s relation. We obtain an explicit basis of these families by using the linear part of Kawashima’s relation, and give interpolations of these families to complex functions.

We recall Hoffman’s algebraic setup with a slightly different convention (see Hoffman [3]). Let $\mathfrak{H}:=\mathbb{Q}\left\langle x,y\right\rangle$ be the noncommutative polynomial ring in two indeterminates $x$ and $y$. We define the $\mathbb{Q}$-linear map $\mathit{Z}:y\mathfrak{H}x\to\mathbb{R}$ by $\mathit{Z}(yx^{k_{1}-1}\cdots yx^{k_{r}-1}):=\zeta(k_{1},\ldots,k_{r})$. For $A\in y\mathfrak{H}$ and $B\in\mathfrak{H}x$, we define a family $f(A,B)\in\mathbb{R}^{\infty}$ by

[TABLE]

Then, the duality formula (Example 1.7) and the derivation relation (Example 1.8) for MZVs can be written as linear relations among $f(A,B)$’s. We define the product $\diamond$ on $\mathfrak{H}$ by

[TABLE]

for $w,w_{1},w_{2}\in\mathfrak{H}$ together with $\mathbb{Q}$-bilinearity.

Remark 1.1*.*

We note that the product $\diamond$ is associative and commutative (see Corollary 2.5).

Let $\tau$ be the anti-automorphism on $\mathfrak{H}$ that interchanges $x$ and $y$.

Theorem 1.2** (Main theorem).**

We have the followings.

1. (i)

For $w_{1},w_{2}\in\mathfrak{H}$, we have

[TABLE] 2. (ii)

For $w\in\mathfrak{H}$, we have

[TABLE] 3. (iii)

For $A\in y\mathfrak{H},B\in\mathfrak{H}x$, there exist unique Dirichlet series $L_{1}(s;A,B)$ and $L_{2}(s;A,B)$ such that

[TABLE]

As a corollary of Theorem 1.2 (i) and (ii), we can obtain an explicit basis of these families.

Corollary 1.3**.**

There exists an explicit basis

[TABLE]

of the $\mathbb{Q}$-linear space spanned by

[TABLE]

In addition, all $\mathbb{Q}$-linear relations of $f(-,-)$ are spanned by the relation in Theorem 1.2 (i).

Remark 1.4*.*

The Dirichlet series $L_{1}(s;A,B)$ and $L_{2}(s;A,B)$ can be continued meromorphically to the whole plane $\mathbb{C}$ and explicitly written by using the multiple zeta functions (for details, see Section 3).

Remark 1.5*.*

Theorem 1.2 (i) is essentially equivalent to the linear part of Kawashima’s relation (see Kawashima [5]). This equivalence will be shown in Section 2.

As a consequence of Corollary 1.3 and Remark 1.5, we obtain the next theorem.

Theorem 1.6** (A special case $A_{6}\subset A_{1}$ of Theorem 4.1).**

If

[TABLE]

then the relation $\sum_{A,B}n_{A,B}\mathit{Z}(AB)=0$ is deduced from the linear part of Kawashima’s relation.

Let us see some examples of $\mathbb{Q}$-linear relations among $f(-,-)$’s, and check what Theorem 1.6 says for those examples.

Example 1.7** (Duality formula).**

The duality formula for MZVs is described as $\mathit{Z}(w)=\mathit{Z}(\tau(w))\,(w\in y\mathfrak{H}x)$. From this, for $A\in y\mathfrak{H}$ and $B\in\mathfrak{H}x$, we have

[TABLE]

Thus, by Theorem 1.6, the duality formula is deduced from the linear part of Kawashima’s relation. This fact is already known (see Kawashima [5, Section 7]).

