A Trigonometric Variant of Kaneko–Tsumura ψ-Values

Abstract

Many variations of the multiple zeta values have been found to play important roles in different branches of mathematics and theoretical physics in recent years, such as the cyclotomic/color version, which appears prominently in the computation of Feynman integrals. In this paper, we introduce a trigonometric variant of the Kaneko–Tsumura $\psi$-function (called the Kaneko–Tsumura $\tilde{\psi }$-function) and discover some nice properties similar to those for ordinary Kaneko–Tsumura $\psi$-values using the method of iterated integrals, which was first studied systematically by K.T. Chen in the 1960s. In particular, we establish some duality formulas involving the Kaneko–Tsumura $\tilde{\psi }$-function and alternating multiple T-values by adapting Yamamoto’s graphical representation method for computing special types of iterated integrals.

1. Introduction

The recent revival in the study of multiple zeta values and their various generalizations has its genesis rooted in their numerous highly non-trivial and often surprising connections with other branches of mathematics and theoretical physics. After the seminal works of Hoffman [1] and Zagier [2] in the early 1990s, many mathematicians as well as physicists have been working in this very fruitful research area. For example, Broadhurst revealed their relations to knot invariants and Feynman diagram computations in [3,4], Brown [5] proved multiple zeta values essentially provide all the periods of the mixed Tate motives over $\mathbb{Z}$, and Zudilin [6] investigated their q-analogs and the connections to q-series.

One of the most recent variations of multiple zeta values is the so-called even–odd variations developed by Hoffman [7], Kaneko and Tsumura [8], and the last two authors of the current paper [9]. These objects have proved to be closely related to the special values we will study in this paper [10,11,12,13,14,15].

We now, first, introduce some basic notations. Let $\mathbb{N}$ be the set of positive integers. We call a finite sequence $k:=(k_1,\dots ,k_r)\in {\mathbb{N}}^r$ a composition, $\left|k\right|:=k_1+\cdots +k_r$ its weight, and r its depth. We say k is admissible if $k_r>1$. Denote by ${\left\{m\right\}}_r$ the sequence of m’s with r repetitions.

In a series of correspondence with Goldbach, Euler [16] first considered double zeta values. In fact, he studied their star version. Recall that for an admissible composition $k=(k_1,\dots ,k_r)$, the multiple zeta values and the multiple zeta-star values are defined by

$$\begin{array}{cc}\hfill & \zeta \left(k\right):=\sum _{0<m_1<\cdots <m_r}{\displaystyle \frac{1}{m_1^{k_1}\cdots m_r^{k_r}}}\mathrm{and}{\zeta }^{★}\left(k\right):=\sum _{0<m_1\le \cdots \le m_r}{\displaystyle \frac{1}{m_1^{k_1}\cdots m_r^{k_r}}},\hfill \end{array}$$

respectively. It is clear that $\zeta \left(k\right)$ can be regarded as a special value ($z\to 1^{-}$) of the multiple polylogarithm function defined by

$$\begin{array}{cc}\hfill & {Li}_k\left(z\right):=\sum _{1\le n_1<\cdots <n_r}{\displaystyle \frac{z^{n_r}}{n_1^{k_1}\cdots n_r^{k_r}}},z\in [-1,1).\hfill \end{array}$$

For a (unnecessarily admissible) composition $k=(k_1,\dots ,k_r)$, the Arakawa–Kaneko zeta function (see [10]) is defined by

$$\begin{array}{c}\hfill \xi (k;s):={\displaystyle \frac{1}{\Gamma \left(s\right)}}\underset{0}{\overset{\infty }{\int }}{\displaystyle \frac{t^{s-1}}{e^t-1}}{Li}_k(1-e^{-t})dt,\Re \left(s\right)>0,\end{array}$$

Further, Kaneko and Tsumura [11] introduced and studied a new kind of Arakawa–Kaneko type function, called the Kaneko–Tsumura η-function:

$$\begin{array}{c}\hfill \eta (k;s):={\displaystyle \frac{1}{\Gamma \left(s\right)}}\underset{0}{\overset{\infty }{\int }}{\displaystyle \frac{t^{s-1}}{1-e^t}}{Li}_k(1-e^t)dt,\Re \left(s\right)>0.\end{array}$$

Many works have already been devoted to the study of Arakawa–Kaneko zeta values and Kaneko–Tsumura $\eta$-values, which are all intimately related to multiple zeta values and some of their variations. For example, in [10,11], the special values of $\xi (k;s)$ and $\eta (k;s)$ at positive integers s are analytically computed and expressed in terms of multiple zeta values and multiple zeta-star values. These values also appear in the computation of some other special constants and polynomials [17,18,19,20] and even unexpectedly in graph theory [21,22].

Using some well-known results for Arakawa–Kaneko zeta values, Kaneko and Ohno [23] proved the following duality property of multiple zeta-star values:

$$\begin{array}{c}\hfill {(-1)}^k{\zeta }^{★}({\left\{1\right\}}_p,k+1)-{(-1)}^p{\zeta }^{★}({\left\{1\right\}}_k,p+1)\in \mathbb{Q}\left[\zeta \left(2\right),\zeta \left(3\right),\zeta \left(4\right),\dots \right],\end{array}$$

which was re-proved by Si and Luo [24] by another method. Here the right-hand side is the $\mathbb{Q}$-algebra generated by the values of the Riemann zeta function at positive integer arguments. Moreover, Kaneko and Tsumura [11] conjectured the following duality formula

$$\begin{array}{c}\hfill \eta (p;k)=\eta (k;p)(k,p\in \mathbb{N}),\end{array}$$

which has been proved by Kawasaki-Ohno [25] and Yamamoto [26]. Recently, Xu [27] and Luo-Si [28] proved two more general duality relations:

$$\begin{array}{c}\hfill \sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1}{k_1,\dots ,k_r\ge 1}}}\eta (k_1,\dots ,k_r;p)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=p+r-1}{k_1,\dots ,k_r\ge 1}}}\eta (k_1,\dots ,k_r;k)\end{array}$$

and

$$\begin{array}{c}\hfill \sum _{j=1}^r\left({\displaystyle \genfrac{}{}{0pt}{}{p+r-j-1}{p-1}}\right)\eta ({\left\{1\right\}}_{j-1},k;p+r-j)=\sum _{j=1}^r\left({\displaystyle \genfrac{}{}{0pt}{}{k+r-j-1}{k-1}}\right)\eta ({\left\{1\right\}}_{j-1},p;k+r-j),\end{array}$$

respectively, where $p,k,r\in \mathbb{N}$.

