On variants of symmetric multiple zeta-star values and the cyclic sum formula

Abstract

The t-adic symmetric multiple zeta values were defined by Jarossay, which have been studied as a real analogue of the ${\varvec{p}}$-adic finite multiple zeta values. In this paper, we consider the star analogues based on several regularization processes of multiple zeta-star values: harmonic regularization, shuffle regularization, and Kaneko–Yamamoto’s type regularization. We also present the cyclic sum formula for t-adic symmetric multiple zeta(-star) values, which is the counterpart of that for ${\varvec{p}}$-adic finite multiple zeta(-star) values obtained by Kawasaki. The proof uses our new relationship that connects the cyclic sum formula for t-adic symmetric multiple zeta-star values and that for the multiple zeta-star values.

1 Introduction

1.1 Multiple zeta(-star) values and their variants

For positive integers $k_1,\ldots ,k_r$ with $k_r \ge 2$, the multiple zeta values (MZVs) and the multiple zeta-star values (MZSVs) are the real numbers defined by

$$\begin{aligned} \zeta (k_1,\ldots , k_r) {:}{=}\sum _{1\le n_1<\cdots <n_r} \frac{1}{n_1^{k_1}\cdots n_r^{k_r}}, \quad \zeta ^\star (k_1,\ldots , k_r){:}{=}\sum _{1\le n_1\le \cdots \le n_r} \frac{1}{n_1^{k_1}\cdots n_r^{k_r}}. \end{aligned}$$

In the following, we call the tuple $(k_1, \ldots , k_r)$ of positive integers an index. We denote the index with $r=0$ by $\varnothing$ and we call it the empty index, and we understand $\zeta (\varnothing )=\zeta ^{\star }(\varnothing )=1$. We also call an index ${\varvec{k}}=(k_1, \ldots , k_r)$ admissible if ${\varvec{k}}=\varnothing$ or $k_r\ge 2$. For an index ${\varvec{k}}=(k_1, \ldots , k_r)$, the quantity $k_1+\cdots +k_r$ is called the weight of ${\varvec{k}}$ and we denote it by ${{\,\mathrm{wt}\,}}({\varvec{k}})$ (we also set ${{\,\mathrm{wt}\,}}(\varnothing ){:}{=}0$).

It is known that there are many variants of MZ(S)Vs. Kaneko and Zagier introduced their two variants, i.e., the finite multiple zeta values $\zeta _{{\mathcal {A}}}({\varvec{k}})$ (FMZVs) and the symmetric multiple zeta values $\zeta _{{\mathcal {S}}}({\varvec{k}})$ (SMZVs). Here, $\zeta _{{\mathcal {A}}}({\varvec{k}})$ is defined as an element in the ring ${\mathcal {A}}{:}{=}(\prod _{p} {\mathbb {Z}}/p{\mathbb {Z}})/(\bigoplus _p {\mathbb {Z}}/p{\mathbb {Z}})$ with p running all the rational primes. On the other hand, $\zeta _{{\mathcal {S}}}({\varvec{k}})$ is defined as an element in the ring ${\mathcal {Z}}/\zeta (2){\mathcal {Z}}$. Here, ${\mathcal {Z}}$ is the ${\mathbb {Q}}$-linear subspace of ${\mathbb {R}}$ spanned by 1 and all multiple zeta values. Note that ${\mathcal {Z}}$ becomes a ${\mathbb {Q}}$-algebra. Kaneko and Zagier conjectured that these two types of finite multiple zeta values satisfy the same algebraic relations over ${\mathbb {Q}}$ (see [5] and [7] for more details on FMZVs and SMZVs).

Recently, Seki [15] introduced the ${\varvec{p}}$-adic finite multiple zeta values $\zeta _{{\widehat{{\mathcal {A}}}}}({\varvec{k}})$ (${\varvec{p}}$-adic FMZVs) as an element in the ring ${\widehat{{\mathcal {A}}}} {:}{=}\varprojlim _n (\prod _{p} {\mathbb {Z}}/p^n{\mathbb {Z}})/(\bigoplus _p {\mathbb {Z}}/p^n{\mathbb {Z}})$. The ${\varvec{p}}$-adic FMZV is a lifting of $\zeta _{{\mathcal {A}}}({\varvec{k}})$, i.e., $\zeta _{{\widehat{{\mathcal {A}}}}}({\varvec{k}})=\zeta _{{\mathcal {A}}}({\varvec{k}}) \bmod {{\varvec{p}}}$ via ${\mathcal {A}}\cong {\widehat{{\mathcal {A}}}}/{\varvec{p}}{\widehat{{\mathcal {A}}}}$, where ${\varvec{p}}{:}{=}((p \bmod {p^n})_p)_n \in {\widehat{{\mathcal {A}}}}$ (see also [14] and [15] for more details on ${\varvec{p}}$-adic FMZVs). In the view point of Kaneko–Zagier conjecture, it is natural to expect that there is also a lifting of $\zeta _{{\mathcal {S}}}({\varvec{k}})$ corresponding to $\zeta _{{\widehat{{\mathcal {A}}}}}({\varvec{k}})$.

1.2 t-Adic symmetric multiple zeta(-star) values

For an index $(k_1,\ldots ,k_r)$, the t-adic symmetric multiple zeta values (t-adic SMZVs) are defined as elements in ${\mathcal {Z}}[[t]]$ by

which were introduced by Jarossay [4] as a counterpart of ${\varvec{p}}$-adic FMZVs. Here, the symbol $\zeta ^\bullet$ on the right-hand side means the regularized values coming from the harmonic ‘$*$’ or the shuffle ‘’ regularizations, i.e., real values obtained by taking constant terms of these regularizations as explained in [3]. Remark that t-adic SMZV is called $\Lambda$-adjoint multiple zeta values in [4] and Jarossay conjectured that $\zeta _{{\widehat{{\mathcal {A}}}}}({\varvec{k}})$ and $\zeta _{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$ satisfy the same relations [4, Conjecture 5.3.2].

In this paper, we consider 3 types of t-adic symmetric multiple zeta-star values (t-adic SMZSVs) as elements in ${\mathcal {Z}}[[t]]$ motivated by the previous works on MZSVs by Muneta [10], Yamamoto [17], and Kaneko–Yamamoto [6]. For an index $(k_{1},\ldots ,k_{r})$, let

Definition 1.1

(t-adic SMZSVs) For an index $(k_{1},\ldots ,k_{r})$, let

where the symbol $\zeta ^{\star ,KY}$ means the shuffle regularized values obtained by Yamamoto’s integral representation of MZSVs (for precise definition, see Sect. 3.3).

Remark 1.2

With a simple calculation, we have

$$\begin{aligned} \zeta ^{\star , \bullet }_{{\widehat{{\mathcal {S}}}}}(k_1, \ldots , k_r)=\sum _{ \begin{array}{c} \square \ \mathrm{{is \ either \ a \ comma \ `,' }} \\ \mathrm{{or \ a \ plus \ `+'}} \end{array} } \zeta _{{\widehat{{\mathcal {S}}}}}^\bullet (k_1\square \cdots \square k_r) \end{aligned}$$

(1)

for .

2 Main results

2.1 Relations among various zeta-star values

The first result we introduce in this paper is the congruence between and $\zeta ^{\star , KY}({\varvec{k}})$.

