On multi-interpolated multiple zeta values

Markus Kuba Markus Kuba

Department Applied Mathematics and Physics
FH - Technikum Wien
Höchstädtplatz 6
1200 Wien, Austria [email protected]

Abstract.

In this note we introduce multi-interpolated multiple zeta values. We provide a basic decomposition of these objects involving ordered partitions. We also obtain identities for special instances of multi-interpolated multiple zeta values $\zeta^{\vec{t}}(\{s\}_{k})$ , generalizing earlier results. Moreover, we introduce a product for multi-interpolated multiple zeta values.

Key words and phrases:

Multi-interpolated multiple zeta values, interpolated multiple zeta values, multiple zeta values

2010 Mathematics Subject Classification:

05A15, 11M32

1. Introduction

The multiple zeta values, henceforth MZVs, are defined by

\zeta(i_{1},\dots,i_{k})=\sum_{\ell_{1}>\cdots>\ell_{k}\geq 1}\frac{1}{\ell_{1}^{i_{1}}\cdots\ell_{k}^{i_{k}}},

with admissible indices $(i_{1},\dots,i_{k})\in\mathbb{N}^{k}$ satisfying $i_{1}\geq 2$ , $i_{j}\geq 1$ for $2\leq j\leq k$ , see Hoffman [7] and Zagier [28]. We refer to $i_{1}+\dots+i_{k}$ as the weight of this MZV, and $k$ as its depth. For a comprehensive overview as well as a great many pointers to the literature we refer to the survey of Zudilin [29]. An important variant of the MZVs are the so-called multiple zeta star values, abbreviated by MZSVs, where equality is allowed:

\begin{split}\zeta^{\star}(i_{1},\dots,i_{k})&=\sum_{\ell_{1}\geq\cdots\geq\ell_{k}\geq 1}\frac{1}{\ell_{1}^{i_{1}}\cdots\ell_{k}^{i_{k}}}.\end{split}

Yamamoto [27] introduced a generalization $\zeta^{t}$ called interpolated multiple zeta values of both $\zeta$ and $\zeta^{\star}$ . Noting that,

\zeta^{\star}(i_{1},\dots,i_{k})=\sum_{\circ=\text{``},\text{''}\text{or}\,\text{``}+\text{''}}\zeta(i_{1}\circ i_{2}\dots\circ i_{k}),

let the parameter $\sigma$ denote the number of plus in the expression $i_{1}\circ i_{2}\dots\circ i_{k}$ . Interpolated multiple zeta values are defined by

\zeta^{t}(i_{1},\dots,i_{k})=\sum_{\circ=\text{``},\text{''}\text{or}\,\text{``}+\text{''}}t^{\sigma}\zeta(i_{1}\circ i_{2}\dots\circ i_{k}).

(1)

Thus, the series $\zeta^{t}(i_{1},\dots,i_{k})$ is a polynomial in $t$ and interpolates between MZVs, $\zeta^{0}=\zeta$ , and MZSVs, $\zeta^{1}=\zeta^{\star}$ . Equivalently, the interpolated MZVs can be defined as

\zeta^{t}(i_{1},\dots,i_{k})=\sum_{\ell_{1}\geq\cdots\geq\ell_{k}\geq 1}\frac{t^{\sigma(\ell_{1},\dots,\ell_{k})}}{\ell_{1}^{i_{1}}\cdots\ell_{k}^{i_{k}}},

(2)

where $\sigma$ is given by the number of equalities:

\sigma(\ell_{1},\dots,\ell_{k})=|\{1\leq r\leq k-1\mid\ell_{r}=\ell_{r+1}\}|.

(3)

It turned out that the interpolated series satisfies many identities generalizing or unifying earlier result for multiple zeta and zeta star values, see for example Yamamoto [27] for a generalization of the sum identity as well as many other results, and also Hoffman and Ihara [12] or Hoffman [8, 11] for further results. We also refer to Tanaka and Wakabayashi [24] for a proof of Kawashima’s relations, to Wakabayashi [26] as well as Li and Qin [19] for Double shuffle and Hoffman’s relations, and to Li [18] for a recent study of algebraic aspects, including extended double shuffle relations, symmetric sum formulas and restricted sum formulas.

Most important, the so-called $t$ -harmonic product $\stackrel{{\scriptstyle t}}{{\ast}}$ was introduced by Yamamoto [27] for the interpolated MZVs, see also [8, 11, 12]. It satisfies

\zeta^{t}\big{(}(i_{1},\dots,i_{k})\stackrel{{\scriptstyle t}}{{\ast}}(j_{1},\dots,j_{\ell})\big{)}=\zeta^{t}(i_{1},\dots,i_{k})\cdot\zeta^{t}(j_{1},\dots,j_{\ell}),

(4)

where the product $\stackrel{{\scriptstyle t}}{{\ast}}$ is actually defined in an algebraic way (see Section 4). Hoffman and Ihara used a general algebra framework, leading, amongst others to expressions for interpolated MZVs $\zeta^{t}(\{s\}_{k})$ in terms of Bell polynomials and ordinary single argument zeta values, $m\geq 1$ . Here and throughout this work $\{s\}_{k}$ means $s$ repeated $k$ times.

In this work we introduce a generalization $\vec{\sigma}$ of the parameter $\sigma$ (3), generalizing interpolated MZVs (2) to what we call multi-interpolated MZVs. This allows to gain more insight into structural decompositions of ordinary interpolated MZVs, as well as a link to the $\mathfrak{t}$ -values of Hoffman [10]. Our results are the following. First, we obtain in Theorem 1 a decomposition of multi-interpolated multiple zeta values using ordered partitions. Second, we obtain several identities for multi-interpolated multiple zeta values, $\zeta^{\vec{t}}(\{s\}_{k})$ , see Theorem 2 and Corollary 1, generalizing earlier results for $\zeta^{t}(\{s\}_{k})$ . Third, as our main result we introduce in Section 4 a product $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ for multi-interpolated multiple zeta values, generalizing (4). Interestingly, it turns out that the product $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ involves a non-commutative variable $\vec{t}$ , in contrast to the earlier $t$ -harmonic product $\stackrel{{\scriptstyle t}}{{\ast}}$ for interpolated MZVs.

2. Multi-interpolated multiple zeta values and related multiple zeta values

Definition 1 (Multi-interpolated multiple zeta values).

Given integers $(i_{1},\dots,i_{k})$ with $i_{1}\geq 2$ , $k\geq 1$ and a sequence of variables $\vec{t}=(t_{1},t_{2},\dots)$ . For our purpose we assume that $t_{j}\in[-1;1]$ , $j\in\mathbb{N}$ . The multi-interpolated multiple zeta value $\zeta^{\vec{t}}(i_{1},\dots,i_{k})$ is defined by

\zeta^{\vec{t}}(i_{1},\dots,i_{k})=\sum_{\ell_{1}\geq\cdots\geq\ell_{k}\geq 1}\frac{\vec{t}^{\vec{\sigma}(\vec{\ell})}}{\ell_{1}^{i_{1}}\dots\ell_{k}^{i_{k}}}=\sum_{\ell_{1}\geq\cdots\geq\ell_{k}\geq 1}\frac{\prod_{j=1}^{\infty}t_{j}^{\sigma_{j}(\ell_{1},\dots,\ell_{k})}}{\ell_{1}^{i_{1}}\dots\ell_{k}^{i_{k}}},

(5)

with parameter $\sigma_{j}$ denoting the number of equalities of the integer $j\in\mathbb{N}$ :

\sigma_{j}(\ell_{1},\dots,\ell_{k})=|\{1\leq r\leq k-1\mid\ell_{r}=\ell_{r+1}=j\}|.

(6)

Remark 1.

Note that the parameters $\sigma_{j}$ refine the parameter $\sigma$ (3) due to the identity

\sigma=\sum_{j\geq 1}\sigma_{j}.

Hence, by setting $t_{j}=t$ , $j\geq 1$ , which we write in a slight abuse of notation simply as $\vec{t}=t$ , we have

\prod_{j=1}^{\infty}t_{j}^{\sigma_{j}(\vec{\ell})}=t^{\sum_{j=1}^{\infty}\sigma_{j}(\vec{\ell})}=t^{\sigma(\vec{\ell})},

(7)

so that $\zeta^{\vec{t}}(i_{1},\dots,i_{k})$ reduces to $\zeta^{t}(i_{1},\dots,i_{k})$ .

Example 1 (Multi-interpolated MZVs: depth one).

For depth one, the multi-interpolated MZVs reduce to ordinary zeta values:

\zeta^{\vec{t}}(i)=\zeta^{t}(i)=\zeta(i),\quad i>1.

Example 2 (Multi-interpolated MZVs: depth two).

The set $\{\ell_{1}\geq\ell_{2}\geq 1\}$ is split into two parts,

\{\ell_{1}\geq\ell_{2}\geq 1\}=\{\ell_{1}>\ell_{2}\geq 1\}\cup\{\ell_{1}=\ell_{2}\geq 1\},

leading to

\displaystyle\zeta^{\vec{t}}(i_{1},i_{2})

\displaystyle=\sum_{\ell_{1}\geq\ell_{2}\geq 1}\frac{\vec{t}^{\vec{\sigma}(\vec{\ell})}}{\ell_{1}^{i_{1}}\ell_{2}^{i_{2}}}=\sum_{\ell_{1}>\ell_{2}\geq 1}\frac{1}{\ell_{1}^{i_{1}}\ell_{2}^{i_{2}}}+\sum_{\ell\geq 1}\frac{t_{\ell}}{\ell^{i_{1}+i_{2}}}=\zeta(i_{1},i_{2})+\sum_{\ell\geq 1}\frac{t_{\ell}}{\ell^{i_{1}+i_{2}}}.

(8)

Here we used $\vec{t}^{\vec{\sigma}(\vec{\ell})}=1$ for $\vec{\ell}\in\{\ell_{1}>\ell_{2}\geq 1\}$ and $\vec{t}^{\vec{\sigma}(\vec{\ell})}=t_{\ell}$ for $\vec{\ell}\in\{\ell=\ell_{1}=\ell_{2}\geq 1\}$ . In the special case $\vec{t}=t$ (7) we reobtain the ordinary interpolated MZVs and the evaluation

\zeta^{t}(i_{1},i_{2})=\zeta(i_{1},i_{2})+t\cdot\zeta(i_{1}+i_{2}).

In our previous example, different basic objects like $\sum_{\ell\geq 1}\frac{t_{\ell}}{\ell^{i_{1}+i_{2}}}$ appeared, compared to the ordinary interpolated MZVs. Thus, in order to analyze multi-interpolated MZVs, we need another generalization, which includes interpolated MZVs (thus, also ordinary MZVs and MZSVs), as well as the atomic parts of the multi-interpolated truncated MZVs.

Definition 2.

