2 Basic properties of the Schur P- and Q-multiple zeta functions

We first review the basic terminology to define Schur P- and Q-multiple zeta functions. A partition $\lambda =(\lambda _{1},\ldots ,\lambda _r)$ is termed strict, if $\lambda _{1}> \lambda _{2} > \cdots > \lambda _r \ge 0$ . Then, we associate the strict partition $\lambda $ with the shifted diagram

$$\begin{align*}SD(\lambda)=\{(i, j)\in {\mathbb Z}^2 ~|~ 1\leq i\leq r, i\leq j\leq \lambda_{i}+i-1\}\end{align*}$$

depicted as a collection of square boxes where the ith row has $\lambda _{i}$ boxes. We say that $(i,j)\in SD(\lambda )$ is a corner of $\lambda $ if $(i+1, j)\notin SD(\lambda )$ and $(i, j+1)\notin SD(\lambda )$ and denote by $SC(\lambda ) \subset SD(\lambda )$ the set of all corners of $\lambda $ ; for example, $SC((4,2,1))=\{(1,4),(3,3)\}$ . For a strict partition $\lambda $ , a shifted tableau $(t_{ij})$ of shape $\lambda $ over a set X is a filling of $SD(\lambda )$ with $t_{ij}\in X$ into the $(i,j)$ box of $SD(\lambda )$ . We denote by $ST(\lambda ,X)$ the set of all shifted tableaux of shape $\lambda $ over X.

Definition 2.2 (Schur P-multiple zeta functions)

For a given set ${ \boldsymbol s}=(s_{ij})\in ST(\lambda ,\mathbb {C})$ of variables, the Schur P-multiple zeta functions of shape $\lambda $ are defined as

(2.1) $$ \begin{align} \zeta_{\lambda}^P({ \boldsymbol s})=\sum_{M\in PSST(\lambda)}\frac{1}{M^{ \boldsymbol s}}, \end{align} $$

where $M^{ \boldsymbol s}=\displaystyle {\prod _{(i, j)\in SD(\lambda )}|m_{ij}|^{s_{ij}}}$ for $M=(m_{ij})\in PSST(\lambda )$ and $|i|=|i'|=i$ .

For example, when $\lambda =(6,5,3,1)$ ,

and

$$ \begin{align*} \frac{1}{M^{\boldsymbol s}}&=\frac{1}{1^{s_{11}}1^{s_{12}}1^{s_{13}}2^{s_{14}}3^{s_{15}}4^{s_{16}}2^{s_{22}}2^{s_{23}}3^{s_{24}}4^{s_{25}}5^{s_{26}}3^{s_{33}}4^{s_{34}}4^{s_{35}}5^{s_{44}}}. \end{align*} $$

Similarly, we define the Schur Q-multiple zeta functions.

Definition 2.3 (Schur Q-multiple zeta functions)

For a given set ${ \boldsymbol s}=(s_{ij})\in ST(\lambda ,\mathbb {C})$ of variables, the Schur Q-multiple zeta functions of shape $\lambda $ are defined to be

(2.2) $$ \begin{align} \zeta_{\lambda}^Q({ \boldsymbol s})=\sum_{M\in QSST(\lambda)}\frac{1}{M^{ \boldsymbol s}}, \end{align} $$

where $M^{ \boldsymbol s}=\displaystyle {\prod _{(i, j)\in SD(\lambda )}|m_{ij}|^{s_{ij}}}$ for $M=(m_{ij})\in QSST(\lambda )$ and $|i|=|i'|=i$ .

For a strict partition $\lambda =(\lambda _{1},\ldots ,\lambda _r)$ , by the definitions of $\zeta _{\lambda }^P$ and $\zeta _{\lambda }^Q$ , we allow the main diagonal entries in tableaux $M\in QSST(\lambda )$ to be marked and obtain

(2.3) $$ \begin{align} \zeta_{\lambda}^Q({ \boldsymbol s})=2^r\zeta_{\lambda}^P({ \boldsymbol s}). \end{align} $$

As in the Introduction, we define the truncated P- and Q-multiple zeta functions:

For a fixed positive integer $N\in \mathbb {N}$ , let $PSST_N(\lambda )$ and $QSST_N(\lambda )$ be the sets of all $(m_{ij})\in PSST(\lambda )$ and $QSST(\lambda )$ such that $m_{ij}\le N$ for all $i,j$ . Then, we define

$$\begin{align*}\zeta_{\lambda}^{P,N}({ \boldsymbol s})=\sum_{M\in PSST_N(\lambda)}\frac{1}{M^{ \boldsymbol s}},\text{ and } \zeta_{\lambda}^{Q,N}({ \boldsymbol s})=\sum_{M\in QSST_N(\lambda)}\frac{1}{M^{ \boldsymbol s}}. \end{align*}$$

In this section, we prove some basic properties of the Schur P- and Q-multiple zeta functions. We first consider the domain of absolute convergence of the series (2.1) and (2.2).

Lemma 2.1 Let

$$\begin{align*}W_{\lambda}^Q = \left\{{\boldsymbol s}=(s_{ij})\in ST(\lambda,\mathbb{C})\,\left|\!\! \begin{array}{l} \textrm{Re}(s_{ij})\ge 1 \text{ for all } (i,j)\in SD(\lambda) \setminus SC(\lambda) \\[3pt] \textrm{Re}(s_{ij})>1 \text{ for all } (i,j)\in SC(\lambda) \end{array} \right. \!\!\!\right\}. \end{align*}$$

Then, the series (2.1) and (2.2) converge absolutely if ${\boldsymbol s}\in W_{\lambda }^Q$ .

Proof By (2.3), it suffices to consider $\zeta _{\lambda }^Q$ . Let $\lambda $ be a strict partition and $SC(\lambda )=\{(i_{1},j_{1}),\ldots ,(i_k,j_k)\}$ where $i_{1}<\cdots <i_k$ and $j_{1}>\cdots >j_k$ . Let $i_0=0$ . Because

$$\begin{align*}\left|\sum_{\substack{M\in QSST(\lambda)\\m_{ij}\le N}}\frac{1}{M^{ \boldsymbol s}}\right|\le \prod_{\ell=1}^k\sum_{\substack{M\in QSST(\lambda_\ell)\\ m_{ij}\le N}}\prod_{(i,j)\in SD(\lambda_\ell)}\frac{1}{|m_{ij}|^{\textrm{Re}(t_{ij,\ell})}}, \end{align*}$$

where $\lambda _\ell =(j_\ell -i_{\ell -1},j_\ell -i_{\ell -1}-1,\ldots ,j_\ell -i_\ell +1)$ and $t_{ij,\ell }=s_{i+i_{\ell -1},j+i_{\ell -1}}$ , we prove that for $\lambda =(\lambda _{1},\ldots ,\lambda _r):=(\lambda _{1},\lambda _{1}-1,\ldots ,\lambda _{1}-r+1)$ ,

(2.4) $$ \begin{align} \sum_{\substack{M\in QSST(\lambda)\\ m_{ij}\le N}}\prod_{(i,j)\in SD(\lambda)}\frac{1}{|m_{ij}|^{\textrm{Re}(s_{ij})}} \end{align} $$

converges absolutely for $\boldsymbol s\in W_{\lambda }^Q$ as $N\rightarrow \infty $ . Rearranging the order of summation, we have

$$\begin{align*}\sum_{\substack{M\in QSST(\lambda)\\ m_{ij}\le N}}\prod_{(i,j)\in SD(\lambda)}\frac{1}{|m_{ij}|^{\textrm{Re}(s_{ij})}} =\sum_{N_{1}=1}^N\left(\sum_{\substack{(m_{ij})\in QSST(\lambda)\\ m_{r\lambda_r}= N_{1}}}\prod_{\substack{(i,j)\in SD(\lambda)\\ (i,j)\neq(r,\lambda_r)}}\frac{1}{|m_{ij}|^{\textrm{Re}(s_{ij})}}\right) \frac{1}{N_{1}^{\textrm{Re}(s_{r\lambda_r})}}. \end{align*}$$

By extending the region of summation and product, it holds that

$$\begin{align*}\sum_{\substack{M\in QSST(\lambda)\\ m_{ij}\le N}}\prod_{(i,j)\in SD(\lambda)}\frac{1}{|m_{ij}|^{\textrm{Re}(s_{ij})}}\le2^{r\lambda_r} \sum_{N_{1}=1}^N\left(\underset{(i,j)\ne (r,\lambda_r)}{\prod^{r}_{i=1}\prod^{\lambda_r}_{j=1}}\sum^{N_{1}}_{m_{ij}=1} \frac{1}{m_{ij}}\right)\frac{1} {N_{1}^{\textrm{Re}(s_{r\lambda_r})}}. \end{align*}$$

Because for any $\varepsilon>0$ , there exists a constant $C_{\varepsilon }>1$ such that

$$\begin{align*}\sum^{N}_{m_{ij}=1}\frac{1}{m_{ij}}<\frac{C_\varepsilon}{2} N^\varepsilon, \end{align*}$$

we can estimate that

$$\begin{align*}\sum_{\substack{M\in QSST(\lambda)\\ m_{ij}\le N}}\prod_{(i,j)\in SD(\lambda)}\frac{1}{|m_{ij}|^{\textrm{Re}(s_{ij})}}\le C_\varepsilon^{r\lambda_r}\sum_{N_{1}=1}^N\frac{N_{1}^{\varepsilon r\lambda_r}} {N_{1}^{\textrm{Re}(s_{r\lambda_r})}}. \end{align*}$$

We can choose a sufficiently small $\varepsilon $ such that $\textrm {Re}(s_{r\lambda _r})-\varepsilon r\lambda _r>1$ . Thus, (2.4) converges absolutely and we obtain the lemma.

We next show that a Schur Q-multiple zeta function can be written as a linear combination of the multiple zeta (star) functions as well as the Schur multiple zeta functions. Indeed, for a strict partition $\lambda $ of n, let $\mathcal {SF}(\lambda )$ be the set of all bijections $f:SD(\lambda )\to \{1,2,\ldots ,n\}$ satisfying the following two conditions:

1. (i) for all i, $f((i,j_{1}))<f((i,j_{2}))$ if and only if $j_{1}<j_{2}$ ,

2. (ii) for all j, $f((i_{1},j))<f((i_{2},j))$ if and only if $i_{1}<i_{2}$ .

For $\boldsymbol s=(s_{ij})\in ST(\lambda ,\mathbb C)$ , put

$$\begin{align*}V(\boldsymbol s)= \left\{\left. \left(s_{f^{-1}(1)},s_{f^{-1}(2)},\ldots,s_{f^{-1}(n)}\right)\in \mathbb C^{n}\,\right|\, f\in \mathcal{SF}(\lambda) \right\}. \end{align*}$$

We write ${\boldsymbol t} \preceq _s \boldsymbol s$ for ${\boldsymbol t}=(t_{1},t_{2},\ldots ,t_m)\in \mathbb C^m$ if there exists $(v_{1},v_{2},\ldots ,v_{n})\in V(\boldsymbol s)$ satisfying the following: for all $1\le k\le m$ , there exist $1\le h_k\le m$ and $l_k\ge 0$ such that

1. (i) $t_k=v_{h_k}+v_{h_k+1}+\cdots +v_{h_k+l_k}$ ,

2. (ii) there are no $(i_{1},i_{2};j_{1},j_{2})$ with $i_{1}<i_{2}$ and $j_{1}<j_{2}$ such that

   $\{s_{i_{1}j_{1}},s_{i_{1}j_{2}},s_{i_{2}j_{2}}\}\subset \{v_{h_k},v_{h_k+1},\ldots ,v_{h_k+l_k}\}$ , and

3. (iii) $\bigsqcup ^{m}_{k=1}\{h_k,h_k+1,\ldots ,h_k+l_k\}=\{1,2,\ldots ,n\}$ (disjoint union).

