Quasi-derivation relations for multiple zeta values revisited

TL;DR

This paper revisits quasi-derivation relations in multiple zeta values, providing a new formula that simplifies proofs and extends the relations to finite multiple zeta values.

Contribution

It introduces a new formula for the quasi-derivation operator, simplifying proofs and extending relations to finite multiple zeta values.

Findings

Simplified proof of quasi-derivation relations.

Extended relations to finite multiple zeta values.

Provided a new formula for the quasi-derivation operator.

Abstract

We take another look at the so-called quasi-derivation relations in the theory of multiple zeta values, by giving a certain formula for the quasi-derivation operator. In doing so, we are not only able to prove the quasi-derivation relations in a simpler manner but also give an analog of the quasi-derivation relations for finite multiple zeta values.

Equations

$$\zeta (k_1,k_2,\dots ,k_r):=\sum _{0<n_1<\cdots <n_r}\frac{1}{n_1^{k_1}{n}_2^{k_2}\cdots n_r^{k_r}}$$

$$Z({\partial }_n(w))=0$$

$$\theta (u)=uz=u(x+y)\text{ for }u=x,y$$

$$\theta (w{w}^{\prime })=\theta (w)w^{\prime }+w\theta (w^{\prime })+cH(w){\partial }_1(w^{\prime })$$

$${\partial }_n^{(c)}:=\frac{1}{(n-1)!}\mathrm{ad}(\theta {)}^{n-1}({\partial }_1).$$

$$Z({\partial }_n^{(c)}(w))=0$$

$$w_{1}\diamond w_{2}:=\phi\bigl{(}\phi(w_{1})*\phi(w_{2})\bigr{)}\quad(w_{1},w_{2}\in\mathfrak{H}),$$

$$\varphi (x)=z=x+y\text{ and }\varphi (y)=-y,$$

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

$$\tilde{\theta }(w):=\theta (w)+cH(w)y(w\in \mathfrak{H}).$$

$$q_n:=\frac{1}{(n-1)!}{\tilde{\theta }}^{n-1}(y).$$

$${\partial }_n^{(c)}(wx)=(w\diamond q_n)x(w\in \mathfrak{H}).$$

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

$$Z\bigl{(}\partial_{n}^{(c)}(ywx)\bigr{)}=Z\bigl{(}(yw\diamond q_{n})x\bigr{)}=Z\bigl{(}\phi\bigl{(}\phi(yw)*\phi(q_{n})\bigr{)}x\bigr{)}=0$$

$$[{\partial }_{n_1}^{(c_1)},{\partial }_{n_2}^{(c_2)}](w)=0$$

$$[{\partial }_{n_1}^{(c_1)},{\partial }_{n_2}^{(c_2)}](wx)$$

$$=((w\diamond q_{n_2})\diamond q_{n_1})x-((w\diamond q_{n_1})\diamond q_{n_2})x$$

$$=0$$

$$z{w}_1\diamond w_2=w_1\diamond z{w}_2=z(w_1\diamond w_2).$$

$${\partial }_1(w)={\partial }_1(z{w}^{\prime })=z{\partial }_1(w^{\prime })=z(w^{\prime }\diamond y-w^{\prime }y)=z{w}^{\prime }\diamond y-z{w}^{\prime }y=w\diamond y-wy.$$

$${\partial }_1(w)$$

$$=y(x{w}^{\prime }\diamond 1)+x(w^{\prime }\diamond y)-x{w}^{\prime }y=x{w}^{\prime }\diamond y-x{w}^{\prime }y$$

$$=w\diamond y-wy.\text{qed}$$

$$\tilde{\theta}(uw)=u\bigl{(}\tilde{\theta}(w)+zw+c(w\diamond y)\bigr{)}.$$

$$\tilde{\theta }(uw)$$

$$=uzw+u\theta (w)+cu{\partial }_1(w)+cuwy+cuH(w)y$$

$$\displaystyle=u\bigl{(}\tilde{\theta}(w)+zw+c(\partial_{1}(w)+wy)\bigr{)}.$$

$$\tilde{\theta }(w_1\diamond w_2)=\tilde{\theta }(w_1)\diamond w_2+w_1\diamond \tilde{\theta }(w_2)$$