For $k\geq 1$ let $j_{1},\dots,j_{k}\geq 0$ denote integers. We introduce multiple zeta values with variables $\vec{t}=(t_{1},t_{2},\dots)$ and admissible indices $(i_{1},\dots,i_{k})$ satisfying $i_{1}\geq 2$ , $i_{j}\geq 1$ for $2\leq j\leq k$ ,

\zeta\big{(}(\vec{t}^{j_{1}},i_{1}),\dots,(\vec{t}^{j_{k}},i_{k})\big{)}=\sum_{\ell_{1}>\cdots>\ell_{k}\geq 1}\frac{t_{\ell_{1}}^{j_{1}}\cdots t_{\ell_{k}}^{j_{k}}}{\ell_{1}^{i_{1}}\cdots\ell_{k}^{i_{k}}}.

Remark 2 (Connection to multiple $\mathfrak{t}$ -values).

The definition above also covers the multiple $\mathfrak{t}$ -values of Hoffman [10]. The $\mathfrak{t}$ -values [22] and multiple $\mathfrak{t}$ -values [10] are defined by

\mathfrak{t}(i_{1},\dots,i_{k})=\sum_{\begin{subarray}{c}\ell_{1}>\cdots>\ell_{k}\geq 1\\ \ell_{i}\text{odd}\end{subarray}}\frac{1}{\ell_{1}^{i_{1}}\cdots\ell_{k}^{i_{k}}}.

(9)

We note in passing that $\mathfrak{t}(i)=(1-2^{-i})\zeta(i)$ . Setting $t_{\ell}=(1-(-1)^{\ell})/2$ , $\ell\geq 1$ , and $j_{1}=\dots=j_{k}=1$ leads to

\zeta\big{(}(\vec{t},i_{1}),\dots,(\vec{t},i_{k})\big{)}=\mathfrak{t}(i_{1},\dots,i_{k}).

Note that for $j_{\ell}=0$ , $1\leq\ell\leq k$ , or $t_{\ell}=1$ , $\ell\geq 1$ , we simply write $i_{\ell}$ instead of $(\vec{1},i_{\ell})$ . The MZSVs with variables $\vec{t}=(t_{1},t_{2},\dots)$ and also the multi-interpolated generalizations $\zeta^{\vec{t}}\big{(}(\vec{t}^{j_{1}},i_{1}),\dots,(\vec{t}^{j_{k}},i_{k})\big{)}$ are defined accordingly. For $t_{\ell_{1}}=-1$ , $\ell_{1}\geq k$ , the value $j_{1}=1$ is also admissible; this leads to alternating MZVs. The generalized MZVs in Definition 2 can be generalized further by setting $\vec{u}_{m}=(u_{m,1},u_{m,2},\dots)$ and we get

\zeta\big{(}(\vec{u}_{1},i_{1}),\dots,(\vec{u}_{k},i_{k})\big{)}=\sum_{\ell_{1}>\cdots>\ell_{k}\geq 1}\frac{u_{1,\ell_{1}}\cdots u_{k,\ell_{k}}}{\ell_{1}^{i_{1}}\cdots\ell_{k}^{i_{k}}},

(10)

such that for $\vec{u}_{m}=\vec{t}^{j_{m}}$ , $1\leq m\leq k$ we reobtain our earlier definition, but for and $\vec{u}_{m}=(x_{m}^{n})_{n\geq 1}$ we obtain multiple polylogarithms. Moreover, mixture models of multiple $\mathfrak{t}$ -values and zeta values can be obtained by suitable choices of $\vec{u}_{m}$ .

Example 3 (Multi-interpolated MZVs: depth three).

We decompose $\zeta^{\vec{t}}(i_{1},i_{2},i_{3})$ into summands by splitting the underlying set into four parts:

	$\displaystyle\{\ell_{1}\geq\ell_{2}\geq\ell_{3}\geq 1\}$	$\displaystyle=\{\ell_{1}>\ell_{2}>\ell_{3}\geq 1\}\cup\{\ell_{1}>\ell_{2}=\ell_{3}\geq 1\}$
		$\displaystyle\quad\cup\{\ell_{1}=\ell_{2}>\ell_{3}\geq 1\}\cup\{\ell_{1}=\ell_{2}=\ell_{3}\geq 1\},$

such that

	$\displaystyle\zeta^{\vec{t}}(i_{1},i_{2},i_{3})$	$\displaystyle=\sum_{\ell_{1}\geq\ell_{2}\geq\ell_{3}\geq 1}\frac{\vec{t}^{\vec{\sigma}(\vec{\ell})}}{\ell_{1}^{i_{1}}\ell_{2}^{i_{2}}\ell_{3}^{i_{3}}}$
		$\displaystyle=\zeta(i_{1},i_{2},i_{3})+\zeta(i_{1},(\vec{t},i_{2}+i_{3}))+\zeta((\vec{t},i_{1}+i_{2}),i_{3})+\zeta((\vec{t}^{2},i_{1}+i_{2}+i_{3})).$

A natural specialization of $\zeta^{\vec{t}}(i_{1},\dots,i_{k})$ are even-odd interpolations.

Example 4 (Even-odd interpolated multiple zeta values).

Given the multi-interpolated MZV $\zeta^{\vec{t}}(i_{1},\dots,i_{k})$ , we choose $t_{2m}=t_{E}$ and $t_{2m-1}=t_{O}$ , $m\geq 1$ , obtaining the even-odd interpolation

\zeta^{t_{E},t_{O}}(i_{1},\dots,i_{k})=\sum_{\ell_{1}\geq\cdots\geq\ell_{k}\geq 1}\frac{t_{E}^{\sigma_{E}(\ell_{1},\dots,\ell_{k})}\cdot t_{O}^{\sigma_{O}(\ell_{1},\dots,\ell_{k})}}{\ell_{1}^{i_{1}}\cdots\ell_{k}^{i_{k}}},

where $\sigma_{E}$ and $\sigma_{O}$ are given by the number of even and odd equalities (6), respectively. Note that here, variants of the multiple $\mathfrak{t}$ -values of Hoffman [10], see (9), naturally appear, as well as mixtures of multiple zeta and $\mathfrak{t}$ -values (or multiple Hurwitz-zeta values); for example, in the case of depth two we obtain from Example 2, (8) the decomposition

\displaystyle\zeta^{\vec{t}}(i_{1},i_{2})

\displaystyle=\zeta(i_{1},i_{2})+t_{E}\cdot\frac{1}{2^{i_{1}+i_{2}}}\zeta(i_{1}+i_{2})+t_{O}\cdot\mathfrak{t}(i_{1}+i_{2}).

2.1. Ordered partitions and multi-interpolated MZVs

Motivated by the special cases of depth one, two and three in Examples 1, 2 and 3, and also the very definition of interpolated MZVs (1), we provide a representation of multi-interpolated MZVs in terms of the values in Definition 2. We start from the representation of interpolated multiple zeta values by ordered partitions $\mathcal{P}(k)$ of the integer $k$ , also called compositions. We associate to each ordered partition $\mathbf{p}=(p_{1},\dots,p_{r})\in\mathcal{P}(k)$ a map from $\mathbb{N}^{k}$ to $\mathbb{N}^{\mathcal{L}(\mathbf{p})}$ , where $\mathcal{L}(.)$ denote the length of the ordered partition:

\mathbf{p}(i_{1},\dots,i_{k})=(\sum_{j_{1}=1}^{P_{1}}i_{j_{1}},\sum_{j_{1}=P_{1}+1}^{P_{2}}i_{j_{1}}\dots,\sum_{j_{r}=P_{r-1}+1}^{P_{r}}i_{j_{r}}).

Here we use the convention $P_{j}=\sum_{\ell=1}^{j}p_{\ell}$ . Then,

\zeta^{t}(i_{1},\dots,i_{k})=\sum_{\mathbf{p}\in\mathcal{P}(k)}t^{k-\mathcal{L}(\mathbf{p})}\zeta(\mathbf{p}(i_{1},\dots,i_{k})),

as each $\mathbf{p}=(p_{1},\dots,p_{r})\in\mathcal{P}(k)$ corresponds to a unique partition of the set

M=\{\ell_{1}\geq\dots\geq\ell_{k}\geq 1\}=\bigcup_{\mathbf{p}\in\mathcal{P}(k)}\mathbf{p}(M)

into subsets

\mathbf{p}(M)=\{\ell_{1}\geq\dots\geq\ell_{k}\geq 1\colon\ell_{1}=\dots=\ell_{P_{1}}>\dots>\ell_{P_{r-1}+1}=\dots=\ell_{P_{r}}\}.

Consequently, we obtain the following representation.

Theorem 1.

The multi-interpolated multiple zeta values $\zeta^{\vec{t}}(i_{1},\dots,i_{k})$ , with $k\in\mathbb{N}$ and arguments $i_{1}>1$ , $i_{2},\dots,i_{k}\in\mathbb{N}$ , can be expressed in terms of ordered partitions $\mathbf{p}\in\mathcal{P}(k)$ :

\displaystyle\zeta^{\vec{t}}(i_{1},\dots,i_{k})=\sum_{\mathbf{p}\in\mathcal{P}(k)}\zeta\Big{(}(\vec{t}^{p_{1}-1},\sum_{j_{1}=1}^{P_{1}}i_{j_{1}})\dots,(\vec{t}^{p_{r}-1},\sum_{j_{r}=P_{r-1}+1}^{P_{r}}i_{j_{r}})\Big{)}.

Proof.

	$\displaystyle\zeta^{\vec{t}}(i_{1},\dots,i_{k})$	$\displaystyle=\sum_{\ell_{1}\geq\cdots\geq\ell_{k}\geq 1}\frac{\vec{t}^{\vec{\sigma}(\vec{\ell})}}{\ell_{1}^{i_{1}}\dots\ell_{k}^{i_{k}}}=\sum_{(\ell_{1},\dots,\ell_{k})\in M}\frac{\vec{t}^{\vec{\sigma}(\vec{\ell})}}{\ell_{1}^{i_{1}}\dots\ell_{k}^{i_{k}}}$
		$\displaystyle=\sum_{\mathbf{p}\in\mathcal{P}(k)}\sum_{(\ell_{1},\dots,\ell_{k})\in\mathbf{p}(M)}\frac{\vec{t}^{\vec{\sigma}(\vec{\ell})}}{\ell_{1}^{i_{1}}\dots\ell_{k}^{i_{k}}}$
		$\displaystyle=\sum_{\mathbf{p}\in\mathcal{P}(k)}\zeta\Big{(}(\vec{t}^{p_{1}-1},\sum_{j_{1}=1}^{P_{1}}i_{j_{1}}),\dots,(\vec{t}^{p_{r}-1},\sum_{j_{r}=P_{r-1}+1}^{P_{r}}i_{j_{r}})\Big{)}.$

∎

3. Multi-interpolated MZVs with repeated arguments

In the following we provide results for $\zeta^{\vec{t}}(\{s\}_{k})$ , generalizing earlier results for $\zeta(\{s\}_{k})$ [3], as well as $\zeta^{t}(\{s\}_{k})$ [12]. Our proofs are based on generating functions and symbolic combinatorial constructions.

Theorem 2.