Here, we note that since $|m_{ij}|=|m_{ij}'|=m_{ij}$ for any positive integer $m_{ij}$ in (2.2), the definition of $\zeta _{\lambda }^Q$ , we have

(2.5) $$ \begin{align} \zeta_{\lambda}^Q({\boldsymbol s}) =\sum_{{\boldsymbol t} \,\preceq_s\, {\boldsymbol s}}2^{m({\boldsymbol t})}\zeta({\boldsymbol t}), \end{align} $$

where $m(\boldsymbol t)$ is a positive integer that depends on the way in which the comma $,$ is changed to the plus $+$ sign. Moreover, by an Inclusion–Exclusion principle, one can also obtain its “dual” expression

(2.6) $$ \begin{align} \zeta_{\lambda}^Q({\boldsymbol s}) =\sum_{{\boldsymbol t} \,\preceq_s\, {\boldsymbol s}}(-1)^{n-\mathrm{dep}({\boldsymbol t})}2^{m({\boldsymbol t})}\zeta^{\star}({\boldsymbol t}), \end{align} $$

where $\mathrm {dep}$ is the number of variables. Combining (2.5) and (2.6) with identity (2.3), we can decompose the Schur P-multiple zeta function into a linear combination of multiple zeta (star) functions defined by

$$\begin{align*}\zeta(s_{1},\ldots,s_r)=\sum_{1\le n_{1}<\cdots< n_r}\frac{1}{n_{1}^{s_{1}}\cdots n_r^{s_r}},\ \zeta^\star(s_{1},\ldots,s_r)=\sum_{1\le n_{1}\le\cdots\le n_r}\frac{1}{n_{1}^{s_{1}} \cdots n_r^{s_r}}. \end{align*}$$

Example 2.2 For ${\boldsymbol s}=(s_{ij})\in ST((3,1),\mathbb {C})$ , we have

$$\begin{align*}V({\boldsymbol s})=\{(s_{11},s_{12},s_{13},s_{22}),(s_{11},s_{12},s_{22},s_{13})\}. \end{align*}$$

One can confirm that ${\boldsymbol t} \preceq _s {\boldsymbol s}$ if and only if ${\boldsymbol t}$ is one of the following:

$$ \begin{align*} & (s_{11},s_{12},s_{13},s_{22}),(s_{11}+s_{12},s_{13},s_{22}), (s_{11},s_{12}+s_{13},s_{22}),(s_{11},s_{12},s_{13}+s_{22}),\\ & (s_{11}+s_{12}+s_{13},s_{22}),(s_{11}+s_{12},s_{13}+s_{22}),(s_{11},s_{12}+s_{13}+s_{22}),\\ & (s_{11},s_{12},s_{22},s_{13}), (s_{11}+s_{12},s_{22},s_{13}),(s_{11},s_{12}+s_{22},s_{13}). \end{align*} $$

This shows that when ${\boldsymbol s}\in W_{(3,1)}^Q$

Example 2.3 It holds that

where $\boldsymbol \ell $ runs over all indices of the form $\boldsymbol \ell = (s_{11}\square s_{12}\square \cdots \square s_{1r})$ in which each $\square $ is filled by a comma $,$ or a plus $+$ sign.

By (2.3), the Schur P-multiple zeta functions can be similarly decomposed into a linear combination of multiple zeta (star) functions.

We next provide a short observation on the relation between Schur Q-multiple zeta values and the Two-One formula conjectured by Ohno and Zudilin [Reference Ohno and Zudilin18], and proved by Zhao [Reference Zhao24].

Theorem 2.4 (Two-One formula [Reference Ohno and Zudilin18, Reference Zhao24])

For a non-negative integer k, we denote $\mu _{2k+1}=(1,\{2\}^k)$ . Then for any admissible index $\boldsymbol k= (k_{1},\ldots ,k_r)$ with odd entries $k_{1},\ldots ,k_r$ , the following identities are valid:

$$ \begin{align*} \zeta^{\star}(\mu_{k_{1}},\ldots,\mu_{k_r}) &=\sum_{\boldsymbol \ell\preceq\boldsymbol k}2^{\mathrm{dep}({\boldsymbol \ell})}\zeta(\boldsymbol \ell),\\ &=\sum_{\boldsymbol \ell\preceq\boldsymbol k}(-1)^{r-\mathrm{dep({\boldsymbol \ell})}}2^{\mathrm{dep}({\boldsymbol \ell})}\zeta^{\star}(\boldsymbol \ell), \end{align*} $$

where the sum $\displaystyle {\sum _{\boldsymbol \ell \preceq \boldsymbol k}}$ extends over all indices of the form $\boldsymbol \ell = (k_{1}\square k_{2}\square \cdots \square k_{r})$ in which each $\square $ is filled by the comma $,$ or the plus $+$ sign.

Combining Theorem 2.4 with Lemma 2.3, we have the following theorem.

Theorem 2.5 For r-tuple $(k_{1},\ldots ,k_r)$ of positive odd integers with $k_r\ge 3$ ,

This theorem establishes a non-trivial identity between a single Schur Q-multiple zeta value and a multiple zeta star value.

Corollary 2.6 For a positive integer $k\ge 4$

Proof By Theorem 2.5 and the sum formula (Theorem 6.1), we have

By (2.5), we have and

3 Pfaffian expression of the Schur Q-multiple zeta functions

The original Schur Q-polynomial is known to have a Pfaffian expression [Reference Macdonald15]. In this section, we provide a Pfaffian expression for the Schur Q-multiple zeta function by following the Stembridge approach [Reference John12]. We first recall the definition of a Pfaffian. Let $\mathfrak S_n$ be the symmetric group of degree n. Then, for a given square matrix ${A=(a_{ij})_{1\leq i,j\leq n}}$ , the determinant $\det (A)$ is defined by

$$\begin{align*}\det (A)=\sum_{\sigma\in \mathfrak S_{n}} \mathrm{sgn}(\sigma)\prod_{i=1}^n a_{i,\sigma(i)},\end{align*}$$

where $\mathrm {sgn}(\sigma )$ is the signature of $\sigma $ .

We derive the Pfaffian by defining a set $\mathfrak F_{2n}$ , a subset of the symmetric group $\mathfrak S_{2n}$ of an even degree,

$$\begin{align*}\mathfrak F_{2n}=\left\{\pi\in\mathfrak S_{2n}\left|\!\!\! \begin{array}{c} \pi(1) < \pi(3) < \cdots < \pi(2n-1),\\ \pi(1) < \pi(2), \pi(3) < \pi(4), \ldots,\pi(2n-1) < \pi(2n) \end{array}\right.\!\!\!\right\}. \end{align*}$$

For an ordered $2n$ -tuple $\boldsymbol v=(v_{1},\ldots ,v_{2n})$ of vertices, we say that a set of edges ${\pi =\{((v_{i},v_{j}),\ldots ,(v_k,v_l))\}}$ on $\boldsymbol v$ is a $1$ -factor if each $v_{i}$ is incident with exactly one edge.

Example 3.1 The following are $1$ -factors of $\{1,2,3,4\}$ .

By convention, we always list the edges of a $1$ -factor $\pi $ in the form $(v_{i},v_{j})$ with $i<j$ . It is known that a bijection can be constructed from $\mathfrak F_{2n}$ to the set of $1$ -factors by $\pi \mapsto \{(v_{\pi (1)},v_{\pi (2)}),\ldots ,(v_{\pi (2n-1)},v_{\pi (2n)})\}$ , and

$$\begin{align*}|\mathfrak F_{2n}|=\frac{(2n)!}{2^nn!}.\end{align*}$$

Then, for a given $2n \times 2n$ upper triangular or anti-symmetric matrix $A=(a_{ij})_{1\leq i,j\leq 2n}$ , the Pfaffian $\mathrm {pf} (A)$ of A is defined by

$$\begin{align*}\mathrm{pf} (A) = \sum_{\pi\in \mathfrak F_{2n}} \mathrm{sgn}(\pi)\prod_{i=1}^n a_{\pi(2i-1),\pi(2i)}. \end{align*}$$

Let $D=(V,E)$ be a directed graph with vertices V and edges E, with the assignment of a direction to each edge with no directed cycles. Multiple edges are allowed. We denote by $u\rightarrow v$ an edge directed from u to v. For any pair of vertices $u,v$ , we denote by $\mathscr P(u, v)$ the set of directed D-paths from u to v on D. If $u=u$ , then $\mathscr P(u, u)$ is a set of a single path of length zero.

Let I and J be ordered sets of vertices of D. Then I is said to be D-compatible with J if, whenever $u<u'$ in I and $v>v'$ in J, every path $P\in \mathscr P(u, v)$ intersects every path $Q \in \mathscr P(u', v')$ . Here, if two paths have a common vertex, we say that they intersect.

For any vertex $u \in V$ and subset $I\subset V$ , let $\mathscr P(u; I)$ denote the set of directed paths from u to any $v\in I$ , and let

$$\begin{align*}Q_I(u) = \sum_{P\in \mathscr P(u; I)} w(P),\end{align*}$$

where w is a particular weight function defined on edges. For any r-tuple ${\boldsymbol u = (u_{1},\ldots ,u_r)}$ of vertices, let $\mathscr P(\boldsymbol u;I)$ be the set of r-tuples of paths $P_{i}\in \mathscr P(u_{i};I)$ . The weight function w is extended to tuples of paths by

$$\begin{align*}w(P_{1},\ldots,P_r)=\prod_{i=1}^rw(P_{i}).\end{align*}$$

Then we define

$$\begin{align*}Q_I(u_{1},\ldots,u_r) = \sum_{(P_{1},\ldots,P_r)\in \mathscr P(\boldsymbol u; I)} w(P_{1},\ldots,P_r). \end{align*}$$

Theorem 3.2 [Reference John12, Theorem 3.1]

Let $\boldsymbol u = (u_{1},\ldots ,u_r)$ be an r-tuple of vertices in a directed acyclic graph D, and assume that r is even. If $I\subset V$ is a totally ordered subset of the vertices such that u is D-compatible with I, then

$$\begin{align*}Q_I(\boldsymbol u)=\mathrm{pf}(Q_I(u_{i},u_{j}))_{1\le i<j\le r}.\end{align*}$$

Stembridge constructed a directed graph D corresponding to the Schur Q-functions [Reference John12]. Moreover, Stembridge applied Theorem 3.2 to obtain the following Pfaffian expression of the Schur Q-polynomial.

