Mixed Parity Variants of Apéry-Type Binomial Series and Level Four Colored Multiple Zeta Values

Abstract

In this paper, we study an Apéry-type series involving the central binomial coefficients ${\sum }_{n_1>\dots >n_d>0}{\displaystyle \frac{1}{4^{n_1}}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)n_1^{-s_1}\dots n_d^{-s_d}$ and its variations where the summation indices may have mixed parities and some or all “>” are replaced by “≥”, as long as the series are defined. These sums have naturally appeared in the calculation of massive Feynman integrals by the work of Jegerlehner, Kalmykov, and Veretin. We show that all these sums can be expressed as $\mathbb{Q}$-linear combinations of the real and/or imaginary parts of the colored multiple zeta values at level four, i.e., special values of multiple polylogarithms at fourth roots of unity. For example, our main theorem shows that when $n_1^{-s_1}$ is replaced by ${\left(2n_1\right)}^{-s_1}$ and other $n_j^{-s_j}$’s are replaced by either ${\left(2n_j\right)}^{-s_j}$ or ${(2n_j+1)}^{-s_j}$, then all the colored multiple zeta values can be chosen to have the same weight $s_1+\dots +s_d$, but the weights of these values are only bounded by $s_1+\dots +s_d$ for general variant Apéry-type series of mixed parities. We also show that the corresponding series where $\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)/4^{n_1}$ is replaced by ${\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)}^2/{16}^{n_1}$ can be expressed in a similar way except for a possible extra factor of $1/\pi$, with the weight of the colored multiple zeta values similarly bounded.

1. Introduction

In [1], we have considered the following Apéry-type inverse binomial series

$$\begin{array}{c}\hfill \sum _{n_1>\dots >n_d>0}{\displaystyle \frac{4^{n_1}}{\left(\genfrac{}{}{0pt}{}{2n_1}{n_1}\right)n_1^{s_1}\dots n_d^{s_d}}}\end{array}$$

for all positive integers $s_1\ge 2,s_2,\dots ,s_d\ge 1$, and their variants with some or all of $n_j$’s replaced by $2n_j\pm 1$ and some or all of “>” replaced by “≥”, as long as the series are defined. By generalizing Akhilesh’s ideas in [2,3] we showed that the above sum can be expressed as $\mathbb{Q}$-linear combinations of the real and/or the imaginary parts of some colored multiple zeta values at level 4, i.e., multiple polylogarithms evaluated at 4th roots of unity. We also proved similar results after replacing $4^{n_1}/\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)$ by ${16}^{n_1}/{\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)}^2$.

In this paper, we will turn to another class of Apéry-type series defined by

$$\begin{array}{c}\hfill \sum _{n_1>\dots >n_d>0}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2n_1}{n_1}\right)}{4^{n_1}{n}_1^{s_1}\dots n_d^{s_d}}}\mathrm{and}\sum _{n_1>\dots >n_d>0}{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2n_1}{n_1}\right)}^2}{{16}^{n_1}{n}_1^{s_1}\dots n_d^{s_d}}}\end{array}$$

(1)

and similar variants with the summation indices having mixed parities. We call these Apéry-type (central) binomial series (not inverse type). Instances of this type already appeared in Leshchiner’s work [4], extending Apéry’s formula. These sums have also appeared in the calculation of massive Feynman integrals [5]. Some of the ideas contained in this paper have already been applied to studying other types of infinite series in [6]. In [7,8], some similar finite sums have also be studied by different methods.

We remark that a particular type of odd variations already appeared implicitly in ([5] A.25). Indeed, it is easy to verify that $\left(2n\right)a_n=(2n-1)a_{n-1}$, so that for any function S

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\sum _{n\ge 0}{\displaystyle \frac{a_n}{{(n+1)}^c}}S\left(n\right)=\sum _{n\ge 1}{\displaystyle \frac{a_{n-1}}{\left(2n\right)n^{c-1}}}S(n-1)=\sum _{n>0}{\displaystyle \frac{a_n}{(2n-1)n^{c-1}}}S(n-1).\end{array}$$

1.1. Colored Multiple Zeta Values

Recall that for any $s=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$ and Nth roots of unity $z_1,\dots ,z_d$ the colored multiple zeta value (CMZV) at level N is defined by

$${Li}_s(z_1,\dots ,z_d):=\sum _{n_1>\dots >n_d>0}{\displaystyle \frac{z_1^{n_1}\dots z_d^{n_d}}{n_1^{s_1}\dots n_d^{s_d}}}.$$

(2)

Conventionally, we call d the depth and $\left|s\right|=s_1+\dots +s_d$ the weight of the CMZV defined by (2).

It is well known that (2) converges if and only if $(s_1,z_1)\ne (1,1)$ (see [9] and [10] Ch. 15), in which case, we say the CMZV is admissible. In fact, a quick calculation shows that if $(s_1,z_1)=(1,1)$, then for any fixed $n_2$, the sum ${\sum }_{n_1>n_2}1/n_1$ diverges as the tail of the harmonic series ${Li}_1\left(1\right)$ which is itself non-admissible, and therefore ${Li}_s(z_1,\dots ,z_d)$ diverges. On the other hand, if $s_1\ge 2$, then (2) converges absolutely. Further, if $s_1=1$ and $z_1\ne 1$ is some Nth root of unity, then one can show the convergence of (2) by applying Abel’s method on summation (see, e.g., [11]).

As an admissible example of (2), the famous Catalan’s constant

$$G:=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k}{{(2k+1)}^2}}={\displaystyle \frac{1}{2i}}\sum _{n=0}^{\infty }{\displaystyle \frac{i^n(1-{(-1)}^n)}{n^2}}={\displaystyle \frac{1}{2i}}({Li}_2\left(i\right)-{Li}_2(-i)),$$

(3)

is the imaginary part of the CMZV ${Li}_2\left(i\right)$ of weight 2 at level 4, where we have set $i=\sqrt{-1}$. We further note that

$${\displaystyle \frac{1}{2i}}\left({Li}_1\left(i\right)-{Li}_1(-i)\right)={\displaystyle \frac{1}{2i}}log{\displaystyle \frac{1-i}{1+i}}={\displaystyle \frac{1}{2i}}logi={\displaystyle \frac{\pi }{4}},$$

which means that $\pi$ is the imaginary part of $4{Li}_1\left(i\right)$. In particular, we see that

$$G\in i{\mathsf{CMZV}}_2^4\mathrm{and}\pi \in i{\mathsf{CMZV}}_1^4,$$

where ${\mathsf{CMZV}}_w^N$ represent the $\mathbb{Q}$-span of all the CMZVs of weight w at level N. Conventionally, we set ${\mathsf{CMZV}}_0=\mathbb{Q}$.

Starting from the 1990s, mathematicians and theoretical physicists have been attracted to the study of colored multiple zeta values due to their frequent and sometimes unexpected appearance in quite a few different branches of mathematics and physics, in particular, the computation of many Feynman diagrams (see, e.g., [12,13,14,15,16,17,18,19,20,21,22,23,24]). We point out also that in the literature, multiple zeta values (and their alternating version) are sometimes called Euler sums (see [25,26]).

1.2. Iterated Integrals

The CMZVs can be expressed by K.T. Chen’s iterated integrals (see [10] Sec. 2.1):

$$\begin{array}{c}\hfill {Li}_s(z_1,\dots ,z_d)={\int }_0^1{\mathtt{a}}^{s_1-1}{\mathtt{x}}_{{\xi }_1}\dots {\mathtt{a}}^{s_d-1}{\mathtt{x}}_{{\xi }_d},\end{array}$$

where $\mathtt{a}=dt/t$, ${\mathtt{x}}_{\xi }=dt/(\xi -t)$, and ${\xi }_j=1/(z_1\dots z_j)$ for all $j=1,\dots ,d$. The theory of iterated integrals was developed firstly by K.T. Chen in the 1960s. It has played important roles in the study of algebraic topology and algebraic geometry in the past half century. Its simplest form is

$$\begin{array}{cc}\hfill {\int }_0^1{f}_1\left(t\right)dt{f}_2\left(t\right)dt\dots f_p\left(t\right)dt=& {\int }_0^1{f}_1\left(t\right)dt\circ f_2\left(t\right)dt\circ \dots \circ f_p\left(t\right)dt\hfill \\ \hfill :=& \underset{1>t_1>\dots >t_p>0}{\int }{f}_1\left(t_1\right)f_2\left(t_2\right)\dots f_p\left(t_p\right)d{t}_{1}d{t}_2\dots d{t}_p.\hfill \end{array}$$

Hence, our conventional order of the iterated integrals is to integrate the last integral over $[0,t]$ first, multiply this function on the penultimate 1-form, integrate over $[0,t]$ again, and then iterate this integrating–multiplying step all the way to the front, with the integration over $[0,1]$ in the last step.

For example, the dilogarithm value ${Li}_2\left(i\right)$ related to Catalan’s constant defined by (3) can be expressed as

$${\int }_0^1{\displaystyle \frac{dx}{x}}{\int }_0^x{\displaystyle \frac{dy}{-i-y}}=\sum _{k=0}^{\infty }{\int }_0^1{\displaystyle \frac{i^{k+1}{x}^{k}dx}{k+1}}=\sum _{k=0}^{\infty }{\displaystyle \frac{i^{k+1}}{{(k+1)}^2}}=\sum _{n=1}^{\infty }{\displaystyle \frac{i^n}{n^2}}={Li}_2\left(i\right).$$

One can further extend the iterated integrals defined on $[0,1]$ to those over any piecewise smooth paths on the complex plane via pull-backs. We refer the interested readers to Chen’s original work [27,28] for more details.

1.3. Recursive Formulas

The key idea to compute the infinite sums in (1) and their even/odd-indexed variations is to derive some recursive formulas for the “n-tails” of a variable version of these series, which are given by the following (see also (14) and (18)–(21)). Here, an n-tail of a series means the complement of its nth partial sum so that for a convergent series, its n-tail vanishes as $n\to \infty$.