Theorem 2.1

For an index ${\varvec{k}}$, we have

By using this theorem, we see that 3 types of the t-adic SMZSVs are equivalent in $({\mathcal {Z}}/\zeta (2){\mathcal {Z}})[[t]]$.

Theorem 2.2

For an index ${\varvec{k}}$, we have

in $({\mathcal {Z}}/\zeta (2){\mathcal {Z}})[[t]]$.

In the sequel, thanks to Theorem 2.2, we denote their $\bmod {\;\zeta (2){\mathcal {Z}}[[t]]}$ reduction by $\zeta ^{\star }_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$.

2.2 Cyclic sum formulas

Our second result is the cyclic sum formula for $\zeta ^{\star , KY}_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$. The original formulas for MZVs and MZSVs were, respectively, obtained by Hoffman and Ohno [2], and Ohno and Wakabayashi. For a positive integer k and a non-negative integer s, we denote $\underbrace{k, \ldots , k}_{s}$ by $\{k\}^s$.

Theorem 2.3

(Cyclic sum formula; Ohno–Wakabayashi [11]) Let k be a positive integer. For a non-empty index ${\varvec{k}}=(k_1,\ldots ,k_r)$ with ${{\,\mathrm{wt}\,}}({\varvec{k}})=k$, we have

$$\begin{aligned}&\sum _{i=1}^r\sum _{j=0}^{k_i-2} \zeta ^\star (j+1, k_{i+1},\ldots , k_r, k_1,\ldots , k_{i-1}, k_i-j) \\&\quad =k\zeta (k+1)-\delta _{{\varvec{k}}, (\{1\}^r)}\cdot k\zeta (k+1). \end{aligned}$$

Here, $\delta _{{\varvec{k}}, (\{1\}^r)}$ is 1 if ${\varvec{k}}=(\{1\}^r)$ and 0 if ${\varvec{k}}\ne (\{1\}^r)$.

Kawasaki proved the analogous formula for ${\varvec{p}}$-adic FMZ(S)Vs in [8]. Here we introduce their counterpart for t-adic SMZ(S)Vs.

Theorem 2.4

Let k be a positive integer. For a non-empty index ${\varvec{k}}=(k_1,\ldots ,k_r)$ with ${{\,\mathrm{wt}\,}}({\varvec{k}})=k$, we have

$$\begin{aligned}&\sum _{i=1}^r \sum _{j=0}^{k_i-2} \zeta ^{\star ,KY}_{{\widehat{{\mathcal {S}}}}} (j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i-j) \\&\quad =\sum _{i=1}^r \sum _{j=0}^\infty \zeta ^{\star ,KY}_{{\widehat{{\mathcal {S}}}}} (j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_i)t^j\\&\qquad \quad +k\zeta ^{\star ,KY}_{{\widehat{{\mathcal {S}}}}}(k+1)-\delta _{{\varvec{k}}, (\{1\}^r)}\cdot (1+(-1)^{k+1})k\zeta (k+1). \end{aligned}$$

Remark 2.5

We shall give the cyclic sum formula for $\zeta _{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$ in Sect. 6. See Theorem 6.1.

Remark 2.6

Sato and the first-named author obtained the formula for refined SMZVs, which is another generalization of the SMZVs.

Let ${\mathfrak {H}}{:}{=}{\mathbb {Q}}\langle x, y \rangle$ be the non-commutative polynomial ring over ${\mathbb {Q}}$ with two variables x and y, and ${\mathfrak {H}}^1{:}{=}{\mathbb {Q}}+y {\mathfrak {H}}\supset {\mathfrak {H}}^0{:}{=}{\mathbb {Q}}+y{\mathfrak {H}}x$ be the ${\mathbb {Q}}$-subalgebras of ${\mathfrak {H}}$. For a positive integer k and an index ${\varvec{k}}$ with ${{\,\mathrm{wt}\,}}({\varvec{k}})=k$, let $w^{\star }({\varvec{k}})$ be the element in ${\mathfrak {H}}^1$ corresponding to Kaneko–Yamamoto’s integral representation of $\zeta ^{\star , KY}({\varvec{k}})$ and let $w^{\star }_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$ be the element in ${\mathfrak {H}}^1[[t]]$ corresponding to that of $\zeta ^{\star , KY}_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$ (for the precise definitions, see Sect. 3.3). Set

$$\begin{aligned} w^{\star }_{\mathrm {CSF}}({\varvec{k}})&{:}{=}\sum _{i=1}^r\sum _{j=0}^{k_i-2}w^{\star }(j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i-j)-kw^{\star }(k+1) \in {\mathfrak {H}}^0, \\ w^{\star }_{\mathrm {CSF}, {\widehat{{\mathcal {S}}}}}({\varvec{k}})&{:}{=}\sum _{i=1}^r\sum _{j=0}^{k_i-2}w^{\star }_{{\widehat{{\mathcal {S}}}}}(j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i-j) \\&\quad -\sum _{i=1}^r\sum _{j=0}^\infty w^{\star }_{{\widehat{{\mathcal {S}}}}}(j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_i)t^j -kw^{\star }_{{\widehat{{\mathcal {S}}}}}(k+1) \in {\mathfrak {H}}^1[[t]]. \end{aligned}$$

Theorem 2.7

Let ${\varvec{k}}=(k_1, \ldots , k_r)$ be an index with ${{\,\mathrm{wt}\,}}({\varvec{k}})=k$. Then we have the following equality in ${\mathfrak {H}}^1[[t]]$:

$$\begin{aligned} w^{\star }_{\mathrm {CSF}, {\widehat{{\mathcal {S}}}}}({\varvec{k}})&=w^\star _{\mathrm {CSF}}({\varvec{k}}) +(-1)^{k+1}\\&\quad \times \sum _{{\varvec{l}}=(l_1, \ldots , l_r) \in {\mathbb {Z}}^r_{\ge 0}} \prod _{j=1}^r\left( {\begin{array}{c}k_j+l_j-1\\ l_j\end{array}}\right) w^{\star }_\mathrm {CSF}(\overline{{\varvec{k}}+{\varvec{l}}}) t^{l_1+\cdots +l_r}. \end{aligned}$$

Here, we set $\overline{{\varvec{k}}+{\varvec{l}}}{:}{=}(k_r+l_r, \ldots , k_1+l_1)$.

Remark 2.8

By Theorem 2.7, we can find Theorem 2.4 easily (see Sect. 5).

The contents of this paper are as follows. In the next section, we introduce the algebraic setup of MZVs and MZSVs, and give the precise definitions of Muneta’s and Kaneko–Yamamoto’s regularized MZSVs. In Sect. 4, we prove the equivalence of the definitions of t-adic SMZSVs. In Sect. 5, we give the proofs of Theorems 2.4 and 2.7. In the final section, we prove the cyclic sum formula for the t-adic SMZV $\zeta _{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$.

3 Preliminaries for the proofs

3.1 Algebraic setup of MZVs

We introduce the algebraic setup of MZVs and MZSVs along with [1]. Let $z_{k}{:}{=}yx^{k-1}$. Note that ${\mathfrak {H}}^1={\mathbb {Q}}\langle z_k \mid k\ge 1 \rangle$. We define the ${\mathbb {Q}}$-linear map $Z :{\mathfrak {H}}^0 \rightarrow {\mathbb {R}}$ by $Z(1){:}{=}1$ and $Z(z_{k_1} \cdots z_{k_r}){:}{=}\zeta (k_1, \ldots , k_r)$ for an admissible index $(k_1, \ldots , k_r)$.