Theorem 3.4 [Reference John12, Theorem 6.1]

Let $\lambda =(\lambda _{1},\ldots ,\lambda _r)$ be a strict partition of even length. Then

$$\begin{align*}Q_{\lambda}=\mathrm{pf}(Q_{(\lambda_{i},\lambda_{j})})_{1\le i<j\le r}.\end{align*}$$

Following the Stembridge approach, we construct a directed graph D corresponding to the Schur Q-multiple zeta functions. We begin with the vertex set of pairs of non-negative integers, and direct an edge $u\rightarrow v$ whenever $u-v = (1, 0), (0, 1)$ , or $(1, 1)$ . Subsequently, we delete the edges $u\rightarrow v$ that contain points whose first coordinates are both zero, as well as those whose second coordinates are both zero. Finally, we divide each of the vertices $(0,j)$ with $j> 1$ into two vertices, say $(0, j)$ and $(0, j+1)'$ , such that the edge $(1,j + 1)\rightarrow (0,j)$ is redirected to $(0, j+1)'$ , whereas the edge $(1, j)\rightarrow (0, j)$ remains intact. Fix a positive integer N and a partition $\lambda $ , and let $\boldsymbol u = (u_{1},\ldots ,u_r)$ be the r-tuple of vertices with $u_{i} = (\lambda _{i}, N)$ . Without loss of generality, we may assume that r is even (if r is odd, set $\lambda _{r+1}= 0$ and $u_{r+1} = (0, N+1)'$ , and replace r with $r+ 1$ ). Let $I_N=\{(0,0),(0,1),(0,2)',(0,2),\ldots ,(0,N)',(0,N),(0,N+1)'\}$ .

For any vertex $u\in V$ , let $\mathscr P_0(u;I)$ be the set of non-intersecting path $P\in \mathscr P(u;I)$ . For any r-tuple $\boldsymbol u = (u_{1},\ldots ,u_r)$ of vertices, let $\mathscr P_0(\boldsymbol u;I)$ be the set of non-intersecting r-tuples of paths $P_{i}\in \mathscr P_{0}(u_{i};I)$ . Then, an element in $QSST_N(\lambda )$ can be identified with a tuple of non-intersecting paths in $\mathscr P_{0}(\boldsymbol u;I_N)$ , and $\boldsymbol u$ is D-compatible with $I_N$ .

Let $v_{i}(P)=(v_{i,j}(P))_{j\ge 0}$ be the sequence of vertices representing a path $P\in \mathscr P_{0}(u_{i};I_N)$ and let $\ell ^i_{xy}$ be the edge $v_{i,j}(=(x,y))\rightarrow v_{i,j+1}$ . If $v_{i,j}(P)-v_{i,j+1}(P)= (1,0)$ or $(1,1)$ , we assign the weight $w(\ell ^i_{xy})=y^{-s_{i,x+i-1}}$ . If $v_{i,j}(P)-v_{i,j+1}(P)= (0,1)$ , we assign the weight $w(\ell ^i_{xy})=1$ . Here, we put $(1,y)-(0,y)'=(1,1)$ for any positive integer y. Then, we define

$$\begin{align*}w(P_{i})=\prod_{\ell_{xy}^i}w(\ell_{xy}^i)\end{align*}$$

(see Example 3.5), and for $(P_{1},\ldots ,P_r)\in \mathscr P(\boldsymbol u;I_N)$ ,

$$\begin{align*}w(P_{1},\ldots,P_r)=\prod_{i=1}^rw(P_{i}).\end{align*}$$

Then, according to the above discussion, we find that

$$\begin{align*}\zeta_{\lambda}^{Q,N}(\boldsymbol s)=\sum_{(P_{1},\ldots,P_r)\in \mathscr P_0(\boldsymbol u;I_N)}w(P_{1},\ldots,P_r).\end{align*}$$

Figure 1:

$(P_{1},P_{2},P_{3},P_{4})$ satisfying the condition in Example 3.5.

For a set X, we define

$$\begin{align*}ST^{\mathrm{diag}}(\lambda,X)=\{(t_{ij})\in W_{\lambda}^Q~|~t_{ij}=t_{1k} \text{ if } j-i=k-1 \text{ for any } k\}.\end{align*}$$

Example 3.5 Let $\lambda =(6,5,3,1)$ and $N=5$ . Then, Figure 1 is a $4$ -tuple of paths $(P_{1},P_{2},P_{3},P_{4})\in \mathscr P(\{u_{1},u_{2}\};I_N)\oplus \mathscr P(\{u_{3},u_{4}\};I_N)$ . Let $(s_{ij})\in ST(\lambda ,\mathbb C)$ . The weights $w(P_{i})$ are

$$ \begin{align*} w(P_{1})&=\frac{1}{1^{s_{11}}1^{s_{12}}1^{s_{13}}2^{s_{14}}3^{s_{15}}4^{s_{16}}}, &&w(P_{2})=\frac{1}{3^{s_{22}}3^{s_{23}}3^{s_{24}}4^{s_{25}}5^{s_{26}}}\\ w(P_{3})&=\frac{1}{2^{s_{33}}2^{s_{34}}5^{s_{35}}},&&w(P_{4})=\frac{1}{4^{s_{44}}}.\end{align*} $$

Theorem 3.6 (Pfaffian expression of the Schur Q-multiple zeta functions)

Let r be an even positive integer. Let $\lambda =(\lambda _{1},\ldots ,\lambda _{r})$ be a strict partition with $\lambda _{i}\ge 0$ . Then for $\boldsymbol s\in ST^{\mathrm {diag}}(\lambda ,\mathbb C)$ ,

$$\begin{align*}\zeta_{\lambda}^Q(\boldsymbol s)=\mathrm{pf}(M_{\lambda}),\end{align*}$$

where $M_{\lambda }=(a_{ij})$ is an $r\times r$ upper triangular matrix with

$$\begin{align*}a_{ij}=\left\{\begin{array}{@{}ll}\zeta_{(\lambda_{i},\lambda_{j})}^Q(\boldsymbol s_{(\lambda_{i},\lambda_{j})}) & \quad \text{for } i<j,\\0 & \quad\text{otherwise,}\end{array}\right. \end{align*}$$

and

where $t_{i}=i+\lambda _{i}-1$ .

Proof We can prove this by following Stembridge’s method (see [Reference John12, Theorem 3.1]). Indeed, a similar discussion is proceeded in terms of appropriate weight corresponding to multiple zeta functions:

By the definition of Pfaffian,

(3.1) $$ \begin{align} \mathrm{pf}(M_{\lambda})=\sum_{\pi\in\mathfrak F_n}\mathrm{ sgn}(\pi)\prod_{(i,j)\in\pi}\zeta_{(\lambda_{i},\lambda_{j})}^Q(\boldsymbol s_{(\lambda_{i},\lambda_{j})}). \end{align} $$

It suffices to show that there exists a sign-reversing summand for each summand resulting from $(P_{1},\ldots ,P_r)$ with at least one pair of intersecting paths.

We consider the right-most intersection point $(p,q)$ appearing in paths $(P_{1},\ldots ,P_r)$ for a 1-factor $\pi $ . For the sake of simplicity, we can assume that for the 1-factor, $\pi $ the two paths $P_{1}$ and $P_{2}$ intersect at $(p,q)$ (Figure 2). Then, the paths $(P_{1},\ldots ,P_r)$ give rise to

$$\begin{align*}S(\pi)=\mathrm{ sgn}(\pi)\prod_{j=1}^{t_{1}}a_{1j}^{-s_{1j}}\prod_{j=2}^{t_{2}}a_{2j}^{-s_{2j}}\prod_{i=3}^r\left(\prod_{j=i}^{t_{i}}a_{ij}^{-s_{ij}}\right),\end{align*}$$

where $a_{ij}$ is the y-coordinate of the corresponding element of $v_{ij}^w(P_{i})$ . On the other hand, we consider the r-tuple of paths $(\overline P_{1},\overline {P}_{2},P_{3},\ldots ,P_r)$ . Here, $\overline P_{i}$ follows $P_{i}$ until it meets the first intersection point $(p,q)$ , whereupon it follows the other path $P_{j}$ to the end (Figure 3).

Figure 2:

$(P_{1},\ldots ,P_r)$ .

Figure 3:

$(\overline P_{1},\overline {P}_{2},P_{3},\ldots ,P_r)$ .

Let $\overline \pi $ be the 1-factor obtained by interchanging $1$ and $2$ . Here, it is necessary to verify that for each 1-factor $(i,j)\in \overline {\pi }$ , the paths $P_{i}$ and $P_{j}$ do not intersect. It suffices to consider the cases involving the modified paths $\overline {P}_{i}$ and $\overline {P}_{j}$ . The definition of v implies that points of intersection other than v do not exist on the right-hand side of v. Hence, the path $P_k$ will intersect $P_{1}$ (resp. $P_{2}$ ) if and only if $P_k$ intersects $\overline P_{2}$ (resp. $\overline {P}_{1}$ ). Thus, we confirm that $\overline {\pi }$ appears in (3.1) and the paths $(\overline {P}_{1},\overline {P}_{2},\ldots ,P_r)$ yield

$$\begin{align*}S(\overline\pi)=\mathrm{ sgn}(\overline\pi)\prod_{j=2}^{p+1}a_{2j}^{-s_{2j}}\prod_{j=p+1}^{t_{1}}a_{1j}^{-s_{1j}}\prod_{j=1}^{p}a_{1j}^{-s_{1j}}\prod_{j=p+2}^{t_{2}}a_{2j}^{-s_{2j}}\prod_{i=3}^r\left(\prod_{j=i}^{t_{i}}a_{ij}^{-s_{ij}}\right).\end{align*}$$

As $\mathrm {sgn}(\pi )\mathrm {sgn}(\overline \pi )=-1$ and $s_{1j}=s_{2(j+1)}$ , one can confirm that

$$\begin{align*}S(\pi)+S(\overline{\pi})=0,\end{align*}$$

and this proves the assertion.

Example 3.7 Let $\lambda =(3,2,1,0)$ . Then, if $(a_{j-i})=(s_{ij})\in ST^{\mathrm {diag}}(\lambda ,\mathbb C)$ ,

As in [Reference Nakasuji and Takeda17], we can consider an extension of Theorem 3.6. In preparation, we define

(3.2) $$ \begin{align} \sum_{\mathrm{diag}}=\sum_{\sigma_{1}\in S_{1}}\cdots\sum_{\sigma_{\lambda_{1}}\in S_{\lambda_{1}}}\sigma_{1}\cdots \sigma_{\lambda_{1}} \end{align} $$

for $S_{j}$ being the set of permutations of the elements of $I(j)=\{(k,l)\in SD(\lambda )~|~l-k\,{=}\,j\}$ . The sum $\displaystyle \sum _{\mathrm {diag}}$ signifies the sum taken over all permutations of all elements on each diagonal $I(j)$ for all j.

We now give an example of (3.2).

Example 3.8 For $\lambda =(3, 2)$ ,

$$ \begin{align*} I(0)&=\{(k, \ell)\in D(\lambda) ~|~ \ell-k=0\}=\{(1,1), (2,2)\},\\ I(1)&=\{(k, \ell)\in D(\lambda) ~|~ \ell-k=1\}=\{(1,2), (2,3)\},\\ I(2)&=\{(k, \ell)\in D(\lambda) ~|~ \ell-k=0\}=\{(1,3)\}. \end{align*} $$

This leads to

$$ \begin{align*} S_0\cong S_{1}\cong {\frak S}_{2}=\{\textrm{id}, \sigma_{1}\},\ S_{2}\cong {\frak S}_{1}=\{\textrm{id}\}, \end{align*} $$

where $\sigma _{1}$ implies the substitution of the first and second components of $I(j)$ for any j. Therefore,

Also, we define a set $W_{\lambda ,H}^Q$ by

$$\begin{align*}W_{\lambda,H}^Q = \left\{{\boldsymbol s}=(s_{ij})\in ST(\lambda,\mathbb{C})\,\left|\!\! \begin{array}{l} \textrm{Re}(s_{ij})\ge 1 \text{ for all } (i,j)\in SD(\lambda) \setminus H(\lambda) \\[3pt] \textrm{Re}(s_{ij})>1 \text{ for all } (i,j)\in H(\lambda) \end{array} \right. \!\!\!\right\}, \end{align*}$$

where $H(\lambda )=\{(i,j)\in SD(\lambda )~|~i-j\in \{k-\lambda _k~|~1\le k\le r\} \}$ . Following the proof of Theorem 3.6 and [Reference Nakasuji and Takeda17, Lemma 3.1], the following theorem can be proved.