Set $a_n\left(x\right)={\displaystyle \frac{1}{4^n}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right)x^{2n}$ and $a_n=a_n\left(1\right)$ for all $n\ge 0$ where $\left({\displaystyle \genfrac{}{}{0pt}{}{0}{0}}\right)=1$. For all $s\ge 1$ and $y\in (-\pi /2,\pi /2)$, we have the following iterated integral expressions of n-tails:

$$\sum _{m>n}{\displaystyle \frac{a_m(siny)}{{\left(2m\right)}^s}}={\int }_0^y{(cottdt)}^{s-1}(1-csctdt\circ sect)a_n(sint)tantdt,$$

(4)

$$\sum _{m>n}{\displaystyle \frac{a_m(siny)}{{(2m+1)}^s}}=cscy{\int }_0^y{(cottdt)}^{s-1}(1-dt\circ csctsect)a_n(sint)sinttantdt,$$

(5)

$$\sum _{m\ge n}{\displaystyle \frac{a_m(siny)}{{(2m+1)}^s}}=cscy{\int }_0^y{(cottdt)}^{s-1}(csct-dt\circ sect)a_n(sint)tantdt,$$

(6)

$$\sum _{m>n}{\displaystyle \frac{a_m(siny)}{2m-1}}=cosy{\int }_0^y{a}_n(sint)tantsectdt,$$

(7)

$$\sum _{m>n}{\displaystyle \frac{a_m(siny)}{{(2m-1)}^s}}=siny{\int }_0^y{(cottdt)}^{s-2}\left({cot}^{2}tdt\right)a_n(sint)tantsectdt(s\ge 2).$$

(8)

In fact, in the above, we have used a notation generalizing that for the iterated integrals. See the beginning of the next section for details. To compute each of the even/odd/mixed variations of Apéry-type binomial series similar to (1), one can use (4)–(8) repeatedly to express it as an iterated integral, and then apply the change of variable $t\to {sin}^{-1}[(1-t^2)/(1+t^2)]$ to convert it to a $\mathbb{Q}$-linear combination of iterated integrals that only involve 1-forms of the shape $dt/t$ and $dt/(\xi -t)$, where ${\xi }^4=1$, so that all the integrals are clearly in ${\mathsf{CMZV}}_{\le \left|s\right|}^4\otimes \mathbb{Q}\left[i\right]$. See the related conversion rules in (46)–(48).

1.4. List of Notations

For the convenience of the readers, we provide the following list of notations we use throughout this paper.

(1) SeriesLet $d\in \mathbb{N}$, $s=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$, and $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n\pm 1$.

$$\begin{array}{cc}\hfill a_n\left(x\right)=& {\displaystyle \frac{1}{4^n}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right)x^{2n},\hfill \\ \hfill \alpha {(\mathit{s};t)}_n:=& \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}\left(t\right)}{{\left(2n_1\right)}^{s_1}\dots {\left(2n_d\right)}^{s_d}}},\hfill \\ \hfill \beta {(\mathit{s};t)}_n:=& \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}\left(t\right)}{{(2n_1+1)}^{s_1}\dots {(2n_d+1)}^{s_d}}},\hfill \\ \hfill {\beta }^{★}{(\mathit{s};t)}_n:=& \sum _{n_1\ge \dots \ge n_d\ge n}{\displaystyle \frac{a_{n_1}\left(t\right)}{{(2n_1+1)}^{s_1}\dots {(2n_d+1)}^{s_d}}},\hfill \\ \hfill \gamma {(\mathit{s};t)}_n:=& \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}\left(t\right)}{{(2n_1-1)}^{s_1}\dots {(2n_d-1)}^{s_d}}},\hfill \\ \hfill \mathrm{\Xi }(\mathit{l};\mathit{s}):=& \sum _{n_1\underset{1}{\succ }\dots \underset{d-1}{\succ }{n}_d\underset{d}{\succ }0}{\displaystyle \frac{a_{n_1}}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}},\hfill \\ \hfill \mathrm{\Psi }(\mathit{l};\mathit{s}):=& \sum _{n_1\underset{1}{\succ }\dots \underset{d-1}{\succ }{n}_d\underset{d}{\succ }0}{\displaystyle \frac{a_{n_1}^2}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}},\hfill \end{array}$$

where $l_j\left(n\right)=2n,2n\pm 1$, $\underset{j}{\succ }$ is “≥” if $l_j\left(n\right)=2n+1$ and is “>” otherwise.

(2) Differential forms. Let $\xi$ and ${\xi }^{\prime }$ be any fourth roots of unity, i.e., ${\xi }^4={{\xi }^{\prime }}^4=1$.

$$\begin{array}{cc}\hfill \mathtt{a}=& {\displaystyle \frac{dt}{t}},\hfill \\ \hfill {\mathtt{x}}_{\xi }=& {\displaystyle \frac{dt}{\xi -t}},{\mathtt{d}}_{\xi ,{\xi }^{\prime }}={\mathtt{x}}_{\xi }-{\mathtt{x}}_{{\xi }^{\prime }},\hfill \\ \hfill \mathtt{y}=& {\mathtt{x}}_{-i}+{\mathtt{x}}_i-{\mathtt{x}}_{-1}-{\mathtt{x}}_1,\hfill \\ \hfill \mathtt{z}=& -\mathtt{a}-{\mathtt{x}}_{-i}-{\mathtt{x}}_i,\hfill \\ \hfill F_s=& {(cottdt)}^{s-1}:=\underset{s-1\mathrm{times}}{\underbrace{(cottdt)\circ \dots \circ (cottdt)}}.\hfill \end{array}$$

(3) Terminology and acronym.

• CMZV: colored multiple zeta value, defined by (2);
• ${\mathsf{CMZV}}_w^N$: the $\mathbb{Q}$-vector space spanned by all the CMZVs of weight w at level N;
• $\mathit{b}\mathit{t}\left(l\right)$: block type of l, defined as $\alpha$, $\beta$, or $\gamma$ if $l\left(n\right)=2n$, $2n+1$, or $2n-1$, respectively;
• $\mathit{b}\mathit{t}(l_1,\dots ,l_d):=(\mathit{b}\mathit{t}\left(l_1\right),\dots ,\mathit{b}\mathit{t}\left(l_d\right))$: we may use this to write $\mathrm{\Xi }\left(\mathit{b}\mathit{t}\right(\mathit{l});\mathit{s})=\mathrm{\Xi }(\mathit{l};\mathit{s})$ and $\mathrm{\Psi }\left(\mathit{b}\mathit{t}\right(\mathit{l});\mathit{s})=\mathrm{\Psi }(\mathit{l};\mathit{s})$;
• $\alpha$-block: a sum of the form ${\sum }_{m>n}{\displaystyle \frac{a_m(sint)}{{\left(2m\right)}^s}}$, or its integral expression (18);
• $\beta$-block: a sum of the form ${\sum }_{m>n}{\displaystyle \frac{a_m(sint)}{{(2m+1)}^s}}$, or its integral expression (19);
• ${\beta }^{★}$-block: a sum of the form ${\sum }_{m\ge n}{\displaystyle \frac{a_m(sint)}{{(2m+1)}^s}}$, or its integral expression (29);
• $\gamma$-block: a sum of the form ${\sum }_{m>n}{\displaystyle \frac{a_m(sint)}{{(2m-1)}^s}}$, or its integral expressions (20) and (21).

1.5. Main Results of the Paper

In the next three sections, we will repeatedly apply the recursive Formulas (4)–(8) in the above to obtain the iterated integral expressions of three general Apéry-type binomial series. In Section 4 and Section 5, we will analyze these iterated integrals carefully to prove one of the main results of this paper relating these sums to the CMZVs at level 4.

Theorem 1 

(cf. Theorems 6 and 8). Let $d\in \mathbb{N}$, $\mathit{s}\in {\mathbb{N}}^d$, and $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n\pm 1$.

(a)

If $\mathit{b}\mathit{t}\left(l_1\right)=\alpha$ and $\mathit{b}\mathit{t}\left(l_j\right)=\alpha$ or β for all $j\ge 2$, then $\mathrm{\Xi }(\mathit{l};\mathit{s})\in {\mathsf{CMZV}}_{\left|s\right|}^4$.

(b)

If $\mathit{b}\mathit{t}\left(l_1\right)=\beta$ and $\mathit{b}\mathit{t}\left(l_j\right)=\alpha$ or β for all $j\ge 2$, then $\mathrm{\Xi }(\mathit{l};\mathit{s})\in i{\mathsf{CMZV}}_{\left|s\right|}^4$.

(c)

If $\mathit{b}\mathit{t}\left(l_1\right)=\gamma$ and $s_1=1$, then $\mathrm{\Xi }(\mathit{l};\mathit{s})=\mathrm{\Xi }(l_2,\dots ,l_d;s_2,\dots ,s_d)$.

(d)

If $\mathit{b}\mathit{t}\left(l_1\right)=\gamma$ and $s_1\ge 2$, then $\mathrm{\Xi }(\mathit{l};\mathit{s})\in {\mathsf{CMZV}}_{\le \left|s\right|-1}^4\otimes \mathbb{Q}\left[i\right]$.

(e)

Moreover, the claim in (d) still holds if one changes any of the strict inequalities $n_j>n_{j+1}$ to $n_j\ge n_{j+1}$ ($n_{d+1}=0$) and vice versa in the definition of $\mathrm{\Xi }(\mathit{l};\mathit{s})$, provided the series is defined.

We then compute many examples (most of which are put in Appendix A) to show how the general procedure works. For the reader’s convenience, we have summarized all the main results of this section and examples of Theorem 8 in Table 1 and Table 2.

In Section 6, we apply the beta integral to derive the iterated integral expressions for the corresponding Apéry-type binomial series in which $a_n^2$ appears. The main result is the following.

Theorem 2 

(cf. Theorem 9). Let $d\in \mathbb{N}$, $\mathit{s}\in {\mathbb{N}}^d$, and $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n\pm 1$.

