1 Introduction
A finite sequence
$\mathbf {k}=(k_1,\ldots ,k_n)$
of positive integers is called an index. The weight, depth and height of the index
$\mathbf {k}$
are defined respectively by
$k_1{\kern-1.5pt}+\cdots +k_n$
, n and the cardinality
$|\{\,j\mid 1\leq j\leq n, k_j\geq 2\}|$
. If
$k_1>1$
, the index
$\mathbf {k}$
is called admissible. For an admissible index
$\mathbf {k}=(k_1,\ldots ,k_n)$
, the multiple zeta value
$\zeta (\mathbf {k})$
and the multiple zeta-star value
$\zeta ^{\star }(\mathbf {k})$
are defined respectively by
$$ \begin{align*} \zeta(\mathbf{k})=\zeta(k_1,\ldots,k_n) & =\sum\limits_{m_1>\cdots>m_n>0}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}},\\ \zeta^{\star}(\mathbf{k})=\zeta^{\star}(k_1,\ldots,k_n) & =\sum\limits_{m_1\geq\cdots\geq m_n>0}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}. \end{align*} $$
A systematic study of multiple zeta values was carried out by Hoffman [Reference Hoffman2] and Zagier [Reference Zagier, Joseph, Mignot, Murat, Prum and Rentschler10]. More results of multiple zeta values can be found in the book [Reference Zhao11] of Zhao.
We focus on Ohno–Zagier type relations. For nonnegative integers
$k,n,s$
, denote by
$I_0(k,n,s)$
the set of admissible indices of weight k, depth n and height s. It is easy to see that
$I_0(k,n,s)$
is nonempty if and only if
$k\geq n+s$
and
$n\geq s\geq 1$
. Using the Gaussian hypergeometric function, Ohno and Zagier proved in [Reference Ohno and Zagier7] that
$$ \begin{align*} \sum\limits_{k\geq n+s,n\geq s\geq 1} & \bigg(\sum\limits_{\mathbf{k}\in I_0(k,n,s)}\zeta(\mathbf{k})\bigg)\,u^{k-n-s}v^{n-s}w^{s-1}\\ &=\frac{1}{uv-w}\bigg\{1-\exp\bigg(\sum\limits_{n=2}^\infty\frac{\zeta(n)}{n}(u^n+v^n-\alpha^n-\beta^n)\bigg)\bigg\}, \end{align*} $$
where
$\alpha $
and
$\beta $
are determined by
$\alpha +\beta =u+v$
and
$\alpha \beta =w$
. Similar studies were carried out on various generalisations of multiple zeta values. For example, Aoki et al. [Reference Aoki, Kombu and Ohno1] gave a similar formula for the sums of multiple zeta-star values involving the generalised hypergeometric function
${}_3F_2$
:
$$ \begin{align*} \sum\limits_{k\geq n+s,n\geq s\geq 1} & \bigg(\sum\limits_{\mathbf{k}\in I_0(k,n,s)}\zeta^{\star}(\mathbf{k})\bigg)\,u^{k-n-s}v^{n-s}w^{2s-2}\\ &=\frac{1}{(1-v)(1-\beta)}\,_3F_2\bigg({1-\beta,1-\beta+u,1\atop 2-v,2-\beta};1\bigg) \end{align*} $$
with
$\alpha $
and
$\beta $
determined by
$\alpha +\beta =u+v$
and
$\alpha \beta =uv-w^2$
. Here, for a positive integer m and complex numbers
$a_1,\ldots ,a_{m+1},b_1,\ldots ,b_m$
with
$b_1,\ldots ,b_m\neq 0,-1,-2,\ldots ,$
the generalised hypergeometric function
${}_{m+1}F_m$
is defined by
$$ \begin{align*} \,_{m+1}F_m\bigg({a_1,\ldots,a_{m+1}\atop b_1,\ldots,b_m};z\bigg)=&\sum\limits_{n=0}^\infty\frac{(a_1)_n\cdots(a_{m+1})_n}{n!(b_1)_n\cdots(b_m)_n}z^n, \end{align*} $$
with the Pochhammer symbol
$(a)_n$
defined by
$$ \begin{align*}(a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}=\begin{cases} 1 & \text{if } n=0,\\ a(a+1)\cdots (a+n-1) & \text{if } n>0. \end{cases}\end{align*} $$
It is known that this formal power series converges absolutely for
$|z|<1$
, and it also converges absolutely for
$|z|=1$
if
$\Re (\sum b_i-\sum a_i)>0$
. If
$m=1$
, we get the Gaussian hypergeometric function.
In [Reference Takeyama9], Takeyama studied the Ohno–Zagier type relation for a level two variant of multiple zeta values, called multiple T-values, introduced by Kaneko and Tsumura [Reference Kaneko, Tsumura, Mishou, Nakamura, Suzuki and Umegaki6]. As a consequence, a weighted sum formula of multiple T-values with fixed weight and depth was given in [Reference Takeyama9]. We consider another level two variant of multiple zeta values, called multiple t-values, introduced by Hoffman in [Reference Hoffman3].