We define the harmonic product $*:{\mathfrak {H}}^1 \times {\mathfrak {H}}^1 \rightarrow {\mathfrak {H}}^1$ and the shuffle product inductively by the following rules:

• (i) $w*1=1*w=w$ for any $w \in {\mathfrak {H}}^1$.

• (ii) $w_1z_k *w_2z_l =(w_1 *w_2z_l)z_k+(w_1z_k *w_2)z_l+(w_1 *w_2)z_{k+l}$ for any $w_1, w_2 \in {\mathfrak {H}}^1$ and $k, l \in {\mathbb {Z}}_{\ge 1}$,

• (i’) .

• (ii’) for any $w_1, w_2 \in {\mathfrak {H}}$ and $u_1, u_2 \in \{x, y\}$.

It is known that the map Z preserves the harmonic product $*$ and the shuffle product . That is, we have

for $w, w' \in {\mathfrak {H}}^0$ (see [1]).

3.2 Regularization of MZVs and MZSVs

Along with [3] and [10], we introduce the harmonic regularized MZV $\zeta ^{*}({\varvec{k}})$ and MZSV $\zeta ^{\star , *}({\varvec{k}})$, also the shuffle regularized MZV and MZSV . For , we denote the ${\mathbb {Q}}$-algebra $({\mathfrak {H}}^1, \bullet )$ and its ${\mathbb {Q}}$-subalgebra $({\mathfrak {H}}^0, \bullet )$ by ${\mathfrak {H}}^1_{\bullet }$ and ${\mathfrak {H}}^0_{\bullet }$, respectively. It is known that ${\mathfrak {H}}^1_{\bullet } \cong {\mathfrak {H}}^0_{\bullet }[y]$ as ${\mathbb {Q}}$-algebra for (See [1] for $\bullet =*$ and [13] for ). Thus, there exists a unique ${\mathbb {Q}}$-linear map $Z^{\bullet } :{\mathfrak {H}}^1_{\bullet } \rightarrow {\mathbb {R}}[T]$ satisfying that $Z^{\bullet }(y)=T$, $Z^{\bullet }|_{{\mathfrak {H}}^0_{\bullet }}=Z$ and $Z^{\bullet }$ preserves the product $\bullet$ on ${\mathfrak {H}}^1$. We denote the image of $Z^{\bullet }$ at $w \in {\mathfrak {H}}^1$ by $Z^{\bullet }(w; T)$. It is known that $Z^{\bullet }(w; T) \in {\mathcal {Z}}[T]$.

For an index ${\varvec{k}}=(k_1, \ldots , k_r)$, we define the harmonic (resp. shuffle) regularized MZV $\zeta ^{*}({\varvec{k}})$ (resp. ) by

Here we set $z_{{\varvec{k}}}{:}{=}z_{k_1}\cdots z_{k_r}$ for an index ${\varvec{k}}=(k_1, \ldots , k_r)$. Let $\sigma :{\mathfrak {H}}\rightarrow {\mathfrak {H}}$ be the ${\mathbb {Q}}$-automorphism defined by $\sigma (x){:}{=}x$ and $\sigma (y){:}{=}x+y$, and $S :{\mathfrak {H}}^1 \rightarrow {\mathfrak {H}}^1$ the ${\mathbb {Q}}$-linear map defined by $S(1){:}{=}1$ and $S(yw){:}{=}y\sigma (w)$ for $w \in {\mathfrak {H}}$. For an index ${\varvec{k}}=(k_1, \ldots , k_r)$, Muneta’s harmonic (resp. shuffle) regularized MZSV $\zeta ^{\star , *}({\varvec{k}})$ (resp. ) is defined by

3.3 Kaneko–Yamamoto’s shuffle regularization of MZSVs

We introduce Kaneko–Yamamoto’s shuffle regularized MZSV $\zeta ^{\star , KY}({\varvec{k}})$ along with [6]. Muneta’s shuffle regularized MZSV is the shuffle regularized values of the sum of MZVs which are obtained by the usual decomposition of MZSV based on series expression. On the other hand, Kaneko–Yamamoto’s shuffle regularized MZSV is coming from the completely different integral expression found by Yamamoto [17]. Yamamoto’s integral expression of MZSV is defined as the certain multiple integral on 2-posets. So, first we recall the definition of 2-posets and the integrals on them.

A 2-poset is a pair $(X, \delta _X)$ consisting of a finite partially ordered set $X=(X, \le )$ and a map $\delta _X : X \rightarrow \{x, y\} \subset {\mathfrak {H}}$ which is called a label map of X. A 2-poset $(X, \delta _X)$ is admissible if $\delta _X(p)=x$ for all maximal elements p in X and $\delta _X(q)=y$ for all minimal elements q in X.

For an admissible 2-poset $(X, \delta _X)$, let

$$\begin{aligned} I(X){:}{=}\int _{\Delta _X}\prod _{p \in X}\omega _{\delta _X(p)}(t_p), \end{aligned}$$

where

$$\begin{aligned} \Delta _X=\{(t_p)_p \in [0, 1]^X \mid t_p<t_q \text { if } p<q \,(p,q\in X)\} \end{aligned}$$

and

$$\begin{aligned} \omega _x(t){:}{=}\frac{dt}{t}, \quad \omega _y(t){:}{=}\frac{dt}{1-t}. \end{aligned}$$

Note that the integral I(X) converges if and only if the 2-poset X is admissible.

We use Hasse diagrams to indicate 2-posets, with vertices $\circ$ and $\bullet$ corresponding to $\delta _X(p)=x$ and y, respectively.

It is known that MZV and MZSV can be written as the integral on a certain 2-poset. Indeed, for an index ${\varvec{k}}=(k_1, \ldots , k_r)$, set

(2)

Then, Yamamoto proved

$$\begin{aligned} \zeta ^\star ({\varvec{k}})=I(X^\star ({\varvec{k}})) \end{aligned}$$

for an admissible index ${\varvec{k}}$ [17, Corollary 1.3]. Note that for the empty 2-poset denoted by $\varnothing$, we set $I(\varnothing ){:}{=}1$.

We recall the algebraic setup of 2-posets. Let ${\mathfrak {P}}$ be the ${\mathbb {Q}}$-algebra generated by the isomorphism classes of 2-posets whose multiplication is given by the disjoint union of 2-posets and ${\mathfrak {P}}^0$ be the ${\mathbb {Q}}$-subalgebra of ${\mathfrak {P}}$ generated by the isomorphism classes of admissible 2-posets. Then the integral I(X) is regarded as the ${\mathbb {Q}}$-algebra map $I :{\mathfrak {P}}^0 \rightarrow {\mathbb {R}}$. Moreover, for a 2-poset $(X, \delta _X)$, set

$$\begin{aligned} W(X){:}{=}\sum _{\begin{array}{c} f :X \rightarrow \{1, 2, \ldots , \#X\} \\ \text {order-preserving bijections} \end{array}} u_1 \cdots u_{\#X}, \end{aligned}$$

where $u_i{:}{=}\delta _X(f^{-1}(i))$. Then, W is the unique ${\mathbb {Q}}$-algebra homomorphism satisfying the following properties:

• (W1) If the 2-poset $X=\{p_1<\cdots <p_k\}$ is totally ordered, the identity $W(X)=\delta _X(p_1) \cdots \delta _X(p_k)$ holds.