(a)

If $\mathit{b}\mathit{t}\left(l_j\right)=\alpha$ or β for all $j\ge 1$, then we have $\mathrm{\Psi }(\mathit{l};\mathit{s})\in {\displaystyle \frac{i}{\pi }}{\mathsf{CMZV}}_{\left|s\right|+1}^4.$

(b)

More generally, if we relax the restriction on the index ordering of $\mathrm{\Psi }(\mathit{l};\mathit{s})$ to allow “$\underset{j}{\succ }$” to be either “≥” or “>” for all $j\ge 2$, provided the series is defined, then the series lies in ${\displaystyle \frac{1}{\pi }}{\mathsf{CMZV}}_{\le \left|s\right|+\eta (s_1,\mathit{l})}^4\otimes \mathbb{Q}\left[i\right]$, where $\eta (s_1,\mathit{l})$ is 0 or $\pm 1$ as defined in (51).

To demonstrate the ideas of Theorem 9, we then compute many examples, most of which are in Appendix B for ease of reading. For the reader’s convenience, we have summarized the results of Theorem 9 and its examples in Table 3 and Table 4.

In the last two sections, we answer some questions in our paper [29], relate our work to some previously known results, and provide a few more enlightening examples. We end the paper by pointing out some possible future research directions.

2. Apéry-Type Central Binomial Series

In this section, we will consider the Apéry-type series defined in (1). We will need to extend Chen’s iterated integrals by combining 1-forms and functions as follows. For any $r\in \mathbb{N}$, 1-forms $f_1\left(t\right)dt,\dots ,f_{r+1}\left(t\right)dt$, and differentiable functions $F_1\left(t\right),\dots ,F_r\left(t\right)$, $G_1\left(t\right),\dots ,G_r\left(t\right)$, we extend the definition of iterated integrals by setting recursively

$$\begin{array}{cc}\hfill & {\int }_a^b\left(\prod _{j=1}^r\circ \left[F_j\left(t\right)+f_j\left(t\right)dt\circ G_j\left(t\right)\right]\right)\circ f_{r+1}\left(t\right)dt\hfill \\ \hfill :=& {\int }_a^b\left(\prod _{j=1}^{r-1}\circ \left[F_j\left(t\right)+f_j\left(t\right)dt\circ G_j\left(t\right)\right]\right)\circ \left(F_r\left(t\right)f_{r+1}\left(t\right)\right)dt\hfill \\ \hfill +& {\int }_a^b\left(\prod _{j=1}^{r-1}\circ \left[F_j\left(t\right)+f_j\left(t\right)dt\circ G_j\left(t\right)\right]\right)\circ f_r\left(t\right)dt\circ G_r\left(t\right)f_{r+1}\left(t\right)dt,\hfill \end{array}$$

where

$$\prod _{j=1}^r\circ {\alpha }_j={\alpha }_1\circ \dots \circ {\alpha }_r.$$

Our first result concerns the n-tails of the Apéry-type binomial series (1). To be consistent with all the major theorems of this paper, we formulate it as an even-indexed variation. For $s\in \mathbb{N}$, we define

$$p_s\left(t\right):=tantdt{\left({\displaystyle \frac{dt}{tant}}\right)}^{s-1}(1-csctdt\circ sect).$$

Theorem 3. 

For all $n\in {\mathbb{N}}_0$, $\mathit{s}=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$, and $y\in [-\pi /2,\pi /2]$ we have

$$\begin{array}{cc}\hfill \alpha {(\mathit{s};siny)}_n:=& \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}(siny)}{{\left(2n_1\right)}^{s_1}\dots {\left(2n_d\right)}^{s_d}}}\hfill \\ \hfill =& coty{\displaystyle \frac{d}{dy}}{\int }_0^y{p}_{s_1}\circ \dots \circ p_{s_d}\circ a_n(sint)tantdt.\hfill \end{array}$$

Here, if $y=\pm \pi /2$, then the right-hand side of the above is defined to be the limit as $y\to \pm \pi /2$.

We will again call the sum $\left|\mathit{s}\right|:=s_1+\dots +s_d$ the weight and d the depth of the series $\alpha (\mathit{s};siny)$, respectively.

Remark 1. 

When $y=\pm \pi /2$, we must take limit $y\to \pm \pi /2$ because $cot(\pm \pi /2)=0$ while the first factor of ${\displaystyle \frac{d}{dy}}(\dots )$ is $tany$ which is undefined at $\pm \pi /2$. However, $cotytany=1$ is well defined at the limit as $y\to \pm \pi /2$.

Proof. 

The proof is in the same spirit as that of ([3] Thm. 4): first express the n-tail $u_n\left(y\right)$ by an integral $v_n\left(y\right)$ (in which the factor $a_n(sint)$ magically appears again) and then iterate the procedure to arbitrary depth, where

$$\begin{array}{cc}\hfill u_n\left(y\right):=& \sum _{m>n}{\displaystyle \frac{a_m(siny)secy}{2m-1}}=tany\sum _{m>n}{\displaystyle \frac{1}{4^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m}{m}}\right){\displaystyle \frac{{sin}^{2m-1}y}{2m-1}},\hfill \\ \hfill v_n\left(y\right):=& {\int }_0^y{a}_n(sint)secttantdt={\int }_0^y\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n-1}t}{4^n}}{tan}^{2}tdt.\hfill \end{array}$$

We claim that $u_n\left(y\right)=v_n\left(y\right)$. Indeed, first, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dy}}(u_{n-1}\left(y\right)-u_n\left(y\right))=& {\displaystyle \frac{1}{4^n}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{d}{dy}}{\displaystyle \frac{{sin}^{2n-1}ytany}{(2n-1)}}\hfill \\ \hfill =& \left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n-1}y}{4^n}}·{\displaystyle \frac{(2n-1)+{sec}^{2}y}{2n-1}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dy}}(v_{n-1}\left(y\right)-v_n\left(y\right))=& {\displaystyle \frac{{sin}^{2n-3}y}{4^n}}{tan}^{2}t\left(4\left({\displaystyle \genfrac{}{}{0pt}{}{2n-2}{n-1}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){sin}^{2}y\right)\hfill \\ \hfill =& {\displaystyle \frac{{sin}^{2n-1}y}{4^n}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){sec}^{2}y\left({\displaystyle \frac{2n}{2n-1}}-{sin}^{2}y\right)\hfill \\ \hfill =& {\displaystyle \frac{{sin}^{2n-1}y}{4^n}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){sec}^{2}y\left({\displaystyle \frac{1}{2n-1}}+{cos}^{2}y\right).\hfill \end{array}$$

Hence, for all $n\ge 1$

$$\begin{array}{c}\hfill u_{n-1}\left(y\right)-u_n\left(y\right)=v_{n-1}\left(y\right)-v_n\left(y\right)\end{array}$$

since clearly the two sides both vanish when $y=0$. Further, for all $y\in (-\pi /2,\pi /2)$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{u}_n\left(y\right)=\underset{n\to \infty }{lim}{v}_n\left(y\right)=0\end{array}$$

since by Stirling’s formula

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{4^n}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right)\sim {\displaystyle \frac{1}{\sqrt{\pi n}}}\mathrm{as}n\to \infty .\end{array}$$

By telescoping, we see immediately that $u_n\left(y\right)=v_n\left(y\right)$, or equivalently,

$$\begin{array}{c}\hfill \sum _{m>n}{\displaystyle \frac{1}{4^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m}{m}}\right){\displaystyle \frac{{sin}^{2m-1}y}{2m-1}}=coty{\int }_0^y\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n-1}t}{4^n}}{tan}^{2}tdt.\end{array}$$

(9)

Differentiating (9), we obtain

$$\begin{array}{c}\hfill \sum _{m>n}{a}_m{sin}^{2m-2}ycosy=\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n}y}{4^n}}secy-{csc}^{2}y{\int }_0^y\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n}t}{4^n}}tantsectdt.\end{array}$$

(10)

Multiplying this by $siny$ and integrating, we have

$$\begin{array}{cc}\hfill \sum _{m>n}{\displaystyle \frac{1}{4^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m}{m}}\right){\displaystyle \frac{{sin}^{2m}y}{2m}}=& {\int }_0^y(1-csctdt\circ sect)\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n}t}{4^n}}tantdt.\hfill \end{array}$$

Repeatedly multiplying these by $coty$ and integrating, we have for all $s\ge 1$

$$\begin{array}{cc}\hfill \sum _{m>n}{\displaystyle \frac{a_m(siny)}{{\left(2m\right)}^s}}=& {\int }_0^y{\left({\displaystyle \frac{dt}{tant}}\right)}^{s-1}(1-csctdt\circ sect)a_n(sint)tantdt.\hfill \end{array}$$

(11)

This shows the $d=1$ case of the theorem. To prove the general depth case, we now apply induction on d. Thus, assuming

$$\alpha {(\mathit{s};siny)}_{n_d}=coty{\displaystyle \frac{d}{dy}}{\int }_0^y{p}_{s_1}\circ \dots \circ p_{s_{d-1}}\circ a_{n_d}(sint)tantdt,$$

we then obtain

$$\begin{array}{cc}\hfill \alpha {(\mathit{s};siny)}_n=& coty{\displaystyle \frac{d}{dy}}{\int }_0^y{p}_{s_1}\circ \dots \circ p_{s_{d-1}}\circ \sum _{n_d>n}{\displaystyle \frac{a_{n_d}(sint)}{{\left(2n_d\right)}^{s_d}}}tantdt\hfill \\ \hfill =& coty{\displaystyle \frac{d}{dy}}{\int }_0^y{p}_{s_1}\circ \dots \circ p_{s_d}\circ a_n(sint)tantdt\hfill \end{array}$$

by taking $m=n_d$ and $s=s_d$ in (11). This completes the proof of the theorem. □

Example 1. 