On the other hand, Kaneko and Tsumura [29] defined a level two analogue of $\xi (k;s)$, which we call the Kaneko–Tsumura $\psi$-function, as follows. For composition k and $\Re \left(s\right)>0$, we write

$$\psi (k;s)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{\displaystyle \frac{\mathrm{A}(k;tanht/2)}{sinh\left(t\right)}}dt,$$

(1)

where for $k=(k_1,\dots ,k_r)$

$$\begin{array}{c}\hfill \mathrm{A}(k;z):=2^r\sum _{{\displaystyle \genfrac{}{}{0pt}{}{1\le n_1<\cdots <n_r}{n_i\equiv i\mathrm{mod}2}}}{\displaystyle \frac{z^{n_r}}{n_1^{k_1}\cdots n_r^{k_r}}},z\in [-1,1).\end{array}$$

In particular, if $s\in \mathbb{N}$ then we call (1) the Kaneko–Tsumura $\psi$-values. Here, $\mathrm{A}(k;z)$ is $2^r$ times the $\mathrm{Ath}(k;z)$ which was introduced in [29]. Some related results for Kaneko–Tsumura $\psi$-values and their related values may be seen in the works of [9,29,30,31,32,33] and the references therein.

Furthermore, Kaneko and Tsumura [8,29] investigated a level two analog of multiple zeta values which are called multiple T-values and defined by

$$\begin{array}{c}\hfill T\left(k\right):=2^r\sum _{{\displaystyle \genfrac{}{}{0pt}{}{0<m_1<\cdots <m_r}{m_i\equiv i\mathrm{mod}2}}}{\displaystyle \frac{1}{m_1^{k_1}\cdots m_r^{k_r}}}=2^r\sum _{0<n_1<\cdots <n_r}{\displaystyle \frac{1}{{(2n_1-1)}^{k_1}{(2n_2-2)}^{k_2}\cdots {(2n_r-r)}^{k_r}}}.\end{array}$$

Subsequently, the last two authors of this paper proved that all the Kaneko–Tsumura $\psi$-values can be expressed in terms of multiple T-values (see [9], Thm. 4.6).

Setting $tanh(t/2)=z$ and $s=p\in \mathbb{N}$ in (1), we can quickly find

$$\begin{array}{cc}\hfill & \psi (k;p)={\displaystyle \frac{{(-1)}^{p-1}}{(p-1)!}}\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{log}^{p-1}\left(\frac{1-z}{1+z}\right)\mathrm{A}(k;z)}{z}}dz\hfill \end{array}$$

(2)

and

$$\begin{array}{cc}\hfill \mathrm{A}(k;z)={\int }_{0<t_1<t_2\cdots <t_{\left|k\right|}<z}{\displaystyle \frac{2d{t}_1}{1-t_1^2}}& \underset{k_1-1}{\underbrace{{\displaystyle \frac{d{t}_2}{t_2}}\cdots {\displaystyle \frac{d{t}_{k_1}}{t_{k_1}}}}}\cdots {\displaystyle \frac{2d{t}_{\left|k\right|-k_r+1}}{1-t_{\left|k\right|-k_r+1}^2}}\underset{k_r-1}{\underbrace{{\displaystyle \frac{d{t}_{\left|k\right|-k_r+2}}{t_{\left|k\right|-k_r+2}}}\cdots {\displaystyle \frac{d{t}_{\left|k\right|}}{t_{\left|k\right|}}}}}.\hfill \end{array}$$

Hence, similar to (2), we can now define the following trigonometric variant of the Kaneko–Tsumura $\psi$-function.

Definition 1.

For a composition $k=(k_1,\dots ,k_r)$ and $\Re \left(s\right)>0$,

$$\begin{array}{c}\hfill \tilde{\psi }(k;s):={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\pi /2}{t}^{s-1}{\displaystyle \frac{B\left(k;tan\frac{t}{2}\right)}{sint}}dt.\end{array}$$

(3)

We call them the Kaneko–Tsumura $\tilde{\psi }$-function. Here for $z\in \left[-1,1\right]$,

$$\begin{array}{c}\hfill \mathrm{B}(k;z):=\left\{\begin{array}{cc}{(-1)}^m\mathrm{i}\mathrm{A}(k;\mathrm{i}z),\hfill & ifr=2m-1;\hfill \\ {(-1)}^m\mathrm{A}(k;\mathrm{i}z),\hfill & ifr=2m.\hfill \end{array}\right.\end{array}$$

In particular, if $s\in \mathbb{N}$ in (3), we call them Kaneko–Tsumura $\tilde{\psi }$-values. Moreover, if $s=p\in \mathbb{N}$, then by applying the change $tan(t/2)=z$ in (3), one obtains

$$\begin{array}{c}\hfill \tilde{\psi }(k;p)={\displaystyle \frac{1}{(p-1)!}}\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{(2arctanz)}^{p-1}\mathrm{B}(k;z)}{z}}dz.\end{array}$$

(4)

Our primary goal in this paper is to establish some explicit formulas involving Kaneko–Tsumua $\psi$-values and $\tilde{\psi }$-values by the method of iterated integrals. In particular, from Section 3, we know that the Kaneko–Tsumura $\tilde{\psi }$-values can be expressed in terms of a $\mathbb{Q}$-linear combinations of multiple $\overline{T}$-values (see (17) for the definition).

2. Main Results

In this section, we will establish some explicit relations between Kaneko–Tsumua $\tilde{\psi }$-values and the alternating multiple T-values by applying the theory of iterated integrals which was first studied by K.T. Chen in the 1960’s [34,35]. For any one-forms $f_1\left(t\right)dt{f}_2\left(t\right)dt,\dots ,f_k\left(t\right)dt$, we define

$${\int }_a^b{f}_1\left(t\right)dt{f}_2\left(t\right)dt\cdots f_k\left(t\right)dt:=\underset{a<t_1<\cdots <t_k<b}{\int }{f}_1\left(t_1\right)f_2\left(t_2\right)\cdots f_k\left(t_k\right)d{t}_{1}d{t}_2\cdots d{t}_k.$$

One of the most important properties of these integrals is the following shuffle product formula (see [35], (151)): for all 1-forms $w_1,\dots ,w_{r+s}$ and any path $\alpha$,

$${\int }_{\alpha }{w}_1\cdots w_r{\int }_{\alpha }{w}_{r+1}\cdots w_{r+s}={\int }_{\alpha }{w}_1\cdots w_rШw_{r+1}\cdots w_{r+s},$$

(5)

where the shuffle product $Ш$ acts as a shuffling of two decks of cards.

We first observe that

$$\begin{array}{cc}\hfill \mathrm{B}(k_1,\dots ,k_r;z)& ={\int }_0^z{\displaystyle \frac{2dt}{1+t^2}}{\left({\displaystyle \frac{dt}{t}}\right)}^{k_1-1}\cdots {\displaystyle \frac{2dt}{1+t^2}}{\left({\displaystyle \frac{dt}{t}}\right)}^{k_r-1}\hfill \\ \hfill & =2^r\sum _{0<n_1<n_2<\cdots <n_r}{\displaystyle \frac{{(-1)}^{n_r-r}{z}^{2n_r-r}}{{(2n_1-1)}^{k_1}{(2n_2-2)}^{k_2}\cdots {(2n_r-r)}^{k_r}}}.\hfill \end{array}$$

(6)

Proposition 1.