The generating function $\Theta(z,\vec{t})=\sum_{k\geq 0}\zeta^{\vec{t}}(\{s\}_{k})z^{k}$ of the multi-interpolated MZVs $\zeta^{\vec{t}}(\{s\}_{k})$ is given by

\Theta(z,\vec{t})=\prod_{m=1}^{\infty}\Big{(}1+\frac{\frac{1}{m^{s}}\cdot z}{1-\frac{1}{m^{s}}zt_{m}}\Big{)}=\exp\Big{(}\sum_{j=1}^{\infty}\frac{z^{j}}{j}\cdot\big{(}\zeta(\vec{t}^{j},js)-\zeta((\vec{t}-\vec{1})^{j},js)\big{)}\Big{)}.

From the generating function we deduce several representations. We collect the definition of the complete Bell polynomials $B_{n}(x_{1},\dots,x_{n})$ , determined by the identity

\exp\Big{(}\sum_{\ell\geq 1}\frac{z^{\ell}}{\ell!}x_{\ell}\Big{)}=\sum_{j\geq 0}\frac{B_{j}(x_{1},\dots,x_{j})}{j!}z^{j},

such that

B_{k}(x_{1},\dots,x_{k})=\sum_{m_{1}+2m_{2}+\dots+km_{k}=k}\frac{k!}{m_{1}!m_{2}!\cdot m_{n}!}\left(\frac{x_{1}}{1!}\right)^{m_{1}}\dots\left(\frac{x_{k}}{k!}\right)^{m_{k}}.

Corollary 1.

The values $\zeta^{\vec{t}}(\{s\}_{k})$ satisfy

\zeta^{\vec{t}}(\{s\}_{k})=\sum_{\ell=0}^{k}\zeta^{\star}(\{(\vec{t},s)\}_{\ell})\cdot\zeta\big{(}\{(\vec{1}-\vec{t},s)\}_{k-\ell}\big{)},

where $\zeta(\{(\vec{t},s)\}_{0})=\zeta^{\star}(\{(\vec{t},s)\}_{0})=1$ .

Moreover,

\zeta^{\vec{t}}(\{s\}_{k})=\frac{1}{k!}B_{k}(x_{1},\dots,x_{k}),

with $x_{j}=(j-1)!\Big{(}\zeta((\vec{t}^{j},js))-\zeta\big{(}((\vec{t}-\vec{1})^{j},js)\big{)}\Big{)}$ , $1\leq j\leq k$ .

Remark 3.

Both expressions are also true for the truncated variants. The Bell polynomial expression also leads to a third representation of $\zeta^{\vec{t}}(\{s\}_{k})$ in terms of a determinant. A determinantal expression for $B_{k}(x_{1},\dots,x_{k})$ is given in [4] based on [5, 15]; another expression is obtained by using modified Bell polynomials $Q_{k}(x_{1},\dots,x_{k})$ , given by

	$\displaystyle Q_{k}(x_{1},\dots,x_{k})$	$\displaystyle=\frac{1}{k!}B_{k}(0!x_{1},1!x_{2},\dots,(k-1)!x_{k})$
		$\displaystyle=\sum_{m_{1}+2m_{2}+\dots+km_{k}=k}\frac{1}{m_{1}!m_{2}!\cdot m_{n}!}\left(\frac{x_{1}}{1}\right)^{m_{1}}\dots\left(\frac{x_{k}}{k}\right)^{m_{k}}$

by MacDonald [20] (see Hoffman [9] for additional properties): with

Q_{k}(x_{1},\dots,x_{k})=\frac{1}{k!}\cdot\left(\begin{matrix}x_{1}&-1&0&\dots&0\\ x_{2}&x_{1}&-2&\dots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{k-1}&x_{k-2}&x_{k-3}&\dots&-(k-1)\\ x_{k}&x_{k-1}&x_{k-2}&\dots&x_{1}\\ \end{matrix}\right).

Proof of Theorem 2.

We use the symbolic constructions [6]: let $\mathcal{Z}_{m}=\{m\}$ be a combinatorial class of size one, $1\leq m\leq n$ . Due to the sequence construction we can describe the class of multisets $\mathcal{B}_{m}$ of $\mathcal{Z}_{m}$ as follows

\mathcal{B}_{m}=\text{SEQ}(\mathcal{Z}_{m})=\{\epsilon\}+\mathcal{Z}_{m}+\mathcal{Z}_{m}\times\mathcal{Z}_{m}+\mathcal{Z}_{m}\times\mathcal{Z}_{m}\times\mathcal{Z}_{m}+\dots;

Thus, the generating function

B_{m}(z)=\sum_{\beta\in\mathcal{B}_{m}}w(\beta)t^{\sigma(\beta)}z^{|\beta|}=1+\sum_{\epsilon\neq\beta\in\mathcal{B}_{m}}w(\beta)t^{|\beta|-1}z^{|\beta|}

with weight $w(\beta)=\frac{1}{m^{s\cdot|\beta|}}$ , is given by

B_{m}(z)=1+\sum_{j=1}^{\infty}t_{m}^{j-1}\frac{1}{(m^{s})^{j}}z^{j}=1+\frac{\frac{1}{m^{s}}z}{1-\frac{1}{m^{s}}\cdot t_{m}z}.

Let

\mathcal{M}_{n,k}=\{\vec{\ell}=(\ell_{k},\ell_{k-1}\dots,\ell_{1})\in\mathbb{N}^{k}\colon 1\leq\ell_{k}\leq\dots\leq\ell_{2}\leq\ell_{1}\leq n\}.

All multisets $\mathcal{M}_{n}=\bigcup_{k=1}^{\infty}\mathcal{M}_{n,k}$ , with $k$ -multisets of $\{1,2,\dots,n\}$ can be combinatorially generated by

\mathcal{M}_{n}=\mathcal{B}_{1}\times\mathcal{B}_{2}\times\dots\times\mathcal{B}_{n}.

Hence, the generating function $\Theta_{n}(z,\vec{t})$ is given by the stated formula. The result for the non-truncated multiple zeta values follow by taking the limit. Then, we use the $\exp-\log$ representation and the expansion of $\ln(1-z)$ to get

	$\displaystyle\Theta(z,\vec{t})$	$\displaystyle=\exp\left(\sum_{m=1}^{\infty}\ln\Big{(}1-\frac{z(t_{m}-1)}{m^{s}}\Big{)}-\ln\Big{(}1-\frac{zt_{m}}{m^{s}}\Big{)}\right)$
		$\displaystyle=\exp\left(\sum_{m=1}^{\infty}\sum_{j=1}^{\infty}\frac{z^{j}}{j}\cdot\frac{t_{m}^{j}-(t_{m}-1)^{j}}{m^{js}}\right)$
		$\displaystyle=\exp\left(\sum_{j=1}^{\infty}\frac{z^{j}}{j}\Big{(}\zeta((\vec{t}^{j},js))-\zeta(((\vec{t}-\vec{1})^{j},js))\Big{)}\right).$

∎

Proof of Corollary 1.

From the expression for $\Theta(z,\vec{t})$ we get

\zeta^{\vec{t}}(\{s\}_{k})=[z^{k}]\Theta(z,\vec{t})=[z^{k}]\left(\prod_{m=1}^{\infty}\Big{(}1+\frac{(1-t_{m})z}{m^{s}}\Big{)}\right)\left(\prod_{m=1}^{\infty}\frac{1}{1-\frac{zt_{m}}{m^{s}}}\right).

The former expression is exactly the generating function of $\zeta(\{(\vec{1}-\vec{t},s)\}_{k})$ , whereas the latter expression is the generating function of $\zeta^{\star}(\{(\vec{t},s)\}_{k})$ . ∎

4. A product for multi-interpolated zeta values

We discuss algebraic properties of the multi-interpolated multiple zeta values. Following Hoffman [8, 11, 12] and Yamamoto [27], let $\mathcal{A}=\{z_{1},z_{2},\dots\}$ denote a countable set of letters. Let $\mathbb{Q}\langle\mathcal{A}\rangle$ denote the rational non-commutative polynomial algebra and $\mathfrak{h}^{1}$ the underlying rational vector space of $\mathbb{Q}\langle\mathcal{A}\rangle$ . There are two products $\ast$ and $\star$ on $\mathfrak{h}^{1}$ defined by

x\ast 1=1\ast x=x,\quad x\star 1=1\star x=x,

For words $x=au$ , $y=bv$ we have

x\ast y=a(u\ast bv)+b(au\ast v)+a\diamond b(u\ast v),

(11)

whereas

x\star y=a(u\star bv)+b(au\star v)-a\diamond b(u\star v).

(12)

Here, $\diamond$ denotes the commutative product

z_{i}\diamond z_{j}=z_{i+j},

(13)

and $z_{i}\diamond 1=1\diamond z_{i}=0$ . The product $\ast$ corresponds to the multiplication of multiple zeta values, whereas the $\star$ product to the multiple zeta star values.

Remark 4.

It is well known that if the $\diamond$ product is defined trivially by $x\diamond y=0$ , then both products reduce to the shuffle product: $\ast=\star=\shuffle$ .

Let $\mathfrak{h}^{0}$ be the subspace of $\mathfrak{h}^{1}$ generated by 1 and monomials that do not start with $z_{1}$ , then the linear map $Z\colon\mathfrak{h}^{0}\to\mathbf{R}$ , defined by

Z(z_{i_{1}}\dots z_{i_{k}})=\zeta(i_{1},\dots,i_{k})

(14)

and $Z(1)=1$ is a homomorphism from $(\mathfrak{h}^{0};\ast)$ to the reals.

Yamamoto [27] introduced the interpolated product $\stackrel{{\scriptstyle t}}{{\ast}}$ , generalizing (11) and (12): $\stackrel{{\scriptstyle 1}}{{\ast}}=\star$ , $\stackrel{{\scriptstyle 0}}{{\ast}}=\ast$ . He also introduced a map $Z^{t}$ from $\mathfrak{h}^{0}$ to $\mathbf{R}$ . It maps a word $x=z_{i_{1}}\dots z_{i_{k}}\in\mathfrak{h}^{0}$ to an interpolated MZV:

Z^{t}(z_{i_{1}}\dots z_{i_{k}})=\zeta^{t}(i_{1},\dots,i_{k}).

Moreover, for $x,y\in\mathfrak{h}^{0}$ the product $\stackrel{{\scriptstyle t}}{{\ast}}$ satisfies the important relation

Z^{t}(x\stackrel{{\scriptstyle t}}{{\ast}}y)=Z^{t}(x)\cdot Z^{t}(y).

The refinement of interpolated MZVs to multi-interpolated MZVs in Definition 1 suggests to look at corresponding generalizations of the map $Z^{t}$ and the product $\stackrel{{\scriptstyle t}}{{\ast}}$ . In the following we introduce a multi-interpolated product $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ , as well as a map $Z^{\vec{t}}$ such that for $x,y\in\mathfrak{h}^{0}$ it holds

Z^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)=Z^{\vec{t}}(x)\cdot Z^{\vec{t}}(y).