If $s_1=\dots =s_d=1$, then we see that

$$\begin{array}{cc}\hfill \alpha (1;siny):=& \sum _{n>0}{\displaystyle \frac{1}{4^n}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n}y}{2n}}={\int }_0^y(csct-cott)dt\hfill \\ \hfill =& {\left.log{\displaystyle \frac{csct}{csct+cott}}\right|}_0^y=log2-log(1+cosy).\hfill \end{array}$$

Thus, $\alpha (1;1)=log2$. When $d=2$, we obtain

$$\begin{array}{cc}\hfill \alpha (1,1;siny):=& {\int }_0^y(csctdt\circ sect-cottdt)(csct-cott)dt\hfill \\ \hfill =& {\int }_0^y(csct-cott)dt(csct-cott)dt+csctdt\circ (sect-1)(csct-cott)dt.\hfill \end{array}$$

Note that

$$\begin{array}{cc}\hfill & {\int }_0^u(sect-1)(csct-cott)dt={\int }_0^u{\displaystyle \frac{{(1-cost)}^{2}dt}{sintcost}}\hfill \\ \hfill =& {\int }_0^u{\displaystyle \frac{(cost-1)dcost}{(1+cost)cost}}={\int }_0^u\left({\displaystyle \frac{2}{1+cost}}-{\displaystyle \frac{1}{cost}}\right)dcost=log{\displaystyle \frac{{(1+cosu)}^2}{4cosu}}.\hfill \end{array}$$

Setting $a=cosy$, we have

$$\begin{array}{cc}\hfill \alpha (1,1;siny):=& {\displaystyle \frac{1}{2}}{\left({\int }_0^y(csct-cott)dt\right)}^2+{\int }_0^{y}log{\displaystyle \frac{{(1+cost)}^2}{4cost}}csctdt\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{log}^2{\displaystyle \frac{2}{1+a}}+{\int }_0^y-{\displaystyle \frac{1}{1-{cos}^{2}t}}log{\displaystyle \frac{{(1+cost)}^2}{4cost}}dcost\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{log}^2{\displaystyle \frac{2}{1+a}}+{\int }_a^1{\displaystyle \frac{1}{1-u^2}}log{\displaystyle \frac{{(1+u)}^2}{4u}}du\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{log}^2{\displaystyle \frac{2}{1+a}}+{\displaystyle \frac{1}{2}}{Li}_2(1-a)-{Li}_2\left({\displaystyle \frac{1-a}{2}}\right)+{\displaystyle \frac{1}{2}}{Li}_2(-a)\hfill \\ \hfill & +log2log(1+a)+{\displaystyle \frac{1}{2}}log\left(a\right)log(1+a)-{\displaystyle \frac{1}{2}}{log}^{2}2-{\displaystyle \frac{1}{2}}{log}^2(1+a)+{\displaystyle \frac{1}{4}}\zeta \left(2\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{Li}_2(1-a)-{Li}_2\left({\displaystyle \frac{1-a}{2}}\right)+{\displaystyle \frac{1}{2}}{Li}_2(-a)+{\displaystyle \frac{1}{2}}log\left(a\right)log(1+a)+{\displaystyle \frac{1}{4}}\zeta \left(2\right).\hfill \end{array}$$

Applying the identity ${Li}_2\left(t\right)+{Li}_2(1-t)+logtlog(1-t)=\zeta \left(2\right)$ with $t=1/2$, we see that

$$\alpha (1,1;1):={\displaystyle \frac{1}{2}}{log}^{2}2+{\displaystyle \frac{1}{4}}\zeta \left(2\right)\approx 0.6514.$$

3. An Odd Variant

In this section, we turn to the Apéry-type binomial series similar to (1) but with summation indices restricted to odd numbers only. Set

$$\begin{array}{c}{\theta }_s\left(t\right)=tantdt{\left({\displaystyle \frac{dt}{tant}}\right)}^{s-1}(1-dt\circ csctsect),\hfill \\ q_s\left(t\right)=sectdt{\left({\displaystyle \frac{dt}{tant}}\right)}^{s-1}(csct-dt\circ sect).\hfill \end{array}$$

Theorem 4. 

For all $n\in {\mathbb{N}}_0$, $\mathit{s}=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$ and $y\in [-\pi /2,\pi /2]$, we have

$$\begin{array}{cc}\hfill \beta {(\mathit{s};siny)}_n:=& \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}(siny)}{{(2n_1+1)}^{s_1}\dots {(2n_d+1)}^{s_d}}}\hfill \\ \hfill =& cscycoty{\displaystyle \frac{d}{dy}}{\int }_0^y{\theta }_{s_1}\circ \dots \circ {\theta }_{s_d}\circ a_n(sint)sinttantdt,\hfill \\ \hfill {\beta }^{★}{(\mathit{s};siny)}_n:=& \sum _{n_1\ge \dots \ge n_d\ge n}{\displaystyle \frac{a_{n_1}(siny)}{{(2n_1+1)}^{s_1}\dots {(2n_d+1)}^{s_d}}}\hfill \\ \hfill =& coty{\displaystyle \frac{d}{dy}}{\int }_0^y{q}_{s_1}\circ \dots \circ q_{s_d}\circ a_n(sint)tantdt.\hfill \end{array}$$

Here, when $y=\pm \pi /2$, we understand the right-hand side of the above as the limit when $y\to \pm \pi /2$.

Remark 2. 

When $y=\pm \pi /2$, we must take limit $y\to \pm \pi /2$ because $cot(\pm \pi /2)=0$ while the first factor of ${\displaystyle \frac{d}{dy}}(\dots )$ is $secy$ which is undefined at $\pm \pi /2$. However, $cotysecy=cscy$ is well defined at the limit $y\to \pm \pi /2$.

Remark 3. 

The introduction of both β-sums and ${\beta }^{★}$ is one of the key ideas of this paper. Their relation is similar to that between the classical multiple zeta values and the multiple zeta star values in which the difference lies in whether we allow summation indices to be equal to each other. Some results look neater and more concise with one type of sums while others are prettier and more manageable with the other.

Proof. 

From (10), we obtain

$$\begin{array}{c}\hfill \sum _{m>n}{a}_m{sin}^{2m-2}ycosy=\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n}y}{4^n}}secy-{csc}^{2}y{\int }_0^y\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n}t}{4^n}}tantsectdt.\end{array}$$

(12)

Multiplying this by ${sin}^{2}y$ on both sides and integrating, we see that

$$\begin{array}{c}\hfill \sum _{m>n}{\displaystyle \frac{1}{4^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m}{m}}\right){\displaystyle \frac{{sin}^{2m+1}y}{2m+1}}={\int }_0^y(1-dt\circ csctsect)\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n+1}t}{4^n}}tantdt.\end{array}$$

Repeatedly multiplying this by $coty$ and integrating, we have for all $s\ge 1$

$$\begin{array}{c}\hfill \sum _{m>n}{\displaystyle \frac{a_m(siny)siny}{{(2m+1)}^s}}={\int }_0^y{\left({\displaystyle \frac{dt}{tant}}\right)}^{s-1}(1-dt\circ csctsect)a_n(sint)sinttantdt.\end{array}$$

(13)

This proves the $d=1$ case of the $\beta$-sum in the theorem

On the other hand, multiplying (12) by ${sin}^{2}y$ and adding ${\displaystyle \frac{1}{4^n}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){sin}^{2n}ycosy$ on both sides, we obtain

$$\begin{array}{c}\sum _{m\ge n}{\displaystyle \frac{1}{4^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m}{m}}\right){sin}^{2m}ycosy=\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n}y}{4^n}}secy-{\int }_0^y\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n}t}{4^n}}tantsectdt.\end{array}$$

Integrating we obtain

$$\begin{array}{c}\sum _{m\ge n}{\displaystyle \frac{1}{4^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m}{m}}\right){\displaystyle \frac{{sin}^{2m+1}y}{2m+1}}={\int }_0^y(csct-dt\circ sect)\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n+1}t}{4^n}}sectdt.\end{array}$$

Repeatedly multiplying these by $coty$ and integrating, we have for all $s\ge 1$

$$\begin{array}{c}\sum _{m\ge n}{\displaystyle \frac{a_m(siny)siny}{{(2m+1)}^s}}={\int }_0^y{\left({\displaystyle \frac{dt}{tant}}\right)}^{s-1}(csct-dt\circ sect)a_n(sint)sintsectdt.\end{array}$$

(14)

This shows that the theorem holds for ${\beta }^{★}$-sum when $d=1$.

To prove the general case of the theorem for the $\beta$- and the ${\beta }^{★}$-sums, we can apply induction on d and using the special case with $m=n_d$ and $s=s_d$ in (13) and (14), respectively. This concludes the proof of the theorem. □

4. Another Odd Variant

In this section, we study another variant of the Apéry-type binomial series similar to (1). Even though these series look very similar to those in the previous section, they are fundamentally different and therefore require different approaches to handle them. Set

$$\begin{array}{c}\hfill {\phi }_s\left(t\right)=\left\{\begin{array}{cc}tantdt,\hfill & \mathrm{if}s=1;\hfill \\ {tan}^{2}tdt\circ {(cottdt)}^{s-2}\circ {cot}^{2}tdt,\hfill & \mathrm{if}s\ge 2.\hfill \end{array}\right.\end{array}$$

Theorem 5. 

For all $n\in {\mathbb{N}}_0$, $\mathit{s}=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$ and $y\in [-\pi /2,\pi /2]$, we have

$$\begin{array}{cc}\hfill \gamma {(\mathit{s};siny)}_n:=& \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}(siny)}{{(2n_1-1)}^{s_1}\dots {(2n_d-1)}^{s_d}}}\hfill \\ \hfill =& cosycoty{\displaystyle \frac{d}{dy}}{\int }_0^y{\phi }_{s_1}\circ \dots \circ {\phi }_{s_d}\circ a_n(sint)tantsectdt.\hfill \end{array}$$

Here, when $y=\pm \pi /2$, we understand the right-hand side of the above as the limit when $y\to \pm \pi /2$.

Proof. 

By (9), we have

$$\begin{array}{c}\hfill \sum _{m>n}{\displaystyle \frac{1}{4^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m}{m}}\right){\displaystyle \frac{{sin}^{2m-1}y}{2m-1}}=coty{\int }_0^y\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n-1}t}{4^n}}{tan}^{2}tdt.\end{array}$$

For any $s\ge 2$, we can repeatedly multiply this equation by $coty$ and integrate to obtain

$$\begin{array}{c}\hfill \sum _{m>n}{\displaystyle \frac{1}{4^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m}{m}}\right){\displaystyle \frac{{sin}^{2m-1}y}{{(2m-1)}^s}}={\int }_0^y{\left({\displaystyle \frac{dt}{tant}}\right)}^{s-2}{\displaystyle \frac{dt}{{tan}^{2}t}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right){\displaystyle \frac{{sin}^{2n-1}t}{4^n}}{tan}^{2}tdt.\end{array}$$

(15)

This proves the $d=1$ case of the theorem. To prove the general case, we can apply induction on d and use the special case with $m=n_d$ and $s=s_d$ in (15). This concludes the proof of the theorem. □

Example 2. 