Theorem 3.9 For any strict partition $\lambda =(\lambda _{1},\ldots ,\lambda _r)$ and $\boldsymbol s\in W_{\lambda ,H}^Q$ , we have

$$\begin{align*}\sum_{\mathrm{diag}}\zeta_{\lambda}^Q(\boldsymbol{s})=\sum_{\mathrm{diag}}\mathrm{pf}(M_{\lambda}),\end{align*}$$

where $M_{\lambda }$ is defined as in Theorem 3.6.

4 Pfaffian expression of the skew type Schur Q-multiple zeta functions

For the strict partitions $\lambda ,\mu $ , we write $\mu \le \lambda $ if $SD(\mu )\subset SD(\lambda )$ . For $\mu \le \lambda $ , the skew shifted diagram of $\lambda /\mu $ is defined as $SD(\lambda /\mu ) = SD(\lambda )\setminus SD(\mu )$ . We use the same notations $ST(\lambda /\mu ,X),ST^{\mathrm {diag}}(\lambda /\mu ,X)$ for a set X, and $PSST(\lambda /\mu )$ as in the previous sections.

Definition 4.1 (skew Schur P- and skew Q-multiple zeta functions)

Let ${\boldsymbol s}=(s_{ij})\in ST(\lambda /\mu ,\mathbb {C})$ . We define skew Schur P- and skew Q-multiple zeta functions associated with $\lambda /\mu $ by

(4.1) $$ \begin{align} \zeta_{\lambda/\mu}^P({\boldsymbol s}) =\sum_{M\in PSST(\lambda/\mu)}\frac{1}{M^{\boldsymbol s}}, \end{align} $$

and

(4.2) $$ \begin{align} \zeta_{\lambda/\mu}^Q({\boldsymbol s}) =\sum_{M\in QSST(\lambda/\mu)}\frac{1}{M^{\boldsymbol s}}. \end{align} $$

Let D be the directed graph defined above and $I_N$ be the same as in the previous section for a fixed positive integer N. We define two sequences of vertices ${\boldsymbol u=(u_{1},\ldots ,u_r)}$ and $\boldsymbol v=(v_{1},\ldots ,v_s)$ by $u_{i}=(\lambda _{i},N)$ and $v_{i}=(\mu _{i},0)$ . We define $\boldsymbol v\oplus I_N$ by the union of $\boldsymbol v$ and $I_N$ , ordered such that each $v_{i}$ precedes each $v\in I_N$ . Then the shifted Young tableaux of shape $\lambda /\mu $ with maximal entry N can be identified with the non-intersecting paths $(P_{1},\ldots ,P_r)$ in $\mathscr P_{0}(\boldsymbol u,\boldsymbol v\oplus I_N)$ , and $\boldsymbol u$ is D-compatible with $\boldsymbol v\oplus I_N$ such that $P_{i}\in \mathscr P(u_{i},v_{i})$ for $1\le i\le s$ and $P_{i}\in \mathscr P(u_{i},I_N)$ for $s< i\le r$ . The weights of paths are defined in the same way as in Section 3.

Example 4.1 Let $\lambda =(6,5,3,1)$ and $\mu =(3,1)$ . Then Figure 4 is a $4$ -tuple of non-intersecting paths $(P_{1},P_{2},P_{3},P_{4})\in \mathscr P_{0}(\boldsymbol u,\boldsymbol v\oplus I_5)$ .

Figure 4:

$(P_{1},P_{2},P_{3},P_{4})$ satisfying the condition in Example 4.1.

Let $(s_{ij})\in ST(\lambda /\mu ,\mathbb C)$ . The weights $w(P_{i})$ are

$$ \begin{align*} w(P_{1})&=\frac{1}{1^{s_{14}}3^{s_{15}}4^{s_{16}}},&&w(P_{2})= \frac{1}{1^{s_{23}}2^{s_{24}}4^{s_{25}}5^{s_{26}}},\\ w(P_{3})&=\frac{1}{2^{s_{33}}3^{s_{34}}5^{s_{35}}},&&w(P_{4})=\frac{1}{4^{s_{44}}}. \end{align*} $$

Then, we find that

$$\begin{align*}\zeta_{\lambda/\mu}^{Q,N}(\boldsymbol s)=\sum_{(P_{1},\ldots,P_r)\in \mathscr P_0(\boldsymbol u;\boldsymbol v\oplus I_N)}w(P_{1},\ldots,P_r).\end{align*}$$

As we proceed with a similar discussion as in Theorem 3.6 for the skew Schur Q-multiple zeta functions, in other word, applying the Stembridge method in [Reference John12, Theorem 3.2] to our case, we have the following result.

Theorem 4.2 (Pfaffian expression of the skew Schur Q-multiple zeta functions)

Let $\lambda =(\lambda _{1},\ldots ,\lambda _{r})$ , $\mu =(\mu _{1},\ldots ,\mu _{s})$ be strict partitions into with $\lambda _{i}\ge 0$ and $2|r+s$ . Then for $\boldsymbol s\in ST^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)$ ,

$$\begin{align*}\zeta_{\lambda/\mu}^Q(\boldsymbol s)=\mathrm{pf}\begin{pmatrix} M_{\lambda} & H_{\lambda,\mu}\\ 0&0 \end{pmatrix} ,\end{align*}$$

where $M_{\lambda }=(a_{ij})$ is an $r\times r$ upper triangular matrix with

$$\begin{align*}a_{ij}=\zeta_{(\lambda_{i},\lambda_{j})}^Q(\boldsymbol s_{(\lambda_{i},\lambda_{j})}), \end{align*}$$

where $t_{i}=i+\lambda _{i}-1$ and $H_{\lambda ,\mu }=(b_{ij})$ is an $r\times s$ matrix with

$$\begin{align*}b_{ij}=\zeta_{(\lambda_{i}-\mu_{s-j+1})}^Q(s_{i,i+j+\mu_s-1},\ldots,s_{i,t_{i}}).\end{align*}$$

Example 4.4 Let $\lambda =(3,2,1)$ and $\mu =(2)$ . Then, if $(a_{j-i})=(s_{ij})\in ST^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)$

5 Outside decomposition

Hamel and Goulden proved a general determinant formula which expressed a Schur function as a determinant of skew Schur functions whose shapes are strips ([Reference Hamel and Goulden9], see also [Reference Chen, Yan and Yang3]). Subsequently, Hamel proved expressions of Schur Q-functions as determinants or Pfaffians associated with the outside decomposition of shifted Young diagrams into strips [Reference Hamel7]. In their study of the multiple zeta function, Bachmann and Charlton proved general Jacobi–Trudi formulas for Schur multiple zeta functions for each outside decomposition. In fact, they proved the Jacobi–Trudi formula for more general functions [Reference Bachmann and Charlton1].

We first review the basic terminology of an outside decomposition given by Hamel and Goulden [Reference Hamel and Goulden9]. For each box $\alpha $ of skew (shifted) diagram of $\lambda /\mu $ , we define the content of $\alpha $ as the quantity $j-i$ where $\alpha $ lies in row i and in column j of the skew (shifted) diagram (conveniently referred to as $(i,j)$ ). A strip in a skew-shaped diagram is a skew (shifted) diagram with an edgewise connected set of boxes that contains no $2\times 2$ block of boxes. In other words, a strip has at most one box on each of its diagonals. We say that the starting box of a strip is the box that is bottommost and leftmost in the strip, and the ending box of a strip is the box which is topmost and rightmost in the strip.

Example 5.1 ( $\lambda =(5,4,2,1)$ )

We provide two examples of an outside decomposition $(\theta _{1},\ldots ,\theta _5)$ of $\lambda $ .

We now define an operation $\theta _{i}\#\theta _{j}$ of strips $\theta _{i}$ and $\theta _{j}$ in the same skew diagram. They are part of an outside decomposition. The following procedure is well-defined by [Reference Hamel8, Property 2.4].

1. Case.1 Suppose $\theta _{i}$ and $\theta _{j}$ have some boxes with the same content. Slide $\theta _{i}$ along top-left-to-bottom-right diagonals so that the box of content k of $\theta _{i}$ is superimposed on the box of content k of $\theta _{j}$ for all $k\in \mathbb Z$ such that both $\theta _{i}$ and $\theta _{j}$ admit a box of content k. We define $\theta _{i}\#\theta _{j}$ to be the diagram obtained from this superposition by taking all boxes between the ending box of $\theta _{i}$ and the starting box of $\theta _{j}$ inclusive.

2. Case.2 Suppose $\theta _{i}$ and $\theta _{j}$ are two disconnected pieces and thus do not have any boxes of the same content. The starting box of one will be to the right and/or above the ending box of the other. To bridge the gap between $\theta _{i}$ and $\theta _{j}$ , insert boxes from the ending box of the one to the starting box of the other such that these inserted boxes follow the approached-from-the-left or approached-from-below arrangement as do other boxes of the same content in the outside decomposition. If there exists content such that the diagram does not include a box with that content (and therefore no determination of the direction from which the box is approached), then arbitrarily choose from which direction boxes of this content should be approached, fix this choice for all boxes of the same content in that particular diagram, and bridge the gap between $\theta _{i}$ and $\theta _{j}$ accordingly. Define $\theta _{i}\#\theta _{j}$ as in Case 1 with the following additional conventions: if the ending box of $\theta _{i}$ is connected via an edge to the starting box of $\theta _{j}$ , and occurs below or to the left of it, then $\theta _{i}\#\theta _{j}=\emptyset $ ; if the ending box of $\theta _{i}$ is not edge connected but occurs below or to the left of the starting box of $\theta _{j}$ , $\theta _{i}\#\theta _{j}$ is undefined.

If $\boldsymbol s=(s_{ij})$ satisfies $s_{ij}=s_{k\ell }$ with $i-j=k-\ell $ , then we may define operation $ \boldsymbol s_{\lambda_{i}}\#\boldsymbol s_{\lambda_{j}} $ of $ \boldsymbol s_{\lambda_{i}} $ and $ \boldsymbol s_{\lambda_{j}} $ in the same manner with the operation $\theta_{i}\#\theta_{j} $ . We note that because $ \boldsymbol s=(s_{ij}) $ have constant entries on the diagonals, this procedure is well-defined.

Example 5.2 For the outside decomposition of the Young diagram $\lambda $ in the figure on the left in Example 5.1, for example, in $\theta _{4}\#\theta _{1}$ , moved below . The approached-from-below arrangement gives . Similarly, we have

where the numbers indicate contents.