(15)

In order to make this precise, we first define the map $Z^{\vec{t}}$ . For this purpose, we introduce a new variable, which we denote with $\vec{t}$ . It does not commute with any letters $z_{i}$ of the alphabet $\mathcal{A}$ . Let $\mathbb{Q}\langle\mathcal{A},\vec{t}\rangle$ denote the algebra of noncommutative polynomials in the letters $z_{1},z_{2},\dots\in\mathcal{A}$ and the symbol $\vec{t}$ , where we denote with $\vec{t}^{m}$ a string consisting of $m$ occurrences of $\vec{t}$ , $m\geq 0$ . We denote with $\mathfrak{h}^{2}$ the subspace, generated by 1 and monomials that do not start with $\vec{t}^{j}z_{1}$ , $j\geq 0$ or end with a letter $\vec{t}$ . Non-empty words $x\in\mathfrak{h}^{2}$ have the form

x=\vec{t}^{m_{1}}z_{i_{1}}\vec{t}^{m_{2}}z_{i_{2}}\dots\vec{t}^{m_{n}}z_{i_{n}},

(16)

with $n\geq 1$ and $m_{1},\dots m_{1}\geq 0$ , where $i_{1}\neq 1$ .

The map $Z^{\vec{t}}$ has domain $\mathfrak{h}^{2}$ and codomain $\mathbf{R}[[\vec{t}]]$ and sends words $x\in\mathfrak{h}^{2}$ (16) to multiple zeta values with variables $t_{1},t_{2},\dots$ , as introduced in Definition 2:

Z^{\vec{t}}(\vec{t}^{m_{1}}z_{i_{1}}\vec{t}^{m_{2}}z_{i_{2}}\dots\vec{t}^{m_{n}}z_{i_{n}})=\zeta^{\vec{t}}\big{(}(\vec{t}^{m_{1}},i_{1}),(\vec{t}^{m_{2}},i_{2}),\dots,(\vec{t}^{m_{n}},i_{n})\big{)}.

(17)

For the structural analysis of $Z^{\vec{t}}$ we will revisit the map $Z$ (14), as well as the ordinary stuffle product $\ast$ (11). We extend the domain and codomain of $Z$ to $\mathfrak{h}^{2}$ and $\mathbf{R}[[\vec{t}]]$ , respectively:

Z(\vec{t}^{m_{1}}z_{i_{1}}\vec{t}^{m_{2}}z_{i_{2}}\dots\vec{t}^{m_{n}}z_{i_{n}})=\zeta\big{(}(\vec{t}^{m_{1}},i_{1}),(\vec{t}^{m_{2}},z_{2})\dots(\vec{t}^{m_{n}},i_{n})\big{)}.

(18)

Example 5.

We emphasize the non-commutativity of the letter $\vec{t}$ with the letters $z_{j}\in\mathcal{A}$ . Let $x=z_{i_{1}}\vec{t}z_{i_{2}}$ and $y=\vec{t}z_{i_{1}}z_{i_{2}}$ . Then,

Z(x)=Z(z_{i_{1}}\vec{t}z_{i_{2}})=\sum_{\ell_{1}>\ell_{2}\geq 1}\frac{t_{\ell_{2}}}{\ell_{1}^{i_{1}}\ell_{2}^{i_{2}}}\neq Z(y)=Z(\vec{t}z_{i_{1}}z_{i_{2}})=\sum_{\ell_{1}>\ell_{2}\geq 1}\frac{t_{\ell_{1}}}{\ell_{1}^{i_{1}}\ell_{2}^{i_{2}}}.

In contrast for $\vec{t}=t$ , see Remark 1, we would have obtained the same result $t\cdot\zeta(i_{1},i_{2})$ , as

z_{i_{1}}tz_{i_{2}}=tz_{i_{1}}z_{i_{2}}.

Next, define how ordinary stuffle product $\ast$ (11) for MZVs acts on words in $\mathfrak{h}^{2}$ . Let $x=\vec{t}^{m_{1}}a_{1}u\in\mathfrak{h}^{2}$ with $u=\vec{t}^{m_{2}}a_{2}\dots\vec{t}^{m_{n}}a_{n}$ and $y=\vec{t}^{k_{1}}b_{1}v\in\mathfrak{h}^{2}$ with $v=\vec{t}^{k_{2}}a_{2}\dots\vec{t}^{k_{r}}b_{r}$ , with $a_{\ell},b_{j}\in\mathcal{A}$ , $1\leq\ell\leq n$ and $1\leq j\leq r$ . Then, $x\ast y$ is defined by

x\ast y=\vec{t}^{m_{1}}a_{1}(u\ast y)+\vec{t}^{k_{1}}b_{1}(x\ast v)+(\vec{t}^{m_{1}}a_{1}\diamond\vec{t}^{k_{1}}b_{1})(u\ast v).

(19)

where we extend the definition of the product $\diamond$ (13) to

\vec{t}^{c}z_{i}\diamond\vec{t}^{d}z_{j}=\vec{t}^{c+d}z_{i+j},\quad c,d\geq 0.

(20)

Next, we introduce the product $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ .

Definition 3 (Product $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ ).

Let $x,y\in\mathfrak{h}^{0}$ denote two words and $\vec{t}$ a variable non-commutative with the letters $z_{k}\in\mathcal{A}$ . If $y=1$ then

x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}1=1\stackrel{{\scriptstyle\vec{t}}}{{\ast}}x=x.

For $x=a\in\mathcal{A}$ and $y=b\in\mathcal{A}$ single letter words we have

x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y=a\stackrel{{\scriptstyle\vec{t}}}{{\ast}}b=ab+ba+(1-2\vec{t})a\diamond b.

For words $x=au\in\mathfrak{h}^{0}$ , $y=bv\in\mathfrak{h}^{0}$ we have

x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y=a(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}bv)+b(au\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)+(1-2\vec{t})a\diamond b(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)+(\vec{t}^{2}-\vec{t})a\diamond b\diamond(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v).

Remark 5.

Our definition looks similar to the definition of Yamamoto [27]. Indeed, when the variable $\vec{t}$ is evaluated into $t$ (7), then $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}\ =\ \stackrel{{\scriptstyle t}}{{\ast}}$ . We emphasize again the key difference, namely the non-commutativity of $\vec{t}$ with the letters $z_{k}\in\mathcal{A}$ as .

Remark 6.

By (5) and (15) we are mainly interested in words $x,y\in\mathfrak{h}^{0}$ . However, we can readily extend the definition of $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ to words $x,y\in\mathfrak{h}^{2}$ , compare with (19):

\begin{split}x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y&=\vec{t}^{m_{1}}a_{1}(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}\vec{t}^{k_{1}}b_{1}v)+\vec{t}^{k_{1}}b_{1}(\vec{t}^{m_{1}}a_{1}u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)+(1-2\vec{t})\vec{t}^{m_{1}+k_{1}}a_{1}\diamond b_{1}(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)\\ &\quad+(\vec{t}^{2}-\vec{t})\vec{t}^{m_{1}+k_{1}}a_{1}\diamond b_{1}\diamond(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v).\end{split}

Example 6.

Let $x=a=z_{i}$ and $y=b=z_{j}$ . Then

x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y=z_{i}\stackrel{{\scriptstyle\vec{t}}}{{\ast}}z_{j}=z_{i}z_{j}+z_{j}z_{i}+(1-2\vec{t})z_{i+j}.

Example 7.

Let $x=a=z_{i}$ and $y=bz_{k}=z_{j}z_{k}$ .

	$\displaystyle x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y$	$\displaystyle=z_{i}z_{j}z_{k}+z_{j}(z_{i}\stackrel{{\scriptstyle\vec{t}}}{{\ast}}z_{k})+(1-2\vec{t})z_{i+j}z_{k}+(\vec{t}^{2}-\vec{t})z_{i+j+k}$
		$\displaystyle=z_{i}z_{j}z_{k}+z_{j}z_{i}z_{k}+z_{j}z_{k}z_{i}+z_{j}(1-2\vec{t})z_{i+k}$
		$\displaystyle\quad+(1-2\vec{t})z_{i+j}z_{k}+(\vec{t}^{2}-\vec{t})z_{i+j+k}.$

In order to obtain the result (15) we need more insight into the map $Z^{\vec{t}}$ . We will realize the map $Z^{\vec{t}}$ using the map $Z$ (14), (18) and a new operator $S^{\vec{t}}$ , generalizing the interpolating operator in [27].

Definition 4 (Multi-interpolation operator).

For the empty word 1 and a single letter $a\in\mathcal{A}$ the multi-interpolation operator $S^{\vec{t}}$ is defined as

S^{\vec{t}}(1)=1,\quad S^{\vec{t}}(a)=a.

For $x=au\in\mathfrak{h}^{0}$ , where $u$ denotes a substring, we set

S^{\vec{t}}(x)=S^{\vec{t}}(au)=aS^{\vec{t}}(u)+\vec{t}a\diamond S^{\vec{t}}(u).

(21)

Remark 7.

Of course, one can also consider words $x\in\mathfrak{h}^{2}$ ,

x=\vec{t}^{m_{1}}a_{1}u,\quad u=\vec{t}^{m_{2}}a_{2}\dots\vec{t}^{m_{n}}a_{n},

where we define

S^{\vec{t}}(x)=S^{\vec{t}}(\vec{t}^{m_{1}}a_{1}u)=\vec{t}^{m_{1}}a_{1}S^{\vec{t}}(u)+\vec{t}^{m_{1}+1}a_{1}\diamond S^{\vec{t}}(u).

(22)

Example 8.

Let $x=z_{i}z_{j}z_{k}$ denote word of length three. Then

	$\displaystyle S^{\vec{t}}(x)$	$\displaystyle=z_{i}S^{\vec{t}}(z_{j}z_{k})+\vec{t}z_{i}\diamond S^{\vec{t}}(z_{j}z_{k})$
		$\displaystyle=z_{i}z_{j}z_{k}+z_{i}\vec{t}z_{j}\diamond z_{k}+\vec{t}z_{i}\diamond z_{j}z_{k}+\vec{t}z_{i}\diamond\vec{t}z_{j}\diamond z_{k}$
		$\displaystyle=z_{i}z_{j}z_{k}+z_{i}\vec{t}z_{j+k}+\vec{t}z_{i+j}z_{k}+\vec{t}^{2}z_{i+j+k}.$

This corresponds exactly to the decomposition of the multi-interpolated multiple zeta value $\zeta^{\vec{t}}(i,j,k)$ in Example 3 and we see that

(Z\circ S^{\vec{t}})(z_{i}z_{j}z_{k})=Z^{\vec{t}}(z_{i}z_{j}z_{k}).

In order to describe $S^{\vec{t}}(x)$ for a word $x$ let $R_{n}$ , $n\in\mathbb{N}$ , denote the set of subsequences $r=(r_{0},\dots,r_{s})$ of $(0,\dots,n)$ such that $r_{0}=0$ and $r_{s}=n$ . For such $r$ and a word $x=a_{1}\dots a_{n}$ , we define the word $\text{Con}_{r}^{\vec{t}}(x)$ with respect to $\vec{t}$ . It is the weighted contraction of $x$ with respect to $r$ , weighted according

\text{Con}_{r}^{\vec{t}}(x)=\vec{t}^{r_{1}-r_{0}-1}b_{1}\cdot\vec{t}^{r_{2}-r_{1}-1}b_{2}\cdots\vec{t}^{r_{s}-r_{s-1}-1}b_{s},\quad b_{i}=a_{r_{i}+1}\diamond\dots\diamond a_{r_{i+1}}.