If $s_1=\dots =s_d=1$, then we see that

$$\begin{array}{cc}\hfill \gamma (\mathit{s};siny):=& \sum _{n_1>\dots >n_d>0}{\displaystyle \frac{a_{n_1}(siny)}{(2n_1-1)\dots (2n_d-1)}}\hfill \\ \hfill =& cosy{\int }_0^y{(tantdt)}^{d-1}\circ tantsectdt=1-cosy\sum _{j=1}^{d-1}{\displaystyle \frac{{log}^j|secy|}{j!}}.\hfill \end{array}$$

In particular, we see that $\gamma ({\left\{1\right\}}^d;1)=1$ for all $d\ge 1$, where ${\left\{s\right\}}^d$ means repeating string s exactly d times.

It turns out that the evaluation $\gamma ({\left\{1\right\}}^d;1)=1$ is just a very special case of the following result on the n-tails $\gamma {({\left\{1\right\}}^d;1)}_n$, which in turn easily leads to Theorem 6.

Corollary 1. 

For all $n\in {\mathbb{N}}_0$ and $d\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill \gamma {({\left\{1\right\}}^d;1)}_n=& \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}}{(2n_1-1)\dots (2n_d-1)}}=a_n.\hfill \end{array}$$

Proof. 

For any $y\in (-\pi /2,\pi /2)$ and $d\in \mathbb{N}$, we set

$$\begin{array}{c}\hfill f_d\left(y\right)={\int }_0^y{(tantdt)}^{d-1}\circ a_n(sint)tantsectdt.\end{array}$$

Since $f_{d+1}^{\prime }\left(y\right)=(tany)f_d\left(y\right)$ if $d\ge 2$, by Theorem 5, we have

$$\begin{array}{cc}\hfill \gamma {(\mathit{s};1)}_n=& \underset{y\to {(\pi /2)}^{-}}{lim}(cosy)f_d\left(y\right)\hfill \\ \hfill =& \underset{y\to {(\pi /2)}^{-}}{lim}{\displaystyle \frac{f_d\left(y\right)}{secy}}=\underset{y\to {(\pi /2)}^{-}}{lim}{\displaystyle \frac{f_d^{\prime }\left(y\right)}{secytany}}=\underset{y\to {(\pi /2)}^{-}}{lim}{\displaystyle \frac{f_{d-1}\left(y\right)}{secy}}\hfill \end{array}$$

by L’Hôpital’s rule. Thus, repeatedly applying L’Hôpital’s rule finally yields

$$\begin{array}{c}\gamma {({\left\{1\right\}}^d;1)}_n=\underset{y\to {(\pi /2)}^{-}}{lim}{\displaystyle \frac{f_1\left(y\right)}{secy}}=\underset{y\to {(\pi /2)}^{-}}{lim}{\displaystyle \frac{f_1^{\prime }\left(y\right)}{secytany}}=\underset{y\to {(\pi /2)}^{-}}{lim}{a}_n(siny)=a_n,\end{array}$$

as desired. □

Theorem 6. 

Suppose $d\in \mathbb{N}$, $s=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$. Let $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n\pm 1$. Then

$$\begin{array}{c}\hfill \sum _{n_0>n_1\succ \dots \succ n_d\succ 0}{\displaystyle \frac{a_{n_0}}{(2n_0-1)l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}=\sum _{n_1\succ \dots \succ n_d\succ 0}{\displaystyle \frac{a_{n_1}}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}},\end{array}$$

where “≻” can be either “≥” or “>”, provided the series is defined.

Proof. 

This follows from Corollary 1 immediately. □

Theorem 6 enables us to consider fewer leading parity cases in Theorem 8.

5. Mixed Parity Variations of Apéry-Type Binomial Series

We now turn to the variants of Apéry-type binomial series in which each summation index is restricted to either even or odd numbers.

5.1. Evaluate Even/Odd Variants of Apéry-Type Series by Iterated Trigonometric Integrals

For any $r\in \mathbb{N}$, 1-forms $f_1\left(t\right)dt,\dots ,f_r\left(t\right)dt$, and any differentiable function $F\left(t\right)$, we extend the iterated integral notation to define

$$\begin{array}{cc}\hfill f_1\left(t\right)dt\circ \overleftarrow{F\left(t\right)}\circ f_2\left(t\right)dt=& F\left(t\right)f_1\left(t\right)dt\circ f_2\left(t\right)dt,\hfill \\ \hfill {\int }_0^y\overleftarrow{F\left(t\right)}\circ f_1\left(t\right)dt\circ \dots \circ f_r\left(t\right)dt=& F\left(y\right){\int }_0^y{f}_1\left(t\right)dt\circ \dots \circ f_r\left(t\right)dt.\hfill \end{array}$$

Define the block type of l by $\mathit{b}\mathit{t}\left(l\right)=\alpha ,\beta$ and $\gamma$ if $l\left(n\right)=2n,2n+1$ and $2n-1$, respectively. For $s\in \mathbb{N}$, set $F_s={(cottdt)}^{s-1}$ and

$$\begin{array}{cc}\hfill {\lambda }_{\alpha ,s}\left(t\right)=& F_s(tantdt-csctdtdsect),\hfill \\ \hfill {\lambda }_{\beta ,s}\left(t\right)=& \overleftarrow{csct}\circ F_s(sinttantdt-dtdsect),\hfill \\ \hfill {\lambda }_{\gamma ,1}\left(t\right)=& \overleftarrow{cost}\circ tantsectdt,\hfill \\ \hfill {\lambda }_{\gamma ,s}\left(t\right)=& \overleftarrow{sint}\circ F_{s-1}{\displaystyle \frac{dt}{{tan}^{2}t}}tantsectdt(s\ge 2).\hfill \end{array}$$

Theorem 7. 

Suppose $d\in \mathbb{N}$ and $\mathit{s}=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$. Let $y\in (-\pi /2,\pi /2)$. Then, for any $l_1\left(n\right)=2n,2n\pm 1$ and $l_2\left(n\right),\dots ,l_d\left(n\right)=2n,2n+1$, we have the n-tails

$$\begin{array}{cc}\hfill & \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}(siny)}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}={\int }_0^y{\lambda }_{bt\left(l_1\right),s_1}\circ \dots {\lambda }_{bt\left(l_d\right),s_d}\circ \overleftarrow{a_n(sint)}.\hfill \end{array}$$

(16)

Using non-trigonometric 1-forms, we obtain, for all $x\in (-1,1)$,

$$\begin{array}{cc}\hfill & \sum _{n_1>\dots >n_d>n}{\displaystyle \frac{a_{n_1}\left(x\right)}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}={\int }_0^x{\Lambda }_{bt\left(l_1\right),s_1}\circ \dots {\Lambda }_{bt\left(l_d\right),s_d}\circ \overleftarrow{a_n\left(t\right)},\hfill \end{array}$$

(17)

where ${\Lambda }_{bt\left(l\right),s}$ are obtained from ${\lambda }_{\mathit{b}\mathit{t}\left(l\right),s}$ by applying the substitution $t\to {sin}^{-1}t$.

Proof. 

By (11), (13), and (15), we readily obtain

$$\begin{array}{c}\sum _{m>n}{\displaystyle \frac{a_m(siny)}{{\left(2m\right)}^s}}={\int }_0^y{F}_s(1-csctdt\circ sect)a_n(sint)tantdt,\hfill \end{array}$$

(18)

$$\begin{array}{c}\sum _{m>n}{\displaystyle \frac{a_m(siny)}{{(2m+1)}^s}}=cscy{\int }_0^y{F}_s(1-dt\circ csctsect)a_n(sint)sinttantdt.\hfill \end{array}$$

(19)

$$\begin{array}{c}\sum _{m>n}{\displaystyle \frac{a_m(siny)}{2m-1}}=cosy{\int }_0^y{a}_n(sint)tantsectdt,\hfill \end{array}$$

(20)

$$\begin{array}{c}\sum _{m>n}{\displaystyle \frac{a_m(siny)}{{(2m-1)}^s}}=siny{\int }_0^y{F}_{s-1}{\displaystyle \frac{dt}{{tan}^{2}t}}{a}_n(sint)tantsectdt(s\ge 2).\hfill \end{array}$$

(21)

Thus,

$$\begin{array}{cc}\hfill {\lambda }_{\alpha ,s}\left(t\right)=& F_s\left(tantdt-csctdtdsect\right)=F_s(csctdt\circ sect-cottdt),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\lambda }_{\beta ,s}\left(t\right)=& \overleftarrow{csct}\circ F_s\left(sinttantdt-dtdsect\right)=\overleftarrow{csct}\circ F_s(dt\circ sect-costdt),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\lambda }_{\gamma ,1}\left(t\right)=& \overleftarrow{cost}\circ dsect,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\lambda }_{\gamma ,s}\left(t\right)=& \overleftarrow{sint}\circ F_{s-1}{\displaystyle \frac{dt}{{tan}^{2}t}}dsect=\overleftarrow{sint}\circ F_{s-1}\left({\displaystyle \frac{dt}{sinttant}}-{\displaystyle \frac{dt}{{tan}^{2}t}}\circ sect\right)(s\ge 2).\hfill \end{array}$$

(25)

Indeed, all the above identities follow directly from the definition. We will demonstrate (22) in detail to illustrate the ideas. We only need to show that for any fixed number c and variable x,

$${\int }_c^{x}tantdt-{\int }_c^x(sect-secc)csctdt={\int }_c^{x}secccsctdt-{\int }_c^{x}cottdt.$$

This is clear when $x=c$. Thus, it suffices to show the two sides have the same derivative, which is straight-forward to verify using the trigonometric identity $tant+cott=sectcsct$. Similarly, (23) follows from the trigonometric identity $sinttant+cost=sect$ while (25) follows from $sect/{tan}^{2}t=1/(sinttant)$.

