# On variants of the Euler sums and symmetric extensions of the Kaneko-Tsumura
conjecture

Weiping Wang$\,{}^{a,}$ Ce Xu$\,{}^{b,}$
$a.$ School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, P.R.
China
$b.$ School of Mathematics and Statistics, Anhui Normal University, Wuhu
241002, P.R. China  _E-mail addresses :_<EMAIL_ADDRESS><EMAIL_ADDRESS>(Weiping Wang).Corresponding author. _E-mail address :_
<EMAIL_ADDRESS>(Ce Xu).

Abstract

By using various expansions of the parametric digamma function and the method
of residue computations, we study three variants of the linear Euler sums,
related Hoffman’s double $t$-values and Kaneko-Tsumura’s double $T$-values,
and establish several symmetric extensions of the Kaneko-Tsumura conjecture.
Some special cases are discussed in detail to determine the coefficients of
involved mathematical constants in the evaluations. In particular, it can be
found that several general convolution identities on the classical Bernoulli
numbers and Genocchi numbers are required in this study, and they are verified
by the derivative polynomials of hyperbolic tangent.

_AMS classification :_ 11M32; 11A07; 40A25; 05A19; 11B68

_Keywords :_ Multiple zeta values; Multiple $t$-values; Multiple $T$-values;
Harmonic numbers; Euler sums; Convolution identities; Bernoulli numbers;
Genocchi numbers

###### Contents

  1. 1 Introduction
  2. 2 Expansions and residue theorem
  3. 3 Symmetric extension on linear $T$-sums and $\tilde{S}$-sums
    1. 3.1 Main theorem on linear $T$-sums and $\tilde{S}$-sums
    2. 3.2 The case of $m=p=1$
    3. 3.3 The case of $m=q$ and $p$ odd
    4. 3.4 More discussions on the case of $m=q$ and $p$ odd
    5. 3.5 The case of $m=q$ and $p$ even
    6. 3.6 Symmetric sum on double $T$-values and double $t$-values
  4. 4 Symmetric extension on linear $R$-sums
    1. 4.1 Main theorem on linear $R$-sums
    2. 4.2 Symmetric sum on double $T$-values
  5. 5 Proof of the required convolution identity
  6. Acknowledgments

## 1\. Introduction

The _alternating multiple zeta values_ (alternating MZVs) are defined by

$\zeta(s_{1},s_{2},\ldots,s_{k};\sigma_{1},\sigma_{2},\ldots,\sigma_{k})=\sum_{n_{1}>n_{2}>\cdots>n_{k}\geq
1}\frac{\sigma_{1}^{n_{1}}\sigma_{2}^{n_{2}}\cdots\sigma_{k}^{n_{k}}}{n_{1}^{s_{1}}n_{2}^{s_{2}}\cdots
n_{k}^{s_{k}}}\,,$ (1.1)

where $s_{j}$ are positive integers, $\sigma_{j}=\pm 1$, for $j=1,2,\ldots,k$,
with $(s_{1},\sigma_{1})\neq(1,1)$. As usual, we can combine the strings of
exponents and signs into a single string, with $s_{j}$ in the $j$th position
when $\sigma_{j}=+1$, and $\bar{s}_{j}$ in the $j$th position when
$\sigma_{j}=-1$. In particular, when $s_{1}>1$, setting $\sigma_{j}=1$,
$j=1,2,\ldots,k$, in Eq. (1.1), we obtain _multiple zeta values_ (MZVs), and
denote them by $\zeta(s_{1},s_{2},\ldots,s_{k})$.

The study of multiple zeta values began in the early 1990s with the works of
Hoffman [13] and Zagier [26], and has attracted a lot of research in the last
two decades. For detailed history and applications, the readers may consult
in, e.g., the book of Zhao [27].

In a recent paper [10], Hoffman introduced and studied an “odd” variant of the
MZVs:

$\displaystyle t(s_{1},s_{2},\ldots,s_{k})$
$\displaystyle=\sum_{\begin{subarray}{c}n_{1}>n_{2}>\cdots>n_{k}\geq 1\\\
n_{i}\ \mathrm{odd}\end{subarray}}\frac{1}{n_{1}^{s_{1}}n_{2}^{s_{2}}\cdots
n_{k}^{s_{k}}}$ $\displaystyle=\sum_{n_{1}>n_{2}>\cdots>n_{k}\geq
1}\frac{1}{(2n_{1}-1)^{s_{1}}(2n_{2}-1)^{s_{2}}\cdots(2n_{k}-1)^{s_{k}}}\,,$

which are called the _multiple $t$-values_ (MtVs). For convenience, let us
call the normalized version

$\displaystyle\tilde{t}(s_{1},s_{2},\ldots,s_{k})$
$\displaystyle=\sum_{n_{1}>n_{2}>\cdots>n_{k}\geq
1}\frac{1}{(n_{1}-1/2)^{s_{1}}(n_{2}-1/2)^{s_{2}}\cdots(n_{k}-1/2)^{s_{k}}}$
$\displaystyle=2^{s_{1}+s_{2}+\cdots+s_{k}}t(s_{1},s_{2},\ldots,s_{k})$

the _multiple $\tilde{t}$-values_. According to the definitions,
$\tilde{t}(s)=2^{s}t(s)=(2^{s}-1)\zeta(s)$ for integer $s\geq 2$, where
$\zeta(s)$ is the _Riemann zeta function_. As showed in [10, Corollaries 4.1
and 4.2], the MtVs are reducible to linear combinations of alternating MZVs.
Due to this fact as well as the congruence condition in the summation, the
MtVs can be regarded as MZVs of level two. In 2020, Kaneko and Tsumura [14]
introduced another variant of MZVs of level two:

$\displaystyle T(s_{1},s_{2},\ldots,s_{k})$
$\displaystyle=2^{k}\sum_{\begin{subarray}{c}n_{1}>n_{2}>\cdots>n_{k}\geq 1\\\
n_{i}\equiv
k-i+1{\@displayfalse\pmod{2}}\end{subarray}}\frac{1}{n_{1}^{s_{1}}n_{2}^{s_{2}}\cdots
n_{k}^{s_{k}}}$ $\displaystyle=2^{k}\sum_{n_{1}>n_{2}>\cdots>n_{k}\geq
1}\frac{1}{(2n_{1}-k)^{s_{1}}(2n_{2}-k+1)^{s_{2}}\cdots(2n_{k}-1)^{s_{k}}}\,,$

which are called the _multiple $T$-values_ (MTVs).

By convention, for the MZVs, MtVs and MTVs, the quantity $k$ is called the “
_depth_ ” and the quantity $w:=s_{1}+s_{2}+\cdots+s_{k}$ is called the “
_weight_ ”.

Let $\mathcal{Z}$ be the space of usual multiple zeta values. In [14,
Conjecture 5.3], Kaneko and Tsumura observed that the following relation
holds:

$\sum_{\begin{subarray}{c}i+j=m\\\ i,j\geq
0\end{subarray}}\binom{p+i-1}{i}\binom{q+j-1}{j}T(p+i,q+j)\in\mathcal{Z}\,,$
(1.2)

for $m,q\geq 1$ and $p\geq 2$, with $m+p+q$ even. That is, these sums are
expressible in terms of MZVs. In 2021, Murakami [18, Theorem 42] proved this
conjecture by using the motivic method employed in [8] (see also [14, Remark
5.6]).

For simplicity, let us define the transformation operator $\lambda_{m}$ by

$\lambda_{m}(\varOmega_{p,q}):=\sum_{\begin{subarray}{c}i+j=m-1\\\ i,j\geq
0\end{subarray}}\binom{p+i-1}{i}\binom{q+j-1}{j}\varOmega_{p+i,q+j}\,,\quad\text{for
}m\geq 1\,,$ (1.3)

which maps a sum $\varOmega_{p,q}$ of two parameters to another one of three
parameters, with $\lambda_{1}(\varOmega_{p,q})=\varOmega_{p,q}$. Then Kaneko
and Tsumura’s conjecture (1.2) can be rewritten as

$\lambda_{m+1}(T(p,q))\in\mathcal{Z}\,,\quad\text{for }m,q\geq 1,\ p\geq 2,\
\text{with }m+p+q\text{ even}.$

Let $H_{n}^{(r)}$ and $h_{n}^{(r)}$ stand for the _generalized harmonic
numbers_ and the _odd harmonic numbers_ of order $r$, respectively, defined by
$H_{0}^{(r)}=h_{0}^{(r)}=0$ and

$H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^{r}}\,,\quad
h_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{(k-1/2)^{r}}\,,\quad\text{for
}n,r=1,2,\ldots,$

with $H_{n}\equiv H_{n}^{(1)}$ and $h_{n}\equiv h_{n}^{(1)}$. The classical
_Euler sums_ are infinite series

$S_{p_{1}p_{2}\cdots
p_{k},q}=\sum_{n=1}^{\infty}\frac{H_{n}^{(p_{1})}H_{n}^{(p_{2})}\cdots
H_{n}^{(p_{k})}}{n^{q}}\,,$

where $p_{1}\leq p_{2}\leq\ldots\leq p_{k}$ and $q\geq 2$, and the linear sums
are of the form $S_{p,q}$. For an early introduction and study on the
evaluations of the classical Euler sums, the readers may consult in Flajolet
and Salvy’s paper [6], and for some recent progress, the readers are referred
to [20, 23, 25] and references therein.

In this paper, using various expansions of the parametric digamma function and
the method of residue computations, we establish symmetric extensions of the
Kaneko-Tsumura conjecture (1.2) on three variants of the linear Euler sums,
defined by

$T_{p,q}:=\sum_{n=1}^{\infty}\frac{h_{n-1}^{(p)}}{(n-1/2)^{q}}\,,\quad\tilde{S}_{p,q}:=\sum_{n=1}^{\infty}\frac{h_{n}^{(p)}}{n^{q}}\,,\quad
R_{p,q}:=\sum_{n=1}^{\infty}\frac{H_{n-1}^{(p)}}{(n-1/2)^{q}}\,,\quad\text{for
}q\geq 2\,,$

respectively, which were introduced and studied in [21, 22]. In particular, we
show that for $m,p\geq 1$ and $q\geq 2$, there hold

$\displaystyle(-1)^{m-1}\lambda_{p}(T_{m,q})+(-1)^{p-1}\lambda_{m}(\tilde{S}_{p,q})\in\mathbb{Q}[\ln(2),\text{zeta
values}]\,,$ (1.4)
$\displaystyle(-1)^{m-1}\lambda_{p}(R_{m,q})+(-1)^{p-1}\lambda_{m}(R_{p,q})\in\mathbb{Q}[\ln(2),\text{zeta
values}]\,.$ (1.5)

In other words, both of the symmetric sums on the left are reducible to
$\ln(2)$ and zeta values. Explicit expressions of these two sums are
presented, and a detailed discussion on some interesting special cases
follows.