(23)

Lemma 1 (Properties - Multi-interpolation operator).

Let $x=a_{1}\dots a_{n}$ . The operator $S^{\vec{t}}$ has the properties

a)

$S^{\vec{t}}(x)=\sum_{r\in R_{n}}\text{Con}_{r}^{\vec{t}}(x)$ ,
b)

$(S^{\vec{t}}-1)^{n}x=0$ ,
c)

Assume that symbols $\vec{t}_{1}$ , $\vec{t}_{2}$ commute with each other, but are non-commutative with respect to letters $z_{k}\in\mathcal{A}$ . Let $\circ$ denote the operator composition, then

$S^{\vec{t}_{1}+\vec{t}_{2}}=S^{\vec{t}_{1}}\circ S^{\vec{t}_{2}}.$
d)

Let $b\in\mathcal{A}$ :

$b\diamond S^{\vec{t}}(x)=S^{\vec{t}}(b\diamond x).$

Remark 8.

Note that for $\vec{t}=t$ the weighted contraction simplifies to

\text{Con}_{r}^{\vec{t}}(x)=t^{\sum_{i=1}^{s}(r_{i}-r_{i-1}-1)}\text{Con}_{r}(x)=t^{n-s}\text{Con}_{r}(x)=t^{\sigma(r)}\text{Con}_{r}(x),

where $\text{Con}_{r}(x)=b_{1}\cdots b_{s}$ denotes the standard contraction operator of [27].

Proof of Lemma 1.

The first part follows directly from the definition. Part (b) is shown using induction. We actually prove the stronger statement that $(S^{\vec{t}}-1)^{n}x=0$ for $x=\vec{t}^{m_{1}}a_{1}\vec{t}^{m_{2}}a_{2}\dots\vec{t}^{m_{n}}a_{n}\in\mathfrak{h}^{2}$ . The statement is obviously true for an empty word $1$ , as well as a single letter word $x=\vec{t}^{m}a$ . Let $x=\vec{t}^{m_{1}}a_{1}u$ with $a_{1}$ a single letter and $u=\vec{t}^{m_{2}}a_{2}\dots\vec{t}^{m_{n}}a_{n}$ the subword. By

(S^{\vec{t}}-1)^{n}x=\vec{t}^{m_{1}}(S^{\vec{t}}-1)^{n-1}(S^{\vec{t}}-1)a_{1}u.

By a slight generalization of part a we observe $(S^{\vec{t}}-1)a_{1}u$ is a combination of words of length less or equal $n-1$ . Hence, by the induction hypothesis the expression is equal to zero. For part (c) we have to argue in combinatorial way, reducing the identity to a counting problem. By definition part a) and (23), we have

S^{\vec{t}_{1}+\vec{t}_{2}}(x)=\sum_{r\in R_{n}}(\vec{t}_{1}+\vec{t}_{2})^{r_{1}-r_{0}-1}b_{1}\cdots(\vec{t}_{1}+\vec{t}_{2})^{r_{s}-r_{s-1}-1}b_{s}.

Assume that $1\leq i\leq s$ and $\rho_{i}=r_{i}-r_{i-1}-1$ . Expanding by the binomial theorem gives

(\vec{t}_{1}+\vec{t}_{2})^{\rho_{i}}=\sum_{\ell=0}^{\rho_{i}}\binom{\rho_{i}}{\ell}\vec{t}_{1}^{\rho_{i}-\ell}\vec{t}_{2}^{\ell}.

For each resulting block $b_{i}=a_{r_{i}+1}\diamond\dots\diamond a_{r_{i+1}}$ , there have to be exactly $\rho_{i}$ contractions leading to it. Out of these $\rho_{i}$ contractions, we can choose $0\leq\ell\leq\rho_{i}$ of them to stem from the application of $S^{\vec{t}_{2}}$ . The subblocks leading to $b_{i}$ all have the common prefactor $\vec{t}_{2}^{\ell}$ , as already $\ell$ contractions have occurred. As there are in total $\rho_{i}$ letters merged into $b_{i}$ , we have $\binom{\rho_{i}}{\ell}$ words in $\text{Con}_{r}^{\vec{t}}(x)$ , with $r\in R_{n}$ , leading after application of $S^{\vec{t}_{1}}$ to the block $b_{i}$ . The application of $S^{\vec{t}_{1}}$ to these words leads to an additional factor $\vec{t}_{1}^{\rho_{i}-\ell}$ , as we contract the remaining letters to obtain $b_{i}$ . Concerning the final statement (d) we observe that

	$\displaystyle S^{\vec{t}}(b\diamond x)$	$\displaystyle=S^{\vec{t}}(b\diamond\vec{t}^{m_{1}}a_{1}u)=S^{\vec{t}}\big{(}\vec{t}^{m_{1}}b\diamond a_{1}u\big{)}$
		$\displaystyle=\vec{t}^{m_{1}}b\diamond a_{1}S^{\vec{t}}(u)+\vec{t}^{m_{1}+1}b\diamond a_{1}\diamond S^{\vec{t}}(u)$
		$\displaystyle=b\diamond\big{(}\vec{t}^{m_{1}}a_{1}S^{\vec{t}}(u)+\vec{t}^{m_{1}+1}a_{1}\diamond S^{\vec{t}}(u)\big{)}=b\diamond S^{\vec{t}}(x).$

∎

We also collect another property of $\mathcal{S}^{\vec{t}}$ when combined a suitably defined derivative.

Definition 5 (Derivative for symbol $\vec{t}$ ).

The action of the differential operator $\frac{d}{d\vec{t}}$ to $x\in\mathfrak{h}^{2}$ is given as follows. For $x=a\in\mathfrak{h}^{0}$ we have

\frac{d}{d\vec{t}}a=0,\quad\frac{d}{d\vec{t}}\vec{t}^{m}a=m\cdot\vec{t}^{m-1}a.

For $x=\vec{t}^{m_{1}}a_{1}u\in\mathfrak{h}^{2}$ with $u=\vec{t}^{m_{2}}a_{2}\dots\vec{t}^{m_{n}}a_{n}$ with $n\geq 2$ we define

\frac{d}{d\vec{t}}x=m_{1}\vec{t}^{m_{1}-1}a_{1}u+\vec{t}^{m_{1}}a_{1}\frac{d}{d\vec{t}}u,

We observe the following.

Lemma 2.

Let $x=a_{1}\dots a_{n}$ . The differential operator $\frac{d}{d\vec{t}}$ acting on $S^{\vec{t}}(x)$ can be decomposed into contractions:

\frac{d}{d\vec{t}}S^{\vec{t}}(x)=\sum_{k=1}^{n-1}S^{\vec{t}}(a_{1}\dots a_{k-1}a_{k}\diamond a_{k+1}a_{k+2}\dots a_{n}).

Proof.

We provide two different proofs. First, we argue in a combinatorial way. By Lemma 1 (a) each word in $S^{\vec{t}}(x)$ has the form

\text{Con}_{r}^{\vec{t}}(x)=\vec{t}^{r_{1}-r_{0}-1}b_{1}\cdot\vec{t}^{r_{2}-r_{1}-1}b_{2}\cdots\vec{t}^{r_{s}-r_{s-1}-1}b_{s},\quad b_{i}=a_{r_{i}+1}\diamond\dots\diamond a_{r_{i+1}}.

Application of the derivative to $\text{Con}_{r}^{\vec{t}}(x)$ leads to $s$ words, each with multiplicative factor $\rho_{i}=r_{i}-r_{i-1}-1$ and the corresponding power of $\vec{t}$ at position $i$ diminished by one. Out of the words in $\sum_{k=1}^{n-1}S^{\vec{t}}(a_{1}\dots a_{k-1}a_{k}\diamond a_{k+1}\dots a_{n})$ exactly the words with $r_{i-1}\leq k\leq r_{i}$ can be mapped to the resulting word in $\frac{d}{d\vec{t}}\text{Con}_{r}^{\vec{t}}(x)$ . Since every word in $\frac{d}{d\vec{t}}\text{Con}_{r}^{\vec{t}}(x)$ is covered like this, we obtain the stated result.

Our second proof uses induction. We use the shorthand notation $a_{i,j}=a_{i}\dots a_{j}$ for $i\leq j$ . We have

\frac{d}{d\vec{t}}S^{\vec{t}}(x)=\frac{d}{d\vec{t}}\Big{(}a_{1}S^{\vec{t}}(a_{2,n})+\vec{t}a_{1}\diamond S^{\vec{t}}(a_{2,n})\Big{)}=a_{1}\frac{d}{d\vec{t}}S^{\vec{t}}(a_{2,n})+\frac{d}{d\vec{t}}\vec{t}a_{1}\diamond S^{\vec{t}}(a_{2,n}).

The term $S^{\vec{t}}(a_{2,n})$ is directly covered by the induction hypothesis. By the ordinary product rule and by Lemma 1 (d) we get further

\frac{d}{d\vec{t}}\vec{t}a_{1}\diamond S^{\vec{t}}(a_{2,n})=S^{\vec{t}}(a_{1}\diamond a_{2}a_{3,n})+\vec{t}\frac{d}{d\vec{t}}S^{\vec{t}}\big{(}(a_{1}\diamond a_{2})a_{3,n}\big{)}.

The induction hypothesis applies to the second summand and we get

	$\displaystyle\frac{d}{d\vec{t}}S^{\vec{t}}(a_{1}\diamond a_{2,n})=\frac{d}{d\vec{t}}S^{\vec{t}}\big{(}(a_{1}\diamond a_{2})a_{3,n}\big{)}$
	$\displaystyle=S^{\vec{t}}\big{(}(a_{1}\diamond a_{2}\diamond a_{3})a_{4,n}\big{)}+\dots+S^{\vec{t}}\big{(}(a_{1}\diamond a_{2})a_{3,n-2}(a_{n-1}\diamond a_{n})\big{)}.$

Collecting all contributions, the induction hypothesis implies that

	$\displaystyle\frac{d}{d\vec{t}}S^{\vec{t}}(x)$	$\displaystyle=a_{1}\sum_{k=2}^{n}S^{\vec{t}}\big{(}a_{2,k-1}(a_{k}\diamond a_{k+1})a{k+2,n}\big{)}+S^{\vec{t}}\big{(}(a_{1}\diamond a_{2})a_{3,n}\big{)}$
		$\displaystyle+\vec{t}\Big{(}S^{\vec{t}}\big{(}(a_{1}\diamond a_{2}\diamond a_{3})a_{4,n}\big{)}+\dots+S^{\vec{t}}\big{(}(a_{1}\diamond a_{2})a_{3,n-2}(a_{n-1}\diamond a_{n})\big{)}\Big{)}.$

On the other hand, we can decompose the righthandside of the equation.