• (W2) If a and b are non-comparable elements of a 2-poset X, the identity $W(X)=W(X^a_b)+W(X^b_a)$ holds. Here, $X^a_b$ (resp. $X^b_a$) denotes the 2-poset which is obtained from X by adjoining the relation $a>b$ (resp. $a<b$) (see [17, Definition 2.2]).

Then we have $W({\mathfrak {P}}^0)={\mathfrak {H}}^0$ and $I=Z\circ W :{\mathfrak {P}}^0 \rightarrow {\mathbb {R}}$. Moreover, we have for 2-posets X and Y. Here, $X\sqcup Y$ denotes the disjoint union of the 2-posets X and Y.

For an index ${\varvec{k}}$, set $w^{\star }({\varvec{k}}){:}{=}W(X^{\star }({\varvec{k}}))$. We define Kaneko–Yamamoto’s shuffle regularized MZSVs $\zeta ^{\star , KY}({\varvec{k}})$ by

Note that, $\zeta ^{\star ,KY}({\varvec{k}})$ is different from and $\zeta ^{\star ,*}({\varvec{k}})$ in general. For example, we have

Moreover, for an index ${\varvec{k}}=(k_1, \ldots , k_r)$, set

$$\begin{aligned} X^{\star }_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})&{:}{=}\sum _{i=0}^{r}(-1)^{k_{i+1}+\cdots +k_r}X^{\star }(k_1, \ldots , k_i)\\&\qquad \sqcup \sum _{l_{i+1}, \ldots , l_r \ge 0}\prod _{j=i+1}^r \left( {\begin{array}{c}k_j+l_j-1\\ l_j\end{array}}\right) \\&\qquad \times X^{\star }(k_r+l_r, \ldots , k_{i+1}+l_{i+1})t^{l_{i+1}+\cdots +l_r} \in {\mathfrak {P}}[[t]]. \end{aligned}$$

We extend W and to the natural ${\mathbb {Q}}$-algebra homomorphisms and , respectively. Then, if we set $w^{\star }_{{\widehat{{\mathcal {S}}}}}({\varvec{k}}){:}{=}W\bigl (X^{\star }_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})\bigr )$ for an index ${\varvec{k}}$, we have

4 Proofs of Theorems 2.1 and 2.2

In this section, we prove Theorems 2.1 and 2.2 .

Proof of Theorem 2.1

Let $\rho , \rho ^{\star } :{\mathbb {R}}[T] \rightarrow {\mathbb {R}}[T]$ be the ${\mathbb {R}}$-linear maps defined by the equalities

$$\begin{aligned} \rho (e^{Tx})=A(x)e^{Tx}, \quad \rho ^{\star }(e^{Tx})=A(-x)^{-1}e^{Tx} \end{aligned}$$

(3)

in ${\mathbb {R}}[T][[x]]$ on which $\rho$ and $\rho ^{\star }$ act coefficientwise, where

$$\begin{aligned} A(x){:}{=}\Gamma (1+x)e^{\gamma x}=\exp \left( \sum _{n=2}^\infty \frac{(-1)^n}{n}\zeta (n)x^n\right) \in {\mathbb {R}}[[x]] \end{aligned}$$

and $\gamma$ is Euler’s constant. Then, it is known that

for an index ${\varvec{k}}$ [3, Theorem 1] and we have

(4)

Moreover, it is known that

(5)

for an index ${\varvec{k}}$ [6, Corollary 4.7]. Thus, since $\rho$ is invertible, by (4) and (5), we have

(6)

On the other hand, since

$$\begin{aligned} \rho ^{\star }\bigl (\rho ^{-1}(e^{Tx})\bigr )=A(x)^{-1}A(-x)^{-1}e^{Tx}=\Gamma (1+x)^{-1}\Gamma (1-x)^{-1}e^{Tx}=\frac{\sin \pi x}{\pi x}e^{Tx} \end{aligned}$$

and

$$\begin{aligned} \frac{\sin \pi x}{\pi x}=\sum _{n=0}^{\infty }(-1)^n\frac{\pi ^{2n}}{(2n+1)!}x^{2n}, \end{aligned}$$

we have

$$\begin{aligned} \rho ^{\star }\bigl (\rho ^{-1}(T^n)\bigr )\equiv T^n \pmod {\zeta (2){\mathcal {Z}}} \end{aligned}$$

(7)

for any non-negative integer n. Therefore, by (6) and (7), we have

for any index ${\varvec{k}}$. This completes the proof of Theorem 2.1 by substituting $T=0$. $\square$

Proof of Theorem 2.2

The former congruence follows from (1) and the fact for an index ${\varvec{k}}$ [12, Proposition 2.1]. The latter congruence follows from Theorem 2.1. $\square$

5 Cyclic sum formula for t-adic SMZSVs

In this section, we give the proofs of Theorems 2.4 and 2.7 by using the theory of Yamamoto integral.

5.1 Preliminary

In this subsection, we state a proposition which leads to Theorem 2.2. For an index ${\varvec{k}}$ with ${{\,\mathrm{wt}\,}}({\varvec{k}})=k$, set

$$\begin{aligned} {\widetilde{w}}^{\star }_{\mathrm {CSF}, {\widehat{{\mathcal {S}}}}}({\varvec{k}})&{:}{=}w^{\star }_{\mathrm {CSF}, {\widehat{{\mathcal {S}}}}}({\varvec{k}})+kw^{\star }_{{\widehat{{\mathcal {S}}}}}(k+1),\\ {\widetilde{w}}^{\star }_{\mathrm {CSF}}({\varvec{k}})&{:}{=}w^{\star }_{\mathrm {CSF}}({\varvec{k}})+kw^{\star }(k+1). \end{aligned}$$

Proposition 5.1

Let ${\varvec{k}}=(k_1, \ldots , k_r)$ be an index with ${{\,\mathrm{wt}\,}}({\varvec{k}})=k$. Then we have

$$\begin{aligned} {\widetilde{w}}^{\star }_{\mathrm {CSF}, {\widehat{{\mathcal {S}}}}}({\varvec{k}})&={\widetilde{w}}^\star _{\mathrm {CSF}}({\varvec{k}}) +(-1)^{k+1}\\&\quad \times \sum _{{\varvec{l}}=(l_1, \ldots , l_r)\in {\mathbb {Z}}^r_{\ge 0}}\prod _{j=1}^r\left( {\begin{array}{c}k_j+l_j-1\\ l_j\end{array}}\right) {\widetilde{w}}^{\star }_\mathrm {CSF}(\overline{{\varvec{k}}+{\varvec{l}}})t^{l_1+\cdots +l_r}. \end{aligned}$$

For the proof of Proposition 5.1, we introduce the cyclic equivalence classes of indices. For positive integers k and r with $r \le k$, set

$$\begin{aligned} I(k, r){:}{=}\{(k_1, \ldots , k_r) \in {\mathbb {Z}}^r_{\ge 1} \mid {{\,\mathrm{wt}\,}}({\varvec{k}})=k\}. \end{aligned}$$