Moreover, by the derivative polynomials of hyperbolic tangent, we obtain a
very general convolution identity for the Bernoulli numbers $B_{n}$ and
Genocchi numbers $G_{n}$, which is used to produce the following one:

$\displaystyle\sum_{i=0}^{q-1}\binom{q-1}{i}\frac{B_{q+i}G_{2n+q-i}}{(q+i)(2n+q-i)}$
$\displaystyle\quad=-\frac{1}{4}\sum_{i=0}^{2n}(-1)^{i}\binom{2n}{i}\frac{G_{q+i}G_{2n+q-i}}{(q+i)(2n+q-i)}+\frac{(-1)^{q}}{q\binom{2q}{q}}\frac{G_{2n+2q}}{2n+2q}\,,\quad\text{for
}n\geq 0\,,\ q\geq 2\,,$ (1.6)

so that the coefficients of various constants in the evaluations of the sums
in (1.4), which correspond to the case of $m=q$ and $p$ odd, can be finally
determined.

On the other hand, by the definitions, the linear $T$-sums and
$\tilde{S}$-sums are associated with Hoffman’s double $t$-values and Kaneko-
Tsumura’s double $T$-values, respectively:

$\displaystyle T_{p,q}$
$\displaystyle=\tilde{t}(q,p)=2^{p+q}t(q,p)=2^{p+q-2}\\{\zeta(q,p)-\zeta(q,\bar{p})-\zeta(\bar{q},p)+\zeta(\bar{q},\bar{p})\\}\,,$
(1.7) $\displaystyle\tilde{S}_{p,q}$ $\displaystyle=2^{p+q-2}T(q,p)$
$\displaystyle=(2^{p+q-1}-1)\\{\zeta(q,p)+\zeta(p+q)\\}+2^{p+q-1}\\{\zeta(\bar{q},p)+\zeta(\overline{p+q})\\}\,,$
(1.8)

and the linear $R$-sums can be expressed in terms of zeta values, $t$-values
and double $T$-values:

$R_{p,q}=\zeta(p)\tilde{t}(q)-\tilde{S}_{q,p}=\zeta(p)\tilde{t}(q)-2^{p+q-2}T(p,q)\,,\quad\text{for
}p,q\geq 2$ (1.9)

(see [21, Eqs. (3.5), (3.7), (3.8) and (3.11)]). Therefore, we can transform
the results on variants of linear Euler sums to those on variants of double
zeta values. For example, a symmetric sum on the double $T$-values can be
obtained directly from (1.5) and (1.9):

$(-1)^{m}\lambda_{p}(T(m,q))+(-1)^{p}\lambda_{m}(T(p,q))\in\mathbb{Q}[\text{zeta
values}]\,,\quad\text{for }m,p,q\geq 2\,,$

which further indicates that the sums $\lambda_{p}(T(p,q))$ are reducible to
zeta values if $p,q\geq 2$. Additionally, we show that for any even weight
$w:=m+q$, with $m,q\geq 2$, the double $t$-values $t(q,m)$ and linear $T$-sums
$T_{m,q}$ are expressible in terms of MZVs.

The paper is organized as follows. In Section 2, we present some expansions of
the parametric digamma function and introduce the residue theorem, which are
used in the establishment of the symmetric extensions of the Kaneko-Tsumura
conjecture (1.2). Section 3 is devoted to the symmetric sums on the linear
$T$-sums and $\tilde{S}$-sums, and Section 4 is devoted to the symmetric sums
on the linear $R$-sums. Finally, in Section 5, we prove the convolution
identity (1.6) on the Bernoulli numbers and Genocchi numbers, which is
required in a proof in Section 3.

## 2\. Expansions and residue theorem

In [24], we introduced a _parametric digamma function_ $\varPsi(-s;a)$ by

$\varPsi(-s;a)+\gamma=\frac{1}{s-a}+\sum_{k=1}^{\infty}\left(\frac{1}{k+a}-\frac{1}{k+a-s}\right)\,,\quad\text{for}\
s\in\mathbb{C}\,,\ a\in\mathbb{C}\setminus\mathbb{Z}^{-}\,,$

where $\gamma$ is the _Euler–Mascheroni constant_ , and
$\mathbb{Z}^{-}:=\\{-1,-2,\ldots\\}$. The function $\varPsi(-s;a)$ is
meromorphic in the entire complex plane with a simple pole at $s=n+a$ for each
nonnegative integer $n$. Here, let

$\varPsi(-s):=\varPsi(-s;-\tfrac{1}{2})+\gamma=\frac{1}{s+1/2}+\sum_{k=1}^{\infty}\left(\frac{1}{k-1/2}-\frac{1}{k-1/2-s}\right)\,.$

By [24, Theorems 2.1–2.3 and Corollary 2.4], the next two lemmas can be
established. Firstly, using the special value of the digamma function
$\psi(1/2)=-2\ln(2)-\gamma$, we have Lemma 2.1.

######  Lemma 2.1.

For integers $n\geq 0$ and $p\geq 2$, the following expansions hold:

$\displaystyle\varPsi(\tfrac{1}{2}-s)\stackrel{{\scriptstyle s\to
n}}{{=}}\frac{1}{s-n}+H_{n}+2\ln(2)+\sum_{j=1}^{\infty}\\{(-1)^{j}H_{n}^{(j+1)}-\zeta(j+1)\\}(s-n)^{j}\,,$
(2.1)
$\displaystyle\frac{\varPsi^{(p-1)}(\tfrac{1}{2}-s)}{(p-1)!}\stackrel{{\scriptstyle
s\to
n}}{{=}}\frac{1}{(s-n)^{p}}+(-1)^{p}\sum_{j=p}^{\infty}\binom{j-1}{p-1}\\{\zeta(j)+(-1)^{j}H_{n}^{(j)}\\}(s-n)^{j-p}\,.$
(2.2)

Next, according to the definitions of the _Hurwitz zeta function_
$\zeta(s,a+1)=\sum_{k=1}^{\infty}\frac{1}{(k+a)^{s}}$ and the _parametric
harmonic numbers_ $H_{n}^{(s)}(a)=\sum_{k=1}^{n}\frac{1}{(k+a)^{s}}$, we have

$\zeta(s,\tfrac{1}{2})=\tilde{t}(s)\,,\quad
H_{n}^{(s)}(\tfrac{1}{2})+2^{s}=h_{n+1}^{(s)}\,,\quad
H_{n}^{(s)}(-\tfrac{1}{2})=h_{n}^{(s)}\,,$

which yield Lemma 2.2.

######  Lemma 2.2.

For integers $n\geq 1$ and $p\geq 2$, the following expansions hold:

$\displaystyle\varPsi(\tfrac{1}{2}-s)\stackrel{{\scriptstyle s\to
n-1/2}}{{=}}h_{n}+\sum_{j=1}^{\infty}\\{(-1)^{j}h_{n}^{(j+1)}-\tilde{t}(j+1)\\}(s-n+\tfrac{1}{2})^{j}\,,$
(2.3)
$\displaystyle\frac{\varPsi^{(p-1)}(\tfrac{1}{2}-s)}{(p-1)!}\stackrel{{\scriptstyle
s\to
n-1/2}}{{=}}(-1)^{p}\sum_{j=p}^{\infty}\binom{j-1}{p-1}\\{\tilde{t}(j)+(-1)^{j}h_{n}^{(j)}\\}(s-n+\tfrac{1}{2})^{j-p}\,,$
(2.4) $\displaystyle\varPsi(\tfrac{1}{2}-s)\stackrel{{\scriptstyle
s\to-(n-1/2)}}{{=}}h_{n-1}+\sum_{j=1}^{\infty}\\{h_{n-1}^{(j+1)}-\tilde{t}(j+1)\\}(s+n-\tfrac{1}{2})^{j}\,,$
(2.5)
$\displaystyle\frac{\varPsi^{(p-1)}(\tfrac{1}{2}-s)}{(p-1)!}\stackrel{{\scriptstyle
s\to-(n-1/2)}}{{=}}(-1)^{p}\sum_{j=p}^{\infty}\binom{j-1}{p-1}\\{\tilde{t}(j)-h_{n-1}^{(j)}\\}(s+n-\tfrac{1}{2})^{j-p}\,.$
(2.6)

Besides the above two lemmas, by computation, we obtain the next one.

######  Lemma 2.3.

For integers $n\geq 1$ and $p\geq 2$, the following expansions hold:

$\displaystyle\varPsi(\tfrac{1}{2}-s)\stackrel{{\scriptstyle
s\to-n}}{{=}}H_{n-1}+2\ln(2)+\sum_{j=1}^{\infty}\\{H_{n-1}^{(j+1)}-\zeta(j+1)\\}(s+n)^{j}\,,$
(2.7)
$\displaystyle\frac{\varPsi^{(p-1)}(\tfrac{1}{2}-s)}{(p-1)!}\stackrel{{\scriptstyle
s\to-n}}{{=}}(-1)^{p-1}\sum_{j=p}^{\infty}\binom{j-1}{p-1}\\{H_{n-1}^{(j)}-\zeta(j)\\}(s+n)^{j-p}\,.$
(2.8)

###### Proof..

It can be found that

$\varPsi(\tfrac{1}{2}-s)\stackrel{{\scriptstyle
s\to-n}}{{=}}2+\sum_{k=0}^{\infty}\left(\frac{1}{k-1/2}-\frac{1}{k+n}-\frac{1}{k+n}\sum_{j=1}^{\infty}\left(\frac{s+n}{k+n}\right)^{j}\right)\,,$

which, together with the infinite series

$\displaystyle\sum_{k=0}^{\infty}\left(\frac{1}{k-1/2}-\frac{1}{k+n}\right)$
$\displaystyle=\sum_{k=0}^{\infty}\left(\frac{1}{k-1/2}-\frac{1}{k+1}\right)+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+n}\right)$
$\displaystyle=-2+2\ln(2)+\gamma+\psi(n)=-2+2\ln(2)+H_{n-1}\,,$

arising from the properties of the digamma function, gives (2.7).
Differentiating (2.7) $p-1$ times with respect to $s$ further leads us to the
second expansion of this lemma. ∎

In particular, if we interpret $\zeta(1):=-2\ln(2)$ and $\tilde{t}(1):=0$
wherever they occur, the expansions (2.2), (2.4), (2.6) and (2.8) hold for
$p=1$. Moreover, it can be found that the expansion (2.4) also hold for $n=0$,
for it coincides with the $n=1$ case of (2.6).