For an admissible index
$\mathbf {k}=(k_1,\ldots ,k_n)$
, the multiple t-value
$t(\mathbf {k})$
and the multiple t-star value
$t^{\star }(\mathbf {k})$
are defined respectively by
$$ \begin{align*} t(\mathbf{k})=t(k_1,\ldots,k_n) & =\sum\limits_{\substack{m_1>\cdots>m_n>0\\ m_i\,\text{odd}}}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}},\\ t^{\star}(\mathbf{k})=t^{\star}(k_1,\ldots,k_n) & =\sum\limits_{\substack{m_1\geq \cdots\geq m_n>0\\ m_i\,\text{odd}}}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}. \end{align*} $$
It is easy to obtain the following iterated integral representations:
$$ \begin{align*} t(\mathbf{k}) & =\int_0^1\bigg(\frac{dt}{t}\bigg)^{k_1-1}\frac{t\,dt}{1-t^2}\cdots \bigg(\frac{dt}{t}\bigg)^{k_{n-1}-1}\frac{t\,dt}{1-t^2}\bigg(\frac{dt}{t}\bigg)^{k_n-1}\frac{dt}{1-t^2},\\ t^{\star}(\mathbf{k}) & =\int_0^1\bigg(\frac{dt}{t}\bigg)^{k_1-1}\frac{dt}{t(1-t^2)}\cdots\bigg(\frac{dt}{t}\bigg)^{k_{n-1}-1}\frac{dt}{t(1-t^2)}\bigg(\frac{dt}{t}\bigg)^{k_n-1}\frac{dt}{1-t^2}, \end{align*} $$
where
$({dt}/{t})^{k_i-1}={{dt}/{t}\cdots {dt}/{t}}\ (k_i-1 \mbox { factors})$
, and for one-forms
$\omega _i(t)=f_i(t)dt$
,
$i=1,2,\ldots ,k$
, we define the iterated integral
$$ \begin{align*}\int_0^z\omega_1(t)\cdots\omega_k(t)=\int_{z>t_1>\cdots>t_k>0}f_1(t_1)\cdots f_k(t_k)\,dt_1\cdots dt_k.\end{align*} $$
We want to study the sum of multiple t-(star) values with fixed weight, depth and height. For nonnegative integers
$k,n,s$
, define
$$ \begin{align*}G_0(k,n,s)=\sum\limits_{\mathbf{k}\in I_0(k,n,s)}t(\mathbf{k}),\quad G_0^{\star}(k,n,s)=\sum\limits_{\mathbf{k}\in I_0(k,n,s)}t^{\star}(\mathbf{k}).\end{align*} $$
Then we obtain Ohno–Zagier type relations which represent the generating functions of
$G_0(k,n,s)$
and
$G_0^{\star }(k,n,s)$
by the generalised hypergeometric function
${}_3F_2$
.
Theorem 1.1. For formal variables
$u,v,w$
,
$$ \begin{align*}\sum\limits_{k\geq n+s,n\geq s\geq 1}G_0(k,n,s)u^{k-n-s}v^{n-s}w^{s-1}=\frac{1}{1-u}\,_3F_2\bigg({\alpha,\beta,1\atop \frac{3-u}{2},\frac{3}{2}};1\bigg),\end{align*} $$
where
$\alpha $
and
$\beta $
are determined by
$\alpha +\beta {\kern-1pt}={\kern-1pt}1-\frac{1}{2}u+\frac{1}{2}v$
and
$\alpha \beta {\kern-1pt}={\kern-1pt}\tfrac 14(1{\kern-1pt}-{\kern-1pt}u+v- uv+w)$
.
Theorem 1.2. For formal variables
$u,v,w$
,
$$ \begin{align*} \sum\limits_{k\geq n+s,n\geq s\geq 1}G_0^{\star}(k,n,s)u^{k-n-s}v^{n-s}w^{s-1}=\frac{1}{1-u-v+uv-w}\,_3F_2 \bigg({\frac{1-u}{2},\frac{1}{2},1\atop \alpha^{\star},\beta^{\star}};1\bigg), \end{align*} $$
where
$\alpha ^{\star }$
and
$\beta ^{\star }$
are determined by
$\alpha ^{\star }+\beta ^{\star }=3-\frac{1}{2}u-\frac{1}{2}v$
and
$\alpha \beta =\tfrac 14(9-3u-3v+uv-w)$
.
From these theorems and using summation formulae for
$\,_3F_2$
, we obtain several corollaries in Section 2. For example, we give a formula for the generating function of sums of multiple t-(star) values of maximal height and a weighted sum formula of sums of multiple t-(star) values with fixed weight and depth. Finally, we prove Theorems 1.1 and 1.2 in Section 3.
2 Applications
2.1 Sums of height one
To save space, we denote a sequence of k repeated n times by
$\{k\}^n$
.