We say two elements of I(k, r) are cyclically equivalent if they are cyclic permutations of each other. That is, if we denote the cyclic permutation $(1 \cdots r)$ of length r by $\tau$, and $(k_1, \ldots , k_r)$ and $(k'_1, \ldots , k'_r)$ are elements in I(k, r), we denote $(k_1, \ldots , k_r) \equiv (k'_1, \ldots , k'_r)$ if there exists $j \in \{1, \ldots , r\}$ such that $k'_i=k_{\tau ^j(i)}$ for all $1\le i \le r$. Let $\Pi (k, r)$ be a set of cyclic equivalence classes of I(k, r). For any $\alpha \in \Pi (k, r)$, set

$$\begin{aligned} {\widetilde{w}}^{\star }_{\mathrm {CSF}, {\widehat{{\mathcal {S}}}}}(\alpha )&{:}{=}\sum _{(k_1, \ldots , k_r) \in \alpha } \sum _{j=0}^{k_r-2} w^{\star }_{{\widehat{{\mathcal {S}}}}}(j+1, k_1, \ldots , k_{r-1}, k_r-j) \\&\quad -\sum _{(k_1, \ldots , k_r) \in \alpha } \sum _{j=0}^{\infty } w^{\star }_{{\widehat{{\mathcal {S}}}}}(j+1, k_1, \ldots , k_r)t^j, \\ {\widetilde{w}}^{\star }_{\mathrm {CSF}}(\alpha )&{:}{=}\sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{j=0}^{k_r-2}w^{\star }(j+1, k_1, \ldots , k_{r-1}, k_r-j) \end{aligned}$$

and

$$\begin{aligned} {\widetilde{u}}^{\star }_{\mathrm {CSF}}(\alpha ; {\varvec{l}})&{:}{=}\sum _{(k_1, \ldots , k_r) \in \alpha } \prod _{s=1}^r\left( {\begin{array}{c}k_s+l_s-1\\ l_s\end{array}}\right) \\&\quad \times \sum _{j=0}^{k_1+l_1-2}w^{\star }(j+1, k_r+l_r, \ldots , k_2+l_2, k_1+l_1-j) \end{aligned}$$

for an ${\varvec{l}}{:}{=}(l_1, \ldots , l_r) \in {\mathbb {Z}}^r_{\ge 0}$. Then, for the proof of Proposition 5.1, it suffices to prove

$$\begin{aligned} {\widetilde{w}}^{\star }_{\mathrm {CSF}, {\widehat{{\mathcal {S}}}}}(\alpha ) ={\widetilde{w}}^{\star }_{\mathrm {CSF}}(\alpha )+(-1)^{k+1}\sum _{{\varvec{l}}=(l_1, \ldots , l_r) \in {\mathbb {Z}}^r_{\ge 0}}{\widetilde{u}}^{\star }_{\mathrm {CSF}}(\alpha ; {\varvec{l}})t^{l_1+\cdots +l_r}. \end{aligned}$$

(8)

Now, we prove (8). For an index ${\varvec{k}}=(k_1, \ldots , k_r)$, set

$$\begin{aligned} F({\varvec{k}}) {:}{=}\sum _{l_1, \ldots , l_r \ge 0} \left\{ \prod _{j=1}^r(-1)^{k_j}\left( {\begin{array}{c}k_j+l_j-1\\ l_j\end{array}}\right) \right\} w^{\star }(k_r+l_r, \ldots , k_1+l_1)t^{l_1+\cdots +l_r}, \end{aligned}$$

and

$$\begin{aligned} A&{:}{=}\sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{\begin{array}{c} a+b=k_r-1 \\ a\ge 0, b\ge 1 \end{array}}\sum _{i=0}^{r-1} w^{\star }(1+a, k_1, \ldots , k_i)F(k_{i+1}, \ldots , k_{r-1}, 1+b),\\ B&{:}{=}\sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{\begin{array}{c} a+b=k_r-1 \\ a\ge 0, b\ge 1 \end{array}} w^{\star }(1+a, k_1, \ldots , k_{r-1}, 1+b),\\ C&{:}{=}\sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{\begin{array}{c} a+b=k_r-1 \\ a\ge 0, b\ge 1 \end{array}} F(1+a, k_1, \ldots , k_{r-1}, 1+b). \end{aligned}$$

Then we have

$$\begin{aligned} \sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{\begin{array}{c} a+b=k_r-1 \\ a\ge 0, b\ge 1 \end{array}} w^{\star }_{{\widehat{{\mathcal {S}}}}}(1+a, k_1, \ldots , k_{r-1}, 1+b) =A+B+C. \end{aligned}$$

5.2 Proofs of Theorems 2.4 and 2.7

In this subsection, we give the proofs of Theorems 2.4 and 2.7. It is easy to see that

$$\begin{aligned} B={\widetilde{w}}^{\star }_{\mathrm {CSF}}(\alpha ). \end{aligned}$$

(9)

We calculate A and C by using the following equality

(10)

Here, c and d are non-negative integers. This equality easily follows from the definition of W.

Lemma 5.2

We have

$$\begin{aligned} A=\sum _{(k_1, \ldots , k_r) \in \alpha }&\Biggl [ \sum _{l=0}^{\infty } \Bigl \{ w^{\star }_{{\widehat{{\mathcal {S}}}}}(1+l, k_1, \ldots , k_r)-F(1+l, k_1, \ldots , k_r) \Bigr \}t^l \\&\quad +F(k_1, \ldots , k_r, 1) \Biggr ]. \end{aligned}$$

Proof

First, from the definition of $F({\varvec{k}})$ and (10), we have

Since

we see that the sum $\sum _{\begin{array}{c} a+b=k_r-1 \\ a\ge 0, b \ge 1 \end{array}}(-1)^{b+1}$ is a telescoping sum. Thus we have

(11)

Next, we divide $A_2$ into $A_{21}$ and $A_{22}$. From the property (W2) of W, we have

(12)

Moreover, from the definition of $F(k_1, \ldots , k_r)$, we see that

$$\begin{aligned} A_{21} =&\sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{i=0}^{r-1}\sum _{l=0}^{\infty } w^{\star }(1+l, k_r, k_1, \ldots , k_i)t^l \cdot F(k_{i+1}, \ldots , k_{r-1})\nonumber \\ =&\sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{l=0}^{\infty } \bigl \{w^{\star }_{{\widehat{{\mathcal {S}}}}}(1+l, k_r, k_1, \ldots , k_{r-1}) \nonumber \\&\qquad \qquad \qquad -w^{\star }(1+l)\cdot F(k_r, k_1, \ldots , k_{r-1})-F(1+l, k_r, k_1, \ldots , k_{r-1})\bigr \}t^l \nonumber \\ =&\sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{l=0}^{\infty } \bigl \{w^{\star }_{{\widehat{{\mathcal {S}}}}}(1+l, k_1, \ldots , k_r) \nonumber \\&\qquad \qquad \qquad -w^{\star }(1+l) \cdot F(k_1, \ldots , k_r)-F(1+l, k_1, \ldots , k_r)\bigr \}t^l. \end{aligned}$$

(13)

Note that the last equality follows from that the considered sum is invariant under the cyclic permutation $\tau$.

Now, we calculate $A_{22}$. From the property (W2) of W, it is easy to see that

(14)

for a positive integer k. By using (14), we have

The second identity above follows from that $A_{22}$ is invariant under the cyclic permutation $\tau$. By changing the variable i to $i+1$, we have

(15)

Then, by comparing the definition of $A_1$ and (15) and using (14), we have

Finally, from the definition of $X^{\star }({\varvec{k}})$, we have

(16)

From (11), (12), (13), and (16), we obtain the desired formula. $\square$

Next, we calculate C.