Finally, due to Flajolet and Salvy’s [6, Lemma 2.1], the following residue
theorem holds.

######  Lemma 2.4.

Let $\xi(s)$ be a kernel function and let $r(s)$ be a rational function which
is $O(s^{-2})$ at infinity. Then

$\sum_{\alpha\in O}{\rm Res}(r(s)\xi(s),\alpha)+\sum_{\beta\in S}{\rm
Res}(r(s)\xi(s),\beta)=0\,,$

where $S$ is the set of poles of $r(s)$ and $O$ is the set of poles of
$\xi(s)$ that are not poles of $r(s)$. Here ${\rm Res}(h(s),\lambda)$ denotes
the residue of $h(s)$ at $s=\lambda$, and the kernel function $\xi(s)$ is
meromorphic in the whole complex plane and satisfies $\xi(s)=o(s)$ over an
infinite collection of circles $|s|=\rho_{k}$ with $\rho_{k}\to+\infty$.

## 3\. Symmetric extension on linear $T$-sums and $\tilde{S}$-sums

### 3.1. Main theorem on linear $T$-sums and $\tilde{S}$-sums

Let us consider the symmetric extension of the Kaneko-Tsumura conjecture on
the linear $T$-sums and $\tilde{S}$-sums.

######  Theorem 3.1.

For integers $m,p\geq 1$ and $q\geq 2$, we have

$\displaystyle(-1)^{m-1}\sum_{\begin{subarray}{c}i+j=p-1\\\ i,j\geq
0\end{subarray}}\binom{m+i-1}{i}\binom{q+j-1}{j}T_{m+i,q+j}$
$\displaystyle\quad+(-1)^{p-1}\sum_{\begin{subarray}{c}i+j=m-1\\\ i,j\geq
0\end{subarray}}\binom{p+i-1}{i}\binom{q+j-1}{j}\tilde{S}_{p+i,q+j}\in\mathbb{Q}[\ln(2),\mathrm{zeta\
values}]\,.$ (3.1)

In particular, the following expression holds:

$\displaystyle(-1)^{m-1}\lambda_{p}(T_{m,q})+(-1)^{p-1}\lambda_{m}(\tilde{S}_{p,q})$
$\displaystyle\quad=(-1)^{m}\lambda_{p}((-1)^{m}\tilde{t}(m)\tilde{t}(q))+(-1)^{p}\lambda_{m}((-1)^{p}\tilde{t}(p)\zeta(q))-\lambda_{q}(\zeta(m)\tilde{t}(p))\,,$
(3.2)

where, by our conventions, $\zeta(1):=-2\ln(2)$ and $\tilde{t}(1):=0$ wherever
they occur.

###### Proof..

To prove this identity, we consider

$\mathcal{G}_{1}(s):=\frac{\varPsi^{(m-1)}(\tfrac{1}{2}-s)\varPsi^{(p-1)}(-s)}{(s+1)^{q}(m-1)!(p-1)!}\,.$

The function $\mathcal{G}_{1}(s)$ has a pole of order $q$ at $s=-1$. By (2.6)
and (2.8), the residue is

${\rm
Res}(\mathcal{G}_{1}(s),-1)=(-1)^{m+p}\lambda_{q}(\zeta(m)\tilde{t}(p))\,.$

Similarly, $\mathcal{G}_{1}(s)$ has poles of order $m$ at $s=n$ and poles of
order $p$ at $s=n-1/2$ for $n\geq 0$. Then by appealing to the expansions
(2.2) and (2.4), the residues are found to be

${\rm Res}(\mathcal{G}_{1}(s),n)=\sum_{\begin{subarray}{c}i+j=m-1\\\ i,j\geq
0\end{subarray}}(-1)^{p+j}\binom{p+i-1}{i}\binom{q+j-1}{j}\frac{\tilde{t}(p+i)+(-1)^{p+i}h_{n+1}^{(p+i)}}{(n+1)^{q+j}}$

and

${\rm
Res}(\mathcal{G}_{1}(s),n-\tfrac{1}{2})=\sum_{\begin{subarray}{c}i+j=p-1\\\
i,j\geq
0\end{subarray}}(-1)^{m+j}\binom{m+i-1}{i}\binom{q+j-1}{j}\frac{\tilde{t}(m+i)+(-1)^{m+i}h_{n}^{(m+i)}}{(n+1/2)^{q+j}}\,,$

respectively. Hence, combining these three residue results, applying Lemma
2.4, and using the definitions of $\zeta(s)$, $\tilde{t}(s)$, $T_{p,q}$ and
$\tilde{S}_{p,q}$, we obtain (3.2), which further gives the statement (3.1)
because $\tilde{t}(s)=(2^{s}-1)\zeta(s)$ for integer $s\geq 2$. ∎

In fact, Theorem 3.1 gives an infinite series identity of weight $w=m+p+q-1$.
Note that, for odd weights, all the linear $T$-sums $T_{p,q}$ and
$\tilde{S}$-sums $\tilde{S}_{p,q}$ are already reducible to $\ln(2)$ and zeta
values [21, Corollaries 3.3 and 3.8]. Therefore, in this case, the statement
(3.1) is somewhat trivial. However, for even weights, the sums $T_{p,q}$ and
$\tilde{S}_{p,q}$ may be only expressible in terms of (alternating) zeta
values and double zeta values (see Eqs. (1.7) and (1.8)), but Theorem 3.1
asserts that the symmetric sums

$(-1)^{m-1}\lambda_{p}(T_{m,q})+(-1)^{p-1}\lambda_{m}(\tilde{S}_{p,q})$

are still reducible to $\ln(2)$ and zeta values. In Sections 3.2 – 3.5, we
present several special cases and related examples of this theorem.

### 3.2. The case of $m=p=1$

When $m=p=1$, Theorem 3.1 reduces to the next result.

######  Corollary 3.2.

For integer $q\geq 2$, the sums $T_{1,q}+\tilde{S}_{1,q}$ are reducible to
$\ln(2)$ and zeta values:

$T_{1,q}+\tilde{S}_{1,q}=\sum_{n=1}^{\infty}\frac{h_{n-1}}{(n-1/2)^{q}}+\sum_{n=1}^{\infty}\frac{h_{n}}{n^{q}}=2\ln(2)\tilde{t}(q)-\sum_{j=1}^{q-2}\zeta(q-j)\tilde{t}(j+1)\,.$
(3.3)

######  Example 3.1.

In Theorem 3.1, replacing $(m,p,q)$ by $(1,1,2)$ – $(1,1,5)$ yields

$\displaystyle T_{1,2}+\tilde{S}_{1,2}=\pi^{2}\ln(2)\,,$ $\displaystyle
T_{1,3}+\tilde{S}_{1,3}=14\ln(2)\zeta(3)-\tfrac{1}{12}\pi^{4}\,,$
$\displaystyle
T_{1,4}+\tilde{S}_{1,4}=-\tfrac{5}{3}\pi^{2}\zeta(3)+\tfrac{1}{3}\pi^{4}\ln(2)\,,$
$\displaystyle
T_{1,5}+\tilde{S}_{1,5}=62\ln(2)\zeta(5)-7\zeta(3)^{2}-\tfrac{1}{30}\pi^{6}\,,$

respectively, which correspond to the cases of $q=2,3,4,5$ of Corollary 3.2.
Note that the two ones corresponding to $q=3,5$ are of even weights, and the
evaluations of the four involved linear $T$-sums and $\tilde{S}$-sums contain
the polylogarithm $\,\mathrm{Li}_{4}(\tfrac{1}{2})$ and alternating double
zeta values $\zeta(\bar{5},1)$:

$\displaystyle
T_{1,3}=-16\,\mathrm{Li}_{4}(\tfrac{1}{2})-\tfrac{2}{3}\ln(2)^{4}+\tfrac{2}{3}\pi^{2}\ln(2)^{2}+\tfrac{23}{360}\pi^{4}\,,$
$\displaystyle
T_{1,5}=-32\zeta(\bar{5},1)+62\ln(2)\zeta(5)+\tfrac{17}{2}\zeta(3)^{2}-\tfrac{73}{1260}\pi^{6}\,,$
$\displaystyle\tilde{S}_{1,3}=16\,\mathrm{Li}_{4}(\tfrac{1}{2})+14\ln(2)\zeta(3)+\tfrac{2}{3}\ln(2)^{4}-\tfrac{2}{3}\pi^{2}\ln(2)^{2}-\tfrac{53}{360}\pi^{4}\,,$
$\displaystyle\tilde{S}_{1,5}=32\zeta(\bar{5},1)-\tfrac{31}{2}\zeta(3)^{2}+\tfrac{31}{1260}\pi^{6}\,.$

These four linear sums have been computed in [21, Examples 3.5 and 3.8]. ∎

### 3.3. The case of $m=q$ and $p$ odd

When $m=q\geq 2$ and $p$ is odd, Theorem 3.1 reduces to

$\displaystyle(-1)^{q-1}\lambda_{p}(T_{q,q})+\lambda_{q}(\tilde{S}_{p,q})$
$\displaystyle\quad=(-1)^{q}\lambda_{p}((-1)^{q}\tilde{t}(q)\tilde{t}(q))+(-1)^{p}\lambda_{q}((-1)^{p}\tilde{t}(p)\zeta(q))-\lambda_{q}(\zeta(q)\tilde{t}(p))\,,$

where the last two terms on the right can be combined into one, as follows:

$\displaystyle(-1)^{p}\lambda_{q}((-1)^{p}\tilde{t}(p)\zeta(q))-\lambda_{q}(\zeta(q)\tilde{t}(p))$
$\displaystyle\quad=\sum_{\begin{subarray}{c}i+j=q-1\\\ i,j\geq
0\end{subarray}}\binom{p+i-1}{i}\binom{q+j-1}{j}((-1)^{i}-1)\tilde{t}(p+i)\zeta(q+j)$
$\displaystyle\quad=-2\sum_{\begin{subarray}{c}i=1\\\ i\
\text{even}\end{subarray}}^{q}\binom{p+i-2}{p-1}\binom{2q-i-1}{q-i}\tilde{t}(p+i-1)\zeta(2q-i)\,.$

Thus, we obtain

$\displaystyle(-1)^{q-1}\lambda_{p}(T_{q,q})+\lambda_{q}(\tilde{S}_{p,q})$
$\displaystyle\quad=(-1)^{q}\lambda_{p}((-1)^{q}\tilde{t}(q)\tilde{t}(q))-2\sum_{k=1}^{[\frac{q}{2}]}\binom{p-2+2k}{p-1}\binom{2q-1-2k}{q-2k}\tilde{t}(p-1+2k)\zeta(2q-2k)\,,$
(3.4)

for integer $q\geq 2$ and odd integer $p\geq 1$. The further special cases of
$p=1,3,5$ of the above identity are of particular interest.