Example 5.3 Let

For the outside decomposition of the shifted Young diagram $\lambda $ in Example 5.1 (the figure on the right in the example),

Hamel [Reference Hamel7] generalized the classical Pfaffian expression of the Schur Q-function involving outside decompositions. To explain the result, we extend the strips of our outside decomposition to the main diagonal. Let $\rho $ be a strip consisting of a single box of content $0$ so that , where the number indicates the content. This allows us to define $\overline {\theta _{i}}=\theta _{i}\#\rho $ . Let $(\overline \theta _p,\overline \theta _q)$ be formed by juxtaposing $\overline \theta _p$ and $\overline \theta _q$ with their boxes of content 0 lying on the main diagonal with that of $\overline \theta _p$ immediately above and to the left of $\overline \theta _q$ .

Example 5.4 The $\overline {\theta }_p$ and $(\overline {\theta }_p,\overline {\theta }_q)$ of the shifted Young diagram $\lambda $ in Example 5.1 (the figure on the right in the example) are $\overline {\theta _p}=\theta _p$ for $1\le p\le 4$ and

where the numbers indicate contents.

Proceeding with the discussion in terms of Schur Q-multiple zeta function following the method in [Reference Hamel7, Theorem 1.4] (cf. [Reference Foley and King4, Theorem 4.3]), we have the theorem below.

Theorem 5.5 Let $\lambda $ and $\mu $ be strict partitions with $\mu \le \lambda $ . Let $\theta =(\theta _{1}, \theta _{2},\ldots , \theta _k, \theta _{k+1},\ldots ,\theta _r)$ be an outside decomposition of $SD(\lambda /\mu )$ , where $\theta _p$ includes a box on the main diagonal of $SD(\lambda /\mu )$ for $1\le p\le k$ and $\theta _p$ does not for $k+1\le p\le r$ . If k is odd, we replace $\theta $ by $(\emptyset ,\theta _{1},\ldots ,\theta _r)$ . Then, for $\boldsymbol s\in ST^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)$ , the Schur Q-multiple zeta functions satisfy the identity

$$\begin{align*}\zeta_{\lambda/\mu}^Q(\boldsymbol s)=\mathrm{pf}\begin{pmatrix}\zeta_{(\overline{\theta}_p,\overline{\theta}_q)}^Q(\boldsymbol s_{(\overline{\theta}_p,\overline{\theta}_q)})&\zeta_{\theta_{i}\#\theta_{r+k+1-j}}^Q(\boldsymbol s_{\theta_{i}\#\theta_{r+k+1-j}})\\-{}^t(\zeta_{\theta_{i}\#\theta_{r+k+1-j}}^Q(\boldsymbol s_{\theta_{i}\#\theta_{r+k+1-j}}))&0\end{pmatrix} \end{align*}$$

with $1 \le p, q \le k$ and $k + 1\le j \le r$ . Here, if $(\overline {\theta }_p,\overline {\theta }_q)$ is not a shifted tableau, then we replace

$$\begin{align*}\zeta_{(\overline{\theta}_p,\overline{\theta}_q)}^Q(\boldsymbol s_{(\overline{\theta}_p,\overline{\theta}_q)})= -\zeta_{(\overline{\theta}_q,\overline{\theta}_p)}^Q(\boldsymbol s_{(\overline{\theta}_q,\overline{\theta}_p)}), \end{align*}$$

and further we put $\zeta_{(\overline{\theta}_p,\overline{\theta}_p)}^Q(\boldsymbol s_{(\overline{\theta}_p,\overline{\theta}_q)})=0 $ .

6 Sum formula

Multiple zeta values of the Euler–Zagier type are well known to satisfy a large number of linear relations among these multiple zeta values, such as the sum formula and the duality formula. The following is the sum formula for multiple zeta values of the Euler–Zagier type.

Theorem 6.1 (Granville [Reference Granville6], Zagier)

For positive integers k and r with $k>r$ , we have

$$ \begin{align*} \sum_{\substack{ k_{1}+\dots+k_{r}=k \\ k_{1},\dots,k_{r-1}\ge1,k_{r}\ge2 }} \zeta(k_{1},\dots,k_{r})&=\zeta(k),\\ \sum_{\substack{ k_{1}+\dots+k_{r}=k \\ k_{1},\dots,k_{r-1}\ge1,k_{r}\ge2 }} \zeta^\star(k_{1},\dots,k_{r})&=\binom{k-1}{r-1}\zeta(k). \end{align*} $$

As in the classical case, we prove the sum formula for a special case of Schur P- and Q-multiple zeta values.

Theorem 6.2 For positive integers k and r with $k>r$ , we have

and

Proof For , let $|\boldsymbol k|=k_{1}+\cdots+k_r $ and $\textrm {dep}(\boldsymbol k)=r$ . By (2.3), it suffices to show the first identity. Example 2.3 leads to

$$ \begin{align*} \sum_{\substack{ k_{1}+\dots+k_{r}=k \\ k_{1},\dots,k_{r-1}\ge1,k_{r}\ge2 }} \zeta_{(r)}^Q\left(\boldsymbol k\right) &=\sum_{\substack{ k_{1}+\dots+k_{r}=k \\ k_{1},\dots,k_{r-1}\ge1,k_{r}\ge2 }}\sum_{\boldsymbol \ell\preceq_s\boldsymbol k}2^{\mathrm{dep}({\boldsymbol \ell})}\zeta(\boldsymbol \ell)\\ &=\sum_{i=1}^r2^{i}\sum_{\substack{ k_{1}+\dots+k_{r}=k \\ k_{1},\dots,k_{r-1}\ge1,k_{r}\ge2 }}\sum_{\substack{ \boldsymbol \ell\preceq_s\boldsymbol k\\ \mathrm{dep}(\boldsymbol \ell)=i}}\zeta(\boldsymbol \ell). \end{align*} $$

For fixed $\boldsymbol \ell $ with $|\boldsymbol \ell |=k$ and $\mathrm {dep}(\boldsymbol \ell )=i$ , we count the number of $\boldsymbol k$ with $ \boldsymbol \ell \preceq _s\boldsymbol k$ with $\boldsymbol k\in ST((r),\mathbb Z)$ . Because $\boldsymbol k$ has to be admissible, it suffices to choose $r-i$ new division points of $\boldsymbol \ell $ out of $(k-1)-(i-1)-1$ possibilities. Therefore,

$$\begin{align*}\#\{\boldsymbol k \in ST((r),\mathbb C)~|~ \boldsymbol \ell\preceq_s\boldsymbol k\}=\binom{k-i-1}{r-i} \end{align*}$$

and we have

$$ \begin{align*} \sum_{\substack{ k_{1}+\dots+k_{r}=k \\ k_{1},\dots,k_{r-1}\ge1,k_{r}\ge2 }} \zeta_{(r)}^Q\left(\boldsymbol k\right) &=\sum_{i=1}^r2^{i}\binom{k-i-1}{r-i}\sum_{\substack{ |\boldsymbol \ell|=k\\ \mathrm{dep}(\boldsymbol \ell)=i}}\zeta(\boldsymbol \ell). \end{align*} $$

The sum formula for multiple zeta values of Euler–Zagier type leads to

$$ \begin{align*} \sum_{\substack{ k_{1}+\dots+k_{r}=k \\ k_{1},\dots,k_{r-1}\ge1,k_{r}\ge2 }} \zeta_{(r)}^Q\left(\boldsymbol k\right) &=\sum_{i=1}^r2^i\binom{k-i-1}{r-i}\zeta(k). \end{align*} $$

This proves the first identity. Dividing both sides by $2$ , we can confirm that the second identity holds. This completes the proof of the theorem.

Example 6.3 For $(k,r)=(5,3)$ , we have

We have the following corollaries of Theorem 6.2. The first is the sum formula in Schur P- or Q-multiple zeta values.

Corollary 6.4 For positive integers k and r with $k>r$ , we have

and

The next corollary is the duality formula for a certain shape and weight. Before we explain the duality property of the Schur Q-multiple zeta function, we review the original duality formula for multiple zeta functions. We denote a string $\underbrace {1,\ldots ,1}_r$ of $1$ ’s by $\{1\}^r$ . Then for an admissible index

$$\begin{align*}{\boldsymbol k}=(\{1\}^{a_{1}-1}, b_{1}+1, \{1\}^{a_{2}-1}, b_{2}+1, \ldots, \{1\}^{a_m-1}, b_m+1) \end{align*}$$

with positive integers $a_{1}, b_{1}, a_{2}, b_{2}, \cdots , a_m, b_m\in {\mathbb Z}_{\geq 1}$ , the following index is referred to as the dual index of ${\boldsymbol k}$ :

$$\begin{align*}{\boldsymbol k^{\dagger}}=(\{1\}^{b_m-1}, a_m+1, \{1\}^{b_{m-1}-1}, a_{m_{1}}+1, \ldots, \{1\}^{b_{1}-1}, a_{1}+1). \end{align*}$$

The duality formula is the following.

Theorem 6.7 (Duality formula [Reference Zagier23])

For any admissible index ${\boldsymbol k}=(k_{1}, \ldots , k_r)$ and its dual index ${\boldsymbol k^{\dagger}}=(k^{\dagger }_{1}., \ldots , k^{\dagger }_s)$ , we have

$$\begin{align*}\zeta(k_{1}, \ldots, k_r)=\zeta(k^{{\dagger}}_{1}, \ldots, k^{\dagger}_s). \end{align*}$$

As a special case of Theorem 6.7, it holds that

$$\begin{align*}\zeta(\{1\}^{k-2},2)=\zeta(k). \end{align*}$$

Taking $\lambda =(k-1)$ and , we have the following formula similar to the above identity.

Corollary 6.8 For positive integers k, we have

and

7 Symplectic Schur multiple zeta functions

First, we review the basic terminology to define symplectic or orthogonal Schur multiple zeta functions. We identify a partition $\lambda $ with the Young diagram

$$\begin{align*}D(\lambda)=\{(i, j)\in {\mathbb Z}^2 ~|~ 1\leq i\leq r, 1\leq j\leq \lambda_{i}\}\end{align*}$$

depicted as a collection of square boxes with the ith row having $\lambda _{i}$ boxes. For a partition $\lambda $ , a Young tableau $(t_{ij})$ of shape $\lambda $ over a set X is a filling of $D(\lambda )$ with $t_{ij}\in X$ into each box $(i,j)$ of $D(\lambda )$ . We denote by $T(\lambda ,X)$ the set of all Young tableaux of shape $\lambda $ over X.

Let $[\overline {N}]$ be the set $\{1,\overline {1},2,\overline {2},\ldots ,N,\overline {N}\}$ with the total ordering $1<\overline {1}<2<\overline {2}<\cdots <N<\overline {N}$ . Then, a symplectic tableau $\boldsymbol t=(t_{ij})\in T(\lambda ,[\overline {N}])$ is obtained by numbering all the boxes of $D(\lambda )$ with letters from $[\overline {N}]$ such that

1. SP1 the entries of $\boldsymbol t$ are weakly increasing along each row of $\boldsymbol t$ ,

2. SP2 the entries of $\boldsymbol t$ are strictly increasing down each column of $\boldsymbol t$ ,

3. SP3 for non-negative integer i, the boxes of content $-i$ contain entries that are greater than or equal to $i+1$ .

We refer to the third condition SP3 as the symplectic condition. We denote by $SP_N(\lambda )$ the set of symplectic tableaux of shape $\lambda $ .