Example 1.8** (Derivation relation; Ihara-Kaneko-Zagier [4]).**

Let $l$ be a positive integer. A derivation $\partial$ on $\mathfrak{H}$ is a $\mathbb{Q}$-linear map $\partial\colon\mathfrak{H}\to\mathfrak{H}$ satisfying Leibniz’s rule $\partial(ww^{\prime})=\partial(w)w^{\prime}+w\partial(w^{\prime})$. We define the derivation $\partial_{l}$ on $\mathfrak{H}$ by $\partial_{l}(x):=y(x+y)^{l-1}x$ and $\partial_{l}(y):=-y(x+y)^{l-1}x$, and we note that $\partial_{l}(x+y)=0$. Then, the derivation relation for MZVs is described as

[TABLE]

From this i.e. $\mathit{Z}(\partial_{l}(A(x+y)^{s}B))=0$, we have

[TABLE]

for $A\in y\mathfrak{H}$ and $B\in\mathfrak{H}x$. Thus, by Theorem 1.6, the derivation relation is deduced from the linear part of Kawashima’s relation. This fact is already known (see Tanaka [7]).

2. Proof of Theorem 1.2 (i)

We define an automorphism $\phi$ on $\mathfrak{H}$ by $\phi(x):=x+y$ and $\phi(y):=-y$. We note that $\phi\circ\phi=\rm{id}$. For $w,w_{1},w_{2}\in\mathfrak{H}$, the harmonic product $\ast$ on $\mathfrak{H}$ is defined by

[TABLE]

together with $\mathbb{Q}$-bilinearity.

Remark 2.1*.*

The definition of the above harmonic product $\ast$ coincides with the one given by Sato and the first-named author in [2]. The harmonic product $\ast$ is associative and commutative. We also note that the definition of the usual harmonic product (see Hoffman [3]) is naturally derived from the above definition.

Theorem 2.2** (Kawashima’s relation; Kawashima [5]).**

For $w_{1},w_{2}\in y\mathfrak{H}$, we have

[TABLE]

Proposition 2.3**.**

For $w_{1},w_{2}\in\mathfrak{H}$, we have

[TABLE]

Proof.

The equality $(x+y)w_{1}\diamond w_{2}=(x+y)(w_{1}\diamond w_{2})$ is evident by the definition of the product $\diamond$. Then, we prove $(x+y)w_{1}\diamond w_{2}=w_{1}\diamond(x+y)w_{2}$ by induction on the degree of $w_{1}$. Due to the symmetry of the definition, we need to prove only for $w_{1}$ starting with $x$. By the definition and the induction hypothesis, we have

[TABLE]

This finishes the proof. ∎

Lemma 2.4**.**

For $w_{1},w_{2}\in\mathfrak{H}$, we have

[TABLE]

Proof.

We prove by induction on the number of indeterminates of $w_{1}$ and $w_{2}$. When $w_{1}=1$ or $w_{2}=1$, the lemma trivially holds. By Proposition 2.3, the definitions of the products $\ast$ and $\diamond$, and the induction hypothesis, for $u_{1},u_{2}\in\{x,y\}$ and $w_{1},w_{2}\in\mathfrak{H}$, we find

[TABLE]

Since the harmonic product $\ast$ is associative and commutative, we easily see the following corollary holds.

Corollary 2.5**.**

The product $\diamond$ is associative and commutative.

Lemma 2.6** (Key lemma for Theorem 1.2 (i)).**

The following vector subspaces $A_{1},\ldots,A_{4}$ of $\mathfrak{H}$ are coincident.

[TABLE]

Proof.

The inclusion relation $A_{4}\subset A_{3}$ is obvious, and $A_{1}=A_{2}$ and $A_{3}\subset A_{4}$ are trivial respectively by Lemma 2.4 and Proposition 2.3.

We also find the inclusion relation $A_{2}\subset A_{4}$ holds by the definition of the product $\diamond\,$;

[TABLE]

Thus, we need to show $A_{4}\subset A_{2}$ i.e.,

[TABLE]

for $w_{1},w_{2}\in\mathfrak{H}$. Let

[TABLE]

and we prove (1) by the induction on $m$. The case $m=0$ is obvious. Assume that $m>0$. Put $w_{1}^{\prime}u_{m}:=w_{1}$ and $w_{2}^{\prime}:=u_{m}w_{2}$. Then we have

[TABLE]

by the induction hypothesis. Thus, we get

[TABLE]

Proof of Theorem 1.2 (i).