######  Corollary 3.3.

For integer $q\geq 2$, we have

$\displaystyle(-1)^{q-1}T_{q,q}+\lambda_{q}(\tilde{S}_{1,q})$
$\displaystyle=(-1)^{q-1}T_{q,q}+\sum_{j=0}^{q-1}\binom{q+j-1}{j}\tilde{S}_{q-j,q+j}$
$\displaystyle=\tilde{t}(q)^{2}-2\sum_{k=1}^{[\frac{q}{2}]}\binom{2q-1-2k}{q-2k}\tilde{t}(2k)\zeta(2q-2k)\,.$

Therefore, the sums $(-1)^{q-1}T_{q,q}+\lambda_{q}(\tilde{S}_{1,q})$ reduce to
rational combinations of $\zeta(q)^{2}$ and $\pi^{2q}$ if $q$ is odd, and to
rational multiples of $\pi^{2q}$ if $q$ is even.

###### Proof..

The final assertion arises from the fact that

$\tilde{t}(2k)\zeta(2q-2k)=(2^{2k}-1)\zeta(2k)\zeta(2q-2k)$

are rational multiples of $\pi^{2q}$. ∎

######  Example 3.2.

In Theorem 3.1, replacing $(m,p,q)$ by $(2,1,2)$ – $(6,1,6)$ yields

$\displaystyle
T_{2,2}-2\tilde{S}_{1,3}-\tilde{S}_{2,2}=-\tfrac{1}{12}\pi^{4}\,,$
$\displaystyle
T_{3,3}+6\tilde{S}_{1,5}+3\tilde{S}_{2,4}+\tilde{S}_{3,3}=49\zeta(3)^{2}-\tfrac{1}{30}\pi^{6}\,,$
$\displaystyle
T_{4,4}-20\tilde{S}_{1,7}-10\tilde{S}_{2,6}-4\tilde{S}_{3,5}-\tilde{S}_{4,4}=-\tfrac{17}{1260}\pi^{8}\,,$
$\displaystyle
T_{5,5}+70\tilde{S}_{1,9}+35\tilde{S}_{2,8}+15\tilde{S}_{3,7}+5\tilde{S}_{4,6}+\tilde{S}_{5,5}=961\zeta(5)^{2}-\tfrac{31}{5670}\pi^{10}\,,$
$\displaystyle
T_{6,6}-252\tilde{S}_{1,11}-126\tilde{S}_{2,10}-56\tilde{S}_{3,9}-21\tilde{S}_{4,8}-6\tilde{S}_{5,7}-\tilde{S}_{6,6}=-\tfrac{691}{311850}\pi^{12}\,,$

respectively. These correspond to the case of $q=2,3,4,5,6$ of Corollary 3.3.
∎

Similarly, when $m=q\geq 2$ and $p=3,5$, Theorem 3.1 reduces to the next two
results.

######  Corollary 3.4.

For integer $q\geq 2$, we have

$\displaystyle(-1)^{q-1}\lambda_{3}(T_{q,q})+\lambda_{q}(\tilde{S}_{3,q})$
$\displaystyle=q(q+1)\tilde{t}(q)\tilde{t}(q+2)-q^{2}\tilde{t}(q+1)^{2}$
$\displaystyle\quad-2\sum_{k=1}^{[\frac{q}{2}]}\binom{2k+1}{2}\binom{2q-1-2k}{q-2k}\tilde{t}(2k+2)\zeta(2q-2k)\,.$

Therefore, the sums
$(-1)^{q-1}\lambda_{3}(T_{q,q})+\lambda_{q}(\tilde{S}_{3,q})$ are reducible to
rational combinations of $\zeta(q)\zeta(q+2)$ and $\pi^{2q+2}$ if $q$ is odd,
and to rational combinations of $\zeta(q+1)^{2}$ and $\pi^{2q+2}$ if $q$ is
even.

######  Corollary 3.5.

For integer $q\geq 2$, we have

$\displaystyle(-1)^{q-1}\lambda_{5}(T_{q,q})+\lambda_{q}(\tilde{S}_{5,q})$
$\displaystyle\quad=2\binom{q+3}{4}\tilde{t}(q)\tilde{t}(q+4)-2q\binom{q+2}{3}\tilde{t}(q+1)\tilde{t}(q+3)+\binom{q+1}{2}^{2}\tilde{t}(q+2)^{2}$
$\displaystyle\quad\quad-2\sum_{k=1}^{[\frac{q}{2}]}\binom{2k+3}{4}\binom{2q-1-2k}{q-2k}\tilde{t}(2k+4)\zeta(2q-2k)\,.$

Therefore, the sums
$(-1)^{q-1}\lambda_{5}(T_{q,q})+\lambda_{q}(\tilde{S}_{5,q})$ are reducible to
rational combinations of $\zeta(q)\zeta(q+4)$, $\zeta(q+2)^{2}$ and
$\pi^{2q+4}$ if $q$ is odd, and to rational combinations of
$\zeta(q+1)\zeta(q+3)$ and $\pi^{2q+4}$ if $q$ is even.

######  Example 3.3.

In Theorem 3.1, replacing $(m,p,q)$ by $(2,3,2)$, $(3,3,3)$, $(4,3,4)$ gives

$\displaystyle
3T_{2,4}+4T_{3,3}+3T_{4,2}-2\tilde{S}_{3,3}-3\tilde{S}_{4,2}=196\zeta(3)^{2}-\tfrac{1}{3}\pi^{6}\,,$
$\displaystyle
2T_{3,5}+3T_{4,4}+2T_{5,3}+2\tilde{S}_{3,5}+3\tilde{S}_{4,4}+2\tilde{S}_{5,3}=868\zeta(3)\zeta(5)-\tfrac{17}{180}\pi^{8}\,,$
$\displaystyle
5T_{4,6}+8T_{5,5}+5T_{6,4}-10\tilde{S}_{3,7}-15\tilde{S}_{4,6}-12\tilde{S}_{5,5}-5\tilde{S}_{6,4}=7688\zeta(5)^{2}-\tfrac{31}{315}\pi^{10}\,,$

respectively. Replacing $(m,p,q)$ by $(2,5,2)$, $(3,5,3)$ yields

$\displaystyle
5T_{2,6}+8T_{3,5}+9T_{4,4}+8T_{5,3}+5T_{6,2}-2\tilde{S}_{5,3}-5\tilde{S}_{6,2}=3472\zeta(3)\zeta(5)-\tfrac{17}{36}\pi^{8}\,,$
$\displaystyle
5T_{3,7}+10T_{4,6}+12T_{5,5}+10T_{6,4}+5T_{7,3}+2\tilde{S}_{5,5}+5\tilde{S}_{6,4}+5\tilde{S}_{7,3}$
$\displaystyle\quad=8890\zeta(3)\zeta(7)+11532\zeta(5)^{2}-\tfrac{31}{135}\pi^{10}\,,$

respectively. These can also be obtained directly from Corollaries 3.4 and
3.5. ∎

### 3.4. More discussions on the case of $m=q$ and $p$ odd

It will be interesting to give an explicit characterization for the
coefficients of various mathematical constants involved in the reduction of
the sums $(-1)^{q-1}\lambda_{p}(T_{q,q})+\lambda_{q}(\tilde{S}_{p,q})$
appeared in (3.4) in the last section.

To do this, let $B_{n}$ be the well-known _Bernoulli numbers_ and $G_{n}$ be
the _Genocchi numbers_ , defined by

$\frac{t}{\mathrm{e}^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}\quad\text{and}\quad\frac{2t}{\mathrm{e}^{t}+1}=\sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}\,,$

respectively (see, for example, [4, Section 1.14]). Then
$G_{n}=2(1-2^{n})B_{n}$ for $n\geq 0$, and $B_{2k+1}=G_{2k+1}=0$ for $k\geq
1$. As mentioned in Section 1, the convolution identity (1.6) on the Bernoulli
numbers and Genocchi numbers can be established, which helps us obtain the
next theorem. Note that the proof of identity (1.6) will be given in the last
section (i.e., Section 5) of the present paper.

######  Theorem 3.6.

For integer $q\geq 2$ and odd integer $p\geq 1$, the following explicit
expression holds:

$\displaystyle(-1)^{q-1}\lambda_{p}(T_{q,q})+\lambda_{q}(\tilde{S}_{p,q})$
$\displaystyle\quad=(-1)^{q-1}\sum_{\begin{subarray}{c}i=0\\\ q+i\
\mathrm{odd}\end{subarray}}^{p-1}\binom{q+i-1}{i}\binom{p+q-2-i}{p-1-i}\tilde{t}(q+i)\tilde{t}(p+q-1-i)$
$\displaystyle\quad\quad+\frac{(-1)^{\frac{p-1}{2}}}{8}\binom{p-2+2q}{p-1}\frac{G_{p-1+2q}}{(p-1+2q)!}(2\pi)^{p-1+2q}\,.$
(3.5)

###### Proof..

By appealing to the values of the Riemann zeta function at even positive
integers:

$\zeta(2k)=\frac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2\cdot(2k)!}\,,$

the second term on the right of Eq. (3.4), abbreviated as $\varOmega_{2}$, can
be rewritten as