For convenience, we call the expression on both sides of (18) (resp. (19), resp. (20), and (21)) an α-block (resp. β-block, resp. γ-block). In (16), each $s_j$ corresponds to (a variation of) such a block. We find that after starting with a block in (18)–(21), we can repeatedly apply (18)–(21) to insert all the middle blocks until the end. This concludes the constructive proof of the theorem. □

5.2. Evaluate Even/Odd Variations of Apéry-Type Binomial Series by CMZVs at Level 4

Let $d\in \mathbb{N}$, $\mathit{s}=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$. Let $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n\pm 1$. Set

$$\begin{array}{c}\hfill \mathrm{\Xi }(\mathit{l};\mathit{s}):=\sum _{n_1\underset{1}{\succ }\dots \underset{d-1}{\succ }{n}_d\underset{d}{\succ }0}{\displaystyle \frac{a_{n_1}}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}},\end{array}$$

(26)

where $\underset{j}{\succ }$ is “≥” if $l_j\left(n\right)=2n+1$ and is “>” otherwise. Due to Theorem 6, we can always assume that if the leading block is a $\gamma$-block, then $s_1\ge 2$.

Theorem 8. 

Let ${\mathsf{CMZV}}_w^4$ be the $\mathbb{Q}$-vector space spanned by all the CMZVs of weight w at level 4. Set

$$\delta \left(l\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}l\left(n\right)=2n;\hfill \\ 1,\hfill & \mathit{otherwise},\hfill \end{array}\right.\nu \left(l\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{l}_1\left(n\right)=2n-1;\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(a)

If $l_1\left(n\right)\ne 2n-1$, then we have $\mathrm{\Xi }(\mathit{l};\mathit{s})\in i^{\delta \left(l_1\right)}{\mathsf{CMZV}}_{\left|s\right|}^4$.

(b)

Suppose $(\mathit{b}\mathit{t}\left(l_1\right),s_1)\ne (\gamma ,1)$. Then, we have $\mathrm{\Xi }(\mathit{l};\mathit{s})\in {\mathsf{CMZV}}_{\left|s\right|-\nu \left(l_1\right)}^4\otimes \mathbb{Q}\left[i\right].$

(c)

Moreover, the claim in (b) still holds if one changes any of the strict inequalities $n_j>n_{j+1}$ in (26) to $n_j\ge n_{j+1}$ ($n_{d+1}=0$) and vice versa, provided the series is defined. In particular,

$$\begin{array}{c}\hfill \sum _{n_1\succ \dots \succ n_d\succ 0}{\displaystyle \frac{a_{n_1}}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}\in {\mathsf{CMZV}}_{\le \left|s\right|-\nu \left(l_1\right)}^4\otimes \mathbb{Q}\left[i\right],\end{array}$$

(27)

where “≻” can be either “≥” or “>”, provided the series is defined.

Proof. 

We first claim that we may reduce the sums in the first two cases to those sums where $l_j\left(n\right)=2n-1$ appears only when $j=1$, if it ever appears. We can prove this by induction on the depth in exactly the same way as was used in the proof of ([1] Thm. 4.2(b)) and thus we omit it here.

(a) Recall that (11) and (14) provide us the following iterative structure:

$$\begin{array}{cc}\hfill \sum _{m>n}{\displaystyle \frac{a_m(siny)}{{\left(2m\right)}^s}}=& {\int }_0^y{F}_s(1-csctdt\circ sect)a_n(sint)tantdt,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \sum _{m\ge n}{\displaystyle \frac{a_m(siny)}{{(2m+1)}^s}}=& cscy{\int }_0^y{F}_s(csct-dt\circ sect)a_n(sint)tantdt,\hfill \end{array}$$

(29)

where $F_s={(cottdt)}^{s-1}$. The key idea is to use (28) and (29) repeatedly to express (26) as an iterated integral and then use the change of variable $t\to {sin}^{-1}[(1-t^2)/(1+t^2)]$ to convert this iterated integral to a $\mathbb{Q}$-linear combination of iterated integrals that are clearly in ${\mathsf{CMZV}}_{\left|s\right|}^4.$

To begin, similarly to the proof of Theorem 7, we call the iterated integral of form (28) in an iteration an α-block, and the iterated integral of form (29) in an iteration a ${\beta }^{★}$-block. The extra function $cscy$ before the ${\beta }^{★}$-block brings the main complication into this process since it changes the shape of the block in front of it.

The first reduction step is that we may assume $\beta$-block does not appear. Indeed, the equality

$$\sum _{m>n}f(m,n)=\sum _{m\ge n}f(m,n)-f(n,n)$$

clearly decomposes all $\beta$-blocks into ${\beta }^{★}$-blocks.

We now assume only $\alpha$- and ${\beta }^{★}$-blocks appear in the sum, since $l_1\left(n\right)\ne 2n-1$ by assumption. Then, there are only four different types of the first block:

    1st-block

$\alpha \circ \alpha$: first block is an $\alpha$-block, followed by another $\alpha$-block or nothing:

$$\begin{array}{cc}\hfill F_s(1-csctdt\circ sect)tantdt=& F_s(tantdt-csctsectdt+csctdt\circ sect)\hfill \\ \hfill =& F_s(csctdt\circ sect-cottdt).\hfill \end{array}$$

(30)

    1st-block

$\alpha \circ {\beta }^{★}$: first block is an $\alpha$-block, followed by a ${\beta }^{★}$-block:

$$\begin{array}{cc}\hfill F_s(1-csctdt\circ sect)sectdt=& F_s(sectdt-csctdtdtant)\hfill \\ \hfill =& F_s(csctdt\circ tant).\hfill \end{array}$$

(31)

    1st-block

${\beta }^{★}\circ \alpha$: first block is a ${\beta }^{★}$-block, followed by an $\alpha$-block:

$$\begin{array}{c}\hfill F_s(csct-dt\circ sect)tantdt=F_s(sectdt-dtdsect)=F_s(dt\circ sect).\end{array}$$

(32)

    1st-block

${\beta }^{★}\circ {\beta }^{★}$: first block is ${\beta }^{★}$-block, followed by another ${\beta }^{★}$-block:

$$\begin{array}{c}\hfill F_s(csct-dt\circ sect)sectdt=F_s(csctsectdt-dtdtant)=F_s(cottdt+dt\circ tant).\end{array}$$

The 1-forms appearing in the above are listed as follows:

$$\begin{array}{c}\hfill csctdt,cottdt,dt.\end{array}$$

(33)

Moreover, the observation below will be crucial later, as follows:

($★$)

The 1-form $dt$ appears only when a ${\beta }^{★}$-block appears first and it always has a trailing $\circ sect$ (resp. $\circ tant$) if it is followed by an $\alpha$-block (resp. ${\beta }^{★}$-block).

Hence, all the blocks after the first may (or may not) be multiplied by either $tant$ or $sect$.

Similarly, for the middle blocks (i.e., neither initial nor end), we have the following four cases:

$$\begin{array}{cccc}& \mathbf{Mid}-\mathbf{block}\alpha \circ \alpha :\hfill & \hfill & \left(1\mathrm{or}sect\right)F_s(csctdt\circ sect-cottdt),\hfill \end{array}$$

(34)

$$\begin{array}{cccc}& \mathbf{Mid}-\mathbf{block}\alpha \circ {\beta }^{★}:\hfill & \hfill & \left(1\mathrm{or}sect\right)F_s(csctdt\circ tant),\hfill \end{array}$$

(35)

$$\begin{array}{cccc}& \mathbf{Mid}-\mathbf{block}{\beta }^{★}\circ \alpha :\hfill & \hfill & \left(1\mathrm{or}tant\right)F_s(dt\circ sect),\hfill \end{array}$$

(36)

$$\begin{array}{cccc}& \mathbf{Mid}-\mathbf{block}{\beta }^{★}\circ {\beta }^{★}:\hfill & \hfill & \left(1\mathrm{or}tant\right)F_s(cottdt+dt\circ tant).\hfill \end{array}$$

(37)

Therefore, the following additional 1-forms may appear:

$$\begin{array}{c}\hfill tantdt,sectcsctdt.\end{array}$$

(38)

We now turn to the ending blocks. Since

$$\begin{array}{cc}\hfill (tantdt-csctdt{\int }_0^{t}secxtanxdx)& =(tantdt-(sect-1)csctdt)=(csct-cott)dt,\hfill \\ \hfill (tantcsctdt-dt{\int }_0^{t}secxtanxdx)& =(sectdt-(sect-1\left)dt\right)=dt,\hfill \end{array}$$

we may have the following forms for the end block:

$$\begin{array}{cccc}& s_d\ge 2,\mathbf{End}-\mathbf{block}\alpha :\hfill & \hfill & \left(1\mathrm{or}sect\right)F_s(csct-cott)dt,\hfill \end{array}$$

(39)

$$\begin{array}{cccc}& s_d\ge 2,\mathbf{End}-\mathbf{block}{\beta }^{★}:\hfill & \hfill & \left(1\mathrm{or}tant\right)F_{s}dt.\hfill \end{array}$$

(40)

If $s=1$, then since

$$\begin{array}{c}\hfill tant(csct-cott)=sect-1,sect(csct-cott)=csct(sect-1)\end{array}$$

we see that the end block has the form

$$\begin{array}{cccc}& s_d=1,\mathbf{End}-\mathbf{block}\alpha :\hfill & \hfill & (csct-cott)dt\mathrm{or}csct(sect-1)dt,\hfill \end{array}$$

(41)

$$\begin{array}{cccc}& s_d=1,\mathbf{End}-\mathbf{block}{\beta }^{★}:\hfill & \hfill & dt\mathrm{or}tantdt.\hfill \end{array}$$

(42)

Under the change of variable $t\to {sin}^{-1}[(1-t^2)/(1+t^2)]$, we have

$$\begin{array}{cc}\hfill dt& \to i{\mathtt{d}}_{-i,i},cottdt\to \mathtt{y},tantdt\to \mathtt{z},\hfill \\ \hfill csctdt& \to {\mathtt{d}}_{-1,1},sectdt\to -\mathtt{a},sectcsctdt\to \mathtt{y}+\mathtt{z},\hfill \end{array}$$