Definition 7.1 (symplectic Schur multiple zeta functions)

For a given set ${ \boldsymbol s}=(s_{ij})\in T(\lambda ,\mathbb {C})$ of variables, the symplectic Schur multiple zeta functions of shape $\lambda $ are defined as

(7.1) $$ \begin{align} \zeta_{\lambda}^{\mathrm{sp},N}({ \boldsymbol s})=\sum_{M\in SP_N(\lambda)}\frac{1}{M^{ \boldsymbol s}}, \end{align} $$

where $M^{ \boldsymbol s}=\displaystyle {\prod _{(i, j)\in D(\lambda )}|m_{ij}|^{s_{ij}}}$ for $M=(m_{ij})\in SP_N(\lambda )$ and $|i|=i,|\overline {i}|=i^{-1}$ .

Hamel constructed a directed graph D corresponding to the symplectic Schur functions [Reference Hamel8] and applied the Stembridge theorem [Reference John12] to obtain the following determinant expression of the symplectic Schur functions.

Theorem 7.1 [Reference Hamel8, Theorem 3.1]

Let $\lambda /\mu $ be a partition of the skew type. Then, for any outside decomposition $(\theta _{1},\ldots ,\theta _r)$ of $\lambda /\mu $ ,

$$\begin{align*}\mathrm{sp}_{\lambda/\mu}=\mathrm{det}(\mathrm{sp}_{\theta_{i}\#\theta_{j}})_{1\le i,j\le r}.\end{align*}$$

Following the Hamel approach, we construct a directed graph D corresponding to the symplectic Schur multiple zeta functions. For a fixed positive integer N, we begin with the y-axis labeled by $1,\overline {1},2,\overline {2},\ldots , N, \overline {N}$ and direct an edge $u\rightarrow v$ whenever $v-u = (0, 1), (0,-1),(1, 0)$ , or $(1, -1)$ . We add four restrictions: A down-vertical step must not precede an up-vertical step, an up-vertical step must not precede a down-vertical step, a down-vertical step must not precede a horizontal step, and an up-vertical step must not precede a diagonal step. Because of the symplectic condition, we add a left boundary in the form of a “backwards lattice path” from $(0,1)$ to $(0,\overline {1})$ to $(0,2)$ to $(-1,2)$ to $(-1,\overline {2})$ to $(-1,3)$ to $(-2,3)$ to $(-2,\overline {3})$ to $(-2,4)$ to $(-3,4)\ldots $ . We indicate this left boundary by the dotted line in Figure 5.

Figure 5:

Left boundary given by the symplectic condition.

Hereinafter, we may omit this left boundary for simplicity.

For a fixed outside decomposition $(\theta _{1},\ldots ,\theta _r)$ of $\lambda /\mu $ , we construct a non-intersecting r-tuple of lattice paths that corresponds to a symplectic tableau of shape $\lambda /\mu $ with the outside decomposition $(\theta _{1},\ldots ,\theta _r)$ , such that the ith path corresponds to the ith strip and begins at $B_{i}$ and ends at $E_{i}$ as described next. Fix points ${B_{i} = (t-s,-(t-s)+1)}$ if the ith strip has the starting box $(s,t)$ on the left perimeter of the diagram and if $t-s\le 0$ (i.e., $B_{i}$ is on the left boundary), or $B_{i} = (t-s,1)$ if the ith strip has the starting box $(s,t)$ on the left perimeter of the diagram and if $t-s>0$ , or $B_{i} = (t-s,\infty )$ if the ith strip has the starting box $(s,t)$ on the bottom perimeter of the diagram ( $B_{i} = (t-s,\infty )$ if both). Fix points $E_{i} = (v-u+1,1)$ if the ith strip has the ending box $(u,v)$ on the top perimeter of the diagram, or $E_{i} = (v-u+1,\infty )$ if the ith strip has the ending box $(u,v)$ on the right perimeter of the diagram ( $E_{i} = (v-u+1,\infty )$ if both).

For the jth strip construct a path starting at $B_{j}$ (termed the starting point) and ending at $E_{j}$ (termed the ending point) as follows: if a box containing i (resp. $\overline {i}$ ) and at coordinates $(a,b)$ in the diagram is approached from the left in the strip, add a horizontal step from $(b-a, i)$ to $(b-a+1, i)$ (resp. $(b-a, \overline {i})$ to $(b-a+1, \overline {i})$ ); if a box containing i (resp. $\overline {i}$ ) and at coordinates $(a,b)$ in the diagram is approached from below in the strip, add a diagonal step from $(b-a, \overline {i})$ to $(b-a+1, i)$ (resp. ${(b-a, i+1)}$ to $(b-a+1, \overline {i})$ ). We note that the physical locations of the termination points of the steps are independent of the outside decomposition and depend only on the contents of the boxes. In Figure 6, the ending points of the steps are first shown alone and then the complete paths for two different outside decompositions are shown. We note that no two paths can have the same starting and/or ending points, because that would imply two boxes of the same content on the same section of the perimeter. Connect these non-vertical steps with vertical steps. This routine is intended to verify that a unique path exists. In the above setup, Hamel showed that the symplectic tableaux of shape $\lambda /\mu $ can be identified with the non-intersecting paths in $\mathscr P_{0}((B_{i});(E_{i}))$ , and $(B_{i})$ is D-compatible with $(E_{i})$ .

Figure 6:

$(P_{1},P_{2},P_{3},P_{4})$ satisfying the condition in Example 7.2.

We next define the weight of each step. Let $v_{i}(P)=(v_{i,j}(P))_{j\ge 0}$ be the sequence of vertices representing the path $P\in \mathscr P_{0}(B_{i};E_{i})$ and let $\ell ^i_{xy}$ be the edge $v_{i,j}(=(x,y))\rightarrow v_{i,j+1}$ . If $v_{i,j}(P)-v_{i,j+1}(P)= (1,0)$ or $(1,-1)$ , we assign the weights $w(\ell ^i_{xy})=|y|^{-s_{pq}}$ with $v_{ij}=(x,y)$ and $(p,q)$ being the jth component of $\theta _{i}$ . If $v_{i,j}(P)-v_{i,j+1}(P)= (0,1)$ or $(0,-1)$ , we assign the weights $w(\ell ^i_{xy})=1$ . Then, we define

$$\begin{align*}w(P_{i})=\prod_{\ell_{xy}^i}w(\ell_{xy}^i),\end{align*}$$

and for an r-tuple of non-intersecting paths of $(P_{1},\ldots ,P_r)$ with $P_{i}\in \mathscr P(B_{i};E_{i})$ ,

$$\begin{align*}w(P_{1},\ldots,P_r)=\prod_{i=1}^rw(P_{i}).\end{align*}$$

Then, owing to the Hamel composition in [Reference Hamel8], we find that

$$\begin{align*}\zeta_{\lambda}^{\mathrm{sp},N}(\boldsymbol s)=\sum_{P_{i}\in \mathscr P(B_{i};E_{i})}w(P_{1},\ldots,P_r).\end{align*}$$

Example 7.2 For $\lambda =(5,3,3,1)$ , let a $4$ -tuple of paths $(P_{1},P_{2},P_{3},P_{4})\in \mathscr P(B_{i};E_{i})$ be given as in Figure 6. For $(s_{ij})\in T(\lambda ,\mathbb C)$ , the weights $w(P_{i})$ are

$$ \begin{align*} w(P_{1})&=1^{s_{11}},&&w(P_{2})=\frac{2^{s_{21}}}{3^{s_{22}}2^{s_{12}}},\\ w(P_{3})&=\frac{3^{s_{32}}3^{s_{33}}}{4^{s_{41}}3^{s_{31}}3^{s_{23}}2^{s_{13}}},&&w(P_{4})=\frac{3^{s_{14}}}{4^{s_{15}}}, \end{align*} $$

and the corresponding symplectic tableau is

As in the proof of Theorem 3.6, proceeding with the discussion in terms of the symplectic Schur multiple zeta function, following the Hamel method in Theorem 7.1, we have the theorem below.

Theorem 7.3 Let $\lambda =(\lambda _{1},\ldots ,\lambda _{r})$ , $\mu =(\mu _{1},\ldots ,\mu _{s})$ be partitions. Then, for $\boldsymbol s\in T^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)$ and any outside decomposition $(\theta _{1},\ldots ,\theta _r)$ of $\lambda /\mu $ ,

$$\begin{align*}\zeta_{\lambda/\mu}^{\mathrm{sp},N}(\boldsymbol s)=\mathrm{det}(\zeta_{\theta_{i}\#\theta_{j}}^{\mathrm{ sp},N}(\boldsymbol s_{(\lambda_{i},\lambda_{j})}))_{1\le i,j\le r}, \end{align*}$$

where $\boldsymbol s_{(\lambda_{i},\lambda_{j})}=\boldsymbol s_{\lambda_{i}}\#\boldsymbol s_{\lambda_{j}} $ .

Example 7.4 Let $\lambda =(3,2)$ and its outside decomposition $(\theta _{1},\theta _{2})$ be depicted as

Then, if $(a_{j-i})=(s_{ij})\in T^{\mathrm {diag}}(\lambda ,\mathbb C)$ ,

Remark 7.5 The function in Example 7.4 satisfies

in general. We note that for $i=-1,0,1,2$ the contents of each are not the same.

8 Orthogonal Schur multiple zeta functions

Hamel also constructed a directed graph D corresponding to the orthogonal Schur functions [Reference Hamel8] and derived the determinant expression of the orthogonal Schur functions. As in Section 7, we construct a directed graph D corresponding to the orthogonal Schur multiple zeta functions. As in Section 7, we prove the results corresponding to the following Hamel result.

We define orthogonal Schur multiple zeta functions. Let $[\overline {N}]^{\infty }$ be the set $\{1,\overline {1},2,\overline {2},\ldots $ , $N,\overline {N},\infty \}$ with the total ordering $1<\overline {1}<2<\overline {2}<\cdots <N<\overline {N}<\infty $ . For a fixed partition $\lambda $ , a so-tableau $\boldsymbol t=(t_{ij})\in T(\lambda ,[\overline {N}]^{\infty })$ is obtained by numbering all the boxes of $D(\lambda )$ with letters from $[\overline {N}]^{\infty }$ such that

1. SO1 the entries of $\boldsymbol t$ are weakly increasing along each row of $\boldsymbol t$ ,

2. SO2 the entries of $\boldsymbol t$ are strictly increasing down each column of $\boldsymbol t$ ,

3. SO3 for non-negative integer i, the boxes of content $-i$ contain entries which are greater than or equal to $i+1$ ,

4. SO4 no two symbols $\infty $ appear in the same row.

One may find that the conditions SO1–SO3 are the same as SP1–SP3. We denote by $SO_N(\lambda )$ the set of so-tableaux of shape $\lambda $ .

Definition 8.1 (orthogonal Schur multiple zeta functions)

For a given set ${ \boldsymbol s}=(s_{ij})\in T(\lambda ,\mathbb {C})$ of variables, the orthogonal Schur multiple zeta functions of shape $\lambda $ are defined as

(8.1) $$ \begin{align} \zeta_{\lambda}^{\mathrm{so},N}({ \boldsymbol s})=\sum_{M\in SO_N(\lambda)}\frac{1}{M^{ \boldsymbol s}}, \end{align} $$

where we set $|\infty |=1$ .

We note that the $\infty $ contributes $1$ to the weight of the tableau. Therefore, they are “dummy elements” in a sense.