For any fixed $r,k\in \mathbb{N}$ and $z\in [0,1]$,

$$\begin{array}{c}\hfill \sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1,}{k_1,\dots ,k_r\ge 1}}}\mathrm{B}(k;z)=\sum _{j=1}^r{\displaystyle \frac{{(-1)}^{j-1}}{(r-j)!}}{(2arctanz)}^{r-j}\mathrm{B}({\left\{1\right\}}_{j-1},k;z)\end{array}$$

(7)

and

$$\begin{array}{c}\hfill \mathrm{B}({\left\{1\right\}}_{r-1},k;z)=\sum _{j=1}^r{\displaystyle \frac{{(-1)}^{j-1}}{(r-j)!}}{(2arctanz)}^{r-j}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_j=k+j-1,}{k_1,\dots ,k_j\ge 1}}}\mathrm{B}(k_1,\dots ,k_j;z).\end{array}$$

(8)

Proof. 

Noting the fact that (6) can be rewritten as

$$\begin{array}{cc}\hfill & \mathrm{B}(k_1,\dots ,k_r;z)\hfill \\ \hfill & ={\displaystyle \frac{2^r}{{\displaystyle \prod _{j=1}^r}(k_j-1)!}}{\int }_{0<t_1<t_2\cdots <t_r<z}{\displaystyle \frac{{log}^{k_1-1}\left(\frac{t_2}{t_1}\right)d{t}_1}{1+t_1^2}}\cdots {\displaystyle \frac{{log}^{k_{r-1}-1}\left(\frac{t_r}{t_{r-1}}\right)d{t}_{r-1}}{1+t_{r-1}^2}}{\displaystyle \frac{{log}^{k_r-1}\left(\frac{z}{t_r}\right)d{t}_r}{1+t_r^2}}.\hfill \end{array}$$

(9)

Summing (9) yields

$$\begin{array}{cc}\hfill & \sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1,}{k_1,\dots ,k_r\ge 1}}}\mathrm{B}(k_1,\dots ,k_r;z)\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!}}{\int }_{0<t_1<t_2\cdots <t_r<z}{\displaystyle \frac{{\left(log\left(\frac{t_2}{t_1}\right)+\cdots +log\left(\frac{t_r}{t_{r-1}}\right)+log\left(\frac{z}{t_r}\right)\right)}^{k-1}}{(1+t_1^2)\cdots (1+t_r^2)}}d{t}_1\cdots d{t}_r\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!}}{\int }_{0<t_1<t_2\cdots <t_r<z}{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t_1}\right)}{(1+t_1^2)\cdots (1+t_r^2)}}d{t}_1\cdots d{t}_r\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!}}{\int }_0^z{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t_1}\right)}{1+t_1^2}}\left\{{\int }_{t_1<t_2\cdots <t_r<z}{\displaystyle \frac{d{t}_2\cdots d{t}_r}{(1+t_2^2)\cdots (1+t_r^2)}}\right\}d{t}_1\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!}}{\int }_0^z{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t_1}\right)}{1+t_1^2}}·{\displaystyle \frac{1}{(r-1)!}}{\left\{{\int }_{t_1}^z{\displaystyle \frac{d{t}_2}{1+t_2^2}}\right\}}^{r-1}d{t}_1\left(\mathrm{by}\right(5\left)\right)\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!(r-1)!}}{\int }_0^z{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t}\right){\left(arctanz-arctant\right)}^{r-1}}{1+t^2}}dt.\hfill \end{array}$$

(10)

On the other hand, letting $k=({\left\{1\right\}}_{r-1},k)$ in (9), we see that

$$\begin{array}{cc}\hfill \mathrm{B}({\left\{1\right\}}_{r-1},k;z)& ={\displaystyle \frac{2^r}{(k-1)!}}{\int }_{0<t_1<t_2\cdots <t_r<z}{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t_r}\right)}{(1+t_1^2)\cdots (1+t_r^2)}}d{t}_1\cdots d{t}_r\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!}}{\int }_0^z{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t_r}\right)}{1+t_r^2}}\left\{{\int }_{0<t_1<\cdots <t_{r-1}<t_r}{\displaystyle \frac{d{t}_1\cdots d{t}_{r-1}}{(1+t_1^2)\cdots (1+t_{r-1}^2)}}\right\}d{t}_r\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!(r-1)!}}{\int }_0^z{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t}\right){\left(arctant\right)}^{r-1}}{1+t^2}}dt.\hfill \end{array}$$

(11)

Hence, combining (10) and (11), we get

$$\begin{array}{cc}\hfill & \sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1,}{k_1,\dots ,k_r\ge 1}}}\mathrm{B}(k_1,\dots ,k_r;z)\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!(r-1)!}}\sum _{j=1}^r{(-1)}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{r-1}{j-1}}\right){(arctanz)}^{r-j}{\int }_0^z{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t}\right){\left(arctant\right)}^{j-1}}{1+t^2}}dt\hfill \\ \hfill & =\sum _{j=1}^r{\displaystyle \frac{{(-1)}^{j-1}}{(r-j)!}}{(2arctanz)}^{r-j}\mathrm{B}({\left\{1\right\}}_{j-1},k;z)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \mathrm{B}({\left\{1\right\}}_{r-1},k;z)\hfill \\ \hfill & ={\displaystyle \frac{2^r}{(k-1)!(r-1)!}}\sum _{j=1}^r{(-1)}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{r-1}{j-1}}\right){(arctanz)}^{r-j}{\int }_0^z{\displaystyle \frac{{log}^{k-1}\left(\frac{z}{t}\right){\left(arctanz-arctant\right)}^{j-1}}{1+t^2}}dt\hfill \\ \hfill & =\sum _{j=1}^r{\displaystyle \frac{{(-1)}^{j-1}}{(r-j)!}}{(2arctanz)}^{r-j}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_j=k+j-1,}{k_1,\dots ,k_j\ge 1}}}\mathrm{B}(k_1,\dots ,k_j;z).\hfill \end{array}$$

This completes the proof of the proposition. □

Proposition 2.

For any fixed $r,k\in \mathbb{N}$ and $z\in [0,1]$,

$$\begin{array}{c}\hfill \sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1}{k_1,\dots ,k_r\ge 1}}}\mathrm{A}\left(k_1,\dots ,k_r;z\right)={(-1)}^{r-1}\sum _{j=1}^r{\displaystyle \frac{{log}^{r-j}\left(\frac{1-z}{1+z}\right)}{(r-j)!}}\mathrm{A}\left({\left\{1\right\}}_{j-1},k;z\right)\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \mathrm{A}\left({\left\{1\right\}}_{r-1},k;z\right)={(-1)}^{r-1}\sum _{j=1}^r{\displaystyle \frac{{log}^{r-j}\left(\frac{1-z}{1+z}\right)}{(r-j)!}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_j=k+j-1,}{k_1,\dots ,k_j\ge 1}}}\mathrm{A}\left(k_1,\dots ,k_j;z\right).\end{array}$$

(13)

Proof. 