$\displaystyle\varOmega_{2}$
$\displaystyle=-2\sum_{k=1}^{[\frac{q}{2}]}\binom{p-2+2k}{p-1}\binom{2q-1-2k}{q-2k}(2^{p-1+2k}-1)\zeta(p-1+2k)\zeta(2q-2k)$
$\displaystyle=\frac{(-1)^{\frac{p-1+2q}{2}}}{2}(2\pi)^{p-1+2q}\sum_{k=1}^{[\frac{q}{2}]}\binom{p-2+2k}{p-1}\binom{2q-1-2k}{q-2k}\frac{(1-2^{p-1+2k})B_{p-1+2k}B_{2q-2k}}{(p-1+2k)!(2q-2k)!}$
$\displaystyle=\frac{(-1)^{\frac{p-1+2q}{2}}}{4}\frac{(2\pi)^{p-1+2q}}{(p-1)!(2q)!}\binom{2q}{q}q^{2}\sum_{k=1}^{[\frac{q}{2}]}\binom{q-1}{2k-1}\frac{G_{p-1+2k}B_{2q-2k}}{(p-1+2k)(2q-2k)}\,.$
(3.6)

By considering the fact $B_{2k+1}=G_{2k+1}=0$ for $k\geq 1$, the convolution
identity (1.6) can be rewritten as

$\displaystyle\sum_{k=1}^{[\frac{q}{2}]}\binom{q-1}{2k-1}\frac{G_{p-1+2k}B_{2q-2k}}{(p-1+2k)(2q-2k)}$
$\displaystyle\quad=\frac{(-1)^{q}}{q\binom{2q}{q}}\frac{G_{p-1+2q}}{p-1+2q}+\frac{1}{4}\sum_{\begin{subarray}{c}i=0\\\
q+i\
\mathrm{even}\end{subarray}}^{p-1}(-1)^{i+1}\binom{p-1}{i}\frac{G_{q+i}G_{p+q-1-i}}{(q+i)(p+q-1-i)}\,,$

where $p\geq 1$ is odd, and $q\geq 2$. Substituting it into the right of
(3.6), and using the relation

$G_{2k}=2(1-2^{2k})B_{2k}=\frac{(-1)^{k}\cdot
4\tilde{t}(2k)\cdot(2k)!}{(2\pi)^{2k}}\,,$

we have

$\displaystyle\varOmega_{2}$
$\displaystyle=\frac{(-1)^{\frac{p-1}{2}}}{8}\binom{p-2+2q}{p-1}\frac{G_{p-1+2q}}{(p-1+2q)!}(2\pi)^{p-1+2q}$
$\displaystyle\quad\quad-\sum_{\begin{subarray}{c}i=0\\\ q+i\
\mathrm{even}\end{subarray}}^{p-1}(-1)^{i}\binom{q+i-1}{i}\binom{p+q-2-i}{p-1-i}\tilde{t}(q+i)\tilde{t}(p+q-1-i)\,.$

On the other hand, the first term on the right of Eq. (3.4) equals

$\displaystyle\varOmega_{1}$
$\displaystyle=(-1)^{q}\lambda_{p}((-1)^{q}\tilde{t}(q)\tilde{t}(q))$
$\displaystyle=(-1)^{q}\sum_{\begin{subarray}{c}i+j=p-1\\\ i,j\geq
0\end{subarray}}\binom{q+i-1}{i}\binom{q+j-1}{j}(-1)^{q+i}\tilde{t}(q+i)\tilde{t}(q+j)$
$\displaystyle=\left\\{\sum_{\begin{subarray}{c}i=0\\\ q+i\
\mathrm{odd}\end{subarray}}^{p-1}+\sum_{\begin{subarray}{c}i=0\\\ q+i\
\mathrm{even}\end{subarray}}^{p-1}\right\\}(-1)^{i}\binom{q+i-1}{i}\binom{p+q-2-i}{p-1-i}\tilde{t}(q+i)\tilde{t}(p+q-1-i)\,.$

Thus, all the summands with even $q+i$ will be eliminated in
$\varOmega_{1}+\varOmega_{2}$, and we obtain (3.5). ∎

By Theorem 3.6, the coefficients of various mathematical constants in the
evaluations of the sums
$(-1)^{q-1}\lambda_{p}(T_{q,q})+\lambda_{q}(\tilde{S}_{p,q})$ can be
completely determined. For example, setting $p=1$ in Eq. (3.5) gives

$(-1)^{q-1}T_{q,q}+\sum_{j=0}^{q-1}\binom{q+j-1}{j}\tilde{S}_{q-j,q+j}=\frac{1-(-1)^{q}}{2}(2^{q}-1)^{2}\zeta(q)^{2}+\frac{G_{2q}}{8\cdot(2q)!}(2\pi)^{2q}\,,$

and setting $p=3$ yields

$\displaystyle(-1)^{q-1}\lambda_{3}(T_{q,q})+\lambda_{q}(\tilde{S}_{3,q})$
$\displaystyle\quad=\frac{1-(-1)^{q}}{2}q(q+1)(2^{q}-1)(2^{q+2}-1)\zeta(q)\zeta(q+2)-\frac{1+(-1)^{q}}{2}q^{2}(2^{q+1}-1)^{2}\zeta(q+1)^{2}$
$\displaystyle\quad\quad-q(2q+1)\frac{G_{2q+2}}{8\cdot(2q+2)!}(2\pi)^{2q+2}\,.$

### 3.5. The case of $m=q$ and $p$ even

Here, we present some interesting cases of Theorem 3.1 with odd weights. For
example, setting $(m,p,q)$ by $(2,1,3)$ and $(3,1,2)$ yields the following two
ones:

$\displaystyle
T_{2,3}-3\tilde{S}_{1,4}-\tilde{S}_{2,3}=-\tfrac{5}{6}\pi^{2}\zeta(3)\,,$
$\displaystyle
T_{3,2}+3\tilde{S}_{1,4}+2\tilde{S}_{2,3}+\tilde{S}_{3,2}=\tfrac{19}{6}\pi^{2}\zeta(3)\,,$

which are reducible to rational multiples of $\pi^{2}\zeta(3)$. Moreover, the
special cases of Theorem 3.1 with the mode of $(m,p,q)=(q,2n,q)$ also deserve
attention.

######  Corollary 3.7.

For integers $q\geq 2$ and $n\geq 1$, the sums
$(-1)^{q-1}\lambda_{2n}(T_{q,q})-\lambda_{q}(\tilde{S}_{2n,q})$ are
expressible in terms of $\zeta(2n+i)\pi^{2q-1-i}$, where $i$ is odd and
satisfies $1\leq i\leq q-1$. In particular, we have

$\displaystyle(-1)^{q-1}\lambda_{2n}(T_{q,q})-\lambda_{q}(\tilde{S}_{2n,q})$
$\displaystyle\quad=-2\sum_{\begin{subarray}{c}i+j=q-1\\\ i\emph{
odd}\end{subarray}}\binom{2n+i-1}{i}\binom{q+j-1}{j}\tilde{t}(2n+i)\zeta(q+j)\,.$
(3.7)

###### Proof..

When $m=q$ and $p=2n$, by Theorem 3.1, the left side of Eq. (3.7) equals

$\text{LHS}=(-1)^{q}\lambda_{2n}((-1)^{q}\tilde{t}(q)\tilde{t}(q))+\lambda_{q}((-1)^{2n}\tilde{t}(2n)\zeta(q))-\lambda_{q}(\zeta(q)\tilde{t}(2n))\,.$

Note that the sum

$\displaystyle(-1)^{q}\lambda_{2n}((-1)^{q}\tilde{t}(q)\tilde{t}(q))$
$\displaystyle\quad=\left\\{\sum_{i=0}^{n-1}+\sum_{i=n}^{2n-1}\right\\}\binom{q+i-1}{q-1}\binom{q+2n-2-i}{q-1}(-1)^{i}\tilde{t}(q+i)\tilde{t}(q+2n-1-i)$

will vanish by changing of the variable $i\to 2n-1-i$ in the second term, and

$\displaystyle\lambda_{q}((-1)^{2n}\tilde{t}(2n)\zeta(q))-\lambda_{q}(\zeta(q)\tilde{t}(2n))$
$\displaystyle\quad=\sum_{\begin{subarray}{c}i+j=q-1\\\ i,j\geq
0\end{subarray}}\binom{2n+i-1}{i}\binom{q+j-1}{j}((-1)^{i}-1)\tilde{t}(2n+i)\zeta(q+j)\,.$

Thus, we obtain Eq. (3.7). Finally, when $i$ is odd, $q+j=2q-1-i$ is even, so
the assertion in the corollary also holds. ∎

It is obvious that there are $[q/2]$ terms in the right side of (3.7). Here,
we present the further cases of $q=2,3,4,5$, which satisfy $[q/2]<3$.

######  Corollary 3.8.

For integer $n\geq 1$, the sums
$\lambda_{2n}(T_{2,2})+\lambda_{2}(\tilde{S}_{2n,2})$ reduce to rational
multiples of $\pi^{2}\zeta(2n+1)$. In particular, we have

$\frac{1}{2}\sum_{\begin{subarray}{c}i+j=2n-1\\\ i,j\geq
0\end{subarray}}(i+1)(j+1)T_{i+2,j+2}+\\{n\tilde{S}_{2n+1,2}+\tilde{S}_{2n,3}\\}=\frac{n(2^{2n+1}-1)}{3}\pi^{2}\zeta(2n+1)\,.$

###### Proof..

It follows from the identity
$\lambda_{2n}(T_{2,2})+\lambda_{2}(\tilde{S}_{2n,2})=4n\zeta(2)\tilde{t}(2n+1)$,
which corresponds to the case $(m,p,q)=(2,2n,2)$. ∎

######  Example 3.4.

Replacing $(m,p,q)$ in Theorem 3.1 by $(2,2,2)$, $(2,4,2)$, $(2,6,2)$ yields

$\displaystyle
T_{2,3}+T_{3,2}+\tilde{S}_{2,3}+\tilde{S}_{3,2}=\tfrac{7}{3}\pi^{2}\zeta(3)\,,$
$\displaystyle
2T_{2,5}+3T_{3,4}+3T_{4,3}+2T_{5,2}+\tilde{S}_{4,3}+2\tilde{S}_{5,2}=\tfrac{62}{3}\pi^{2}\zeta(5)\,,$
$\displaystyle
3T_{2,7}+5T_{3,6}+6T_{4,5}+6T_{5,4}+5T_{6,3}+3T_{7,2}+\tilde{S}_{6,3}+3\tilde{S}_{7,2}=127\pi^{2}\zeta(7)\,.$

These can also be obtained from Corollary 3.8 by setting $n=1,2,3$. ∎

The following three identities give special cases of
$(m,p,q)=(3,2n,3),(4,2n,4),(5,2n,5)$, respectively, and can be derived
directly from Corollary 3.7.