Setting
$w=0$
in Theorem 1.1,
$$ \begin{align*}\alpha,\beta=\frac{1-u}{2},\frac{1+v}{2}.\end{align*} $$
Hence, we get the following result, which gives the generating function of height one multiple t-values.
Corollary 2.1 [Reference Hoffman3, Theorem 5.1]
We have
$$ \begin{align*}\sum\limits_{k\geq n+1,n\geq 1}t(k-n+1,\{1\}^{n-1})u^{k-n-1}v^{n-1}=\frac{1}{1-u}\,_3F_2\bigg({\frac{1-u}{2},\frac{1+v}{2},1\atop \frac{3-u}{2},\frac{3}{2}};1\bigg).\end{align*} $$
Similarly, setting
$w=0$
in Theorem 1.2,
$$ \begin{align*}\alpha^{\star},\beta^{\star}=\frac{3-u}{2},\frac{3-v}{2}.\end{align*} $$
Therefore, we obtain the generating function of height one multiple t-star values.
Corollary 2.2. We have
$$ \begin{align*}\sum\limits_{k\geq n+1,n\geq 1}t^{\star}(k-n+1,\{1\}^{n-1})u^{k-n-1}v^{n-1}=\frac{1}{(1-u)(1-v)}\,_3F_2\bigg({\frac{1-u}{2},\frac{1}{2},1\atop \frac{3-u}{2},\frac{3-v}{2}};1\bigg).\end{align*} $$
Note that using [Reference Hoffman3, Lemma 5.2], for any integer
$m\geq 2$
,
$$ \begin{align*} \sum\limits_{n=1}^\infty t(m,\{1\}^{n-1})v^{n-1}=\,_{m+1}F_{m}\bigg({\frac{1+v}{2},1,\{\frac{1}{2}\}^{m-1}\atop \{\frac{3}{2}\}^m};1\bigg),\\ \sum\limits_{n=1}^\infty t^{\star}(m,\{1\}^{n-1})v^{n-1}=\frac{1}{1-v}\,_{m+1}F_{m}\bigg({1,\{\frac{1}{2}\}^{m}\atop \frac{3-v}{2},\{\frac{3}{2}\}^{m-1}};1\bigg). \end{align*} $$
2.2 Sums of maximal height
Setting
$v=0$
in Theorem 1.1, we get the generating function of sums of multiple t-values of maximal height. By the symmetric sum formula [Reference Hoffman3, Theorem 3.2], the sum of multiple t-values with fixed weight, depth and maximal height can be represented by t-values. Here we give a closed formula for the generating function of the sums of maximal height.
Corollary 2.3. For formal variables
$u,w$
,
$$ \begin{align} 1+\sum\limits_{k\geq 2n,n\geq 1}G_0(k,n,n)u^{k-2n}w^n=\exp\bigg\{\sum\limits_{n=2}^\infty\frac{t(n)}{n}(u^n-x^n-y^n)\bigg\}, \end{align} $$
where x and y are determined by
$x+y=u$
and
$xy=w$
.
Proof. Setting
$v=0$
in Theorem 1.1, we get
$\alpha +\beta =1-\frac{1}{2}u$
and
$\alpha \beta =\tfrac 14(1-u+w)$
. Let
$x=1-2\alpha $
and
$y=1-2\beta $
, then
$x+y=u$
and
$xy=w$
. Using the summation formula [Reference Prudnikov, Brychkov and Marichev8, 7.4.4.28],
$$ \begin{align} \,_3F_2\bigg({a,b,1\atop c,2+a+b-c};1\bigg) =\frac{1+a+b-c}{(1+a-c)(1+b-c)}\bigg(1-c+\frac{\Gamma(c)\Gamma(1+a+b-c)}{\Gamma(a)\Gamma(b)}\bigg). \end{align} $$
With
$a=\alpha $
,
$b=\beta $
and
$c=\tfrac 32$
,
$$ \begin{align*} \frac{1}{1-u}\,_3F_2\bigg({\alpha,\beta,1\atop \frac{3-u}{2},\frac{3}{2}};1\bigg) &=\frac{1}{1-u}\frac{\frac{1-u}{2}}{(\alpha-\frac{1}{2})(\,\beta-\frac{1}{2})}\bigg(-\frac{1}{2}+\frac{\Gamma(\frac{3}{2})\Gamma(\frac{1-u}{2})}{\Gamma(\alpha)\Gamma(\,\beta)}\bigg)\\ &=-\frac{1}{w}+\frac{\sqrt{\pi}}{w}\frac{\Gamma(\frac{1-u}{2})}{\Gamma(\frac{1-x}{2})\Gamma(\frac{1-y}{2})}. \end{align*} $$
Using the duplication formula
$$ \begin{align*}\Gamma\bigg(\frac{1}{2}-\frac{z}{2}\bigg)=\frac{\sqrt{\pi}2^{z}\Gamma(1-z)}{\Gamma(1-\frac{z}{2})}\end{align*} $$
and the expansion
$$ \begin{align} \Gamma(1-z)=\exp\bigg(\gamma z+\sum\limits_{n=2}^\infty\frac{\zeta(n)}{n}z^n\bigg), \end{align} $$
where
$\gamma $
is Euler’s constant,
$$ \begin{align*}\Gamma\bigg(\frac{1-z}{2}\bigg)=\sqrt{\pi}2^z\exp\bigg\{\frac{\gamma z}{2}+\sum\limits_{n=2}^\infty \frac{\zeta(n)}{n}(1-2^{-n})z^n\bigg\}.\end{align*} $$
Since
$t(n)=(1-2^{-n})\zeta (n)$
, we find that (see also [Reference Hoffman3, Theorem 3.3])
$$ \begin{align} \Gamma\bigg(\frac{1-z}{2}\bigg)=\sqrt{\pi}2^z\exp\bigg\{\frac{\gamma z}{2}+\sum\limits_{n=2}^\infty \frac{t(n)}{n}z^n\bigg\}. \end{align} $$
Now it is easy to finish the proof.