The proof is completely similar to the proof of Proposition 1. We leave the details to the interested reader. □

Remark 1.

A similar evaluation of Propositions 1 and 2 for multiple polylogarithm functions can be found in [28] (Lem. 2.1).

Theorem 1.

For any positive integers $r,k$ and p, we have

$$\begin{array}{c}\hfill \psi ({\left\{1\right\}}_{r-1},k;p)=\sum _{j=1}^r\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_j=k+j-1,}{k_1,\dots ,k_j\ge 1}}}{(-1)}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{p+r-j-1}{p-1}}\right)\psi (k_1,\dots ,k_j;p+r-j),\end{array}$$

(14)

$$\begin{array}{c}\hfill \sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1,}{k,\dots ,k_r\ge 1}}}\psi (k_1,\dots ,k_r;p)=\sum _{j=1}^r{(-1)}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{p+r-j-1}{p-1}}\right)\psi ({\left\{1\right\}}_{j-1},k;p+r-j).\end{array}$$

(15)

Proof. 

Multiplying (12) and (13) by ${log}^{p-1}\left({\displaystyle \frac{1-z}{1+z}}\right)/z$ and applying (2), we quickly arrive at Formulas (14) and (15). □

Theorem 2.

For any positive integers $r,k$ and p, we have

$$\begin{array}{cc}\hfill & \tilde{\psi }({\left\{1\right\}}_{r-1},k;p)=\sum _{j=1}^r\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_j=k+j-1,}{k_1,\dots ,k_j\ge 1}}}{(-1)}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{p+r-j-1}{p-1}}\right)\tilde{\psi }(k_1,\dots ,k_j;p+r-j),\hfill \\ \hfill & \sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1,}{k,\dots ,k_r\ge 1}}}\tilde{\psi }(k_1,\dots ,k_r;p)=\sum _{j=1}^r{(-1)}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{p+r-j-1}{p-1}}\right)\tilde{\psi }({\left\{1\right\}}_{j-1},k;p+r-j).\hfill \end{array}$$

Proof. 

By the definition of $\tilde{\psi }$-value and using (7) and (8), we see that

$$\begin{array}{cc}\hfill & \tilde{\psi }({\left\{1\right\}}_{r-1},k;p)\hfill \\ & ={\displaystyle \frac{1}{(p-1)!}}\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{(2arctanz)}^{p-1}\mathrm{B}({\left\{1\right\}}_{r-1},k;z)}{z}}dz\hfill \\ \hfill & =\sum _{j=1}^r{\displaystyle \frac{{(-1)}^{j-1}}{(r-j)!(p-1)!}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_j=k+j-1,}{k_1,\dots ,k_j\ge 1}}}\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{(2arctanz)}^{p+r-j-1}\mathrm{B}(k_1,\dots ,k_j;z)}{z}}dz\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1,}{k,\dots ,k_r\ge 1}}}\tilde{\psi }(k_1,\dots ,k_r;p)\hfill \\ \hfill & =\sum _{{\displaystyle \genfrac{}{}{0pt}{}{k_1+\cdots +k_r=k+r-1,}{k,\dots ,k_r\ge 1}}}{\displaystyle \frac{1}{(p-1)!}}\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{(2arctanz)}^{p-1}\mathrm{B}(k_1,\dots ,k_r;z)}{z}}dz\hfill \\ \hfill & =\sum _{j=1}^r{\displaystyle \frac{{(-1)}^{j-1}}{(r-j)!(p-1)!}}\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{(2arctanz)}^{p+r-j-1}\mathrm{B}({\left\{1\right\}}_{j-1},k;z)}{z}}dz.\hfill \end{array}$$

Thus, an elementary calculation leads to the desired evaluations. □

Remark 2.

A similar evaluation of Theorems 1 and 2 for the Arakawa–Kaneko zeta function and Kaneko–Tsumura $\eta$-function can be found in [28] (Thms. 1.1 and 1.2).

Lemma 1

(cf. [35], (1.6.1-2)). If $f_i(i=1,\cdots ,m)$ are integrable real functions, then the following identity holds:

$$g\left(f_1,\dots ,f_m\right)+{(-1)}^{m}g\left(f_m,\dots ,f_1\right)=\sum _{j=1}^{m-1}{(-1)}^{j-1}g(f_1,f_2,\dots ,f_{m-j})g(f_m,f_{m-1},\dots ,f_{m-j+1}),$$

where $g\left(f_1,\dots ,f_m\right)$ is defined by

$$g\left(f_1,\dots ,f_m\right):=\underset{0<t_1<\cdots <t_m<1}{\int }{f}_1\left(t_1\right)\cdots f_m\left(t_m\right)d{t}_1\cdots d{t}_m.$$

Theorem 3.

For any positive integers $r,k$ and p, we have

$$\begin{array}{cc}\hfill & \tilde{\psi }({\left\{1\right\}}_{r-1},k+1;p+1)-{(-1)}^k\tilde{\psi }({\left\{1\right\}}_{p-1},k+1;r+1)\hfill \\ \hfill & ={(-1)}^{r+p}\sum _{j=1}^k{(-1)}^{j-1}\overline{T}({\left\{1\right\}}_{r-1},k-j+2)\overline{T}({\left\{1\right\}}_{p-1},j+1),\hfill \end{array}$$

(16)

where $\overline{T}(k_1,\dots ,k_{r-1},k_r)$ stands for a kind of alternating, multiple T-values [27,32,36], which is defined for positive integers $k_1,\dots ,k_r$ by

$$\begin{array}{c}\hfill \overline{T}(k_1,\dots ,k_{r-1},k_r):=2^r\sum _{0<n_1<\cdots <n_r}{\displaystyle \frac{{(-1)}^{n_r}}{{(2n_1-1)}^{k_1}\cdots {(2n_{r-1}-r+1)}^{k_{r-1}}{(2n_r-r)}^{k_r}}}.\end{array}$$

(17)

We call them multiple $\overline{T}$-values (M$\overline{\mathrm{T}}$Vs).

Proof. 