######  Corollary 3.9.

For integer $n\geq 1$, we have

$\displaystyle\lambda_{2n}(T_{3,3})-\lambda_{3}(\tilde{S}_{2n,3})=-12n\zeta(4)\tilde{t}(2n+1)\,,$
$\displaystyle\lambda_{2n}(T_{4,4})+\lambda_{4}(\tilde{S}_{2n,4})=40n\zeta(6)\tilde{t}(2n+1)+\tfrac{1}{3}(2n+2)(2n+1)(2n)\zeta(4)\tilde{t}(2n+3)\,,$
$\displaystyle\lambda_{2n}(T_{5,5})-\lambda_{5}(\tilde{S}_{2n,5})=-140n\zeta(8)\tilde{t}(2n+1)-\tfrac{5}{3}(2n+2)(2n+1)(2n)\zeta(6)\tilde{t}(2n+3)\,.$

Therefore, the sums
$(-1)^{q-1}\lambda_{2n}(T_{q,q})-\lambda_{q}(\tilde{S}_{2n,q})$ reduce to
rational multiples of $\pi^{4}\zeta(2n+1)$ if $q=3$. Moreover, they are
reducible to combinations of $\pi^{6}\zeta(2n+1)$ and $\pi^{4}\zeta(2n+3)$ if
$q=4$, and to combinations of $\pi^{8}\zeta(2n+1)$ and $\pi^{6}\zeta(2n+3)$ if
$q=5$.

######  Example 3.5.

Replacing $(m,p,q)$ by $(3,2,3)$, $(3,4,3)$ in Theorem 3.1 yields

$\displaystyle
T_{3,4}+T_{4,3}-2\tilde{S}_{2,5}-2\tilde{S}_{3,4}-\tilde{S}_{4,3}=-\tfrac{14}{45}\pi^{4}\zeta(3)\,,$
$\displaystyle
5T_{3,6}+9T_{4,5}+9T_{5,4}+5T_{6,3}-3\tilde{S}_{4,5}-6\tilde{S}_{5,4}-5\tilde{S}_{6,3}=-\tfrac{62}{15}\pi^{4}\zeta(5)\,,$

respectively. Replacing $(m,p,q)$ by $(4,2,4)$, $(5,2,5)$ yields

$\displaystyle
T_{4,5}+T_{5,4}+5\tilde{S}_{2,7}+5\tilde{S}_{3,6}+3\tilde{S}_{4,5}+\tilde{S}_{5,4}=\tfrac{31}{45}\pi^{4}\zeta(5)+\tfrac{2}{27}\pi^{6}\zeta(3)\,,$
$\displaystyle
T_{5,6}+T_{6,5}-14\tilde{S}_{2,9}-14\tilde{S}_{3,8}-9\tilde{S}_{4,7}-4\tilde{S}_{5,6}-\tilde{S}_{6,5}=-\tfrac{248}{945}\pi^{6}\zeta(5)-\tfrac{14}{675}\pi^{8}\zeta(3)\,,$

respectively. ∎

### 3.6. Symmetric sum on double $T$-values and double $t$-values

By the relations

$\tilde{S}_{p,q}=2^{p+q-2}T(q,p)\quad\text{and}\quad
T_{p,q}=\tilde{t}(q,p)=2^{p+q}t(q,p)\,,$

we can transform Theorem 3.1 into the following one on Hoffman’s double
$t$-values and Kaneko-Tsumura’s double $T$-values.

######  Theorem 3.10.

For integers $m,p\geq 1$ and $q\geq 2$, we have

$(-1)^{m-1}2^{m+p+q-1}\lambda_{p}(t(q,m))+(-1)^{p-1}2^{m+p+q-3}\lambda_{m}(T(q,p))\in\mathbb{Q}[\ln(2),\mathrm{zeta\
values}]\,.$

In particular, the following expression holds:

$\displaystyle(-1)^{m-1}2^{m+p+q-1}\lambda_{p}(t(q,m))+(-1)^{p-1}2^{m+p+q-3}\lambda_{m}(T(q,p))$
$\displaystyle\quad=(-1)^{m}\lambda_{p}((-1)^{m}\tilde{t}(m)\tilde{t}(q))+(-1)^{p}\lambda_{m}((-1)^{p}\tilde{t}(p)\zeta(q))-\lambda_{q}(\zeta(m)\tilde{t}(p))\,,$
(3.8)

where, by our conventions, $\zeta(1):=-2\ln(2)$ and $\tilde{t}(1):=0$ wherever
they occur.

###### Proof..

It follows from the transformation formulas

$\lambda_{p}(T_{m,q})=2^{m+p+q-1}\lambda_{p}(t(q,m))\,,\quad\lambda_{m}(\tilde{S}_{p,q})=2^{m+p+q-3}\lambda_{m}(T(q,p))\,,$
(3.9)

and Eq. (3.2). ∎

In the next corollary, we show that by Theorem 3.10, similar results to (1.2)
also hold for double $t$-values, linear $T$-sums and linear $\tilde{S}$-sums,
from which we can further show a reduction property of the double $t$-values
and linear $T$-sums.

######  Corollary 3.11.

For any even weight $w:=m+q$, where $m,q\geq 2$, the double $t$-values
$t(q,m)$ and linear $T$-sums $T_{m,q}$ are expressible in terms of MZVs.

###### Proof..

According to the Kaneko-Tsumura conjecture (1.2) (see also [18, Theorem 42]),
for integers $m,q\geq 2$ and $p\geq 1$, with $m+p+q-1$ even, the sums
$\lambda_{m}(T(q,p))\in\mathcal{Z}$, so
$\lambda_{m}(\tilde{S}_{p,q})\in\mathcal{Z}$ by (3.9). On the other hand, if
$m,q\geq 2$, we have

$(-1)^{m-1}2^{m+p+q-1}\lambda_{p}(t(q,m))+(-1)^{p-1}2^{m+p+q-3}\lambda_{m}(T(q,p))\in\mathbb{Q}[\mathrm{zeta\
values}]\,,$

and $\ln(2)$ does not appear in the evaluations of the sums on the left. Thus,
when $m,q\geq 2$, $p\geq 1$, and $m+p+q-1$ is even, we have

$\lambda_{p}(t(q,m)),\ \lambda_{p}(T_{m,q})\in\mathcal{Z}\,.$ (3.10)

The final assertion of this corollary follows by setting $p=1$. ∎

######  Remark 3.12.

Corollary 3.11 can be compared with [21, Corollary 3.3], where the latter
shows that $T_{p,q}$ and $t(q,p)$ are reducible to $\ln(2)$ and zeta values if
$p\geq 1$, $q\geq 2$, and $p+q$ is odd. Corollary 3.11 is also a special case
of Murakami’s recent result [18, Theorem 1], which shows that when all
$s_{j}\geq 2$, the MtVs $t(s_{1},\ldots,s_{k})$ are expressible in terms of
MZVs. ∎

## 4\. Symmetric extension on linear $R$-sums

### 4.1. Main theorem on linear $R$-sums

In this section, we present a symmetric extension of the Kaneko-Tsumura
conjecture (1.2) on the linear $R$-sums:

$R_{p,q}:=\sum_{n=1}^{\infty}\frac{H_{n-1}^{(p)}}{(n-1/2)^{q}}\,,\quad\text{for
}q\geq 2\,.$

######  Theorem 4.1.

For integers $m,p\geq 1$ and $q\geq 2$, we have

$\displaystyle(-1)^{m-1}\sum_{\begin{subarray}{c}i+j=p-1\\\ i,j\geq
0\end{subarray}}\binom{m+i-1}{i}\binom{q+j-1}{j}R_{m+i,q+j}$
$\displaystyle\quad+(-1)^{p-1}\sum_{\begin{subarray}{c}i+j=m-1\\\ i,j\geq
0\end{subarray}}\binom{p+i-1}{i}\binom{q+j-1}{j}R_{p+i,q+j}\in\mathbb{Q}[\ln(2),\mathrm{zeta\
values}]\,.$ (4.1)

In particular, the following expression holds:

$\displaystyle(-1)^{m-1}\lambda_{p}(R_{m,q})+(-1)^{p-1}\lambda_{m}(R_{p,q})$
$\displaystyle\quad=\binom{m+p+q-2}{q-1}\tilde{t}(m+p+q-1)+(-1)^{m}\lambda_{p}((-1)^{m}\zeta(m)\tilde{t}(q))$
$\displaystyle\quad\quad+(-1)^{p}\lambda_{m}((-1)^{p}\zeta(p)\tilde{t}(q))-\lambda_{q}(\tilde{t}(m)\tilde{t}(p))\,,$
(4.2)

where, by our conventions, $\zeta(1):=-2\ln(2)$ and $\tilde{t}(1):=0$ wherever
they occur.

###### Proof..

The proof of this theorem is similar to that of Theorem 3.1. Now, consider the
function

$\mathcal{G}_{2}(s):=\frac{\varPsi^{(m-1)}(\tfrac{1}{2}-s)\varPsi^{(p-1)}(\tfrac{1}{2}-s)}{(s+\tfrac{1}{2})^{q}(m-1)!(p-1)!}\,.$

It is obvious that the only singularities are poles at $s=-1/2$ and $s=n$ for
$n\geq 0$. By (2.4), the pole at $-1/2$ has order $q$, and the residue is

${\rm
Res}(\mathcal{G}_{2}(s),-\tfrac{1}{2})=(-1)^{m+p}\lambda_{q}(\tilde{t}(m)\tilde{t}(p))\,.$

Next, by (2.2), the pole at a nonnegative integer $n$ has order $m+p$, and the
residue is

$\displaystyle{\rm Res}(\mathcal{G}_{2}(s),n)$
$\displaystyle\quad=(-1)^{m+p-1}\binom{m+p+q-2}{q-1}\frac{1}{(n+\tfrac{1}{2})^{m+p+q-1}}$
$\displaystyle\quad\quad+(-1)^{m+p-1}\sum_{\begin{subarray}{c}i+j=p-1\\\
i,j\geq
0\end{subarray}}\binom{m+i-1}{i}\binom{q+j-1}{j}\frac{(-1)^{i}\zeta(m+i)+(-1)^{m}H_{n}^{(m+i)}}{(n+\tfrac{1}{2})^{q+j}}$
$\displaystyle\quad\quad+(-1)^{m+p-1}\sum_{\begin{subarray}{c}i+j=m-1\\\
i,j\geq
0\end{subarray}}\binom{p+i-1}{i}\binom{q+j-1}{j}\frac{(-1)^{i}\zeta(p+i)+(-1)^{p}H_{n}^{(p+i)}}{(n+\tfrac{1}{2})^{q+j}}\,.$

Hence, summing these two contributions, considering the definitions of
$\tilde{t}(s)$ and $R_{p,q}$, and doing some transformations, we obtain the
desired formula (4.2), and therefore the statement. ∎

Now, let us briefly discuss some special cases of this theorem. Setting
$m=p=1$ in Eq. (4.2) yields an expression of the sums $R_{1,q}$, and an
alternate way to obtain this expression is to set $a=-1/2$ in [24, Theorem
3.2].