	$\displaystyle\sum_{k=1}^{n-1}S^{\vec{t}}\big{(}a_{1,k-1}(a_{k}\diamond a_{k+1})a_{k+2,n}\big{)}=S^{\vec{t}}((a_{1}\diamond a_{2})a_{3,n})$
	$\displaystyle\quad+\sum_{k=2}^{n-1}S^{\vec{t}}\big{(}a_{1,k-1}(a_{k}\diamond a_{k+1})a_{k+2,n}\big{)}$

Furthermore,

	$\displaystyle S^{\vec{t}}\big{(}a_{1,k-1}a_{k}\diamond a_{k+1,n}\big{)}$	$\displaystyle=a_{1}S^{\vec{t}}\big{(}a_{2,k-1}(a_{k}\diamond a_{k+1})a_{k+2,n}\big{)}$
		$\displaystyle\quad+\vec{t}S^{\vec{t}}\big{(}(a_{1}\diamond a_{2})a_{3,k-1}(a_{k}\diamond a_{k+1})a_{k+2,n}\big{)}.$

This provides the induction step and thus the stated result. ∎

A direct application of Lemma 2 is a generalization of the so-called alternating sum formula [14]:

\sum_{k=0}^{n}(-1)^{k}(a_{1}\cdots a_{k})\ast S(a_{n}\cdots a_{k+1})=0.

(24)

This formula was extended to interpolated MZVs [27, Proposition 3.7]

\sum_{k=0}^{n}(-1)^{k}S^{t}(a_{1}\cdots a_{k})\ast S^{1-t}(a_{n}\cdots a_{k+1})=0.

Theorem 3 (Alternating sum formula).

The multi-interpolation operator $S^{\vec{t}}$ satisfies

\sum_{k=0}^{n}(-1)^{k}S^{\vec{t}}(a_{1}\cdots a_{k})\ast S^{\vec{1}-\vec{t}}(a_{n}\cdots a_{k+1})=0.

Proof.

We use induction with respect to $n$ . For $n=1$ we have

S^{\vec{t}}(a_{1})\cdot 1-S^{\vec{1}-\vec{t}}(a_{1})\cdot 1=a_{1}-a_{1}=0.

For $n=2$ observe that

	$\displaystyle S^{\vec{t}}(a_{1}a_{2})\cdot 1-S^{\vec{t}}(a_{1})S^{\vec{1}-\vec{t}}(a_{2})+S^{\vec{1}-\vec{t}}(a_{2}a_{1})$
	$\displaystyle\quad=a_{1}a_{2}+\vec{t}a_{1}\diamond a_{2}-\big{(}a_{1}a_{2}+a_{2}a_{1}-a_{1}\diamond a_{2}\big{)}+a_{2}a_{1}+(\vec{1}-\vec{t})a_{2}\diamond a_{1}=0.$

Assume now that $n\geq 2$ . We observe that the left-hand side is given by a sum of words of the form $x=\vec{t}^{m}_{1}b_{1}\dots\vec{t}^{m}_{k}b_{k}$ , with $1\leq k\leq n$ plus words without any powers of $\vec{t}$ . By the original alternating sum formula (24) we know that for $\vec{t}=0$ the truth of the statement. If there are words of a specific length $k$ of the form $x=\vec{t}^{m}_{1}b_{1}\dots\vec{t}^{m}_{k}b_{k}$ , such that their sum does not vanish, then the derivative $\frac{d}{d\vec{t}}$ results in words of the form $\vec{t}^{m}_{1}b_{1}\dots\vec{t}^{m_{r}-1}b_{r}\dots\vec{t}^{m}_{k}b_{k}$ , $1\leq r\leq k$ , which cannot vanish either. Hence, it suffices to prove that

\frac{d}{d\vec{t}}\sum_{k=0}^{n}(-1)^{k}S^{\vec{t}}(a_{1,k})\ast S^{\vec{1}-\vec{t}}(A_{n,k+1})=0,

where used again the shorthand notation $a_{i,j}=a_{i}\dots a_{j}$ for $i\leq j$ and also $A_{j,i}=a_{j}\dots a_{i}$ . We can now proceed identically to [27] and obtain for this derivative of the left-hand side by Lemma 2 and the Leibniz rule

	$\displaystyle\sum_{k=2}^{n}(-1)^{k}\sum_{i=1}^{k-1}S^{\vec{t}}(a_{1,i-1}a_{i}\diamond a_{i+1}a_{i+2,k})\ast S^{\vec{1}-\vec{t}}(A_{n,k+1})$
	$\displaystyle\quad-\sum_{k=0}^{n-2}(-1)^{k}\sum_{i=k+1}^{n-1}S^{\vec{t}}(a_{1,k})\ast S^{\vec{1}-\vec{t}}(A_{n,i+2}a_{i+1}\diamond a_{i}A_{i-1,k})$
	$\displaystyle=-\sum_{i=1}^{n-1}\bigg{(}\sum_{k=0}^{i-1}(-1)^{k}S^{\vec{t}}(a_{1,k})\ast S^{\vec{1}-\vec{t}}(A_{n,i+2}a_{i+1}\diamond a_{i}A_{i-1,k})$
	$\displaystyle\quad+\sum_{k=i+1}^{n}(-1)^{k-1}S^{\vec{t}}(a_{1,i-1}a_{i}\diamond a_{i+1}a_{i+2,k})\ast S^{\vec{1}-\vec{t}}(A_{n,k+1}).\bigg{)}$

By the induction hypothesis the sums inside the brackets vanish, which proves the stated result. ∎

Now we turn to the statement of our main result.

Theorem 4.

Let $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ denote the commutative product on the algebra $\mathbb{Q}\langle\mathcal{A}\rangle[\vec{t}]$ . The map $S^{\vec{t}}$ is an isomorphism from $(\mathfrak{h}^{2},\stackrel{{\scriptstyle\vec{t}}}{{\ast}})$ to $(\mathfrak{h}^{2},\stackrel{{\scriptstyle\vec{t}}}{{\ast}})$ , such that

S^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)=S^{\vec{t}}(x)\ast S^{\vec{t}}(y).

The map $Z^{\vec{t}}$ (17) can be decomposed into the map $Z$ (14), (18) and the multi-interpolation operator $S^{\vec{t}}$ (21), such that $Z^{\vec{t}}=Z\circ S^{\vec{t}}$ . Moreover,

Z^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)=Z^{\vec{t}}(x)\cdot Z^{\vec{t}}(y).

Remark 9.

If we define $\zeta^{\vec{t}}\big{(}(i_{1},\dots,i_{k})\stackrel{{\scriptstyle t}}{{\ast}}(j_{1},\dots,j_{\ell})\big{)}$ as $Z^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$ with $x=z_{i_{1}}\dots z_{i_{k}}$ and $y=z_{j_{1}}\dots z_{j_{\ell}}$ , then we write

\zeta^{\vec{t}}\big{(}(i_{1},\dots,i_{k})\stackrel{{\scriptstyle t}}{{\ast}}(j_{1},\dots,j_{\ell})\big{)}=\zeta^{\vec{t}}(i_{1},\dots,i_{k})\cdot\zeta^{\vec{t}}(j_{1},\dots,j_{\ell}),

compare with (4).

Example 9.

Let $x=a=z_{i}$ and $y=b=z_{j}$ . By Example 6 we get

	$\displaystyle Z^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$	$\displaystyle=Z^{\vec{t}}(z_{i}z_{j})+Z^{\vec{t}}(z_{j}z_{i})+Z^{\vec{t}}\big{(}(1-2\vec{t})z_{i+j}\big{)}$
		$\displaystyle=\zeta^{\vec{t}}(i,j)+\zeta^{\vec{t}}(j,i)+\sum_{\ell\geq 1}\frac{1-2t_{\ell}}{\ell^{i+j}}$
		$\displaystyle=\zeta(i,j)+\zeta\big{(}(i+j)\vec{t}\big{)}+\zeta(j,i)+\zeta\big{(}(i+j)\vec{t}\big{)}+\sum_{\ell\geq 1}\frac{1-2t_{\ell}}{\ell^{i+j}}$
		$\displaystyle=\zeta(i,j)+\zeta(j,i)+\zeta(i+j)=\zeta^{\vec{t}}(i)\zeta^{\vec{t}}(j)=Z^{\vec{t}}(z_{i})\cdot Z^{\vec{t}}(z_{j}),$

since $\zeta^{\vec{t}}(i)=\zeta(i)$ and $\zeta^{\vec{t}}(j)=\zeta(j)$ .

Example 10.

Let $x=a=z_{i}$ and $y=bz_{k}=z_{j}z_{k}$ . By Example 7 we get

	$\displaystyle Z^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$	$\displaystyle=\zeta(i,j,k)+\zeta(i+j,k)+\zeta(j,i,k)+\zeta(j,i+k)+\zeta(j,i,k)$
		$\displaystyle\quad+\zeta(i,\vec{t}(j+k))+\zeta(\vec{t}(i+j+k))+\zeta(\vec{t}(j+k),i),$

which is exactly the product of $Z^{\vec{t}}(z_{i})=\zeta(i)$ and $Z^{\vec{t}}(z_{j}z_{k})=\zeta(j,k)+\zeta((\vec{t},j+k))$ .

Proof.

By induction with respect to the number of letters $z_{k}\in\mathcal{A}$ , te multi-interpolation map $S^{\vec{t}}$ is injective. It is also surjective, as $S^{\vec{t}}\big{(}S^{-\vec{t}}(x)\big{)}=S^{0}(x)=x$ . Once, the homomorphism is established, the result for $Z^{\vec{t}}$ follows directly by Theorem 1 and Lemma 1, as well as an application of $Z$ . In order to prove the homomorphism, we follow closely the strategy of [27] and also use results of, so one should give much credit to these works. For the sake of simplicity, we present the proof solely for words $x,y\in\mathfrak{h}^{0}$ , as the general case $x,y\in\mathfrak{h}^{2}$ is more involved, to the additional occurrences (16) of $\vec{t}$ . We proceed by induction with respect to the total length of the words. For $x=1$ or $y=1$ or both, this is obviously true. For $x=a$ and $y=b$ , corresponding to Example 6, we have

	$\displaystyle S^{\vec{t}}(a\stackrel{{\scriptstyle\vec{t}}}{{\ast}}b)$	$\displaystyle=S^{\vec{t}}\big{(}ab+ba+(1-2\vec{t})a\diamond b\big{)}$
		$\displaystyle=ab+\vec{t}a\diamond b+ba+\vec{t}b\diamond a+(1-2\vec{t})a\diamond b$
		$\displaystyle=ab+ba+2\vec{t}a\diamond b+a\diamond b-2\vec{t}a\diamond b$
		$\displaystyle=ab+ba+a\diamond b=S^{\vec{t}}(a)\ast S^{\vec{t}}(b).$

Assume that $x=au$ and $y=bv$ . On one hand, we have

S^{\vec{t}}(x)\ast S^{\vec{t}}(y)=\big{(}aS^{\vec{t}}(u)+\vec{t}a\diamond S^{\vec{t}}(u)\big{)}\ast\big{(}bS^{\vec{t}}(v)+\vec{t}b\diamond S^{\vec{t}}(v)\big{)}.