$$\tilde{\theta }(1\diamond 1)=\tilde{\theta }(1)=0=\tilde{\theta }(1)\diamond 1+1\diamond \tilde{\theta }(1).$$

$$\displaystyle\tilde{\theta}(zw^{\prime}_{1}\diamond w_{2})=\tilde{\theta}(z(w^{\prime}_{1}\diamond w_{2}))=z\bigl{(}\tilde{\theta}(w^{\prime}_{1}\diamond w_{2})+z(w_{1}^{\prime}\diamond w_{2})+c(w_{1}^{\prime}\diamond w_{2}\diamond y)\bigr{)}.$$

$$\tilde{\theta }(z{w}_1^{\prime })\diamond w_2+z{w}_1^{\prime }\diamond \tilde{\theta }(w_2)$$

$$\displaystyle=z\bigl{(}\tilde{\theta}(w_{1}^{\prime})+zw_{1}^{\prime}+c(w_{1}^{\prime}\diamond y)\bigr{)}\diamond w_{2}+z\bigl{(}w_{1}^{\prime}\diamond\tilde{\theta}(w_{2})\bigr{)}$$

$$\displaystyle=z\bigl{(}\tilde{\theta}(w_{1}^{\prime})\diamond w_{2}+w_{1}^{\prime}\diamond\tilde{\theta}(w_{2})+z(w_{1}^{\prime}\diamond w_{2})+c(w_{1}^{\prime}\diamond w_{2}\diamond y)\bigr{)}.$$

$$\tilde{\theta }(z{w}_1^{\prime }\diamond w_2)=\tilde{\theta }(z{w}_1^{\prime })\diamond w_2+z{w}_1^{\prime }\diamond \tilde{\theta }(w_2).$$

Full text

1. Introduction

The quasi-derivation relations in the theory of multiple zeta values is a generalization, proposed by the first-named author and established by T. Tanaka, of a set of linear relations known as derivation relations, which we are first going to recall.

We use Hoffman’s algebraic setup ([5]) with a slightly different convention. Let $\mathfrak{H}:=\mathbb{Q}\left\langle x,y\right\rangle$ be the noncommutative polynomial algebra in two indeterminates $x$ and $y$. This was introduced in order to encode multiple zeta values in the way the monomial $yx^{k_{1}-1}yx^{k_{2}-1}\cdots yx^{k_{r}-1}$ corresponds to the multiple zeta value

[TABLE]

when $k_{r}>1$, which is a real number as the limiting value of a convergent series. If we denote by $Z$ the $\mathbb{Q}$-linear map from $y\mathfrak{H}x$ to $\mathbb{R}$ assigning each monomial $yx^{k_{1}-1}yx^{k_{2}-1}\cdots yx^{k_{r}-1}$ to $\zeta(k_{1},\dots,k_{r})$, the derivation relations state that

[TABLE]

for all $n\geq 1$ and $w\in y\mathfrak{H}x$. Here the operator $\partial_{n}$ is a $\mathbb{Q}$-linear derivation on $\mathfrak{H}$ determined uniquely by $\partial_{n}(x)=y(x+y)^{n-1}x$ and $\partial_{n}(y)=-y(x+y)^{n-1}x$. Set $z=x+y$, so that $\partial_{n}(z)=0$. We use this repeatedly in the sequel.

In order to introduce the quasi-derivation relations, we first define a $\mathbb{Q}$-linear map $\theta:=\theta^{(c)}\colon\mathfrak{\mathfrak{H}\to\mathfrak{H}}$ with a parameter $c\in\mathbb{Q}$ (we often drop $c$ from the notation) by setting

[TABLE]

and requiring

[TABLE]

for $w,w^{\prime}\in\mathfrak{H}$, where $H$ is the $\mathbb{Q}$-linear map from $\mathfrak{H}$ to itself defined by $H(w)=\deg(w)\cdot w$ for any monomial $w\in\mathfrak{H}$ ($\deg(w)$ is the degree of $w$). This is well defined because $H$ is a derivation on $\mathfrak{H}$. Now we define the quasi-derivation map $\partial_{n}^{(c)}$. Write $\mathrm{ad}(\theta)$ the adjoint operator by $\theta$, i.e., $\mathrm{ad}(\theta)(\partial):=[\theta,\partial]=\theta\partial-\partial\theta$.