Lemma 5.3

We have

$$\begin{aligned} C&=(-1)^{k+1}\sum _{{\varvec{l}}=(l_1, \ldots , l_r) \in {\mathbb {Z}}^r_{\ge 0}} {\widetilde{u}}^{\star }_{\mathrm {CSF}}(\alpha ; {\varvec{l}})t^{l_1+\cdots +l_r}\\&\quad + \sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{l=0}^{\infty }F(1+l, k_1, \ldots , k_r)t^l -\sum _{(k_1, \ldots , k_r) \in \alpha }F(k_1, \ldots , k_r, 1). \end{aligned}$$

Proof

From the definition of $F({\varvec{k}})$, we have

$$\begin{aligned} C&=\sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{\begin{array}{c} a+b=k_r-1 \\ a\ge 0, b\ge 0 \end{array}}\sum _{l_1, \ldots , l_{r-1}, a', b' \ge 0} \left( {\begin{array}{c}a+a'\\ a\end{array}}\right) \left( {\begin{array}{c}b+b'\\ b\end{array}}\right) (-1)^{k_r+1}t^{a'+b'}\\&\qquad \times \prod _{j=1}^{r-1}\left( {\begin{array}{c}k_j+l_j-1\\ l_j\end{array}}\right) (-1)^{k_j}t^{l_j}\\&\qquad \times w^{\star }(1+b+b', k_{r-1}+l_{r-1}, \ldots , k_1+l_1, 1+a+a') \\&\qquad -\sum _{(k_1, \ldots , k_r) \in \alpha }F(k_1, \ldots , k_r, 1)\\&=\sum _{(k_1, \ldots , k_r) \in \alpha } \sum _{\begin{array}{c} a+b=k_r-1 \\ a \ge 0, b \ge 0 \end{array}} \sum _{\begin{array}{c} l_1, \ldots , l_{r-1} \ge 0 \\ a' \ge a, b' \ge b \end{array}} \left( {\begin{array}{c}a'\\ a\end{array}}\right) \left( {\begin{array}{c}b'\\ b\end{array}}\right) (-1)^{k_r+1}t^{a'+b'-k_r+1}\\&\qquad \times \prod _{j=1}^{r-1}\left( {\begin{array}{c}k_j+l_j-1\\ l_j\end{array}}\right) (-1)^{k_j}t^{l_j}\\&\qquad \times w^{\star }(1+b', k_{r-1}+l_{r-1}, \ldots , k_1+l_1, 1+a')-\sum _{(k_1, \ldots , k_r) \in \alpha }F(k_1, \ldots , k_r, 1). \end{aligned}$$

Set $l_r{:}{=}a'+b'-(k_r-1)$. Then, by Chu–Vandermonde identity, we have

$$\begin{aligned}&\sum _{\begin{array}{c} a+b=k_r-1 \\ a \ge 0, b \ge 0 \end{array}}\sum _{a' \ge a, b' \ge b} \left( {\begin{array}{c}a'\\ a\end{array}}\right) \left( {\begin{array}{c}b'\\ b\end{array}}\right) t^{a'+b'-(k_r-1)}\\&\qquad \times w^{\star }(1+b', k_{r-1}+l_{r-1}, \ldots , k_1+l_1, 1+a')\\&\quad =\sum _{l_r \ge 0}\sum _{a'=0}^{k_r+l_r-1}\sum _{a=0}^{k_r-1} \left( {\begin{array}{c}a'\\ a\end{array}}\right) \left( {\begin{array}{c}k_r+l_r-1-a'\\ k_r-1-a\end{array}}\right) t^{l_r}\\&\qquad \times w^{\star }(k_r+l_r-a', k_{r-1}+l_{r-1}, \ldots , k_1+l_1, 1+a')\\&\quad =\sum _{l_r \ge 0}\sum _{a'=0}^{k_r+l_r-1} \left( {\begin{array}{c}k_r+l_r-1\\ l_r\end{array}}\right) t^{l_r} w^{\star }(k_r+l_r-a', k_{r-1}+l_{r-1}, \ldots , k_1+l_1, 1+a')\\&\quad =\sum _{l_r \ge 0}\sum _{\begin{array}{c} a'+b'=k_r+l_r-1 \\ a' \ge 0, b' \ge 0 \end{array}} \left( {\begin{array}{c}k_r+l_r-1\\ l_r\end{array}}\right) t^{l_r} w^{\star }(1+b', k_{r-1}+l_{r-1}, \ldots , k_1+l_1, 1+a'). \end{aligned}$$

Therefore, we have

$$\begin{aligned} C&=-\sum _{(k_1, \ldots , k_r) \in \alpha } \sum _{l_1, \ldots , l_r \ge 0} \sum _{\begin{array}{c} a'+b'=k_r-1+l_r \\ a' \ge 0, b' \ge 0 \end{array}} \prod _{j=1}^{r}\left( {\begin{array}{c}k_j+l_j-1\\ l_j\end{array}}\right) (-1)^{k_j}t^{l_j}\\&\quad \times w^{\star }(1+b', k_{r-1}+l_{r-1}, \ldots , k_1+l_1, 1+a')-\sum _{(k_1, \ldots , k_r) \in \alpha }F(k_1, \ldots , k_r, 1). \end{aligned}$$

Moreover, since

$$\begin{aligned}&w^{\star }(k_r+l_r, \ldots , k_1+l_1, 1) \\&\quad =\sum _{l, l_0 \ge 0}(-1)^{l}\left( {\begin{array}{c}l+l_0\\ l_0\end{array}}\right) t^{l+l_0} w^{\star }(k_r+l_r, \ldots , k_1+l_1, 1+l+l_0), \end{aligned}$$

we obtain

$$\begin{aligned} C&=(-1)^{k+1}\sum _{{\varvec{l}}=(l_1, \ldots , l_r) \in {\mathbb {Z}}^r_{\ge 0}} {\widetilde{u}}^{\star }_{\mathrm {CSF}}(\alpha ; {\varvec{l}})t^{l_1+\cdots +l_r}\\&\quad + \sum _{(k_1, \ldots , k_r) \in \alpha }\sum _{l=0}^{\infty }F(1+l, k_1, \ldots , k_r)t^l-\sum _{(k_1, \ldots , k_r) \in \alpha }F(k_1, \ldots , k_r, 1), \end{aligned}$$

which completes the proof. $\square$

Proof of Proposition 5.1

From (9), Lemmas 5.2 and 5.3, we obtain the formula (8), which leads to Proposition 5.1. $\square$

Proof of Theorem 2.7

From Proposition 5.1, it suffices to prove that

$$\begin{aligned} kw^{\star }_{{\widehat{{\mathcal {S}}}}}(k+1)&=kw^{\star }(k+1)+(-1)^{k+1}\\&\quad \times \sum _{l_1, \ldots , l_r \ge 0}(k+l) \prod _{j=1}^r \left( {\begin{array}{c}k_j+l_j-1\\ l_j\end{array}}\right) w^{\star }(k+l+1)t^{l}, \end{aligned}$$

where $l{:}{=}l_1+\cdots +l_r$. This equality follows immediately from the fact that $w^{\star }(n)=z_n=yx^{n-1}$ for a positive integer n and an easy calculation. $\square$

Proof of Theorem 2.4

Theorem 2.4 follows immediately from applying to the both hand sides of Theorem 2.7, substituting $T=0$, and using the cyclic sum formula for MZSVs [11]. Note that $w^{\star }_{\mathrm {CSF}}(\{1\}^k)=-kw^{\star }(k+1)=-kz_{k+1}$. $\square$

6 Cyclic sum formula for t-adic SMZVs

In this section, we prove the cyclic sum formula for $\zeta _{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$ from that for $\zeta ^{\star }_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})$.