First, by definition:

$$\begin{array}{c}\hfill B({\left\{1\right\}}_{r-1},k+1;x)=2^r{\int }_0^x{\left({\displaystyle \frac{dt}{1+t^2}}\right)}^r{\left({\displaystyle \frac{dt}{t}}\right)}^k={\displaystyle \frac{2^r}{r!}}{\int }_0^x{\displaystyle \frac{{arctan}^{r}t}{t}}dt{\left({\displaystyle \frac{dt}{t}}\right)}^{k-1}.\end{array}$$

Hence, setting $k=({\left\{1\right\}}_{r-1},k+1)$ in (4) and replacing p with $p+1$, we get

$$\begin{array}{c}\hfill \tilde{\psi }({\left\{1\right\}}_{r-1},k+1;p+1)={\displaystyle \frac{2^{p+r}}{p!r!}}{\int }_0^1{\displaystyle \frac{{arctan}^{r}t}{t}}dt{\left({\displaystyle \frac{dt}{t}}\right)}^{k-1}{\displaystyle \frac{{arctan}^{p}t}{t}}dt.\end{array}$$

Using Lemma 1 and noting the fact that

$$\begin{array}{c}\hfill {\int }_0^1{\displaystyle \frac{{arctan}^{p}t}{t}}dt{\left({\displaystyle \frac{dt}{t}}\right)}^{k-1}={\displaystyle \frac{p!{(-1)}^p}{2^p}}\overline{T}({\left\{1\right\}}_{p-1},k+1),\end{array}$$

we see that

$$\begin{array}{cc}\hfill & {\int }_0^1{\displaystyle \frac{{arctan}^{r}t}{t}}dt{\left({\displaystyle \frac{dt}{t}}\right)}^{k-1}{\displaystyle \frac{{arctan}^{p}t}{t}}dt+{(-1)}^{k+1}{\int }_0^1{\displaystyle \frac{{arctan}^{p}t}{t}}dt{\left({\displaystyle \frac{dt}{t}}\right)}^{k-1}{\displaystyle \frac{{arctan}^{r}t}{t}}dt\hfill \\ \hfill & ={\displaystyle \frac{p!r!}{2^{p+r}}}{(-1)}^{p+r}\sum _{j=1}^k{(-1)}^{j-1}\overline{T}({\left\{1\right\}}_{r-1},k-j+2)\overline{T}({\left\{1\right\}}_{p-1},j+1).\hfill \end{array}$$

Hence, the theorem follows immediately. □

Corollary 1.

For positive integers $r,k$ and p, we have

$$\begin{array}{cc}\hfill & {(-1)}^r\sum _{j=0}^p{\displaystyle \frac{{\left(\frac{\pi }{2}\right)}^{p-j}}{(p-j)!}}\overline{T}({\left\{1\right\}}_{r-1},k+2,{\left\{1\right\}}_j)-{(-1)}^{k+p}\sum _{j=0}^r{\displaystyle \frac{{\left(\frac{\pi }{2}\right)}^{r-j}}{(r-j)!}}\overline{T}({\left\{1\right\}}_{p-1},k+2,{\left\{1\right\}}_j)\hfill \\ \hfill & ={(-1)}^{r+p}\sum _{j=1}^k{(-1)}^{j-1}\overline{T}({\left\{1\right\}}_{r-1},k-j+2)\overline{T}({\left\{1\right\}}_{p-1},j+1).\hfill \end{array}$$

(18)

Proof. 

Noting the facts that

$$\begin{array}{c}\hfill {\int }_0^x{\left({\displaystyle \frac{dt}{1+t^2}}\right)}^r={\displaystyle \frac{{(arctanx)}^r}{r!}}\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_x^1{\left({\displaystyle \frac{dt}{1+t^2}}\right)}^r={\displaystyle \frac{{\left(\frac{\pi }{4}-arctanx\right)}^r}{r!}},\end{array}$$

we find

$$\begin{array}{cc}\hfill & \tilde{\psi }({\left\{1\right\}}_{r-1},k+1;p+1)\hfill \\ \hfill & ={\displaystyle \frac{2^{p+r}}{p!r!}}{\int }_0^1{\displaystyle \frac{{arctan}^{r}t}{t}}dt{\left({\displaystyle \frac{dt}{t}}\right)}^{k-1}{\displaystyle \frac{{arctan}^{p}t}{t}}dt\hfill \\ \hfill & ={\displaystyle \frac{2^{p+r}}{p!}}{\int }_0^1{\left({\displaystyle \frac{dt}{1+t^2}}\right)}^r{\left({\displaystyle \frac{dt}{t}}\right)}^k{\displaystyle \frac{{\left(arctant-\frac{\pi }{4}+\frac{\pi }{4}\right)}^p}{t}}dt\hfill \\ \hfill & ={\displaystyle \frac{2^{p+r}}{p!}}\sum _{j=0}^p\left({\displaystyle \genfrac{}{}{0pt}{}{p}{j}}\right){(-1)}^j{\left({\displaystyle \frac{\pi }{4}}\right)}^{p-j}{\int }_0^1{\left({\displaystyle \frac{dt}{1+t^2}}\right)}^r{\left({\displaystyle \frac{dt}{t}}\right)}^k{\displaystyle \frac{{\left(\frac{\pi }{4}-arctant\right)}^j}{t}}dt\hfill \\ \hfill & ={\displaystyle \frac{2^{p+r}}{p!}}\sum _{j=0}^p\left({\displaystyle \genfrac{}{}{0pt}{}{p}{j}}\right){(-1)}^j{\left({\displaystyle \frac{\pi }{4}}\right)}^{p-j}j!{\int }_0^1{\left({\displaystyle \frac{dt}{1+t^2}}\right)}^r{\left({\displaystyle \frac{dt}{t}}\right)}^{k+1}{\left({\displaystyle \frac{dt}{1+t^2}}\right)}^j\hfill \\ \hfill & =\sum _{j=0}^p{(-1)}^j{\displaystyle \frac{{\left(\frac{\pi }{2}\right)}^{p-j}}{(p-j)!}}{\int }_0^1{\left({\displaystyle \frac{2dt}{1+t^2}}\right)}^r{\left({\displaystyle \frac{dt}{t}}\right)}^{k+1}{\left({\displaystyle \frac{2dt}{1+t^2}}\right)}^j\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & ={(-1)}^r\sum _{j=0}^p{\displaystyle \frac{{\left(\frac{\pi }{2}\right)}^{p-j}}{(p-j)!}}\overline{T}({\left\{1\right\}}_{r-1},k+2,{\left\{1\right\}}_j).\hfill \end{array}$$

(20)

Here, in the last step, we have used the iterated integral expression of M$\overline{\mathrm{T}}$Vs

$$\begin{array}{c}\hfill \overline{T}(k_1,\dots ,k_r):={(-1)}^r{\int }_0^1{\displaystyle \frac{2dt}{1+t^2}}{\left({\displaystyle \frac{dt}{t}}\right)}^{k_1-1}\cdots {\displaystyle \frac{2dt}{1+t^2}}{\left({\displaystyle \frac{dt}{t}}\right)}^{k_r-1}.\end{array}$$

Finally, substituting (19) into (16) yields the desired evaluation. □

It is not hard to derive some explicit relations of M$\overline{\mathrm{T}}$Vs from (18). For example, setting $r=p=k=1$ in (18), we immediately produce the identity

$$\begin{array}{c}\hfill \overline{T}(3,1)=-{\displaystyle \frac{1}{2}}{\overline{T}}^2\left(2\right)-{\displaystyle \frac{\pi }{2}}\overline{T}\left(3\right).\end{array}$$

Remark 3.