Definition 1.1**.**

For each positive integer $n$ and any rational number $c$, we define a $\mathbb{Q}$-linear map $\partial_{n}^{(c)}\colon\mathfrak{H}\to\mathfrak{H}$ by

[TABLE]

Then the quasi-derivation relations of Tanaka [13] is stated as

[TABLE]

for all $n\geq 1$, $c\in\mathbb{Q}$, and $w\in y\mathfrak{H}x$. Our aim in this paper is to take another look at this relation, or rather at the operator $\partial_{n}^{(c)}$.

Remark 1.2*.*

1. We have changed the definition of $\theta=\theta^{(c)}$ by shifting the original ([8, 13]) by the derivation $w\to[z,w]/2=(zw-wz)/2$. However, we can check that this does not change $\partial_{n}^{(c)}(w)$. Note also that the convention of the order of the product in $\mathfrak{H}$ there is opposite from ours.

2. As noted in [6], the special case $c=0$ gives the original derivation $\partial_{n}$: $\partial_{n}=\partial_{n}^{(0)}$. This together with works of Connes-Moscovicci [1, 2] motivated us to define $\partial_{n}^{(c)}(w)$ in [8].

3. From $\theta(z^{r})=rz^{r+1}\,(r\geq 1)$ and $\partial_{n}(z)=0$, we see that $\partial_{n}^{(c)}(wz)=\partial_{n}^{(c)}(w)z$ and $\partial_{n}^{(c)}(zw)=z\partial_{n}^{(c)}(w)$. We need to use this at several points later.

2. Main Theorem

We present a formula for $\partial_{n}^{(c)}(w)$ when $w$ is in $\mathfrak{H}x$. To describe the formula, we define a product $\diamond$ on $\mathfrak{H}$ introduced in Hirose-Murahara-Onozuka [3] by

[TABLE]

where $\phi$ is an involutive automorphism of $\mathfrak{H}$ determined by

[TABLE]

and $*$ is the harmonic product on $\mathfrak{H}$ (see [5, 4] for the precise definition of $*$). This is an associative and commutative binary operation with $1\diamond w=w\diamond 1=w$ for any $w\in\mathfrak{H}$. In [3], the definition of $\diamond$ is given in an inductive manner like the definition of $*$ in [4]. Later we only use the shuffle-type equality

[TABLE]

which holds for any $w_{1},w_{2}\in\mathfrak{H}$.

We define a specific element $q_{n}=q_{n}^{(c)}$ in $\mathfrak{H}$ for each $n\geq 1$ as follows.

Definition 2.1**.**

Let $\tilde{\theta}=\tilde{\theta}^{(c)}$ be the map from $\mathfrak{H}$ to itself given by

[TABLE]

For each positive integer $n$, we define

[TABLE]

We thus have $q_{1}=y$ and $q_{n}=\tilde{\theta}(q_{n-1})/(n-1)$ for $n\geq 2$.

Note that $q_{n}=q_{n}^{(c)}$ is in $y\mathfrak{H}$, as can be seen inductively by the definition. We shall give an explicit formula for $q_{n}$ in the next section. Here is our main theorem.

Theorem 2.2**.**

For all $n\geq 1$ and $c\in\mathbb{Q}$, we have

[TABLE]

Assuming the theorem, it is straightforward to deduce the quasi-derivation relations from Kawashima’s relations (strictly speaking, its “linear part”). Recall the linear part of Kawashima’s relations [11] asserts that

[TABLE]

for any $w_{1},w_{2}\in y\mathfrak{H}$. Using this and the definition (1) of $\diamond$, we see that

[TABLE]

because both $\phi(yw)$ and $\phi(q_{n})$ are in $y\mathfrak{H}$. This is the quasi-derivation relations.