(43)

where ${\mathtt{x}}_{\xi }=dt/(\xi -t)$, ${\mathtt{d}}_{\xi ,{\xi }^{\prime }}={\mathtt{x}}_{\xi }-{\mathtt{x}}_{{\xi }^{\prime }}$, $\mathtt{y}={\mathit{x}}_{-i}+{\mathtt{x}}_i-{\mathtt{x}}_{-1}-{\mathtt{x}}_1$ and $\mathtt{z}=-\mathtt{a}-{\mathtt{x}}_{-i}-{\mathtt{x}}_i$. We see that under the above change of variable, which reverses the order of the 1-forms, the 1-form $\mathit{a}$ does not appear at the end (see (33)). On the other hand, the only 1-forms that can appear at the beginning are

$$\begin{array}{c}\hfill dt\to i{\mathtt{d}}_{-i,i},tantdt\to \mathtt{z},(csct-cott)dt\to {\mathtt{d}}_{-1,1}-\mathtt{y}=2{\mathtt{x}}_{-1}-{\mathtt{x}}_{-i}-{\mathtt{x}}_i,\\ \hfill sectdt\to -\mathtt{a},csct(sect-1)dt\to \mathtt{y}+\mathtt{z}-{\mathtt{d}}_{-1,1}=-\mathtt{a}-2{\mathtt{x}}_{-1}.\end{array}$$

The key observation is that ${\mathtt{x}}_1$ does not appear at the beginning. Consequently, all the iterated integrals are convergent and provide the real or the imaginary part of some admissible CMZVs at level 4.

To determine exactly whether it is the real or the imaginary part, we need to count the number of $i{\mathtt{d}}_{-i,i}$’s, the only 1-form that has the imaginary coefficient i, which is produced only by the original 1-form $dt$ in the trigonometric iterated integral expression. To do this, we break into two cases, guided by the crucial observation ($★$) above. To save space, we denote by $N\left(dt\right)$ the number of 1-form $dt$ in the trigonometric iterated integral expression of the sum in (26).

(A) The starting block is ${\beta }^{★}$. When followed by another ${\beta }^{★}$-block, the trailing $tant$ can only be combined with two other forms, namely, $cottdt$ or $dt\circ tant$ (see (37)), which produces $d{t}^2$ or $dt(tantdt)\circ tant$. Repeating this until the end block if no $\alpha$-block appears, or until a transition ${\beta }^{★}$-$\alpha$ block chain emerges (36), we see that either (i) there are even number of $dt$’s without trailing $\circ tant$ or (ii) there are odd number of $dt$’s with a trailing $\circ tant$. If $\alpha$-block does not appear at all, then the end block is given by (40) ($s=s_d\ge 2$) and (42) ($s=s_d=1$) so that the parity changes in case (i) or the parity does not change in case (ii). If there is a transition ${\beta }^{★}$-$\alpha$ block chain, then there is a trailing $\circ sect$ produced while there are odd number of $dt$’s in front of $\circ sect$. By case (B) below, we see that the number of $dt$’s after this transition must be even. To summarize, we see that $N\left(dt\right)$ is always odd if the starting block is ${\beta }^{★}$.

(B) The starting block is $\alpha$. If there is no ${\beta }^{★}$-block, then clearly $N\left(dt\right)=0$ by (30), (34), (39), and (41). Suppose ${\beta }^{★}$-block does appear. By (36) and (37), we see that this block produces either

(i)

an even number of $dt$’s with a trailing $\circ sect$ when followed by an $\alpha$-block, or

(ii)

an even number of $dt$’s with a trailing $\circ tant$ when followed by a ${\beta }^{★}$-block, or

(iii)

an odd number of $dt$’s without a trailing function.

Case (i) leads to no $dt$’s until the next ${\beta }^{★}$-block appears. For the other two cases, we can argue exactly as in case (A) above and show that these repeat until the end or until a transition $\alpha$-${\beta }^{★}$ block chain (back to case (i)). Repeating the above argument in the three cases (i)–(iii), we see that until the end, we still have the same three cases. If the end block is $\alpha$, then we must be back in case (i) and by (39) and (41), this block does not produce the 1-form $dt$ so that $N\left(dt\right)$ is even. If the end the block is ${\beta }^{★}$, then it has the form $(tant)F_{s}dt$ in case (ii) and $F_{s}dt$ in case (iii). In case (ii), either two $dt$’s or no $dt$ is produced so it does not change the parity of $N\left(dt\right)$. In case (iii), one $dt$ is produced which changes $N\left(dt\right)$ to even. To summarize, we see that $N\left(dt\right)$ is always even if the starting block is $\alpha$.

Furthermore, (b) and (c) can be proved by the same proof as that for ([1] Thm. 4.2(b–d)). We may first reduce the general case to the case where if $\gamma$-block appears then it only appears as the first block, in which case we can assume the weight of this block $s\ge 2$ by Theorem 6. We now further consider two subcases: (i) the second block is an $\alpha$-block of weight b and (ii) the second block is a ${\beta }^{★}$-block of weight b. Thus, we have two kinds of iterated integrals to consider, as follows:

$$\begin{array}{cc}\hfill \left(i\right):& {\int }_0^{\pi /2}{F}_{s-1}{\displaystyle \frac{dt}{{tan}^{2}t}}tantsectdt{F}_{b-1}(csct-cott)dt\dots ,\hfill \\ \hfill \left(ii\right):& {\int }_0^{\pi /2}{F}_{s-1}{\displaystyle \frac{dt}{{tan}^{2}t}}{sec}^{2}tdt{F}_{b-1}dt\dots .\hfill \end{array}$$

The claim in the theorem then follows immediately from the computation in Examples A5 and A6. Indeed, from the examples, we see that the claims in the theorem hold when the first two blocks are given by either $\gamma$-$\alpha$-block or $\gamma$-${\beta }^{★}$-block chains. But if there are more blocks after these two, the argument in case (a) applies since these additional ones are either $\alpha$-blocks or ${\beta }^{★}$-blocks.

This completes the proof of the theorem. □

Remark 4. 

The theorem generalizes the first inclusion relation in ([29] Thm. 9.6).

Corollary 2. 

Suppose $d\in \mathbb{N}$, $s=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$. Let $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n+1$. Then, we have

$$\begin{array}{cc}\hfill \sum _{n_1>n_2\succ \dots \succ n_d\underset{d}{\succ }0}{\displaystyle \frac{a_{n_1}}{n_1^{s_1}{l}_2{\left(n_2\right)}^{s_2}\dots l_d{\left(n_d\right)}^{s_d}}}\in & {\mathsf{CMZV}}_{\left|s\right|}^4,\hfill \\ \hfill \sum _{n_1\ge n_2\succ \dots \succ n_d\underset{d}{\succ }0}{\displaystyle \frac{a_{n_1}}{{(2n_1+1)}^{s_1}{l}_2{\left(n_2\right)}^{s_2}\dots l_d{\left(n_d\right)}^{s_d}}}\in & i{\mathsf{CMZV}}_{\left|s\right|}^4,\hfill \\ \hfill \sum _{n_1\succ \dots \succ n_d\underset{d}{\succ }0}{\displaystyle \frac{a_{n_1}}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}\in & {\mathsf{CMZV}}_{\left|s\right|}^4\otimes \mathbb{Q}\left[i\right],\hfill \end{array}$$

(44)

where $\underset{d}{\succ }$ is “≥” or “>” depending on whether $l_d\left(n\right)=2n+1$ or $2n$, and all other ≻’s are either “≥” or “>”, as long as the series are defined.

Proof. 

The proof easily follows from the fact that using the Principle of Inclusion and Exclusion, we may convert ≥ to > (and vice versa) using the partial fractions. □

Remark 5. 

Since the proofs of Theorem 8 and Corollary 2 are both constructive, we see that every sum of the form (26), or more generally of the form (44), can be computed exactly in terms of the CMZVs at level 4.

We summarize the results of Theorems 6 and 8 and Corollary 2 in Table 1. Set

$$\begin{array}{cc}\hfill \mathrm{\Xi }(\mathit{l};\mathit{s};\mathrm{Rel}):=& \sum _{n_1\succ \dots \succ n_d\succ 0}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2n_1}{n_1}\right)}{4^{n_1}{l}_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}},\hfill \end{array}$$

where $\mathrm{Rel}=\{\succ ,\dots ,\succ \}$ represents any fixed choice of all the relations “≻” which can be either “≥” or “>”, provided the series is defined. Also, put ${\mathit{l}}^{\prime }=(l_2,\dots ,l_d)$ and $s^{\prime }=(s_2,\dots ,s_d)$.

Table 1. Summary of Theorem 8, where $\alpha /\beta$ means either $\alpha$ or $\beta$.

$\mathit{b}\mathit{t}({\mathit{l}}_1,\dots ,{\mathit{l}}_{\mathit{d}})$	$\mathbf{Rel}$	$\mathbf{\Xi }(\mathit{l};\mathit{s};\mathbf{Rel})\in$
$(\alpha ,{\{\alpha /\beta \}}^{d-1})$	$\{>,\succ ,\dots ,\succ ,\underset{d}{\succ }\}$	${\mathsf{CMZV}}_{\left|s\right|}^4$
$(\beta ,{\{\alpha /\beta \}}^{d-1})$	$\{\ge ,\succ ,\dots ,\succ ,\underset{d}{\succ }\}$	$i{\mathsf{CMZV}}_{\left|s\right|}^4$
$\left({\{\alpha /\beta \}}^d\right)$	$\{\succ ,\dots ,\succ \}$	${\mathsf{CMZV}}_{\left|s\right|}^4\otimes \mathbb{Q}\left[i\right]$
$(\gamma ,{\{\alpha /\beta /\gamma \}}^{d-1})$, ($s_1=1$)	$\{>,{\mathrm{Rel}}^{\prime }\}$	$=\mathrm{\Xi }({\mathit{l}}^{\prime };{\mathit{s}}^{\prime };{\mathrm{Rel}}^{\prime })$
$(\gamma ,{\{\alpha /\beta /\gamma \}}^{d-1})$, ($s_1\ge 2$)	$\{\underset{1}{\succ },\dots ,\underset{d}{\succ }\}$	$i{\mathsf{CMZV}}_{\left|s\right|-1}^4$
$(\alpha /\beta ,{\{\alpha /\beta /\gamma \}}^{d-1})$	$\{\succ ,\dots ,\succ \}$	${\mathsf{CMZV}}_{\le \left|s\right|}^4\otimes \mathbb{Q}\left[i\right]$
$(\gamma /\beta ,{\{\alpha /\beta /\gamma \}}^{d-1})$, ($s_1\ge 2$)	$\{\succ ,\dots ,\succ \}$	${\mathsf{CMZV}}_{\le \left|s\right|-1}^4\otimes \mathbb{Q}\left[i\right]$

Table 1. Summary of Theorem 8, where $\alpha /\beta$ means either $\alpha$ or $\beta$.