Similarly, we have a formula for the generating function of the sums of multiple t-star values of maximal height.
Corollary 2.4. For formal variables
$u,w$
,
$$ \begin{align} 1+\sum\limits_{k\geq 2n,n\geq 1}G_0^{\star}(k,n,n)u^{k-2n}w^n=\exp\bigg\{\sum\limits_{n=2}^\infty\frac{t(n)}{n}((x^\star)^n+(y^{\star})^n-u^n)\bigg\}, \end{align} $$
where
$x^{\star }$
and
$y^{\star }$
are determined by
$x^{\star }+y^{\star }=u$
and
$x^{\star }y^{\star }=-w$
.
Proof. Setting
$v=0$
in Theorem 1.2, we obtain
$\alpha ^{\star }+\beta ^{\star }=3-\frac{1}{2}u$
and
$\alpha ^{\star }\beta ^{\star }=\tfrac 14 (9-3u-w)$
. Let
$x^{\star }=3-2\alpha ^{\star }$
and
$y^{\star }=3-2\beta ^{\star }$
, so that
$x^{\star }+y^{\star }=u$
and
$x^{\star }y^{\star }=-w$
. Using the summation formula (2.2) with
$a=\frac{1}{2}(1-u)$
,
$b=\tfrac 12$
and
$c=\alpha ^{\star }$
,
$$ \begin{align*} \frac{1}{1-u-w}\,_3F_2\bigg({\frac{1-u}{2},\frac{1}{2},1\atop \alpha^{\star},\beta^{\star}};1\bigg) &=\frac{1}{1-u-w}\frac{-(\,\beta^{\star}-1)}{(\alpha^{\star}-\frac{3}{2})(\,\beta^{\star}-\frac{3}{2})}\bigg(1-\alpha^{\star}+\frac{\Gamma(\alpha^{\star})\Gamma(\,\beta^{\star}-1)}{\Gamma(\frac{1-u}{2})\Gamma(\frac{1}{2})}\bigg)\\ &=-\frac{1}{w}+\frac{1}{w\sqrt{\pi}}\frac{\Gamma(\frac{1-x^{\star}}{2})\Gamma(\frac{1-y^{\star}}{2})}{\Gamma(\frac{1-u}{2})}. \end{align*} $$
Now the result follows from (2.4).
$$ \begin{align*}\bigg(1+\sum\limits_{k\geq 2n,n\geq 1}G_0(k,n,n)u^{k-2n}w^n\bigg)\bigg(1+\sum\limits_{k\geq 2n,n\geq 1}(-1)^nG_0^{\star}(k,n,n)u^{k-2n}w^n\bigg)=1.\end{align*} $$
Also, setting
$u=0$
in (2.1) and (2.5),
$$ \begin{align*} 1+\sum\limits_{n=1}^{\infty}t(\{2\}^n)w^n & =\exp\bigg\{\sum\limits_{n=1}^\infty\frac{(-1)^{n-1}}{n}t(2n)w^n\bigg\},\\ 1+\sum\limits_{n=1}^{\infty}t^{\star}(\{2\}^n)w^n & =\exp\bigg\{\sum\limits_{n=1}^\infty\frac{1}{n}t(2n)w^n\bigg\}. \end{align*} $$
These formulae can also be deduced from the identities [Reference Ihara, Kajikawa, Ohno and Okuda4, Reference Ihara, Kaneko and Zagier5]
$$ \begin{align*} \frac{1}{1-z_kw}=\exp_{\ast}\bigg(\sum\limits_{n=1}^\infty \frac{(-1)^{n-1}}{n}z_{nk}w^n\bigg), \quad \bigg(\frac{1}{1+z_kt}\bigg)\ast S\bigg(\frac{1}{1-z_kt}\bigg)=1, \end{align*} $$
in the harmonic shuffle algebra.