Noticing that $B({\left\{1\right\}}_r;z)={\displaystyle \frac{2^r}{r!}}{arctan}^r\left(z\right)$, we get

$$\begin{array}{c}\hfill \tilde{\psi }({\left\{1\right\}}_r;p+1)={\displaystyle \frac{2^{p+r}}{p!r!}}{\int }_0^1{\displaystyle \frac{{arctan}^{r+p}t}{t}}dt.\end{array}$$

By similar arguments, as used in the proofs of Theorem 3 and Corollary 1, we can find that Theorem 3 and Corollary 1 also hold for $k=0$.

Now, for ${\mathit{k}}_r=(k_1,\dots ,k_r)\in {\mathbb{N}}^r$ and $|{\mathit{k}}_r|:=k_1+\cdots +k_r$, we adopt the following notations:

$$\begin{array}{cc}\hfill & {\overrightarrow{\mathit{k}}}_r^{-}:=(k_1,\dots ,k_{r-1},k_r-1),{\overleftarrow{\mathit{k}}}_r^{-}:=(k_r,\dots ,k_2,k_1-1),\hfill \\ \hfill & \overrightarrow{{\mathit{k}}_j}:=(k_1,k_2,\dots ,k_j),\overleftarrow{{\mathit{k}}_j}:=(k_r,k_{r-1},\dots ,k_{r+1-j}),\hfill \\ \hfill & \left|\overrightarrow{{\mathit{k}}_j}\right|:=k_1+k_2+\cdots +k_j,\left|\overleftarrow{{\mathit{k}}_j}\right|:=k_r+k_{r-1}+\cdots +k_{r+1-j}\hfill \end{array}$$

with $\overrightarrow{{\mathit{k}}_0}=\overleftarrow{{\mathit{k}}_0}:=\varnothing$ and $\left|\overrightarrow{{\mathit{k}}_0}\right|=\left|\overleftarrow{{\mathit{k}}_0}\right|:=0$. Next, we prove a more general duality formula.

Theorem 4.

For positive integers $p, q, r$, and $\mathit{k}=(k_1,\dots ,k_r)$, with $k_1,k_2,\dots ,k_r\in \mathbb{N}\setminus \left\{1\right\}$, we have

$$\begin{array}{cc}\hfill & \tilde{\psi }\left({\left\{1\right\}}_{q-1},{\overrightarrow{\mathit{k}}}_r^{-};p+1\right)-{(-1)}^{\left|\mathit{k}\right|}\tilde{\psi }\left({\left\{1\right\}}_{p-1},{\overleftarrow{\mathit{k}}}_r^{-};q+1\right)\hfill \\ \hfill & ={(-1)}^{p+q+r}\sum _{j=0}^{r-1}{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_j\mid }\sum _{i=1}^{k_{r-j}-2}{(-1)}^i\overline{T}\left({\left\{1\right\}}_{p-1},{\overleftarrow{\mathit{k}}}_j,i+1\right)\overline{T}\left({\left\{1\right\}}_{q-1},{\overrightarrow{\mathit{k}}}_{r-j-1},k_{r-j}-i\right)\hfill \\ \hfill & -{(-1)}^{p+q+r}\sum _{j=0}^{r-2}{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_{j+1}\mid }\left\{\begin{array}{c}\overline{T}\left({\left\{1\right\}}_{p-1},{\overleftarrow{\mathit{k}}}_{j+1}\right)\overline{T}\left({\left\{1\right\}}_{q-1},{\overrightarrow{\mathit{k}}}_{r-j-1},1\right)\hfill \\ -\overline{T}\left({\left\{1\right\}}_{p-1},{\overleftarrow{\mathit{k}}}_{j+1},1\right)\overline{T}\left({\left\{1\right\}}_{q-1},{\overrightarrow{\mathit{k}}}_{r-j-1}\right)\hfill \end{array}\right\}.\hfill \end{array}$$

(21)

Proof. 

Noting that $B({\left\{1\right\}}_r;z)={\displaystyle \frac{2^r}{r!}}{arctan}^r\left(z\right)$ and using (4), we can find that

$$\begin{array}{cc}\hfill & \tilde{\psi }({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-1;p+1)\hfill \\ & =\underset{0}{\overset{1}{\int }}{\displaystyle \frac{B({\left\{1\right\}}_p;z)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-1;z)}{z}}dz.\hfill \end{array}$$

(22)

Applying

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dz}}B(k_1,\dots ,k_{r-1},k_r;z)=\left\{\begin{array}{c}{\displaystyle \frac{1}{z}}B(k_1,\dots ,k_{r-1},k_r-1;z)(k_r\ge 2),\\ {\displaystyle \frac{2}{1+z^2}}B(k_1,\dots ,k_{r-1};z)(k_r=1)\end{array}\right.\end{array}$$