######  Corollary 4.2.

For integer $q\geq 2$, the linear sums $R_{1,q}$ are reducible to $\ln(2)$ and
zeta values:

$R_{1,q}=\sum_{n=1}^{\infty}\frac{H_{n-1}}{(n-1/2)^{q}}=\frac{q}{2}\tilde{t}(q+1)-2\ln(2)\tilde{t}(q)-\frac{1}{2}\sum_{j=1}^{q-2}\tilde{t}(q-j)\tilde{t}(j+1)\,.$
(4.3)

######  Example 4.1.

The first few sums are

$\displaystyle R_{1,2}=7\zeta(3)-\pi^{2}\ln(2)\,,$ $\displaystyle
R_{1,3}=-14\ln(2)\zeta(3)+\tfrac{1}{8}\pi^{4}\,,$ $\displaystyle
R_{1,4}=62\zeta(5)-\tfrac{7}{2}\pi^{2}\zeta(3)-\tfrac{1}{3}\pi^{4}\ln(2)\,,$
$\displaystyle
R_{1,5}=-62\ln(2)\zeta(5)-\tfrac{49}{2}\zeta(3)^{2}+\tfrac{1}{12}\pi^{6}\,,$

which can also be computed from Theorem 4.1 by replacing $(m,p,q)$ by
$(1,1,2)$ – $(1,1,5)$. The values of more sums can be obtained by specifying
the parameter $q$ directly. ∎

More generally, setting $m=p\geq 2$ in Eq. (4.2), we have

######  Corollary 4.3.

For integers $p,q\geq 2$, the sums $\lambda_{p}(R_{p,q})$ are reducible to
zeta values:

$\displaystyle\lambda_{p}(R_{p,q})$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=p-1\\\ i,j\geq
0\end{subarray}}\binom{p+i-1}{i}\binom{q+j-1}{j}R_{p+i,q+j}$
$\displaystyle=\frac{(-1)^{p-1}}{2}\left\\{\binom{2p+q-2}{q-1}\tilde{t}(2p+q-1)-\lambda_{q}(\tilde{t}(p)\tilde{t}(p))\right\\}-\lambda_{p}((-1)^{p}\zeta(p)\tilde{t}(q))\,.$

As illustrated in [21, Eq. (3.9)] and [22, Corollary 3.2], besides the linear
sums $R_{1,q}$, the sums $R_{p,q}$ with $p+q$ odd are also reducible to
$\ln(2)$ and zeta values. Therefore, we show here some more special cases of
Theorem 4.1 and Corollary 4.3 with even weights.

######  Example 4.2.

Setting $(m,p,q)$ by $(2,2,3)$, $(2,2,5)$ in Theorem 4.1 or Corollary 4.3, we
have

$\displaystyle
3R_{2,4}+2R_{3,3}=3\sum_{n=1}^{\infty}\frac{H_{n-1}^{(2)}}{(n-1/2)^{4}}+2\sum_{n=1}^{\infty}\frac{H_{n-1}^{(3)}}{(n-1/2)^{3}}=112\zeta(3)^{2}-\tfrac{1}{6}\pi^{6}\,,$
$\displaystyle
5R_{2,6}+2R_{3,5}=1798\zeta(3)\zeta(5)-\tfrac{17}{72}\pi^{8}\,.$

However, by (1.8) and (1.9), it can be found that (alternating) double zeta
values appear in the evaluations of the involved $R$-sums:

$\displaystyle
R_{2,4}=128\zeta(\bar{5},1)+\zeta(3)^{2}-\tfrac{1}{210}\pi^{6}\,,$
$\displaystyle
R_{3,3}=-192\zeta(\bar{5},1)+\tfrac{109}{2}\zeta(3)^{2}-\tfrac{8}{105}\pi^{6}\,,$
$\displaystyle
R_{2,6}=768\zeta(\bar{7},1)+289\zeta(6,2)-864\zeta(3)\zeta(5)+\tfrac{59}{525}\pi^{8}\,,$
$\displaystyle
R_{3,5}=-1920\zeta(\bar{7},1)-\tfrac{1445}{2}\zeta(6,2)+3059\zeta(3)\zeta(5)-\tfrac{2011}{5040}\pi^{8}\,.$

See also the evaluations in [21, Example 3.13], which are obtained by colored
multiple zeta values. Similarly, more relations can be established. For
example, let $(m,p,q)$ by $(3,3,5)$ and $(4,4,5)$. Then we have

$\displaystyle
5R_{3,7}+5R_{4,6}+2R_{5,5}=-3810\zeta(3)\zeta(7)-5704\zeta(5)^{2}+\tfrac{31}{270}\pi^{10}\,,$
$\displaystyle
7R_{4,8}+12R_{5,7}+10R_{6,6}+4R_{7,5}=64640\zeta(5)\zeta(7)-\tfrac{691}{9450}\pi^{12}\,,$

respectively. ∎

######  Example 4.3.

Finally, we present another two special cases:

$\displaystyle
9R_{2,10}+2R_{3,9}=58254\zeta(3)\zeta(9)+94488\zeta(5)\zeta(7)-\tfrac{691}{3780}\pi^{12}\,,$
$\displaystyle
42R_{4,10}+56R_{5,9}+35R_{6,8}+10R_{7,7}=1802808\zeta(5)\zeta(9)+1614170\zeta(7)^{2}-\tfrac{5461}{14175}\pi^{14}\,,$

which correspond to the cases of $(m,p,q)=(2,3,8),(4,5,6)$ of Theorem 4.1,
respectively. ∎

### 4.2. Symmetric sum on double $T$-values

By substituting the relations (1.9), we can use Theorem 4.1 to establish a
symmetric extension of the Kaneko-Tsumura conjecture (1.2) on the double
$T$-values.

######  Theorem 4.4.

For integers $m,p,q\geq 2$, we have

$(-1)^{m}\lambda_{p}(T(m,q))+(-1)^{p}\lambda_{m}(T(p,q))\in\mathbb{Q}[\mathrm{zeta\
values}].$

In particular, the following expression holds:

$\displaystyle
2^{m+p+q-3}\\{(-1)^{m}\lambda_{p}(T(m,q))+(-1)^{p}\lambda_{m}(T(p,q))\\}$
$\displaystyle\quad=\binom{m+p+q-2}{q-1}\tilde{t}(m+p+q-1)+(-1)^{m}\lambda_{p}((1+(-1)^{m})\zeta(m)\tilde{t}(q))$
$\displaystyle\quad\quad+(-1)^{p}\lambda_{m}((1+(-1)^{p})\zeta(p)\tilde{t}(q))-\lambda_{q}(\tilde{t}(m)\tilde{t}(p))\,.$
(4.4)

Taking $m=p$ in Theorem 4.4 further yields the next corollary.

######  Corollary 4.5.

For integers $p,q\geq 2$, the sums $\lambda_{p}(T(p,q))$ are reducible to zeta
values:

$\displaystyle\lambda_{p}(T(p,q))$
$\displaystyle=\frac{(-1)^{p}}{2^{2p+q-2}}\binom{2p+q-2}{q-1}\tilde{t}(2p+q-1)+\frac{1}{2^{2p+q-3}}\lambda_{p}((1+(-1)^{p})\zeta(p)\tilde{t}(q))$
$\displaystyle\quad-\frac{(-1)^{p}}{2^{2p+q-2}}\lambda_{q}(\tilde{t}(p)\tilde{t}(p))\,.$

By specifying the parameters, a series of relations on double $T$-values can
be established.

######  Example 4.4.

For example, in Theorem 4.1, replacing $(m,p,q)$ by $(5,4,2)$, $(5,6,2)$ gives
two relations on the linear $R$-sums:

$\displaystyle
5R_{4,6}+12R_{5,5}+15R_{6,4}+10R_{7,3}=70\zeta(3)\zeta(7)+4216\zeta(5)^{2}-\tfrac{31}{630}\pi^{10}\,,$
$\displaystyle
3R_{5,7}+10R_{6,6}+18R_{7,5}+21R_{8,4}+14R_{9,3}=-10872\zeta(5)\zeta(7)+98\zeta(3)\zeta(9)+\tfrac{691}{56700}\pi^{12}\,;$

while for the double $T$-values, substituting the same parameters into Theorem
4.4 yields

$\displaystyle
5T(4,6)+12T(5,5)+15T(6,4)+10T(7,3)=-\tfrac{961}{64}\zeta(5)^{2}+\tfrac{1}{4608}\pi^{10}\,,$
$\displaystyle
3T(5,7)+10T(6,6)+18T(7,5)+21T(8,4)+14T(9,3)=\tfrac{11811}{1024}\zeta(5)\zeta(7)-\tfrac{1}{92160}\pi^{12}\,,$

respectively. ∎

## 5\. Proof of the required convolution identity

In this section, we give the proof of the convolution identity (1.6) on the
Bernoulli numbers and Genocchi numbers used in the discussions of Section 3.4.

Firstly, it can be found that there are polynomials $P_{n}(y)$ of degree $n+1$
with integer coefficients, so that

$P_{n}(\tanh(t))=\mathcal{D}_{t}^{n}\tanh(t)\,,\quad\text{for
}n=0,1,2,\ldots,$

where $\mathcal{D}_{t}$ is the derivative operator defined by
$\mathcal{D}_{t}f(t)=f^{\prime}(t)$. In particular,

$P_{0}(y)=y\,,\quad P_{1}(y)=1-y^{2}\,,\quad P_{2}(y)=-2y+2y^{3}\,,\quad
P_{3}(y)=-2+8y^{2}-6y^{4}\,,$

and

$P_{n+1}(y)=(1-y^{2})P_{n}^{\prime}(y)\,,\quad\text{for }n\geq 0\,.$

These polynomials are called the _derivative polynomials of hyperbolic
tangent_. Note that the higher derivatives of $\coth(t)$ are formed in the
same pattern as those of $\tanh(t)$. Therefore, $P_{n}(y)$ are also the
derivative polynomials of $\coth(t)$.