Let $U=S^{\vec{t}}(u)$ and $V=S^{\vec{t}}(v)$ . Thus,

\displaystyle S^{\vec{t}}(x)\ast S^{\vec{t}}(y)=\big{(}aU+\vec{t}a\diamond U\big{)}\ast\big{(}bV+\vec{t}b\diamond V\big{)}.

Expanding the term above gives

	$\displaystyle S^{\vec{t}}(x)\ast S^{\vec{t}}(y)$	$\displaystyle=(aU)\ast(bV)+(aU)\ast(\vec{t}b\diamond V)$		(25)
		$\displaystyle\quad+(\vec{t}a\diamond U)\ast(bV)+(\vec{t}a\diamond U)\ast(\vec{t}b\diamond V).$

On the other hand, by definition of $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ we have

	$\displaystyle S^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$	$\displaystyle=S^{\vec{t}}\big{(}a(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}bv)\big{)}+S^{\vec{t}}\big{(}b(au\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)\big{)}+S^{\vec{t}}\big{(}(1-2\vec{t})a\diamond b(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)\big{)}$
		$\displaystyle\quad+S^{\vec{t}}\big{(}(\vec{t}^{2}-\vec{t})a\diamond b\diamond(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)\big{)}.$

By the definition of $S^{\vec{t}}$ we get further

	$\displaystyle S^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$	$\displaystyle=aS^{\vec{t}}\big{(}u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}bv\big{)}+\vec{t}a\diamond S^{\vec{t}}\big{(}u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}bv\big{)}$
		$\displaystyle\quad+bS^{\vec{t}}\big{(}au\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v\big{)}+\vec{t}b\diamond S^{\vec{t}}\big{(}au\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v\big{)}$
		$\displaystyle\quad+(1-2\vec{t})a\diamond bS^{\vec{t}}\big{(}u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v\big{)}+(1-2\vec{t})\vec{t}a\diamond b\diamond S^{\vec{t}}\big{(}u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v\big{)}$
		$\displaystyle\quad+S^{\vec{t}}\big{(}(\vec{t}^{2}-\vec{t})a\diamond b\diamond(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)\big{)}.$

Simplification by Lemma 1 gives

	$\displaystyle S^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$	$\displaystyle=aS^{\vec{t}}\big{(}u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}bv\big{)}+\vec{t}a\diamond S^{\vec{t}}\big{(}u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}bv\big{)}$
		$\displaystyle\quad+bS^{\vec{t}}\big{(}au\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v\big{)}+\vec{t}b\diamond S^{\vec{t}}\big{(}au\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v\big{)}$
		$\displaystyle\quad+(1-2\vec{t})a\diamond bS^{\vec{t}}\big{(}u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v\big{)}-\vec{t}^{2}a\diamond b\diamond S^{\vec{t}}(u\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v)\big{)}.$

Let $U=S^{\vec{t}}(u)$ and $V=S^{\vec{t}}(v)$ . By the induction hypothesis, we get further

	$\displaystyle S^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$	$\displaystyle=a\big{(}U\ast S^{\vec{t}}(bv)\big{)}+\vec{t}a\diamond\big{(}U\ast S^{\vec{t}}(bv)\big{)}$
		$\displaystyle\quad+bS^{\vec{t}}(au)\ast S^{\vec{t}}(v)+\vec{t}b\diamond S^{\vec{t}}\big{(}au\stackrel{{\scriptstyle\vec{t}}}{{\ast}}v\big{)}$
		$\displaystyle\quad+(1-2\vec{t})(a\diamond b)(U\ast V)-\vec{t}^{2}a\diamond b\diamond S^{\vec{t}}(U\ast V).$

Using again the definition of $S^{\vec{t}}$ , we obtain

	$\displaystyle S^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$	$\displaystyle=a\big{(}U\ast(bV)\big{)}+a\big{(}U\ast(\vec{t}b\diamond V)\big{)}$
		$\displaystyle\quad+\vec{t}a\diamond(U\ast bV)+\vec{t}a\diamond\big{(}U\ast(\vec{t}b\diamond V)\big{)}$
		$\displaystyle\quad+b\big{(}(aU)\ast V\big{)}+b\big{(}(\vec{t}a\diamond U)\ast V\big{)}$
		$\displaystyle\quad+\vec{t}b\diamond\big{(}(aU)\ast V\big{)}+\vec{t}b\diamond\big{(}(\vec{t}a\diamond U)\ast V\big{)}$
		$\displaystyle\quad+(1-2\vec{t})(a\diamond b)(U\ast V)-\vec{t}^{2}(a\diamond b\diamond(U\ast V)\big{)}.$

We use the following identities of [14] (see also [27]):

	$\displaystyle(a\diamond U)\ast(bV)$	$\displaystyle=a\diamond\big{(}U\ast(bV)\big{)}+b\big{(}(a\diamond U)\ast V\big{)}-(a\diamond b)(U\ast V),$
	$\displaystyle(aU)\ast(b\diamond V)$	$\displaystyle=b\diamond\big{(}(aU)\ast V\big{)}+a\big{(}U\ast(b\diamond V)\big{)}-(a\diamond b)(U\ast V),$
	$\displaystyle(a\diamond U)\ast(b\diamond V)$	$\displaystyle=a\diamond\big{(}U\ast(b\diamond V)\big{)}+b\diamond\big{(}(a\diamond U)\ast V\big{)}-(a\diamond b)\diamond(U\ast V).$

They imply that

	$\displaystyle\vec{t}a\diamond\big{(}U\ast(bV)\big{)}+b\big{(}(\vec{t}a\diamond U)\ast V\big{)}=(\vec{t}a\diamond U)\ast(bV)+\vec{t}(a\diamond b)(U\ast V),$
	$\displaystyle\vec{t}b\diamond\big{(}(aU)\ast V\big{)}+a\big{(}U\ast(\vec{t}b\diamond V)\big{)}=(aU)\ast(\vec{t}b\diamond V)+\vec{t}(a\diamond b)(U\ast V),$
	$\displaystyle\vec{t}a\diamond\big{(}U\ast(\vec{t}b\diamond V)\big{)}+\vec{t}b\diamond\big{(}(\vec{t}a\diamond U)\ast V\big{)}$
	$\displaystyle\qquad=(\vec{t}a\diamond U)\ast(\vec{t}b\diamond V)+\vec{t}^{2}(a\diamond b)\diamond(U\ast V).$

Together with the definition

(aU)\ast(bV)=a\big{(}U\ast(bV)\big{)}+b((aU)\ast V)+(a\diamond b)(U\ast V),

we get the desired result

	$\displaystyle S^{\vec{t}}(x\stackrel{{\scriptstyle\vec{t}}}{{\ast}}y)$	$\displaystyle=(\vec{t}a\diamond U)\ast(bV)+(aU)\ast(\vec{t}b\diamond V)$
		$\displaystyle\qquad+(\vec{t}a\diamond U)\ast(\vec{t}b\diamond V)+(aU)\ast(bV),$

which equals $S^{\vec{t}}(x)\ast S^{\vec{t}}(y)$ (25). In the general case, instead of $x=au$ and $y=bv$ we have $x=\vec{t}^{m}au\in\mathfrak{h}^{2}$ with $u=\vec{t}^{m_{2}}a_{2}\dots\vec{t}^{m_{n}}a_{n}$ and $y=\vec{t}^{k}bv\in\mathfrak{h}^{2}$ with $v=\vec{t}^{k_{2}}a_{2}\dots\vec{t}^{k_{r}}b_{r}$ , with $a,b,a_{\ell},b_{j}\in\mathcal{A}$ , $2\leq\ell\leq n$ and $2\leq j\leq r$ . We can proceed very similar so the arguments are omitted. ∎

5. Outlook and summary

5.1. Open problems: properties of multi-interpolation

At the moment, it seems difficult to translate more properties for $\zeta^{t}(\vec{i})$ to $\zeta^{\vec{t}}(\vec{i})$ . Yamamoto established, amongst many other things, the sum property for interpolated multiple zeta values:

\sum_{\begin{subarray}{c}k_{1}\geq 2,k_{i}\geq 1\\ \sum_{\ell=1}^{n}k_{\ell}=k\end{subarray}}\zeta^{t}(k_{1},\dots,k_{n})=\zeta(k)\cdot\sum_{j=0}^{n-1}\binom{k-1}{j}t^{j}(1-t)^{n-1-j},

(26)

with $k>n$ . For $\zeta^{\vec{t}}(\vec{i})$ we expect that only the simplest case $n=2$ can be easily treated in full generality:

\sum_{k_{1}=2}^{k-1}\zeta^{\vec{t}}(k_{1},k-k_{1})=\zeta(k)+(k-2)\zeta(k\vec{t}).

It is certainly desirable to obtain purely algebraic proofs of Theorem 2 and Corollary 1. Moreover, as the new definitions combine both interpolated MZVs as well as multiple $\mathfrak{t}$ -values, it is of interest to look deeper into a common framework for these objects.

5.2. Outlook: multi-interpolation based on ordered partitions

One can also look at different kinds interpolations. Extraction of the coefficient of $t^{i}$ in Yamamoto’s sum formula implies Yamamoto’s refined identity (see also [18, 19]):

\sum_{\begin{subarray}{c}k_{1}\geq 2,k_{i}\geq 1\\ \sum_{\ell=1}^{n}k_{\ell}=k\end{subarray}}\sum_{\begin{subarray}{c}\mathbf{p}_{n}\in\mathcal{P}(n)\\ n-\mathcal{L}(\mathbf{p}_{n})=i\end{subarray}}\zeta\big{(}\mathbf{p}(k_{1},\dots,k_{n})\big{)}=\zeta(k)\cdot\binom{k-n+i-1}{k-n-1}.

(27)

For example,

\sum_{\begin{subarray}{c}k_{1}\geq 2,k_{i}\geq 1\\ \sum_{\ell=1}^{n}k_{\ell}=k\end{subarray}}\Big{(}\zeta\big{(}k_{1}+k_{2},\dots,k_{n})+\dots+\zeta\big{(}k_{1},\dots,k_{n-1}+k_{n})\big{)}=(k-n)\zeta(k).

This motivates a different definition of multi-interpolated MZVs, based on ordered partitions:

\zeta^{(\vec{u})}(k_{1},\dots,k_{n})=\sum_{\mathbf{p}_{n}\in\mathcal{P}(n)}\bigg{(}\prod_{m=1}^{\infty}u_{m-1}^{N_{m}(\mathbf{p}_{n})}\bigg{)}\zeta(\mathbf{p}_{n}(\vec{k})),

(28)

where $N_{m}(.)$ counts the number of parts of size $m$ in an ordered partition. Here we set $u_{0}=1$ . Since

\sum_{m=1}^{\infty}(m-1)\cdot N_{m}(\mathbf{p}_{n})=n-\mathcal{L}(\mathbf{p}_{n})

we reobtain for $u_{m}=t^{m}$ the ordinary interpolated MZVs. For example,

\zeta^{(\vec{u})}(k_{1},k_{2},k_{3})=u_{0}^{3}\zeta(k_{1},k_{2},k_{3})+u_{0}u_{1}\big{(}\zeta(k_{1}+k_{2},k_{3})+\zeta(k_{1},k_{2}+k_{3})\big{)}+u_{2}\zeta(k_{1}+k_{2}+k_{3}).