2.3 A weighted sum formula
Let
$I_0(k,n)$
be the set of admissible indices of weight k and depth n, and define
$$ \begin{align*}G_0(k,n)=\sum\limits_{\mathbf{k}\in I_0(k,n)}t(\mathbf{k}),\quad G_0^{\star}(k,n)=\sum\limits_{\mathbf{k}\in I_0(k,n)}t^{\star}(\mathbf{k}),\end{align*} $$
which are the sums of multiple t-values and multiple t-star values with fixed weight k and depth n, respectively. By setting
$w=uv$
and then
$v=2u$
or
$v=-2u$
in Theorems 1.1 and 1.2, we obtain the following results.
Proposition 2.5. For a formal variable u,
$$ \begin{align} \sum\limits_{k=2}^\infty\bigg(\sum\limits_{n=1}^{k-1}2^{n-1}G_0(k,n)\bigg)u^{k-2} & =\sum\limits_{k=2}^\infty\bigg(\sum\limits_{n=1}^{k-1}(-1)^{k-n-1}2^{n-1}G_0^{\star}(k,n)\bigg)u^{k-2}\nonumber\\ & = 2^{2u}t(2)\exp\bigg(\sum\limits_{n=2}^\infty\frac{4(2^{n-1}-1)}{n(2^n-1)}t(n)u^n\bigg). \end{align} $$
Proof. If
$w=uv$
in Theorem 1.1,
$$ \begin{align*}\alpha,\beta=\frac{1-u+v}{2},\frac{1}{2}.\end{align*} $$
Hence, we get the expression for the generating function of sums of multiple t-values with fixed weight and depth:
$$ \begin{align*}\sum\limits_{k>n\geq 1}G_0(k,n)u^{k-n-1}v^{n-1}=\frac{1}{1-u}\,_3F_2\bigg({\frac{1-u+v}{2},\frac{1}{2},1\atop \frac{3-u}{2},\frac{3}{2}};1\bigg).\end{align*} $$
Setting
$v=2u$
,
$$ \begin{align*}\sum\limits_{k>n\geq 1}2^{n-1}G_0(k,n)u^{k-2}=\frac{1}{1-u}\,_3F_2\bigg({\frac{1+u}{2},\frac{1}{2},1\atop \frac{3-u}{2},\frac{3}{2}};1\bigg).\end{align*} $$
Using Dixon’s summation formula [Reference Prudnikov, Brychkov and Marichev8, 7.4.4.21]: for
$\Re (a-2b-2c)>-2$
,
$$ \begin{align*} \,_3F_2\bigg({a,b,c\atop 1+a-b,1+a-c};1\bigg) =\frac{\sqrt{\pi}}{2^a}\frac{\Gamma(1+a-b)\Gamma(1+a-c)\Gamma(1+\frac{a}{2}-b-c)}{\Gamma(\frac{1+a}{2})\Gamma(1+\frac{a}{2}-b)\Gamma(1+\frac{a}{2}-c)\Gamma(1+a-b-c)}, \end{align*} $$
with
$a=1$
,
$b=\tfrac 12$
and
$c=\frac{1}{2}(1+u)$
, we get
$$ \begin{align*} \frac{1}{1-u}\,_3F_2\bigg({\frac{1+u}{2},\frac{1}{2},1\atop \frac{3-u}{2},\frac{3}{2}};1\bigg) =\frac{1}{1-u} \frac{\sqrt{\pi}}{2}\frac{\Gamma(\frac{3}{2})\Gamma(\frac{3-u}{2})\Gamma(\frac{1-u}{2})}{\Gamma(1-\frac{u}{2})\Gamma(1-\frac{u}{2})} =\frac{\pi}{8}\frac{\Gamma(\frac{1-u}{2})^2}{\Gamma(1-\frac{u}{2})^2}. \end{align*} $$
Then using (2.3), (2.4), the relation
$\zeta (n)=(1-2^{-n})^{-1}t(n)$
and
$t(2)={\pi ^2}/{8}$
, we get
$$ \begin{align*}\sum\limits_{k=2}^\infty\bigg(\sum\limits_{n=1}^{k-1}2^{n-1}G_0(k,n)\bigg)u^{k-2}=2^{2u}t(2)\exp\bigg(\sum\limits_{n=2}^\infty\frac{4(2^{n-1}-1)}{n(2^n-1)}t(n)u^n\bigg).\end{align*} $$
Similarly, setting
$w=uv$
in Theorem 1.2,
$$ \begin{align*}\alpha^{\star},\beta^{\star}=\frac{3-u-v}{2},\frac{3}{2}.\end{align*} $$
Hence,
$$ \begin{align*}\sum\limits_{k>n\geq 1}G_0^{\star}(k,n)u^{k-n-1}v^{n-1}=\frac{1}{1-u-v}\,_3F_2\bigg({\frac{1-u}{2},\frac{1}{2},1\atop \frac{3-u-v}{2},\frac{3}{2}};1\bigg).\end{align*} $$
Let
$v=-2u$
. Then
$$ \begin{align*}\sum\limits_{k>n\geq 1}(-2)^{n-1}G_0^{\star}(k,n)u^{k-2}=\frac{1}{1+u}\,_3F_2\bigg({\frac{1-u}{2},\frac{1}{2},1\atop \frac{3+u}{2},\frac{3}{2}};1\bigg).\end{align*} $$
Now it is easy to finish the proof.