The concepts of derivative polynomials for tangent and secant were introduced
by Hoffman [12, 11], but the study of these polynomials goes back to Knuth and
Buckholtz [15], and Krishnamachary and Bhimasena Rao [16]. More results on
these polynomials as well as their hyperbolic analogs can be found in, for
example, the papers due to Boyadzhiev [2], Chu and Wang [3], Cvijović [5],
Hetyei [9], and Ma [17].

Next, from the series expansion of the hyperbolic tangent, we have

$\tanh(t)=\sum_{n=1}^{\infty}\frac{2^{2n}(2^{2n}-1)B_{2n}t^{2n-1}}{(2n)!}=-\sum_{n=1}^{\infty}G_{2n}\frac{(2t)^{2n-1}}{(2n)!}=1-\sum_{k=0}^{\infty}\frac{G_{k+1}}{k+1}\frac{(2t)^{k}}{k!}\,,$

which further gives

$P_{n}(\tanh(t))=-\sum_{k=0}^{\infty}\frac{G_{k+n+1}}{k+n+1}\frac{2^{k+n}t^{k}}{k!}+\delta_{n,0}\,,$
(5.1)

where $\delta_{n,k}$ is the Kronecker delta. Chu and Wang [3, Section 2.4]
used the method of linearization to establish the expansion

$P_{m}(y)P_{n}(y)=-\rho_{m,n}^{(0)}P_{m+n+1}(y)-\sum_{k=1}^{[(m+n)/2]}\rho_{m,n}^{(k)}\frac{2^{2k}B_{2k}}{2k}P_{m+n+1-2k}(y)+\chi(m=n=0)\,,$

where $\chi$ is used in place of the Iverson bracket, and defined by

$\chi(\text{true})=1\quad\text{and}\quad\chi(\text{false})=0\,,$

and

$\rho_{m,n}^{(k)}=\left\\{\begin{array}[]{cc}\displaystyle(-1)^{m}\binom{n}{m+n+1-2k}+(-1)^{n}\binom{m}{m+n+1-2k}\,,&k\geq
1\,,\\\\[8.61108pt]
\displaystyle\frac{m!n!}{(m+n+1)!}\,,&k=0\,,\end{array}\right.$ (5.2)

with $m,n\geq 0$. Then

$\displaystyle[t^{l}]P_{m}(\tanh(t))P_{n}(\tanh(t))$
$\displaystyle\quad=\frac{2^{l+m+n+1}}{l!}\left\\{\rho_{m,n}^{(0)}\frac{G_{l+m+n+2}}{l+m+n+2}+\sum_{k=1}^{[(m+n)/2]}\rho_{m,n}^{(k)}\frac{B_{2k}G_{l+m+n-2k+2}}{(2k)(l+m+n-2k+2)}\right\\}$
$\displaystyle\quad\quad+\chi(l=m=n=0)\,,$ (5.3)

and the following theorem can be established.

######  Theorem 5.1.

For integers $n,\alpha,\gamma\geq 0$ and $\delta,\varepsilon=0,1$, there holds
the convolution identity

$\displaystyle\sum_{k=0}^{n}\binom{2n+2-\delta-\varepsilon}{2k+1-\delta}\frac{G_{2k+2\alpha+2}}{k+\alpha+1}\frac{G_{2n-2k+2\gamma+2}}{n-k+\gamma+1}$
$\displaystyle\quad=4\rho_{2\alpha+\delta,2\gamma+\varepsilon}^{(0)}\frac{G_{2n+2\alpha+2\gamma+4}}{n+\alpha+\gamma+2}+2\sum_{k=1}^{\alpha+\gamma+[(\delta+\varepsilon)/2]}\rho_{2\alpha+\delta,2\gamma+\varepsilon}^{(k)}\frac{B_{2k}}{k}\frac{G_{2n-2k+2\alpha+2\gamma+4}}{n-k+\alpha+\gamma+2}\,,$

where $\rho_{m,n}^{(k)}$ is defined in (5.2).

###### Proof..

Using Eq. (5.1), and doing some elementary reduction, we have

$P_{2\alpha+\delta}(\tanh(t))=-\sum_{k=0}^{\infty}\frac{G_{2k+2\alpha+2}}{k+\alpha+1}\frac{2^{2k+2\alpha}t^{2k+1-\delta}}{(2k+1-\delta)!}\,,$

for $\alpha\geq 0$ and $\delta=0,1$. Applying Eq. (5.3) to
$[t^{2n+2-\delta-\varepsilon}]P_{2\alpha+\delta}(\tanh(t))P_{2\gamma+\varepsilon}(\tanh(t))$,
and considering the above expansion as well as the fact

$\chi(2n+2-\delta-\varepsilon=2\alpha+\delta=2\gamma+\varepsilon=0)=0\,,$

we obtain the desired convolution formula. ∎

Finally, the identity (1.6) can be verified by combining Theorem 5.1 with Eq.
(5.3).

######  Theorem 5.2.

For integers $n\geq 0$ and $q\geq 2$, there holds the convolution identity

$\sum_{i=0}^{q-1}\binom{q-1}{i}\frac{B_{q+i}G_{2n+q-i}}{(q+i)(2n+q-i)}+\frac{1}{4}\sum_{i=0}^{2n}(-1)^{i}\binom{2n}{i}\frac{G_{q+i}G_{2n+q-i}}{(q+i)(2n+q-i)}=\frac{(-1)^{q}}{q\binom{2q}{q}}\frac{G_{2n+2q}}{2n+2q}\,.$

###### Proof..

Let us prove Theorem 5.2 according to the parity of the parameter $q$. In
Theorem 5.1, setting $\alpha=\gamma=l$ with $l\geq 0$ and
$\delta=\varepsilon=1$, we have

$\displaystyle\sum_{k=0}^{n}\binom{2n}{2k}\frac{G_{2k+2l+2}G_{2n-2k+2l+2}}{(k+l+1)(n-k+l+1)}$
$\displaystyle\quad=\frac{4}{(4l+3)\binom{4l+2}{2l+1}}\frac{G_{2n+4l+4}}{n+2l+2}-4\sum_{k=1}^{2l+1}\binom{2l+1}{4l+3-2k}\frac{B_{2k}G_{2n-2k+4l+4}}{k(n-k+2l+2)}$
$\displaystyle\quad=\frac{4}{(l+1)\binom{4l+4}{2l+2}}\frac{G_{2n+4l+4}}{n+2l+2}-4\sum_{k=0}^{l}\binom{2l+1}{2k}\frac{B_{2k+2l+2}G_{2n-2k+2l+2}}{(k+l+1)(n-k+l+1)}\,,$

for $n,l\geq 0$, which is just the $q=2l+2$ case of Theorem 5.2. Similarly, in
Theorem 5.1, let $\alpha=\gamma=l+1$ with $l\geq 0$ and
$\delta=\varepsilon=0$, and do some transformation. Then we can show that when
$q=2l+3$ and $n\geq 1$, Theorem 5.2 is still true. Hence, it suffices to show
that when $q=2l+3$ and $n=0$, the result holds. In fact, according to (5.1),
we have

$[t^{0}]P_{n}(\tanh(t))=-\frac{2^{n}G_{n+1}}{n+1}+\delta_{n,0}\,.$

Then by setting $m=n=2l+2$, for $l\geq 0$, we obtain from (5.3) that

$[t^{0}]P_{2l+2}(\tanh(t))^{2}=0=2^{4l+5}\left\\{\rho_{2l+2,2l+2}^{(0)}\frac{G_{4l+6}}{4l+6}+\sum_{k=1}^{2l+2}\rho_{2l+2,2l+2}^{(k)}\frac{B_{2k}G_{4l-2k+6}}{(2k)(4l-2k+6)}\right\\}\,.$

This further gives

$\sum_{k=0}^{l}\binom{2l+2}{2k+1}\frac{B_{2l+2k+4}G_{2l-2k+2}}{(l+k+2)(l-k+1)}=-\frac{2}{(2l+3)\binom{4l+6}{2l+3}}\frac{G_{4l+6}}{2l+3}\,,$
(5.4)

which is just the $q=2l+3$ and $n=0$ case of the desired result. ∎

######  Example 5.1.

In Theorem 5.2, setting further $n=0,1$ gives

$\displaystyle\sum_{i=0}^{q-1}\binom{q-1}{i}\frac{B_{q+i}G_{q-i}}{(q+i)(q-i)}=\frac{(-1)^{q}}{q\binom{2q}{q}}\frac{G_{2q}}{2q}-\frac{G_{q}^{2}}{4q^{2}}\,,$
$\displaystyle\sum_{i=0}^{q-1}\binom{q-1}{i}\frac{B_{q+i}G_{q+2-i}}{(q+i)(q+2-i)}=\frac{(-1)^{q}}{q\binom{2q}{q}}\frac{G_{2q+2}}{2q+2}-\frac{G_{q}G_{q+2}}{2q(q+2)}+\frac{G_{q+1}^{2}}{2(q+1)^{2}}\,,$

for $q\geq 2$; while setting $q=3,4$ yields

$\displaystyle\sum_{i=0}^{2n}(-1)^{i}\binom{2n}{i}\frac{G_{i+3}G_{2n+3-i}}{(i+3)(2n+3-i)}=-\frac{G_{2n+6}}{15(2n+6)}+\frac{G_{2n+2}}{15(2n+2)}\,,$
$\displaystyle\sum_{i=0}^{2n}(-1)^{i}\binom{2n}{i}\frac{G_{i+4}G_{2n+4-i}}{(i+4)(2n+4-i)}=\frac{G_{2n+8}}{70(2n+8)}+\frac{G_{2n+4}}{30(2n+4)}-\frac{G_{2n+2}}{21(2n+2)}\,,$

for $n\geq 0$. More special cases can be obtained from Theorems 5.1 and 5.2 by
specifying the parameters. ∎

For various other convolution identities on the Bernoulli numbers
(polynomials), Euler numbers (polynomials) and Genocchi numbers, the readers
are referred to, for example, the works of Agoh and Dilcher [1], Chu and Wang
[3], Gessel [7], Pan and Sun [19], and some further generalizations of their
results.

## Acknowledgments

The first author Weiping Wang is supported by the National Natural Science
Foundation of China (under Grant 11671360). The corresponding author Ce Xu is
supported by the National Natural Science Foundation of China (under Grant
12101008) and the Scientific Research Foundation for Scholars of Anhui Normal
University.

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