Theorem 8.1 [Reference Hamel8, Theorem 3.2]

Let $\lambda /\mu $ be a partition of the skew type. Then, for any outside decomposition $(\theta _{1},\ldots ,\theta _r)$ of $\lambda /\mu $ ,

$$\begin{align*}\mathrm{so}_{\lambda/\mu}=\mathrm{det}(\mathrm{so}_{\theta_{i}\#\theta_{j}})_{1\le i,j\le r}.\end{align*}$$

As in the symplectic Schur multiple zeta functions, we consider the y-axis with $1,\overline {1},2,\overline {2},\ldots ,N,\overline {N},\infty $ . We define lattice paths with five types of permissible steps. These steps are the four steps in Section 7, and up-diagonal steps from height $\overline {N}$ to height $\infty $ that increase the x- and y-coordinates by $1$ , respectively. We distinguish between horizontal steps at integer levels and horizontal steps at $\infty $ . The steps are subject to the same restrictions as in Section 7 plus the following additional restrictions: an up-vertical step must not precede a horizontal step at $\infty $ , and a down-vertical step must not precede an up-diagonal step. We also require that all steps between lines $x=c$ and $x=c+1$ for all c are either

1. (1) horizontal at $\infty $ or down-diagonal, or

2. (2) horizontal at integer levels or up-diagonal.

Determining whether the steps are of type (1) or (2) depends on the outside decomposition: if boxes of content c are approached from the left, then the steps between $x=c$ and $x=c+1$ must be of type (2); if the boxes of content c are approached from below, then the steps between $x=c$ and $x=c+1$ must be of type (1). We fix beginning points $B_{i}$ and ending points, $E_{i}$ as in Section 7 with the adjustment that the y-coordinate of the highest points is $\infty +1$ instead of $\infty $ . Given $\boldsymbol s\in SO(\lambda /\mu ,\mathbb C)$ with an outside decomposition, we can construct an r-tuple of non-intersecting lattice paths. For each strip, construct a path as follows: if a box contains i or $\overline {i}$ , place a step as in the proof of Section 7. If a box contains $\infty $ , is at coordinates $(a,b)$ in the diagram, and is approached from the left in the strip, add an up-diagonal step from $(a-b, \overline {N})$ to $(a-b+1,\infty )$ ; if it is approached from below, add a horizontal step from $(a-b, \infty )$ to $(a-b+1,\infty )$ . We connect these non-vertical paths with vertical paths. The weights of paths are defined in the same way as in Section 7. Note that we put $w(\ell _{xy}^i)=1$ if $y=\infty $ .

Then, owing to the Hamel composition [Reference Hamel8], we find that

$$\begin{align*}\zeta_{\lambda}^{\mathrm{so},N}(\boldsymbol s)=\sum_{P_{i}\in \mathscr P(B_{i};E_{i})}w(P_{1},\ldots,P_r).\end{align*}$$

Example 8.2 For $\lambda =(5,3,3,1)$ , let $(s_{ij})\in T(\lambda ,\mathbb C)$ . For a 4-tuple of paths ( $P_1,P_2,P_3,P_4$ ) given in Figure 7, the weights $w(P_{i})$ are

$$ \begin{align*} w(P_{1})&=1^{s_{11}},&&w(P_{2})=\frac{2^{s_{21}}}{3^{s_{22}}2^{s_{12}}},\\ w(P_{3})&=\frac{3^{s_{32}}3^{s_{33}}}{3^{s_{31}}3^{s_{23}}2^{s_{13}}},&&w(P_{4})=3^{s_{24}}3^{s_{14}}, \end{align*} $$

Figure 7:

$(P_{1},P_{2},P_{3},P_{4})$ satisfying the condition in Example 8.2.

and the corresponding orthogonal tableau is

As in the previous sections, proceeding with the discussion in terms of the orthogonal Schur multiple zeta function following the Hamel method in Theorem 8.1, we have the theorem below.

Theorem 8.3 Let $\lambda =(\lambda _{1},\ldots ,\lambda _{r})$ , $\mu =(\mu _{1},\ldots ,\mu _{s})$ be partitions. Then, for ${\boldsymbol s\in T^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)}$ and any outside decomposition $(\theta _{1},\ldots ,\theta _r)$ of $\lambda /\mu $ ,

$$\begin{align*}\zeta_{\lambda/\mu}^{\mathrm{so},N}(\boldsymbol s)=\mathrm{det}(\zeta_{\theta_{i}\#\theta_{j}}^{\mathrm{ so},N}(\boldsymbol s_{(\lambda_{i},\lambda_{j})}))_{1\le i,j\le r}, \end{align*}$$

where $ \boldsymbol s_{(\lambda_{i},\lambda_{j})}=\boldsymbol s_{\lambda_{i}}\#\boldsymbol s_{\lambda_{j}} $ .

Example 8.4 Let $\lambda =(3,2)$ and its outside decomposition $(\theta _{1},\theta _{2})$ be depicted as

Then, if $(a_{j-i})=(s_{ij})\in T^{\mathrm {diag}}(\lambda ,\mathbb C)$ , we obtain

9 Decomposition of symplectic and orthogonal multiple zeta functions

In this section, we express a symplectic and an orthogonal multiple zeta function as a linear combination of the truncated multiple zeta functions. Analogous to the method of the proof of (2.5) and (2.6), by the Inclusion–Exclusion principle, we may find the following decompositions.

Theorem 9.1 For any positive integer N and $\boldsymbol s\in T(\lambda ,\mathbb C)$ , the function $\zeta _{\lambda }^{\square ,N}(\boldsymbol s)$ for ${\square \in \{\mathrm {sp},\mathrm {so}\}}$ can be decomposed as a sum of truncated multiple zeta functions: for a positive integer N,

$$\begin{align*}\zeta^N(s_{1},\ldots,s_r)=\sum_{1\le n_{1}<\cdots< n_r\le N}\frac{1}{n_{1}^{s_{1}}\cdots n_r^{s_r}}. \end{align*}$$

Example 9.2 For any positive integer N and $a,b\in \mathbb C$ , we have

Example 9.3 For any positive integer N and $a,b\in \mathbb C$ , we have

Note that in the case of $\zeta _{(\{1\}^r)}^{\mathrm {sp},N}$ , we have

where $\displaystyle {\sum _{\mathrm {sign}}}$ means the summation over all cases of plus-minus signs and the last term $(\cdots )$ is caused from $\zeta ^N$ ’s whose elements contain at least two different $(s_{i}-s_{i+1})$ ’s like $\zeta ^N(s_{1}-s_{2},-s_{3},s_{4}-s_{5},s_6-s_7,s_{8})$ .

If we use the decompositions by rows as an outside decomposition of $\lambda /\mu $ , then for any $\boldsymbol s\in T^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)$ , $\zeta _{\lambda /\mu }^{\mathrm {sp},N}(\boldsymbol s)$ and $\zeta _{\lambda /\mu }^{\mathrm {so},N}(\boldsymbol s)$ appear to be decomposed into a sum of $\zeta _{(\{1\}^r)}^{\mathrm {sp},N}$ and $\zeta _{(\{1\}^r)}^{\mathrm {so},N}$ , respectively. As in Remark 7.5, we note that the outside decomposition and operation $\theta _{i}\#\theta _{j}$ retains the content and two different functions may be associated with the same shape $\lambda =(\{1\}^r)$ and the same variable $\boldsymbol s=(s_{ij})$ .

Similarly, we attain the following results, in which we decompose the symplectic zeta function into the sum of truncated multiple zeta star functions: for a positive integer N,

$$\begin{align*}\zeta^{\star N}(s_{1},\ldots,s_r)=\sum_{1\le n_{1}<\cdots< n_r}\frac{1}{n_{1}^{s_{1}}\cdots n_r^{s_r}},\ \zeta^\star(s_{1},\ldots,s_r)=\sum_{1\le n_{1}\le\cdots\le n_r}\frac{1}{n_{1}^{s_{1}} \cdots n_r^{s_r}} \end{align*}$$

Theorem 9.4 For any positive integer N and $s_{i}\in \mathbb C$ , we have

and for $r\ge 2$

where $ \displaystyle{\sum_{\mathrm{sign}}} $ means the summation over all cases of plus-minus signs and $\boldsymbol \ell $ runs over all indices of the form $ \boldsymbol \ell_R = (\pm s_{1}\square \pm s_{2}\square\cdots\square \pm s_{R}) $ in which each $\square $ is filled by the comma, or the plus sign $+$ . If $\square =+$ then $\pm s_{j}\square\pm s_{j+1} $ is assigned $s_{j+1}-s_{j}$ and the square is not filled with consecutive plus signs $+$ .

Example 9.5 ( $r\le 2$ )

For any positive integer N and $a,b\in \mathbb C$ , we have

10 Schur quasi-symmetric functions

We here investigate the quasi-symmetric functions, introduced by Gessel [Reference Gessel5], related to symmetric multiple zeta functions defined in this article. We note that the Schur type quasi-symmetric function was discussed in [Reference Nakasuji, Phuksuwan and Yamasaki16].

10.1 Quasi-symmetric functions

Let ${\boldsymbol t}=(t_{1},t_{2},\ldots )$ be variables and $\mathfrak {P}$ the subalgebra of $\mathbb {Z}[\![t_{1},t_{2},\ldots \,]\!]$ consisting of all formal power series with integer coefficients of bounded degree. We refer to ${p=p({\boldsymbol t})\in \mathfrak {P}}$ as a quasi-symmetric function if the coefficient of $t^{\gamma _{1}}_{k_{1}}t^{\gamma _{2}}_{k_{2}}\cdots t^{\gamma _n}_{k_n}$ of p is the same as that of $t^{\gamma _{1}}_{h_{1}}t^{\gamma _{2}}_{h_{2}}\cdots t^{\gamma _n}_{h_n}$ of p whenever $k_{1}<k_{2}<\cdots <k_n$ and $h_{1}<h_{2}<\cdots <h_n$ . The algebra of all quasi-symmetric functions is denoted by $\mathrm {Qsym}$ . For a composition ${{\boldsymbol \gamma }=(\gamma _{1},\gamma _{2},\ldots ,\gamma _n)}$ of a positive integer, define the monomial quasi-symmetric function $M_{\boldsymbol \gamma }$ and the essential quasi-symmetric function $E_{\boldsymbol \gamma }$ , respectively, by

$$\begin{align*}M_{\boldsymbol \gamma}=\sum_{m_{1}<m_{2}<\cdots<m_n}t_{m_{1}}^{\gamma_{1}}t_{m_{2}}^{\gamma_{2}}\cdots t^{\gamma_n}_{m_n},\quad E_{\boldsymbol \gamma}=\sum_{m_{1}\le m_{2}\le \cdots \le m_n}t_{m_{1}}^{\gamma_{1}}t_{m_{2}}^{\gamma_{2}}\cdots t^{\gamma_n}_{m_n}. \end{align*}$$

We know that these respective functions form the integral basis of $\mathrm {Qsym}$ . Notice that

(10.1) $$ \begin{align} E_{\boldsymbol \gamma} =\sum_{{\boldsymbol \delta} \,\preceq\, {\boldsymbol \gamma}}M_{\boldsymbol \delta}, \quad M_{\boldsymbol \gamma} =\sum_{{\boldsymbol \delta} \,\preceq\, {\boldsymbol \gamma}}(-1)^{n-\ell({\boldsymbol \delta})}E_{\boldsymbol \delta}. \end{align} $$