Expanding the right-hand side of (2.6) gives the following weighted sum formula.
Corollary 2.6. For any integer
$k\geq 2$
,
$$ \begin{align*} \sum\limits_{n=1}^{k-1} & 2^{n-1}G_0(k,n) =\sum\limits_{n=1}^{k-1}(-1)^{k-n-1}2^{n-1}G_0^{\star}(k,n)\\ & =\sum\limits_{\substack{n+n_1+\cdots+n_m=k-2\\ n,m\geq 0,n_1,\ldots,n_m\geq 2}}\frac{2^{n+2m}(2^{n_1-1}-1)\cdots(2^{n_m-1}-1)}{n!m!n_1\cdots n_m(2^{n_1}-1)\cdots(2^{n_m}-1)}t(2)t(n_1)\cdots t(n_m)\log^n 2. \end{align*} $$
3 Proofs of Theorems 1.1 and 1.2
The proofs of Theorems 1.1 and 1.2 are similar to that of the Ohno–Zagier relation for multiple zeta values in [Reference Ohno and Zagier7].
As in [Reference Hoffman3], for an index
$\mathbf {k}=(k_1,\ldots ,k_n)$
, define
$$ \begin{align*} &\mathcal{L}_{\mathbf{k}}(z)=\mathcal{L}_{k_1,\ldots,k_n}(z)=\sum\limits_{\substack{m_1>\cdots>m_n>0\\ m_i\,\text{odd}}}\frac{z^{m_1}}{m_1^{k_1}\cdots m_n^{k_n}},\\ &\mathcal{L}_{\mathbf{k}}^{\star}(z)=\mathcal{L}_{k_1,\ldots,k_n}^{\star}(z)=\sum\limits_{\substack{m_1\geq\cdots\geq m_n>0\\ m_i\,\text{odd}}}\frac{z^{m_1}}{m_1^{k_1}\cdots m_n^{k_n}}. \end{align*} $$
Then
$\mathcal {L}_{\mathbf {k}}(z)$
and
$\mathcal {L}_{\mathbf {k}}^{\star }(z)$
converge absolutely for
$|z|<1$
. If
$k_1>1$
,
$\mathcal {L}_{\mathbf {k}}(1)=t(\mathbf {k})$
and
$\mathcal {L}_{\mathbf {k}}^{\star }(1)=t^{\star }(\mathbf {k})$
. From [Reference Hoffman3, Lemma 5.1],
$$ \begin{align} \frac{d}{dz}\mathcal{L}_{k_1,\ldots,k_n}(z)=\begin{cases} \displaystyle\frac{1}{z}\mathcal{L}_{k_1-1,k_2,\ldots,k_n}(z) & \text{if } k_1>1,\\ \displaystyle\frac{z}{1-z^2}\mathcal{L}_{k_2,\ldots,k_n}(z) & \text{if } n\geq 2, k_1=1,\\ \displaystyle\frac{1}{1-z^2} & \text{if } n=k_1=1. \end{cases} \end{align} $$
Similarly,
$$ \begin{align} \frac{d}{dz}\mathcal{L}_{k_1,\ldots,k_n}^{\star}(z)=\begin{cases} \displaystyle\frac{1}{z}\mathcal{L}_{k_1-1,k_2,\ldots,k_n}^{\star}(z) & \text{if } k_1>1,\\ \displaystyle\frac{1}{z(1-z^2)}\mathcal{L}_{k_2,\ldots,k_n}^{\star}(z) & \text{if } n\geq 2, k_1=1,\\ \displaystyle\frac{1}{1-z^2} & \text{if } n=k_1=1. \end{cases} \end{align} $$
One can also obtain (3.1) and (3.2) from the following iterated integral representations:
$$ \begin{align*} &\mathcal{L}_{k_1,\ldots,k_n}(z)=\int_0^z\bigg(\frac{dt}{t}\bigg)^{k_1-1}\frac{t\,dt}{1-t^2}\cdots\bigg(\frac{dt}{t}\bigg)^{k_{n-1}-1}\frac{t\,dt}{1-t^2}\bigg(\frac{dt}{t}\bigg)^{k_n-1}\frac{dt}{1-t^2},\\ &\mathcal{L}_{k_1,\ldots,k_n}^{\star}(z)=\int_0^z\bigg(\frac{dt}{t}\bigg)^{k_1-1}\frac{dt}{t(1-t^2)}\cdots\bigg(\frac{dt}{t}\bigg)^{k_{n-1}-1}\frac{dt}{t(1-t^2)}\bigg(\frac{dt}{t}\bigg)^{k_n-1}\frac{dt}{1-t^2}. \end{align*} $$
3.1 Proof of Theorem 1.1
For nonnegative integers
$k,n,s$
, denote by
$I(k,n,s)$