Another immediate corollary to the theorem is the commutativity of the operators $\partial_{n}^{(c)}$, that is, $\partial_{n_{1}}^{(c_{1})}$ and $\partial_{n_{2}}^{(c_{2})}$ commute with each other for any $n_{1},n_{2}\geq 1$ and $c_{1},c_{2}\in\mathbb{Q}$. This was proved in [13] but the argument was quite involved. Here we may show

[TABLE]

first for $w\in\mathfrak{H}x$ as

[TABLE]

because the product $\diamond$ is associative and commutative, and then for the general case by induction on the degree of $w$ by noting $\partial_{n}^{(c)}(wz)=\partial_{n}^{(c)}(w)z$ as remarked before.

Proof of Theorem 2.2.

We need some lemmas. Recall $z=x+y$.

Lemma 2.3**.**

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

[TABLE]

Proof.

This follows from $\phi(z)=x,\,\phi(x)=z$ and $xw_{1}*w_{2}=w_{1}*xw_{2}=x(w_{1}*w_{2})$. See also [3]. ∎

Lemma 2.4**.**

For $w\in\mathfrak{H}$, we have $\partial_{1}(w)=w\diamond y-wy$.

Proof.

We proceed by induction on $\deg(w)$. The case $\deg(w)=0$ is obvious because $\partial_{1}(1)=0$. Suppose $\deg(w)\geq 1$. By linearity, it is enough to prove the equation when $w$ is of the form $zw^{\prime}$ and $xw^{\prime}$. If $w=zw^{\prime}$, we have, by using the induction hypothesis and Lemma 2.3,

[TABLE]

When $w=xw^{\prime}$, we similarly compute (using equation (2))

[TABLE]

Lemma 2.5**.**

For $u\in\mathbb{Q}x+\mathbb{Q}y$, we have

[TABLE]

Proof.

We only need to show the equation for $u=x\text{ and }y$. By the definition of $\tilde{\theta}$, we have

[TABLE]

From Lemma 2.4, we complete the proof. ∎

We need one more preparatory result, which may be of interest in its own right.

Proposition 2.6**.**

The $\mathbb{Q}$-linear map $\tilde{\theta}$ is a derivation on $\mathfrak{H}$ with respect to the product $\diamond$, i.e., the equation

[TABLE]

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

Proof.

We prove this by induction on $\deg(w_{1})+\deg(w_{2})$. The case $\deg(w_{1})+\deg(w_{2})=0$ holds trivially:

[TABLE]

When $\deg(w_{1})+\deg(w_{2})\geq 1$, we first prove when $w_{1}$ is of the form $w_{1}=zw_{1}^{\prime}$. By the definition of $\tilde{\theta}$ and Lemmas 2.3 and 2.5, we have

[TABLE]

On the other hand, we have

[TABLE]

Hence by the induction hypothesis we obtain

[TABLE]

Since the binary operator $\diamond$ is commutative and bilinear, it suffices then to prove equation (3) only in the case where $w_{1}=xw_{1}^{\prime}$ and $w_{2}=yw_{2}^{\prime}$. By using equation (2) and Lemma 2.5, we have

[TABLE]

From these, we see by the induction hypothesis that

[TABLE]

holds. ∎

Now we prove Theorem 2.2 by induction on $n$. When $n=1$, we have

[TABLE]

by Lemma 2.4. When $n\geq 2$, we have

[TABLE]

By the induction hypothesis, we have

[TABLE]

and

[TABLE]

We therefore obtain by Proposition 2.6

[TABLE]

which completes the proof. ∎

3. Explicit formula for $q_{n}$

We now describe the element $q_{n}=q_{n}^{(c)}$ in an explicit manner. For any index $\boldsymbol{l}=(l_{1},\dots,l_{s})\in\mathbb{N}^{s}$, we define $a(\boldsymbol{l})=a(l_{1},\dots,l_{s})\in\mathbb{Q}$ (or $\in\mathbb{Z}[c]$ if we view $c$ as a variable) inductively by $a(1):=1$ and

[TABLE]

where

[TABLE]

Proposition 3.1**.**

For $n\geq 1$, we have

[TABLE]

where the sum runs over all indices $\boldsymbol{l}=(l_{1},\dots,l_{s})\in\mathbb{N}^{s}$ of any length $s$ and of weight $|\boldsymbol{l}|:=l_{1}+\cdots+l_{s}=n$, and $w(\boldsymbol{l})=\phi(yx^{l_{1}-1}\cdots yx^{l_{s}-1})=(-1)^{s}yz^{l_{1}-1}\cdots yz^{l_{s}-1}$.