By Theorem 2.2 and Lemma 2.6, we have the desired result. ∎

3. Proofs of Theorem 1.2 (ii) and (iii)

To prove Theorem 1.2 (iii), we need to show Theorem 3.5 and 3.7. For positive integers $k_{1},\ldots,k_{r}$, we denote by $F_{(k_{1},\ldots,k_{r})}(z)$ the Kawashima function as defined by Yamamoto in [8].

Lemma 3.1** ([8, Proposition 2.6]).**

For non-negative integer $N$ and positive integers $k_{1},\ldots,k_{r}$, the function $F_{(k_{1},\ldots,k_{r})}(z)$ is holomorphic at $z=N$ and

[TABLE]

We define an automorphism $S_{1}$ on $\mathfrak{H}$ by $S_{1}(x):=x$ and $S_{1}(y):=x+y$, and a $\mathbb{Q}$-linear map $S$ on $y\mathfrak{H}$ by $S(yw):=yS_{1}(w)\,\,(w\in\mathfrak{H})$.

Lemma 3.2**.**

For positive integers $k_{1},\ldots,k_{r}$, the coefficients of $z^{1}$ in the Taylor expansion of $F_{(k_{1},\ldots,k_{r})}(z)$ at $z=0$ is given by

[TABLE]

Proof.

This is a special case of [8, Proposition 3.1]. ∎

Lemma 3.3** ([8, Proposition 2.9]).**

For positive integers $r$ and $k_{1},\ldots,k_{r}$, we have

[TABLE]

Lemma 3.4** (Key lemma for Theorem 3.5).**

For positive integers $k_{1},\ldots,k_{r}$, we have

[TABLE]

Proof.

For non-negative integer $n$, we denote by $A_{(k_{1},\ldots,k_{r})}^{(j)}(n)$ the coefficients of $z^{j}$ in the Taylor expansion of $F_{(k_{1},\ldots,k_{r})}(z+n)$. By replacing $z$ with $z+n$ and seeing the coefficients of $z^{1}$ of Lemma 3.3, we have

[TABLE]

By using this equation repeatedly and putting $n=0$, we obtain

[TABLE]

Since

[TABLE]

by Lemma 3.2, the claim is proved. ∎

Theorem 3.5**.**

For $A\in y\mathfrak{H}$ and $B\in\mathfrak{H}x$, there exist Dirichlet series $L_{1}(s)$ and $L_{2}(s)$ such that $L_{1}(s)$ and $L_{2}(s)$ converge for all $s\in\mathbb{Z}_{\geq 0}$, and

[TABLE]

for all $s\in\mathbb{Z}_{\geq 0}$.

Proof.

It is enough to only consider the case $A=y$ and $B=(x+y)^{k_{1}-1}x\cdots(x+y)^{k_{r}-1}x\,\,\,(r\in\mathbb{Z}_{\geq 0},k_{1},\ldots,k_{r}\in\mathbb{Z}_{\geq 1})$ from Theorem 1.2 (i). Let $s$ be a non-negative integer. We note that

[TABLE]

Thus, by Lemma 3.4, we have

[TABLE]

Hence, the Dirichlet series

[TABLE]

where

[TABLE]

satisfy the condition. ∎

Remark 3.6*.*

By the proof of Theorem 3.5, the Dirichlet series $L_{1}(s)$ and $L_{2}(s)$ are explicitly written by the multiple zeta functions. Hence, these series can be continued meromorphically to the whole plane $\mathbb{C}$ (for details, see Akiyama-Egami-Tanigawa [1] and Zhao [9]).

Theorem 3.7**.**

Let $L_{1}(s)=\sum_{n=1}^{\infty}a_{n}/n^{s}$ and $L_{2}(s)=\sum_{n=1}^{\infty}b_{n}/n^{s}$ be two Dirichlet series. If $L_{1}(s)$ and $L_{2}(s)$ converge for all $s\in\mathbb{Z}_{\geq 0}$, and $L_{1}(s)+sL_{2}(s)=0$ for all $s\in\mathbb{Z}_{\geq 0}$, then we have $a_{i}=b_{i}=0$ for all $i$.

Proof.