the set of indices of weight k, depth n and height s, and define the sums
$$ \begin{align*}G(k,n,s;z)=\sum\limits_{\mathbf{k}\in I(k,n,s)}\mathcal{L}_{\mathbf{k}}(z),\quad G_0(k,n,s;z)=\sum\limits_{\mathbf{k}\in I_0(k,n,s)}\mathcal{L}_{\mathbf{k}}(z).\end{align*} $$
If the index set is empty, the sum is treated as zero. We also set
$G(0,0,0;z)=1$
. Note that if
$k\geq n+s$
and
$n\geq s\geq 1$
,
$$ \begin{align*}G_0(k,n,s;1)=G_0(k,n,s).\end{align*} $$
For integers
$k,n,s$
, using (3.1), we have the following identities:
-
(1) if
$k\geq n+s$
and
$n\geq s\geq 1$
, (3.3)
$$ \begin{align} \frac{d}{dz} & G_0(k,n,s;z) \nonumber\\ & =\frac{1}{z}[G(k-1,n,s-1;z)+G_0(k-1,n,s;z)-G_0(k-1,n,s-1;z)]; \end{align} $$
-
(2) if
$k\geq n+s$
,
$n\geq s\geq 0$
and
$n\geq 2$
, (3.4)
$$ \begin{align} \frac{d}{dz}[G(k,n,s;z)-G_0(k,n,s;z)]=\frac{z}{1-z^2}G(k-1,n-1,s;z). \end{align} $$
We define the generating functions
$$ \begin{align*} \Phi(z)&=\Phi(u,v,w;z)=\sum\limits_{k,n,s\geq 0}G(k,n,s;z)u^{k-n-s}v^{n-s}w^{s},\\ &=1+\mathcal{L}_1(z)v+\sum\limits_{k\geq 2}\mathcal{L}_k(z)u^{k-2}w+\sum\limits_{\substack{k\geq n+s\\ n\geq s\geq 0,n\geq 2}}G(k,n,s;z)u^{k-n-s}v^{n-s}w^{s},\\ \Phi_0(z)&=\Phi_0(u,v,w;z)=\sum\limits_{k,n,s\geq 0}G_0(k,n,s;z)u^{k-n-s}v^{n-s}w^{s-1}\\ &=\sum\limits_{\substack{k\geq n+s\\ n\geq s\geq 1}}G_0(k,n,s;z)u^{k-n-s}v^{n-s}w^{s-1}. \end{align*} $$
$$ \begin{align*} &\frac{d}{dz}\Phi_0(z)=\frac{1}{vz}(\Phi(z)-1-w\Phi_0(z))+\frac{u}{z}\Phi_0(z),\\ &\frac{d}{dz}(\Phi(z)-w\Phi_0(z))=\frac{vz}{1-z^2}\Phi(z)+\frac{v}{1+z}. \end{align*} $$
Eliminating
$\Phi (z)$
, we obtain the differential equation satisfied by
$\Phi _0(z)$
.
Proposition 3.1.
$\Phi _0=\Phi _0(z)$
satisfies the following differential equation:
$$ \begin{align} z(1-z^2)\Phi_0"+[(1-u)(1-z^2)-vz^2]\Phi_0'+(uv-w)z\Phi_0=1. \end{align} $$
We want to find the unique power series solution
$\Phi _0(z)=\sum _{n=1}^\infty a_nz^n$
. From (3.5), we see that
$a_1={1}/({1-u})$
,
$a_2=0$
and
$$ \begin{align*}a_{n+1}=\frac{(n-1)(n-2)+(1-u+v)(n-1)-(uv-w)}{(n+1-u)(n+1)}a_{n-1},\quad n\geq 2.\end{align*} $$
Hence, for any
$n\geq 1$
, we have
$a_{2n}=0$
and
$$ \begin{align*} a_{2n+1}=\frac{(n-\frac{1}{2})(n-1)+(\frac{1}{2}-\frac{u}{2}+\frac{v}{2})(n-\frac{1}{2})-\frac{1}{4}(uv-w)}{(n+\frac{1}{2}-\frac{u}{2})(n+\frac{1}{2})}a_{2n-1}.\end{align*} $$
Since
$(\alpha -1)+(\,\beta -1)=-1-\frac{1}{2}u+\frac{1}{2}v$
and
$(\alpha -1)(\,\beta -1)=\tfrac 14(1+u-v-uv+w)$
,
$$ \begin{align*} a_{2n+1}=\frac{(n+\alpha-1)(n+\beta-1)}{(n+\frac{1}{2}-\frac{u}{2})(n+\frac{1}{2})}a_{2n-1}=\frac{(\alpha)_n(\,\beta)_n}{(\frac{3-u}{2})_n(\frac{3}{2})_n}\frac{1}{1-u}. \end{align*} $$
Therefore, we can represent
$\Phi _0(z)$
by the generalised hypergeometric function
$\,_3F_2$
as displayed in the following theorem.