Theorem 6.1

For a non-empty index ${\varvec{k}}=(k_1, \ldots , k_r)$, we have

$$\begin{aligned}&\sum _{i=1}^r \sum _{j=0}^{k_i-2} \zeta _{{\widehat{{\mathcal {S}}}}} (j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i-j) \\&=\sum _{i=1}^r \sum _{j=0}^\infty \left( \zeta _{{\widehat{{\mathcal {S}}}}} (k_i+j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1})\right. \\&\quad \left. +\zeta _{{\widehat{{\mathcal {S}}}}} (j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_i) \right) t^j \\&\quad +\sum _{i=1}^r \zeta _{{\widehat{{\mathcal {S}}}}} (k_{i+1}, \ldots , k_r, k_1, \ldots ,k_{i-1}, k_i+1). \end{aligned}$$

6.1 Preliminary

Let ${\mathcal {R}}$ be the ${\mathbb {Q}}$-vector space generated by all the indices. For an index ${\varvec{k}}=(k_1, \ldots ,k_r)$, set

$$\begin{aligned} {\varvec{k}}^{\star }{:}{=}\sum _{ \square =,\text { or } +} (k_1 \square \cdots \square k_r) \in {\mathcal {R}}. \end{aligned}$$

Lemma 6.2

For a non-empty index ${\varvec{k}}=(k_1, \ldots , k_r)$, we have

$$\begin{aligned} \sum _{m=0}^{r-1} (-1)^m \sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (k_1\square \cdots \square k_r)^\star =(k_1,\ldots ,k_r). \end{aligned}$$

Proof

We prove this lemma by induction on r. The case $r=1$ is trivial. In the general case, by the induction hypothesis, we have

$$\begin{aligned} \text {L.H.S.}&=\sum _{m=0}^{r-2} (-1)^m \sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (k_1,k_2\square \cdots \square k_r)^\star \\&\quad -\sum _{m=0}^{r-2} (-1)^m \sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (k_1+k_2\square \cdots \square k_r)^\star \\&=\sum _{ \triangle =,\text { or } + } (k_1\triangle k_2,\ldots ,k_r) -(k_1+k_2,k_3,\ldots ,k_r) \\&=\text {R.H.S.} \end{aligned}$$

$\square$

Hereafter, we want to calculate a sum over an index set such like

$$\begin{aligned} \{(k_{i+1}, \ldots , k_r, k_1, \ldots , k_i) \mid 1 \le i \le r\}. \end{aligned}$$

However, the same indices may appear more than once in such an expression. To treat such overlaps properly, we introduce the following abuse of notation. For $M{:}{=}\sum _{{\varvec{k}}} c_{{\varvec{k}}} \cdot {\varvec{k}}\in {\mathcal {R}}$, we set

$$\begin{aligned} \sum _{{\varvec{k}}\in M}f({\varvec{k}}){:}{=}\sum _{{\varvec{k}}} c_{{\varvec{k}}} \cdot f({\varvec{k}}). \end{aligned}$$

For example, for $M=(2,3)+2(5, 7)$, we have $\sum _{(k_1, k_2)\in M}(k_1, k_2+1)=(2,4)+2(5,8)$.

For a non-empty index ${\varvec{k}}=(k_{1},\ldots ,k_{r})$ and $m\in {\mathbb {Z}}_{\ge 0}$ with $m \le r-1$, set

$$\begin{aligned} S_m({\varvec{k}}) {:}{=}\sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (k_{1}\square \cdots \square k_r \square ), \end{aligned}$$

where

$$\begin{aligned} (k_1\square _1 \cdots \square _{r-1} k_r \square _r){:}{=}(k_{j+1}\square _{j+1} \cdots \square _{r-1} k_r \square _r k_1 \square _1 \cdots \square _{j-1} k_j) \end{aligned}$$

with $1 \le j \le r$ such that $\square _j=\ , \$. This definition depends on the choice of j, but hereafter we only consider sums such that this ambiguity does not concern. We note that all indices appearing in $S_m({\varvec{k}})$ have the same depth$(=r-m)$.

The following lemma is obvious, so we omit the proof.

Lemma 6.3

For a non-empty index ${\varvec{k}}=(k_{1},\ldots ,k_{r})$ and $m\in {\mathbb {Z}}_{\ge 0}$ with $m\le r-1$, we have

$$\begin{aligned}&\sum _{i=1}^{r-m} \sum _{(l_1,\ldots ,l_{r-m})\in S_m({\varvec{k}})} (l_{i+1},\ldots ,l_{r-m},l_1,\ldots ,l_i) \\&\quad =\sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } \sum _{i=1}^{r} (k_{i+1}\square \cdots \square k_r\square k_1 \square \cdots \square k_i). \end{aligned}$$

Remark 6.4

Similar to Lemma 6.3, we also have

$$\begin{aligned}&\sum _{i=1}^{r-m} \sum _{(l_1,\ldots ,l_{r-m})\in S_m({\varvec{k}})} (l_{i+1},\ldots ,l_{r-m},l_1,\ldots ,l_i)^\star \\&\quad =\sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } \sum _{i=1}^{r} (k_{i+1}\square \cdots \square k_r\square k_1 \square \cdots \square k_i)^\star . \end{aligned}$$

6.2 Proof of the theorem

To prove Theorem 6.1 from Theorem 2.4, we need Propositions 6.5 and 6.6 .

Proposition 6.5

For a non-empty index ${\varvec{k}}=(k_1, \ldots ,k_r)$ and a non-negative integer j, we have

$$\begin{aligned}&\sum _{i=1}^{r} \bigl \{(j+1+k_i,k_{i+1},\ldots ,k_r,k_1,\ldots ,k_{i-1}) +(j+1,k_{i},\ldots ,k_r,k_1,\ldots , k_{i-1})\bigr \} \\&\quad =\sum _{m=0}^{r-1} (-1)^m \sum _{i=1}^{r-m} \sum _{(l_1,\ldots ,l_{r-m})\in S_m({\varvec{k}})} (j+1,l_{i+1},\ldots ,l_{r-m},l_1,\ldots ,l_i)^\star . \end{aligned}$$

Proof

By Lemma 6.2, we have

$$\begin{aligned}&(j+1+k_{i},k_{i+1},\ldots ,k_r, k_1,\ldots ,k_{i-1}) +(j+1 ,k_{i},\ldots ,k_r,k_1,\ldots , k_{i-1}) \\&\quad =\sum _{m=0}^{r-1} (-1)^m \sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (j+1,k_{i}\square \cdots \square k_r\square k_1 \square \cdots \square k_{i-1})^\star . \end{aligned}$$

Then, by Remark 6.4, we have

$$\begin{aligned} \text {L.H.S.}&=\sum _{m=0}^{r-1} (-1)^m \sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } \sum _{i=1}^{r} (j+1, k_{i}\square \cdots \square k_r\square k_1 \square \cdots \square k_{i-1})^\star \\&=\text {R.H.S.} \end{aligned}$$