and integrating by parts, we get

$$\begin{array}{cc}\hfill & \tilde{\psi }({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-1;p+1)\hfill \\ & =B({\left\{1\right\}}_{p-1},2;1)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-1;1)\hfill \\ & -\underset{0}{\overset{1}{\int }}{\displaystyle \frac{B({\left\{1\right\}}_{p-1},2;z)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-2;z)}{z}}dz\hfill \\ \hfill & =\cdots \hfill \\ \hfill & =\sum _{i=1}^{k_r-2}{(-1)}^{i-1}B({\left\{1\right\}}_{p-1},i+1;1)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-i;1)\hfill \\ \hfill & +{(-1)}^{k_r-2}\underset{0}{\overset{1}{\int }}{\displaystyle \frac{B({\left\{1\right\}}_{p-1},k_r-1;z)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},1;z)}{z}}dz\hfill \\ \hfill & =\sum _{i=1}^{k_r-2}{(-1)}^{i-1}B({\left\{1\right\}}_{p-1},i+1;1)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-i;1)\hfill \\ \hfill & +{(-1)}^{k_r}\left\{{\displaystyle \genfrac{}{}{0pt}{}{B({\left\{1\right\}}_{p-1},k_r;1)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},1;1)}{-2\underset{0}{\overset{1}{\int }}\frac{B({\left\{1\right\}}_{p-1},k_r;z)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1};z)}{1+z^2}dz}}\right\}\hfill \\ \hfill & =\sum _{i=1}^{k_r-2}{(-1)}^{i-1}B({\left\{1\right\}}_{p-1},i+1;1)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-i;1)\hfill \\ \hfill & +{(-1)}^{k_r}\left\{{\displaystyle \genfrac{}{}{0pt}{}{B({\left\{1\right\}}_{p-1},k_r;1)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},1;1)}{-B({\left\{1\right\}}_{p-1},k_r,1;1)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1};1)}}\right\}\hfill \\ \hfill & +{(-1)}^{k_r}\underset{0}{\overset{1}{\int }}{\displaystyle \frac{B({\left\{1\right\}}_{p-1},k_r,1;z)B({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-2},k_{r-1}-1;z)}{z}}dz\hfill \\ \hfill & =\cdots \hfill \\ \hfill & =\sum _{j=0}^{r-2}{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_j\mid }\sum _{i=1}^{k_{r-j}-2}{(-1)}^{i-1}B({\left\{1\right\}}_{p-1},k_r,k_{r-1},\dots ,k_{r+1-j},i+1;1)\hfill \\ & \times B({\left\{1\right\}}_{q-1},k_1,k_2,\dots ,k_{r-j-1},k_{r-j}-i;1)\hfill \\ \hfill & +\sum _{j=0}^{r-2}{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_{j+1}\mid }\left\{\begin{array}{c}B({\left\{1\right\}}_{p-1},k_r,k_{r-1},\dots ,k_{r-j};1)B({\left\{1\right\}}_{q-1},k_1,k_2,\dots ,k_{r-j-1},1;1)\hfill \\ -B({\left\{1\right\}}_{p-1},k_r,k_{r-1},\dots ,k_{r-j},1;1)B({\left\{1\right\}}_{q-1},k_1,k_2,\dots ,k_{r-j-1};1)\hfill \end{array}\right\}\hfill \\ \hfill & +{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_{r-1}\mid }\underset{0}{\overset{1}{\int }}{\displaystyle \frac{B({\left\{1\right\}}_{p-1},k_r,k_{r-1},\dots ,k_2,1;z)B({\left\{1\right\}}_{q-1},k_1-1;z)}{z}}dz\hfill \\ \hfill & =\sum _{j=0}^{r-1}{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_j\mid }\sum _{i=1}^{k_{r-j}-2}{(-1)}^{i-1}B({\left\{1\right\}}_{p-1},k_r,k_{r-1},\dots ,k_{r+1-j},i+1;1)\hfill \\ & \times B({\left\{1\right\}}_{q-1},k_1,k_2,\dots ,k_{r-j-1},k_{r-j}-i;1)\hfill \\ \hfill & +\sum _{j=0}^{r-2}{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_{j+1}\mid }\left\{\begin{array}{c}B({\left\{1\right\}}_{p-1},k_r,k_{r-1},\dots ,k_{r-j};1)B({\left\{1\right\}}_{q-1},k_1,k_2,\dots ,k_{r-j-1},1;1)\hfill \\ -B({\left\{1\right\}}_{p-1},k_r,k_{r-1},\dots ,k_{r-j},1;1)B({\left\{1\right\}}_{q-1},k_1,k_2,\dots ,k_{r-j-1};1)\hfill \end{array}\right\}\hfill \\ \hfill & +{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_r\mid }\underset{0}{\overset{1}{\int }}{\displaystyle \frac{B({\left\{1\right\}}_{p-1},k_r,k_{r-1},\dots ,k_2,k_1-1;z)B({\left\{1\right\}}_q;z)}{z}}dz.\hfill \end{array}$$

Finally, using $B(k_1,\dots ,k_r;1)={(-1)}^r\overline{T}(k_1,\dots ,k_r)$, we arrive at Formula (22). □

By a similar argument as used in the proof of (19), we obtain

$$\begin{array}{cc}\hfill & \tilde{\psi }({\left\{1\right\}}_{q-1},k_1,\dots ,k_{r-1},k_r-1;p+1)\hfill \\ & ={(-1)}^{q+r-1}\sum _{j=0}^p{\displaystyle \frac{{\left(\frac{\pi }{2}\right)}^{p-j}}{(p-j)!}}\overline{T}({\left\{1\right\}}_{q-1},k_1,\dots ,k_r,{\left\{1\right\}}_j).\hfill \end{array}$$

(23)

Hence, substituting (23) into (21) yields the following corollary:

Corollary 2.

For positive integers $p, q, r$, and $\mathit{k}=(k_1,\dots ,k_r)$ with $k_1,k_2,\dots ,k_r\in \mathbb{N}\setminus \left\{1\right\}$,

$$\begin{array}{cc}\hfill & {(-1)}^{q+r-1}\sum _{j=0}^p{\displaystyle \frac{{\left(\frac{\pi }{2}\right)}^{p-j}}{(p-j)!}}\overline{T}\left({\left\{1\right\}}_{q-1},{\mathit{k}}_r,{\left\{1\right\}}_j\right)-{(-1)}^{\left|\mathit{k}\right|+p+r-1}\sum _{j=0}^q{\displaystyle \frac{{\left(\frac{\pi }{2}\right)}^{q-j}}{(q-j)!}}\overline{T}\left({\left\{1\right\}}_{p-1},\overleftarrow{{\mathit{k}}_r},{\left\{1\right\}}_j\right)\hfill \\ \hfill & ={(-1)}^{p+q+r}\sum _{j=0}^{r-1}{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_j\mid }\sum _{i=1}^{k_{r-j}-2}{(-1)}^i\overline{T}\left({\left\{1\right\}}_{p-1},{\overleftarrow{\mathit{k}}}_j,i+1\right)\overline{T}\left({\left\{1\right\}}_{q-1},{\overrightarrow{\mathit{k}}}_{r-j-1},k_{r-j}-i\right)\hfill \\ \hfill & -{(-1)}^{p+q+r}\sum _{j=0}^{r-2}{(-1)}^{\mid {\overleftarrow{\mathit{k}}}_{j+1}\mid }\left\{\begin{array}{c}\overline{T}\left({\left\{1\right\}}_{p-1},{\overleftarrow{\mathit{k}}}_{j+1}\right)\overline{T}\left({\left\{1\right\}}_{q-1},{\overrightarrow{\mathit{k}}}_{r-j-1},1\right)\hfill \\ -\overline{T}\left({\left\{1\right\}}_{p-1},{\overleftarrow{\mathit{k}}}_{j+1},1\right)\overline{T}\left({\left\{1\right\}}_{q-1},{\overrightarrow{\mathit{k}}}_{r-j-1}\right)\hfill \end{array}\right\}.\hfill \end{array}$$

(24)

Clearly, setting $(q,r)\to (r,1)$ and $k_1=k+2$ in (24) yields (18).

3. Associated Multiple Integrals and 2-Labeled Posets

In [37], Yamamoto provided a graphical representation of the iterated integrals of multiple zeta values and their star versions by using 2-posets, which enabled him to discover a few very elegant relations among these values. In this section, we will follow his approach closely and represent M$\overline{\mathrm{T}}$Vs using a slightly modified version of these 2-posets. In this way, it is much easier to see many relations between M$\overline{\mathrm{T}}$Vs.

Let’s begin by generalizing Yamamoto’s definition of 2-poset so that we can adapt it to the M$\overline{\mathrm{T}}$V setting.

Definition 2.

Let $\kappa \ne 0$ be a real number and let $X=(X,\le )$ be any partially ordered finite set. Let ${\delta }_X:X⟶\{0,\kappa \}$ be a label map. We then say the pair $(X,{\delta }_X)$ is a 2-poset targeting at $\{0,\kappa \}$. We often call X itself a 2-poset if no confusion arises.

A 2-poset $(X,{\delta }_X)$ is called admissible if ${\delta }_X\left(x\right)=0$ for all maximal elements $x\in (X,\le )$ and ${\delta }_X\left(x\right)=\kappa$ for all minimal elements $x\in (X,\le )$.