Note that this new definition (28) of multi-interpolated MZVs can be obtained from our previous Definitions 1 and 2 by application of a formal map $V$ , whose action is given as follows¹¹1Actually, $V$ can be rigorously defined algebraically.:

V\big{(}\zeta(i_{1}\vec{t}^{j_{1}},\dots,i_{k}\vec{t}^{j_{k}})\big{)}=\zeta(i_{1},\dots,i_{k})\cdot\prod_{\ell=1}^{k}u_{j_{\ell}}.

This implies that

\zeta^{(\vec{u})}(i_{1},\dots,i_{k})=V\big{(}\zeta^{\vec{t}}(i_{1},\dots,i_{k})\big{)}.

However, in general

V\big{(}\zeta^{\vec{t}}(i_{1},\dots,i_{k})\cdot\zeta^{\vec{t}}(j_{1},\dots,j_{m})\big{)}\neq V\big{(}\zeta^{\vec{t}}(i_{1},\dots,i_{k})\big{)}\cdot V\big{(}\zeta^{\vec{t}}(j_{1},\dots,j_{m})\big{)}.

It remains to be seen if there exists a suitably defined harmonic product $\ast^{(\vec{u})}$ , or even a more general product.

5.3. Further variations

Another generalization of $\zeta^{t}(k_{1},\dots,k_{n})$ are interpolated Schur multiple zeta values, as introduced in [2] (see also [21] for Schur multiple zeta values). We note that it is possible to introduce multi-interpolated Schur multiple zeta values, unifying interpolated Schur multiple zeta values and multi-interpolated multiple zeta values: the parameters $v(\mathbf{m})$ , counting the vertical equalities, and $h(\mathbf{m})$ , counting the horizontal equalities can be refined by taking into account the values of the equal entries, leading to $\vec{t}^{\vec{v}(\mathbf{m})}(\vec{1}-\vec{t})^{\vec{h}(\mathbf{m})}$ (see [2], Definition 2.3).

Finally, following Ohno and Wayama’s [23] generalization of the Arakawa-Kaneko multiple zeta function [1], we note that a multi-interpolated Arakawa-Kaneko multiple zeta function can be defined as follows:

\xi^{\vec{t}}(k_{1},\dots,k_{r};s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}z^{s-1}\frac{\text{Li}^{\vec{t}}_{k_{1},\dots,k_{r}}(1-e^{-z})}{e^{z}-1}dz,

were $\text{Li}^{\vec{t}}_{k_{1},\dots,k_{r}}(z)$ denotes a multi-interpolated multi-polylogarithm defined by

\text{Li}^{\vec{t}}_{k_{1},\dots,k_{r}}(z)=\sum_{\ell_{1}\geq\cdots\geq\ell_{r}\geq 1}\frac{z^{\ell_{1}}\cdot\vec{t}^{\vec{\sigma}(\vec{\ell})}}{\ell_{1}^{k_{1}}\dots\ell_{r}^{k_{r}}}.

By setting $\vec{t}=t$ we reobtain the $t$ -Arakawa–Kaneko multiple zeta functions of [23].

It remains to be seen if multi-interpolated objects, like multi-interpolated Schur multiple zeta values or multi-interpolated Arakawa-Kaneko multiple zeta functions, still have interesting structural properties.

5.4. Summary

In this note we introduced a multi-interpolated multiple zeta value $\zeta^{\vec{t}}(\vec{i})$ with variables $\vec{t}=(t_{1},t_{2},\dots)$ , generalized the ordinary interpolated multiple zeta value $\zeta^{t}(\vec{i})$ , case $t_{\ell}=t$ , $1\leq\ell$ . A few properties of $\zeta^{\vec{t}}(\vec{i})$ where established in this note, in particular, formulas for $\zeta^{\vec{t}}(\{s\}_{k})$ , as well as a harmonic product $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ such that $\zeta^{\vec{t}}((i_{1},\dots,i_{m})\stackrel{{\scriptstyle\vec{t}}}{{\ast}}(j_{1},\dots,j_{k}))=\zeta^{\vec{t}}(i_{1},\dots,i_{m})\cdot\zeta^{\vec{t}}(j_{1},\dots,j_{k})$ . We proposed several open problems and pointed out different variants of multi-interpolated MZVs.

References

[1] T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153, 189–209, 1999.
[2] H. Bachmann, Interpolated Schur multiple zeta values. J. Aust. Math. Soc. 104, 289–307, 2018.
[3] J. Borwein, D. Bradley and D. Broadhoarst, Evaluations of $k$ -fold Euler/Zagier sums: a compendium of results for arbitrary $k$ , Electronic journal of combinatorics. vol. 4 (2), R5, 21 pp., 1997.
[4] C. B. Collins, The role of Bell polynomials in integration. Journal of Computational and Applied Mathematics 131, 195–222, 2001.
[5] L. Comtet, Advanced Combinatorics. D. Reidel, Dordrecht, 1974 (Translation of Analyse Combinatoire, Tomes I et II, Presses Universitaires de France, Paris, 1970).
[6] P. Flajolet and R. Sedgewick, Analytic Combinatorics. Cambridge Univ. Press, Cambridge, UK, 2009.
[7] M. E. Hoffman, Multiple harmonic series. Pacific J. Math. 152, 275-290, 1992.
[8] M. E. Hoffman, Quasi-shuffle products. Journal of Algebraic Combinatorics 11, 49–68, 2000.
[9] M. E. Hoffman, Harmonic-number summation identities, symmetric functions, and multiple zeta values. Ramanujan J. 42, 501–526, 2017.
[10] M. E. Hoffman, An odd variant of multiple zeta values. Communications in Number Theory and Physics 13, 529–567, 2019.
[11] M. E. Hoffman, Quasi-shuffle algebras and applications, Algebraic Combinatorics, Resurgence, Moulds and Applications, Vol. 2 (IRMA Lectures in Mathematics and Theoretical Physics vol. 32), F. Chapoton et. al. (eds.), European Math. Soc. Publ. House, Berlin, 327–348, 2020.
[12] M. E. Hoffman and K. Ihara, Quasi-shuffle products revisited. Journal of Algebra 481, 293–326, 2017.
[13] M. E. Hoffman and I. Mező, Zeros of the digamma function and its Barnes G-function analogue. Integral Transforms and Special Functions 28, 846–858, 2017.
[14] K. Ihara, J. Kajikawa, Y. Ohno and J. Okuda, Multiple zeta values vs. multiple zeta-star values. J. Alg. 332, 187–208, 2011.
[15] V.F. Ivanov, Problem. Amer. Math. Monthly, 65, 212, 1958.
[16] M. Kuba, On multisets, interpolated multiple zeta values and limit laws. Submitted.
[17] M. Kuba and A. Panholzer, A Note on Harmonic number identities, Stirling series and multiple zeta values. International journal of number theory, Vol. 15, No. 07, 1323–1348, 2019.
[18] Z. Li, Algebraic relations of interpolated multiple zeta values. Submitted.
[19] Z. Li and C. Qin, Some relations of interpolated multiple zeta values. Internat. J. Math. 28 (5), 25 pp., 2017.
[20] I. G. MacDonald, Symmetric Functions and Hall Polynomials. 2nd ed., Clarendon Press, Oxford, 1995.
[21] M. Nakasuji, O. Phuksuwan and Y. Yamasaki, On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions. Advances in Mathematics 333, 570–619, 2018,
[22] N. Nielsen, Handbuch der Theorie der Gammafunktion, B. G. Teubner, Leipzig, 1906; reprinted in Die Gammafunktion, Chelsea, New York, 1965.
[23] Y. Ohno and H. Wayama, Interpolation between Arakawa–Kaneko and Kaneko–Tsumura Multiple Zeta Functions. Advanced Studies in Pure Mathematics 84, 361–366, 2020.
[24] T. Tanaka and N. Wakabayashi, Kawashima’s relations for interpolated multiple zeta values. J. Algebra 447, 424–431, 2016.
[25] C. Vignat and T. Wakhare, Multiple zeta values for classical special functions. Submitted.
[26] N. Wakabayashi, Double shuffle and Hoffman’s relations for interpolated multiple zeta values. Int. J. Number Theory 13 (9), 2245–2251, 2017.
[27] S. Yamamoto, Interpolation of multiple zeta and zeta-star values. J. Algebra 385, 102–114, 2013.
[28] D. Zagier, Values of zeta functions and their applications. First European Congress of Mathematics (Paris, 1992), vol. II (A. Joseph et al., eds.), (Progr. Math., vol. 120) Birkhäuser, Boston 1994, pp. 497–512.
[29] W. Zudilin, Algebraic relations for multiple zeta values. Russian Math. Surveys 58:1 1–29, 2003.

On multi-interpolated multiple zeta values

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

2. Multi-interpolated multiple zeta values and related multiple zeta values

Definition 1 (Multi-interpolated multiple zeta values).

Remark 1.

Example 1 (Multi-interpolated MZVs: depth one).

Example 2 (Multi-interpolated MZVs: depth two).

Definition 2.

Remark 2 (Connection to multiple 𝔱\mathfrak{t}-values).

Example 3 (Multi-interpolated MZVs: depth three).

Example 4 (Even-odd interpolated multiple zeta values).

2.1. Ordered partitions and multi-interpolated MZVs

Theorem 1.

Proof.

3. Multi-interpolated MZVs with repeated arguments

Theorem 2.

Corollary 1.

Remark 3.

Proof of Theorem 2.

Proof of Corollary 1.

4. A product for multi-interpolated zeta values

Remark 4.

Example 5.

Definition 3 (Product ∗t→\stackrel{{\scriptstyle\vec{t}}}{{\ast}}).

Remark 5.

Remark 6.

Example 6.

Example 7.

Definition 4 (Multi-interpolation operator).

Remark 7.

Example 8.

Lemma 1 (Properties - Multi-interpolation operator).

Remark 8.

Proof of Lemma 1.

Definition 5 (Derivative for symbol t→\vec{t}).

Lemma 2.

Proof.

Theorem 3 (Alternating sum formula).

Proof.

Theorem 4.

Remark 9.

Example 9.

Example 10.

Proof.

5. Outlook and summary

5.1. Open problems: properties of multi-interpolation

5.2. Outlook: multi-interpolation based on ordered partitions

5.3. Further variations

5.4. Summary

References

Remark 2 (Connection to multiple $\mathfrak{t}$ -values).

Definition 3 (Product $\stackrel{{\scriptstyle\vec{t}}}{{\ast}}$ ).

Definition 5 (Derivative for symbol $\vec{t}$ ).