Theorem 3.2. We have
$$ \begin{align*}\Phi_0(k,n,s;z)=\frac{z}{1-u}\,_3F_2\bigg({\alpha,\beta,1\atop \frac{3-u}{2},\frac{3}{2}};z^2\bigg).\end{align*} $$
3.2 Proof of Theorem 1.2
Similarly, for nonnegative integers
$k,n,s$
, we define the sums
$$ \begin{align*}G^{\star}(k,n,s;z)=\sum\limits_{\mathbf{k}\in I(k,n,s)}\mathcal{L}_{\mathbf{k}}^{\star}(z),\quad G_0^{\star}(k,n,s;z)=\sum\limits_{\mathbf{k}\in I_0(k,n,s)}\mathcal{L}_{\mathbf{k}}^{\star}(z)\end{align*} $$
with
$G^{\star }(0,0,0;z)=1$
. Using (3.2), we have the following identities:
-
(1) if
$k\geq n+s$
and
$n\geq s\geq 1$
, (3.6)
$$ \begin{align} &\frac{d}{dz} G_0^{\star}(k,n,s;z)\nonumber\\ &\quad =\frac{1}{z}[G^{\star}(k-1,n,s-1;z)+G_0^{\star}(k-1,n,s;z)-G_0^{\star}(k-1,n,s-1;z)]; \end{align} $$
-
(2) if
$k\geq n+s$
,
$n\geq s\geq 0$
and
$n\geq 2$
, (3.7)
$$ \begin{align} \frac{d}{dz}[G^{\star}(k,n,s;z)-G_0^{\star}(k,n,s;z)]=\frac{1}{z(1-z^2)}G^{\star}(k-1,n-1,s;z). \end{align} $$
We define the generating functions
$$ \begin{align*} \Phi^{\star}(z) & =\Phi^{\star}(u,v,w;z)=\sum\limits_{k,n,s\geq 0}G^{\star}(k,n,s;z)u^{k-n-s}v^{n-s}w^{s},\\ \Phi_0^{\star}(z) & =\Phi_0^{\star}(u,v,w;z)=\sum\limits_{k,n,s\geq 0}G_0^{\star}(k,n,s;z)u^{k-n-s}v^{n-s}w^{s-1}. \end{align*} $$
Then using (3.2), (3.6) and (3.7),
$$ \begin{align*} &\frac{d}{dz}\Phi_0^{\star}(z)=\frac{1}{vz}(\Phi^{\star}(z)-1-w\Phi_0^{\star}(z))+\frac{u}{z}\Phi_0^{\star}(z),\\ &\frac{d}{dz}(\Phi^{\star}(z)-w\Phi_0^{\star}(z))=\frac{v}{z(1-z^2)}\Phi^{\star}(z)-\frac{v}{z(1+z)}. \end{align*} $$
Eliminating
$\Phi ^{\star }(z)$
, we get the differential equation satisfied by
$\Phi _0^{\star }(z)$
.
Proposition 3.3.
$\Phi _0^{\star }=\Phi _0^{\star }(z)$
satisfies the following differential equation:
$$ \begin{align} z^2(1-z^2)(\Phi_0^{\star})"+[(1-u)z(1-z^2)-vz](\Phi_0^{\star})'+(uv-w)\Phi_0^{\star}=z. \end{align} $$
Assume that
$\Phi _0^{\star }(z){\kern-1pt}={\kern-2pt}\sum _{n=1}^\infty{\kern-1pt} a_n^{\star }z^n$
. Using (3.8), we find that
$a_1^{\star }{\kern-1pt}={\kern-1pt}{1}/({1{\kern-1pt}-{\kern-1pt}u{\kern-1pt}-{\kern-1pt}v+uv{\kern-1pt}-{\kern-1pt}w})$
,
$a_2^{\star }=0$
and
$$ \begin{align*}a_{n}^{\star}=\frac{(n-2)(n-2-u)}{n(n-1)+n(1-u-v)+uv-w}a_{n-2}^{\star},\quad n\geq 3.\end{align*} $$
Hence, for any
$n\geq 1$
, we have
$a_{2n}^{\star }=0$
and
$$ \begin{align*} a_{2n+1}^{\star}=\frac{(n-\frac{1}{2}-\frac{u}{2})(n-\frac{1}{2})}{(n+\alpha^{\star}-1)(n+\beta^{\star}-1)}a_{2n-1}^{\star}=\frac{(\frac{1-u}{2})_n(\frac{1}{2})_n}{(\alpha^{\star})_n(\,\beta^{\star})_n} \frac{1}{1-u-v+uv-w}.\end{align*} $$
Therefore, we have the following theorem.
Theorem 3.4. We have
$$ \begin{align*}\Phi_0^{\star}(k,n,s;z)=\frac{z}{1-u-v+uv-w}\,_3F_2\bigg({\frac{1-u}{2},\frac{1}{2},1\atop \alpha^{\star},\beta^{\star}};z^2\bigg).\end{align*} $$