Definition 3.

Let X be an admissible 2-poset X targeting at $\{0,-1\}$ and set ${\Delta }_X=\left\{{\left(t_x\right)}_x\in {[0,1]}^X|t_x<t_y\mathrm{if}x<y\right\}$. We define the (multiple) integral associated with X by

$$I\left(X\right)={\int }_{{\Delta }_X}\prod _{x\in X}{\omega }_{{\delta }_X\left(x\right)}\left(t_x\right),$$

where

$${\omega }_0\left(t\right)={\displaystyle \frac{dt}{t}},{\omega }_{-1}\left(t\right)={\displaystyle \frac{2dt}{1+t^2}}.$$

We put $I(\varnothing ):=1$ for the empty 2-poset ∅.

Proposition 3.

For non-comparable elements a and b of a 2-poset X, $X_a^b$ denotes the 2-poset that is obtained from X by adjoining the relation $a<b$. If X is an admissible 2-poset, then the 2-poset $X_a^b$ and $X_b^a$ are admissible and

$$I\left(X\right)=I\left(X_a^b\right)+I\left(X_b^a\right).$$

Proof. 

This is clear from the definition. □

Note that the admissibility of a 2-poset corresponds to the convergence of the associated integrals. In what follows, we adopt the usual Hasse diagrams to indicate 2-posets, with vertices ∘ and • corresponding to $\delta \left(x\right)=0$ and $\delta \left(x\right)=-1$, respectively. For example, the diagram

represents the admissible 2-poset $X=\{x_1,x_2,x_3,x_4,x_5\}$, with the order $x_1<x_2>x_3<x_4<x_5$ and label $({\delta }_X\left(x_1\right),\dots ,{\delta }_X\left(x_5\right))=(-1,0,-1,0,0)$. To save space, for a composition $\mathit{k}=(k_1,\dots ,k_r)$ of positive integers, we write for the diagrams

By the definition of the multiple $\overline{T}$-values, we have

$$\begin{array}{cc}\hfill & \overline{T}\left(\mathit{k}\right)={(-1)}^r\underset{0}{\overset{1}{\int }}{\displaystyle \frac{2dt}{1+t^2}}{\left({\displaystyle \frac{dt}{t}}\right)}^{k_1-1}\cdots {\displaystyle \frac{2dt}{1+t^2}}{\left({\displaystyle \frac{dt}{t}}\right)}^{k_r-1}.\hfill \end{array}$$

Hence, using this notation of associated integrals, we can verify that

(25)

Thus, we may deduce from (4) that

since there are exactly $p!$ ways to impose a total order on the p black vertices. By Proposition 3, this implies that $\tilde{\psi }(k_1,\dots ,k_r;p+1)$ can be expressed as a finite sum of M$\overline{\mathrm{T}}$Vs. For example, using the shuffle relations of iterated integral (5), we have

$$\begin{array}{cc}\hfill & \mathrm{B}(1;z)\mathrm{B}(k_1,\dots ,k_r;z)\hfill \\ & =\left({\int }_0^z{\displaystyle \frac{2dt}{1+t^2}}\right)\left({\int }_0^z{\displaystyle \frac{2dt}{1+t^2}}{\left({\displaystyle \frac{dt}{t}}\right)}^{k_1-1}\cdots {\displaystyle \frac{2dt}{1+t^2}}{\left({\displaystyle \frac{dt}{t}}\right)}^{k_r-1}\right)\hfill \\ \hfill & =\mathrm{B}(1,k_1,\dots ,k_r;z)+\sum _{l=1}^r\sum _{{\displaystyle \genfrac{}{}{0pt}{}{i+j=k_l-1,}{i,j\ge 0}}}\mathrm{B}(k_1,\dots ,k_{l-1},i+1,j+1,k_{l+1},\dots ,k_{r-1},k_r;z).\hfill \end{array}$$

(26)

Hence, for $p=1$, we obtain

$$\begin{array}{cc}\hfill \tilde{\psi }(\mathit{k};2)& ={\int }_0^1{\displaystyle \frac{\mathrm{B}(1;z)\mathrm{B}(k_1,\dots ,k_r;z)}{z}}dz\hfill \\ \hfill & ={(-1)}^{r+1}\overline{T}(1,k_1,\dots ,k_{r-1},k_r+1)+{(-1)}^{r+1}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{i+j=k_r-1,}{i,j\ge 0}}}\overline{T}(k_1,\dots ,k_{r-1},i+1,j+2)\hfill \\ \hfill & +{(-1)}^{r+1}\sum _{l=1}^{r-1}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{i+j=k_l-1,}{i,j\ge 0}}}\overline{T}(k_1,\dots ,k_{l-1},i+1,j+1,k_{l+1},\dots ,k_{r-1},k_r+1).\hfill \end{array}$$

(27)

On the other hand, letting $p=q=1$ and $k_r\to k_r+1$ in (23), one obtains

$$\begin{array}{c}\hfill \tilde{\psi }(\mathit{k};2)={(-1)}^r{\displaystyle \frac{\pi }{2}}\overline{T}(k_1,\dots ,k_{r-1},k_r+1)+{(-1)}^r\overline{T}(k_1,\dots ,k_{r-1},k_r+1,1).\end{array}$$

(28)

Comparing (27) and (28), we find the following identity of alternating multiple T-values

$$\begin{array}{c}{\displaystyle \frac{\pi }{2}}\overline{T}(k_1,\dots ,k_{r-1},k_r+1)+\overline{T}(k_1,\dots ,k_{r-1},k_r+1,1)+\overline{T}(1,k_1,\dots ,k_{r-1},k_r+1)\hfill \\ \hfill +\sum _{l=1}^r\sum _{{\displaystyle \genfrac{}{}{0pt}{}{i+j=k_l-1,}{i,j\ge 0}}}\overline{T}(k_1,\dots ,k_{l-1},i+1,j+1,k_{l+1},\dots ,k_{r-1},k_r+1)=0,\end{array}$$

where, if $l=r$, then no component appears after $j+1$ in the last sum above.

4. Concluding Remarks

We have studied the special values of the Kaneko–Tsumura $\psi$-functions and its closely related cousin—the Kaneko–Tsumura $\tilde{\psi }$-functions—in this paper. By using the method of iterated integrals first developed by K.T. Chen in 1960s and Yamamoto’s recently discovered graphical representation, we have found some nontrivial relations between these values and M$\overline{\mathrm{T}}$Vs. As corollaries, we can then discover a few intricate relations among M$\overline{\mathrm{T}}$Vs themselves. Previously, we studied the sum formula or the duality formula among Kaneko–Tsumura $\psi$-values in [9,32]. One may wonder if there are some forms of the sum formula or the duality formula among Kaneko–Tsumura $\tilde{\psi }$-values themselves, or similar formulas for M$\overline{\mathrm{T}}$Vs themselves. We hope to return to these problems in the future.