Proof.

Let $q^{\prime}_{n}$ denote the right-hand side of (4). We prove (4) by induction on $n$. When $n=1$, we easily see $q_{1}^{\prime}=y$.

Suppose $n\geq 2$. We want to show that $q^{\prime}_{n}=\tilde{\theta}(q^{\prime}_{n-1})/(n-1)$. Since $\theta(z^{m})=mz^{m+1}$ and $\partial_{1}(z)=0$, we have

[TABLE]

and so

[TABLE]

Since $cH(yz^{k_{1}-1}\cdots yz^{k_{r}-1})y=c(k_{1}+\cdots+k_{r})yz^{k_{1}-1}\cdots yz^{k_{r}-1}y$, we finally obtain for $\boldsymbol{k}=(k_{1},\ldots,k_{r})$

[TABLE]

If we write

[TABLE]

we see from this that the coefficient $a^{\prime}(\boldsymbol{l})$ of $w(\boldsymbol{l})=(-1)^{s}yz^{l_{1}-1}\cdots yz^{l_{s}-1}$ is given exactly by $a(\boldsymbol{l})$ as defined recursively. ∎

4. Quasi-derivation relations for finite multiple zeta values

In this section, we briefly discuss how the quasi-derivation relations look like for “finite” multiple zeta values. There are two versions, denoted $\zeta_{\mathcal{A}}(k_{1},\ldots,k_{r})$ and $\zeta_{\mathcal{S}}(k_{1},\ldots,k_{r})$, of “finite” analogues of multiple zeta values. The former lives in the $\mathbb{Q}$-algebra $\mathcal{A}:=\prod_{p}\mathbb{F}_{p}/\bigoplus_{p}\mathbb{F}_{p}$ and the latter the quotient $\mathbb{Q}$-algebra of classical multiple zeta values modulo the ideal generated by $\zeta(2)$. It is conjectured that the two versions satisfy completely the same relations, and there is a conjectural isomorphism between two $\mathbb{Q}$-algebras generated by those two versions. For more on finite multiple zeta values, see for instance [9].

Denote by $\mathit{Z}_{\mathcal{F}}$ the $\mathbb{Q}$-linear map from $y\mathfrak{H}$ to either algebra assigning the monomial $yx^{k_{1}-1}\cdots yx^{k_{r}-1}$ to $\zeta_{\mathcal{A}}(k_{1},\ldots,k_{r})$ or $\zeta_{\mathcal{S}}(k_{1},\ldots,k_{r})$. Then the derivation relations for finite multiple zeta values established by the second-named author [12] is the relation

[TABLE]

that holds for all $w\in y\mathfrak{H}x$.

As a consequence of our Theorem 2.2, we have the following.

Theorem 4.1** (Quasi-derivation relations for finite multiple zeta values).**

For all $n\geq 1$ and $c\in\mathbb{Q}$, we have

[TABLE]

Proof.

This is almost immediate from Theorem 2.2 if one notes $\mathit{Z}_{\mathcal{F}}\circ\phi=\mathit{Z}_{\mathcal{F}}$ and $\mathit{Z}_{\mathcal{F}}$ is a $*$-homomorphism (for these, see [7, 9, 10]). ∎

Remark 4.2*.*

When $c=0$, we can easily compute that $q_{n}^{(0)}=yz^{n-1}$. Since \mathit{Z}_{\mathcal{F}}(yz^{n-1})=\mathit{Z}_{\mathcal{F}}\bigl{(}\phi(yz^{n-1})\bigr{)}=-\mathit{Z}_{\mathcal{F}}(yx^{n-1})=-\zeta_{\mathcal{F}}(n)=0 for $\mathcal{F}=\mathcal{A}$ or $S$, we see that Theorem 4.1 generalizes the derivation relations (5).

Acknowledgement

This work was supported by JSPS KAKENHI Grant Numbers JP16H06336.