The relation between the multiple zeta values and quasi-symmetric functions was studied by Hoffman [Reference Hoffman10] (remark that the notations used for the multiple zeta (star) function in [Reference Hoffman10] are different from ours, which are $\zeta (s_n,s_{n-1},\ldots ,s_{1})$ and $\zeta ^{\star }(s_n,s_{n-1},\ldots ,s_{1})$ , respectively). Let $\mathfrak {H}=\mathbb {Z}\langle x,y\rangle $ be the noncommutative polynomial algebra over $\mathbb {Z}$ . We can define a commutative and associative multiplication $\ast $ , known as the harmonic product, on $\mathfrak {H}$ . We refer to $(\mathfrak {H},\ast )$ as (integral) harmonic algebra. Let $\mathfrak {H}^{1}=\mathbb {Z}1+y\mathfrak {H}$ , which is a subalgebra of $\mathfrak {H}$ . Notice that every $w\in \mathfrak {H}^{1}$ can be written as an integral linear combination of $z_{\gamma _{1}}z_{\gamma _{2}}\cdots z_{\gamma _n}$ where $z_{\gamma }=yx^{\gamma -1}$ for $\gamma \in \mathbb {N}$ . For each $N\in \mathbb {N}$ , define the homomorphism $\phi _N:\mathfrak {H}^{1}\to \mathbb {Z}[t_{1},t_{2},\ldots ,t_N]$ by $\phi _N(1)=1$ and

$$\begin{align*}\phi_N(z_{\gamma_{1}}z_{\gamma_{2}}\cdots z_{\gamma_n}) = \begin{cases} \displaystyle{\sum_{m_{1}<m_{2}<\cdots<m_n\le N}t_{m_{1}}^{\gamma_{1}}t_{m_{2}}^{\gamma_{2}}\cdots t^{\gamma_n}_{m_n}} & n\le N, \\ 0 & \text{otherwise}, \end{cases} \end{align*}$$

and extend it additively to $\mathfrak {H}^{1}$ . There exists a unique homomorphism $\phi :\mathfrak {H}^{1}\to \mathfrak {P}$ such that $\pi _N\phi =\phi _N$ , where $\pi _N$ is the natural projection from $\mathfrak {P}$ to $\mathbb {Z}[t_{1},t_{2},\ldots ,t_N]$ . We have $\phi (z_{\gamma _{1}}z_{\gamma _{2}}\cdots z_{\gamma _n})=M_{(\gamma _{1},\gamma _{2},\ldots ,\gamma _n)}$ . Moreover, as described in [Reference Hoffman10], $\phi $ is an isomorphism between $\mathfrak {H}^1$ and $\mathrm {Qsym}$ .

Let e be the function that sends $t_{i}$ to $\frac {1}{i}$ . Moreover, define $\rho _N:\mathfrak {H}^{1}\to \mathbb {R}$ by $\rho _N=e\phi _N$ . For a composition ${\boldsymbol \gamma }$ , we have

$$\begin{align*}\rho_N\phi^{-1}(M_{\boldsymbol \gamma})=\zeta^{N}({\boldsymbol \gamma}), \quad \rho_N\phi^{-1}(E_{\boldsymbol \gamma})=\zeta^{\star N}({\boldsymbol \gamma}). \end{align*}$$

We define the map $\rho :\mathfrak {H}^1\to \mathbb {R}^{\mathbb {N}}$ by $\rho (w)=(\rho _N(w))_{N\ge 1}$ for $w\in \mathfrak {H}^{1}$ . Notice that if ${w\in \mathfrak {H}^0=\mathbb {Z}1+y\mathfrak {H}x}$ , which is a subalgebra of $\mathfrak {H}^1$ , then we may understand that $\rho (w)=\lim _{N\to \infty }\rho _N(w)\in \mathbb {R}$ . In particular, for a composition ${\boldsymbol \gamma }=(\gamma _{1},\gamma _{2},\ldots ,\gamma _n)$ with $\gamma _n\ge 2$ , we have

(10.2) $$ \begin{align} \rho\phi^{-1}(M_{\boldsymbol \gamma}) =\zeta({\boldsymbol \gamma}), \quad \rho\phi^{-1}(E_{\boldsymbol \gamma}) =\zeta^{\star}({\boldsymbol \gamma}). \end{align} $$

10.2 Schur P- and Q-type quasi-symmetric functions

Now, the following Schur P- and Q-type quasi-symmetric functions are easily defined. For strict partitions $\lambda $ and $\mu $ , and ${\boldsymbol s}=(s_{ij})\in ST(\lambda /\mu ,\mathbb {C})$ , we define Schur P- and Q-type quasi-symmetric functions associated with $\lambda /\mu $ by

(10.3) $$ \begin{align} S_{\lambda/\mu}^P({\boldsymbol s}) =\sum_{M\in PSST(\lambda/\mu)}\displaystyle{\prod_{(i, j)\in SD(\lambda)}t_{|m_{ij}|}^{s_{ij}}}, \end{align} $$

and

(10.4) $$ \begin{align} S_{\lambda/\mu}^Q({\boldsymbol s}) =\sum_{M\in QSST(\lambda/\mu)}\displaystyle{\prod_{(i, j)\in SD(\lambda)}t_{|m_{ij}|}^{s_{ij}}}. \end{align} $$

Theorem 10.1 Let $\lambda =(\lambda _{1},\ldots ,\lambda _{r})$ , $\mu =(\mu _{1},\ldots ,\mu _{s})$ be strict partitions into with ${\lambda _{i}\ge 0}$ and $2|r+s$ . Then, for $\boldsymbol s\in ST^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)$ ,

$$\begin{align*}S_{\lambda/\mu}^Q(\boldsymbol s)=\mathrm{pf}\begin{pmatrix}M_{\lambda} &H_{\lambda,\mu}\\0&0\end{pmatrix}, \end{align*}$$

where $M_{\lambda }=(a_{ij})$ is an $r\times r$ upper triangular matrix with

$$\begin{align*}a_{ij}=S_{(\lambda_{i},\lambda_{j})}^Q(\boldsymbol s_{(\lambda_{i},\lambda_{j})}), \end{align*}$$

where $t_{i}=i+\lambda _{i}-1$ and $H_{\lambda }=(b_{ij})$ is an $r\times s$ matrix with

$$\begin{align*}b_{ij}=S_{(\lambda_{i}-\mu_s-j+1)}^Q(s_{i(i+j+\mu_s-1)},\ldots,s_{it_{i}}).\end{align*}$$

Theorem 10.2 (cf. [Reference Foley and King4, Theorem 4.3],[Reference Hamel7, Theorem 1.4])

Let $\lambda $ and $\mu $ be strict partitions with $\mu \le \lambda $ . Let $\theta =(\theta _{1}, \theta _{2},\ldots , \theta _k, \theta _{k+1},\ldots ,\theta _r)$ be an outside decomposition of $SD(\lambda /\mu )$ , where $\theta _p$ includes a box on the main diagonal of $SD(\lambda /\mu )$ for $1\le p\le k$ and $\theta _p$ does not for $k+1\le p\le r$ . If k is odd, we replace $\theta $ by $(\emptyset ,\theta _{1},\ldots ,\theta _r)$ . Then, the Schur Q-type quasi-symmetric functions satisfy the identity

$$\begin{align*}S_{\lambda/\mu}^Q(\boldsymbol s)=\mathrm{pf}\begin{pmatrix}S_{(\overline{\theta}_p,\overline{\theta}_q)}^Q(\boldsymbol s_{(\overline{\theta}_p,\overline{\theta}_q)})&S_{\theta_{i}\#\theta_{r+k+1-j}}^Q(\boldsymbol s_{\theta_{i}\#\theta_{r+k+1-j}})\\-{}^t(S_{\theta_{i}\#\theta_{r+k+1-j}}^Q(\boldsymbol s_{\theta_{i}\#\theta_{r+k+1-j}}))&0\end{pmatrix} \end{align*}$$

with $1 \le p, q \le k$ and $k + 1\le j \le r$ . Here

$$\begin{align*}S_{(\overline{\theta}_p,\overline{\theta}_q)}^Q(\boldsymbol s_{(\overline{\theta}_p,\overline{\theta}_q)})= -S_{(\overline{\theta}_q,\overline{\theta}_p)}^Q(\boldsymbol s_{(\overline{\theta}_q,\overline{\theta}_p)}) \end{align*}$$

and $S_{(\overline{\theta}_p,\overline{\theta}_p)}^Q(\boldsymbol s_{(\overline{\theta}_p,\overline{\theta}_q)})=0 $ .

10.3 Symplectic type and orthogonal type quasi-symmetric functions

Similarly, we define the following symplectic quasi-symmetric functions and orthogonal quasi-symmetric functions. For partitions $\lambda $ and $\mu $ , and ${\boldsymbol s}=(s_{ij})\in T(\lambda /\mu ,\mathbb {C})$ , we define symplectic quasi-symmetric functions and orthogonal quasi-symmetric functions associated with $\lambda /\mu $ by

(10.5) $$ \begin{align} S_{\lambda/\mu}^{\mathrm{sp},N}({\boldsymbol s}) =\sum_{M\in SP_N(\lambda/\mu)}\displaystyle{\prod_{(i, j)\in D(\lambda)}t_{|m_{ij}|}^{s_{ij}}}, \end{align} $$

and

(10.6) $$ \begin{align} S_{\lambda/\mu}^{\mathrm{so},N}({\boldsymbol s}) =\sum_{M\in SO_N(\lambda/\mu)}\displaystyle{\prod_{(i, j)\in D(\lambda)}t_{|m_{ij}|}^{s_{ij}}}. \end{align} $$

Theorem 10.3 Let $\lambda =(\lambda _{1},\ldots ,\lambda _{r})$ , $\mu =(\mu _{1},\ldots ,\mu _{s})$ be partitions. Then, for ${\boldsymbol s\in T^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)}$ and any outside decomposition $(\theta _{1},\ldots ,\theta _r)$ of $\lambda /\mu $ ,

$$\begin{align*}S_{\lambda/\mu}^{\mathrm{sp},N}(\boldsymbol s)=\mathrm{det}(S_{\theta_{i}\#\theta_{j}}^{\mathrm{sp},N}(\boldsymbol s_{(\lambda_{i},\lambda_{j})}))_{1\le i,j\le r}, \end{align*}$$

where $\boldsymbol s_{(\lambda_{i},\lambda_{j})}=\boldsymbol s_{\lambda_{i}}\#\boldsymbol s_{\lambda_{j}} $ .

Theorem 10.4 Let $\lambda =(\lambda _{1},\ldots ,\lambda _{r})$ , $\mu =(\mu _{1},\ldots ,\mu _{s})$ be partitions. Then, for ${\boldsymbol s\in T^{\mathrm {diag}}(\lambda /\mu ,\mathbb C)}$ and any outside decomposition $(\theta _{1},\ldots ,\theta _r)$ of $\lambda /\mu $ ,

$$\begin{align*}S_{\lambda/\mu}^{\mathrm{so},N}(\boldsymbol s)=\mathrm{det}(S_{\theta_{i}\#\theta_{j}}^{\mathrm{so},N}(\boldsymbol s_{(\lambda_{i},\lambda_{j})}))_{1\le i,j\le r}, \end{align*}$$

where $\boldsymbol s_{(\lambda_{i},\lambda_{j})}=\boldsymbol s_{\lambda_{i}}\#\boldsymbol s_{\lambda_{j}} $ .