$\square$

Proposition 6.6

For a positive integer k and an index ${\varvec{k}}=(k_1, \ldots , k_r)$ with ${{\,\mathrm{wt}\,}}({\varvec{k}})=k$, we have

$$\begin{aligned}&\sum _{m=0}^{r-1}(-1)^m\sum _{i=1}^{r-m}\sum _{(l_1, \ldots , l_{r-m}) \in S_m({\varvec{k}})}\sum _{j=0}^{l_i-1} (j+1, l_{i+1}, \ldots , l_{r-m}, l_1, \ldots , l_{i-1}, l_i-j)^{\star }\\&\quad =\sum _{i=1}^r\sum _{j=0}^{k_i-1}(j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i-j)-(-1)^rk \cdot (k+1)^{\star }. \end{aligned}$$

Proof

Set $\alpha ({\varvec{k}}){:}{=}\sum _{i=1}^r(k_{i+1}, \ldots , k_r, k_1, \ldots , k_i)$. Then we have

$$\begin{aligned}&\text {L.H.S.} \\&\quad =\sum _{m=0}^{r-1}(-1)^m\sum _{(k'_1, \ldots , k'_r) \in \alpha ({\varvec{k}})}\sum _{i=1}^r \sum _{j=0}^{k'_1+\cdots +k'_i-1}\sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m-(i-1) \end{array} }\\&\qquad \qquad \qquad \qquad (j+1, k'_{i+1}\square \cdots \square k'_r, k'_1+\cdots +k'_i-j)^{\star }\\&\quad =\sum _{m=0}^{r-1}(-1)^m\sum _{(k'_1, \ldots , k'_r) \in \alpha ({\varvec{k}})}\sum _{i=1}^r \sum _{s=1}^i\sum _{j=0}^{k'_s-1}\sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m-(i-1) \end{array} }\\&\qquad \qquad \qquad \qquad (j+1+k'_{s+1}+\cdots +k'_i, k'_{i+1}\square \cdots \square k'_r, k'_1+\cdots +k'_s-j)^{\star }\\&\quad =\sum _{m=0}^{r-1}(-1)^m\sum _{(k'_1, \ldots , k'_r) \in \alpha ({\varvec{k}})}\sum _{0 \le a \le b<r} \sum _{j=0}^{k'_r-1}\sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m-r-a+b+2 \end{array} }\\&\qquad \qquad \qquad (j+1+k'_{1}+\cdots +k'_{a}, k'_{a+1}\square \cdots \square k'_{b}, k'_{b+1}+\cdots +k'_{r}-j)^{\star }\\&\quad =\sum _{m=0}^{r-1}(-1)^m\sum _{(k'_1, \ldots , k'_r) \in \alpha ({\varvec{k}})} \sum _{j=0}^{k'_r-1}\sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (j+1 \square k'_1 \square \cdots \square k'_{r-1} \square k'_r-j)^{\star }\\&\quad =\sum _{(k'_1, \ldots , k'_r) \in \alpha ({\varvec{k}})}\sum _{j=0}^{k'_r-1} \bigl \{(j+1, k'_1, \ldots , k'_{r-1}, k'_r-j)\\&\qquad \qquad \qquad \qquad \qquad \qquad -(-1)^r(j+1+k'_1+\cdots +k'_r-j)\bigr \}\\&=\text {R.H.S.}, \end{aligned}$$

which completes the proof. $\square$

Proposition 6.7

For a non-empty index ${\varvec{k}}=(k_1, \ldots , k_r)$, we have

$$\begin{aligned}&\sum _{m=0}^{r-1}(-1)^m \sum _{(l_1, \ldots , l_{r-m}) \in S_m({\varvec{k}})}\sum _{i=1}^{r-m} (l_i, \ldots , l_{r-m}, l_1, \ldots , l_{i-1}, 1)^{\star }\\&\quad = \sum _{i=1}^r\bigl \{(k_{i+1}, \ldots , k_r, k_1, \ldots , k_i, 1)+(k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i+1)\bigr \}. \end{aligned}$$

Proof

By Lemma 6.3 and an easy calculation, we have

$$\begin{aligned} \text {L.H.S.}&=\sum _{m=0}^{r-1}(-1)^m\sum _{i=1}^{r}\sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (k_{i+1}\square \cdots \square k_r \square k_1 \square \cdots \square k_i, 1)^{\star }\\&=\sum _{i=1}^r\sum _{m=0}^r(-1)^m \sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (k_{i+1} \square \cdots k_r \square k_1 \square \cdots \square k_i \square 1)^{\star }\\&\quad +\sum _{i=1}^r \sum _{m=0}^{r-1} (-1)^m \sum _{ \begin{array}{c} \square =,\text { or } + \\ \#(+)=m \end{array} } (k_{i+1} \square \cdots k_r \square k_1 \square \cdots \square k_{i-1}\square k_i+1)^{\star }. \end{aligned}$$

By Lemma 6.2, this coincides with

$$\begin{aligned} \sum _{i=1}^r\bigl \{(k_{i+1}, \ldots , k_r, k_1, \ldots , k_i, 1)+(k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i+1)\bigr \}=\text {R.H.S.}, \end{aligned}$$

which completes the proof. $\square$

Proof of Theorem 6.1

For an index ${\varvec{k}}=(k_1, \ldots , k_r)$, set

$$\begin{aligned}&\mathrm {CSF}_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})\\&=\sum _{i=1}^r \sum _{j=0}^{k_i-2} \zeta _{{\widehat{{\mathcal {S}}}}} (j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i-j) \\&\quad -\sum _{i=1}^r \sum _{j=0}^\infty \left( \zeta _{{\widehat{{\mathcal {S}}}}} (k_i+j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1})\right. \\&\quad \left. +\zeta _{{\widehat{{\mathcal {S}}}}} (j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_i) \right) t^j \\&\quad -\sum _{i=1}^r \zeta _{{\widehat{{\mathcal {S}}}}} (k_{i+1}, \ldots , k_r, k_1, \ldots ,k_{i-1}, k_i+1) \end{aligned}$$

and

$$\begin{aligned}&\mathrm {CSF}^{\star }_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})\\&=\sum _{i=1}^{r}\sum _{j=0}^{k_i-2} \zeta ^{\star }_{{\widehat{{\mathcal {S}}}}}(j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_{i-1}, k_i-j)\\&\quad -\sum _{i=1}^{r}\sum _{j=0}^{\infty } \zeta ^{\star }_{{\widehat{{\mathcal {S}}}}}(j+1, k_{i+1}, \ldots , k_r, k_1, \ldots , k_i)t^j -{{\,\mathrm{wt}\,}}({\varvec{k}})\zeta ^{\star }_{{\widehat{{\mathcal {S}}}}}({{\,\mathrm{wt}\,}}({\varvec{k}})+1). \end{aligned}$$

Note that by Theorem 2.4, we have $\mathrm {CSF}^{\star }_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})=0$ for any non-empty index ${\varvec{k}}$. Therefore, by Propositions 6.5, 6.6, and 6.7, we see that

$$\begin{aligned} \mathrm {CSF}_{{\widehat{{\mathcal {S}}}}}({\varvec{k}})=\sum _{m=0}^{r-1}(-1)^m \sum _{(l_1, \ldots , l_{r-m}) \in S_m({\varvec{k}})} \mathrm {CSF}^{\star }_{{\widehat{{\mathcal {S}}}}}(l_1, \ldots , l_{r-m})=0, \end{aligned}$$

which completes the proof of Theorem 6.1. $\square$