If the claim in the theorem does not hold, there exists a positive integer $N$ such that

[TABLE]

where $a_{N}+sb_{N}\neq 0$. Here, we split the sum as

[TABLE]

where

[TABLE]

Since $L_{1}(s)$ and $L_{2}(s)$ are convergent at $s=0$, $a_{n}=O(1)$ and $b_{n}=O(1)$ hold. Hence for large real $s$, we have

[TABLE]

Therefore we have

[TABLE]

Hence there exists $\sigma_{0}\in\mathbb{R}$ such that $|I_{1}(s)|>|I_{2}(s)|$ holds for any real $s>\sigma_{0}$. On the other hand, we have $I_{1}(s)+I_{2}(s)=0$ for any non-negative integer $s$. This is a contradiction, and the theorem is proved. ∎

Proof of Theorem 1.2 (iii).

By Theorems 3.5 and 3.7, we have the desired result. ∎

Now, we prove Theorem 1.2 (ii). We need the following lemma (for a proof, see e.g., Panzer [6, Lemma 3.3.5]).

Lemma 3.8**.**

The set of complex functions

[TABLE]

on $\{|z|<1\}$ is $\mathbb{C}$-linearly independent, where ${\rm Li}_{\Bbbk}(z)$ are the multiple polylogarithms defined by

[TABLE]

Proof of Theorem 1.2 (ii).

Take $w\in\mathfrak{H}x$ such that $f(y,w)=0$. For $N\in\mathbb{Z}_{>0}$, we define a linear map $h_{N}:y\mathfrak{H}\to\mathbb{Q}$ by

[TABLE]

By the proof of Theorem 3.5, there exist two Dirichlet series

[TABLE]

such that $L_{1}(w;s)$ and $L_{2}(w;s)$ converge for all $s\in\mathbb{Z}_{\geq 0}$,

[TABLE]

for all $N\in\mathbb{Z}_{>0}$, and

[TABLE]

for all $s\in\mathbb{Z}_{\geq 0}$. Furthermore, by Theorem 3.7, $c_{2}(w;N)=0$ for all $N\in\mathbb{Z}_{>0}$. Thus

[TABLE]

where $n(k_{1},\dots,k_{r})$ are coefficients of $yx^{k_{1}-1}\cdots yx^{k_{r}-1}$ in $\tau\circ S_{1}^{-1}(w)$. Therefore by Lemma 3.8, $n(\Bbbk)=0$ for all $\Bbbk$. Now Theorem 1.2 (ii) is proved since

[TABLE]

4. Some equivalences of families of relations

In this section, we prove Theorem 4.1 which gives characterizations of the linear part of Kawashima’s relation. Roughly speaking, $A_{1}$ is the set of ($\mathbb{Q}$-linear sums of) the linear part of Kawashima’s relation, and $A_{5},A_{6}$ are sets of $\mathbb{Q}$-linear relations which are extendable to $\mathbb{Q}$-linear relations between families $\{f(A,B)\mid A\in y\mathfrak{H},B\in\mathfrak{H}x\}$. Theorem 1.6 is a special case $A_{6}\subset A_{1}$ of Theorem 4.1.

Theorem 4.1**.**

The following vector subspaces $A_{1},\ldots,A_{6}$ of $\mathfrak{H}$ are coincident.

[TABLE]

Proof.

The equality of $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$ was proved in Lemma 2.6. The inclusion relation $A_{4}\subset A_{6}\subset A_{5}$ is obvious by Theorem 1.2 (i). Thus, we need to show $A_{5}\subset A_{3}$. Take $M\in\mathbb{Z}_{\geq 0},n_{i}\in\mathbb{Q},A_{i}\in y\mathfrak{H}$, and $B_{i}\in\mathfrak{H}x$ such that

[TABLE]

Put $yw_{1}^{(i)}:=A_{i}$ and $w_{2}^{(i)}x:=B_{i}$. Then, by Theorem 1.2 (i), we have

[TABLE]

Thus, by Theorem 1.2 (ii), we have

[TABLE]

Therefore, we get

[TABLE]

Acknowledgements

The authors would like to thank Professor Masanobu Kaneko for valuable comments. This work was supported by JSPS KAKENHI Grant Numbers JP18J00982, JP18K13392.