$\mathit{b}\mathit{t}({\mathit{l}}_1,\dots ,{\mathit{l}}_{\mathit{d}})$	$\mathbf{Rel}$	$\mathbf{\Xi }(\mathit{l};\mathit{s};\mathbf{Rel})\in$
$(\alpha ,{\{\alpha /\beta \}}^{d-1})$	$\{>,\succ ,\dots ,\succ ,\underset{d}{\succ }\}$	${\mathsf{CMZV}}_{\left|s\right|}^4$
$(\beta ,{\{\alpha /\beta \}}^{d-1})$	$\{\ge ,\succ ,\dots ,\succ ,\underset{d}{\succ }\}$	$i{\mathsf{CMZV}}_{\left|s\right|}^4$
$\left({\{\alpha /\beta \}}^d\right)$	$\{\succ ,\dots ,\succ \}$	${\mathsf{CMZV}}_{\left|s\right|}^4\otimes \mathbb{Q}\left[i\right]$
$(\gamma ,{\{\alpha /\beta /\gamma \}}^{d-1})$, ($s_1=1$)	$\{>,{\mathrm{Rel}}^{\prime }\}$	$=\mathrm{\Xi }({\mathit{l}}^{\prime };{\mathit{s}}^{\prime };{\mathrm{Rel}}^{\prime })$
$(\gamma ,{\{\alpha /\beta /\gamma \}}^{d-1})$, ($s_1\ge 2$)	$\{\underset{1}{\succ },\dots ,\underset{d}{\succ }\}$	$i{\mathsf{CMZV}}_{\left|s\right|-1}^4$
$(\alpha /\beta ,{\{\alpha /\beta /\gamma \}}^{d-1})$	$\{\succ ,\dots ,\succ \}$	${\mathsf{CMZV}}_{\le \left|s\right|}^4\otimes \mathbb{Q}\left[i\right]$
$(\gamma /\beta ,{\{\alpha /\beta /\gamma \}}^{d-1})$, ($s_1\ge 2$)	$\{\succ ,\dots ,\succ \}$	${\mathsf{CMZV}}_{\le \left|s\right|-1}^4\otimes \mathbb{Q}\left[i\right]$

5.3. Typical Examples of Even/Odd Variations of Apéry-Type Binomial Series

In this subsection, we will compute a few examples of the form

$$\begin{array}{cc}\hfill \mathrm{\Xi }\left(bt\right(\mathit{l});\mathit{s})=\mathrm{\Xi }(\mathit{l};\mathit{s}):=& \sum _{n_1\underset{1}{\succ }\dots \underset{d-1}{\succ }{n}_d\underset{d}{\succ }0}{\displaystyle \frac{a_{n_1}}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}},a_{n_1}={\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2n_1}{n_1}\right)}{4^{n_1}}}.\hfill \end{array}$$

We first convert these even/odd variations of Apéry-type binomial series to the CMZVs at level 4 by Theorem 8, and then we use Au’s package [30] to simplify the expressions. We also note that under the change of variable

$$\begin{array}{c}\hfill t\to {sin}^{-1}t\mathrm{followed}\mathrm{by}t\to {\displaystyle \frac{1-t^2}{1+t^2}}\end{array}$$

(45)

we have

$$\begin{array}{cc}\hfill cottdt\to {\omega }_0:={\displaystyle \frac{dt}{t}}\to \mathtt{y},& \hfill csctdt\to {\omega }_3:={\displaystyle \frac{dt}{t\sqrt{1-t^2}}}\to {\mathtt{d}}_{-1,1},\end{array}$$

(46)

$$\begin{array}{cc}\hfill dt\to {\omega }_1:={\displaystyle \frac{dt}{\sqrt{1-t^2}}}\to i{\mathtt{d}}_{-i,i},& \hfill sectcsctdt\to {\omega }_{20}:={\displaystyle \frac{dt}{t(1-t^2)}}\to \mathtt{y}+\mathtt{z},\end{array}$$

(47)

$$\begin{array}{cc}\hfill tantdt\to {\omega }_2:={\displaystyle \frac{tdt}{1-t^2}}\to \mathtt{z},& \hfill sectdt\to {\omega }_8:={\displaystyle \frac{dt}{1-t^2}}\to -\mathtt{a}.\end{array}$$

(48)

Example 3. 

For a single α-block or ${\beta }^{★}$-block, and any $s\in \mathbb{N}$, we have (39) and (40)

$$\begin{array}{c}\mathrm{\Xi }(\alpha ;s)=\sum _{n\ge 0}{\displaystyle \frac{a_n}{{\left(2n\right)}^s}}={\int }_0^{\pi /2}{F}_s(csct-cott)dt={(-1)}^s{\int }_0^1(2{\mathit{x}}_{-1}-{\mathit{x}}_{-i}-{\mathit{x}}_i){\mathit{y}}^{s-1},\hfill \\ \mathrm{\Xi }(\beta ;s)=\sum _{n\ge 0}{\displaystyle \frac{a_n}{{(2n+1)}^s}}={\int }_0^{\pi /2}{F}_{s}dt={(-1)}^{s}i{\int }_0^1{\mathit{d}}_{-i,i}{\mathit{y}}^{s-1}\hfill \end{array}$$

by (46)–(48). Here, we point out that after change of variable (45), we need to multiply ${(-1)}^w$ when reversing the interval back to $[0,1]$, where w is the weight of the value, i.e., the number of 1-forms in the iterated integral. In particular, when $s\le 3$, we have

$$\begin{array}{cccc}\hfill & \mathrm{\Xi }(\alpha ;1)=log2,\hfill & \mathrm{\Xi }(\alpha ;2)={\displaystyle \frac{{\pi }^2-12{log}^{2}2}{24}},\hfill & \mathrm{\Xi }(\alpha ;3)={\displaystyle \frac{4{log}^{3}2-{\pi }^{2}log2+6\zeta \left(3\right)}{24}},\hfill \end{array}$$

(49)

$$\begin{array}{cccccc}& \mathrm{\Xi }(\beta ;1)={\displaystyle \frac{\pi }{2}},\hfill & & \mathrm{\Xi }(\beta ;2)={\displaystyle \frac{\pi log2}{2}},\hfill & & \mathrm{\Xi }(\beta ;3)={\displaystyle \frac{\pi }{48}}\left({\pi }^2+12{log}^{2}2\right).\hfill \end{array}$$

(50)

Example 4. 

For a double ${\beta }^{★}$-block, for any $s\in \mathbb{N}$ we have, by (37) and (40)

$$\begin{array}{cc}\hfill \mathrm{\Xi }(\beta ,\beta ;s,1)=& \sum _{n_1\ge n_2\ge 0}{\displaystyle \frac{a_{n_1}}{{(2n_1+1)}^s(2n_2+1)}}\hfill \\ \hfill =& {\int }_0^{\pi /2}{F}_s(cottdtdt+dttantdt)={(-1)}^{s-1}i{\int }_0^1({\mathit{d}}_{-i,i}\mathit{y}+\mathit{z}{\mathit{d}}_{-i,i}){\mathit{y}}^{s-1}.\hfill \end{array}$$

When $s\le 3$, we obtain

$$\begin{array}{cc}\hfill \mathrm{\Xi }(\beta ,\beta ;1,1)=& \pi log2,\hfill \\ \hfill \mathrm{\Xi }(\beta ,\beta ;2,1)=& {\displaystyle \frac{3\pi {log}^{2}2}{4}},\hfill \\ \hfill \mathrm{\Xi }(\beta ,\beta ;3,1)=& {\displaystyle \frac{\pi }{48}}\left({\pi }^{2}log2+16{log}^{3}2+3\zeta \left(3\right)\right).\hfill \end{array}$$

Example 5. 

We now consider the series given by a ${\beta }^{★}$-α-block chain. For any $s\in \mathbb{N}$, we obtain

$$\begin{array}{cc}\hfill \mathrm{\Xi }(\beta ,\alpha ;s,1)=& \sum _{n_1\ge n_2>0}{\displaystyle \frac{a_{n_1}}{{(2n_1+1)}^s\left(2n_2\right)}}={\int }_0^{\pi /2}{F}_{s}dt\circ sect\sum _{n_2>0}{\displaystyle \frac{a_{n_2}(sint)}{2n_2}}\left(\mathrm{by}\left(32\right)\right)\hfill \\ \hfill =& {\int }_0^{\pi /2}{F}_{s}dt(sectcsctdt-csctdt)={(-1)}^{s}i{\int }_0^1(\mathit{a}+2{\mathit{x}}_{-1}){\mathit{d}}_{-i,i}{\mathit{y}}^{s-1}.\hfill \end{array}$$

by (41). When $s=1$, we obtain

$$\begin{array}{cc}\hfill \mathrm{\Xi }(\beta ,\alpha ;1,1)=& 2\mathrm{Im}(2{Li}_{1,1}(-1,-i)-{Li}_2\left(i\right))=2G-{\displaystyle \frac{1}{2}}\pi log2\approx 0.7431,\hfill \\ \hfill \mathrm{\Xi }(\beta ,\alpha ;2,1)=& {\displaystyle \frac{3{\pi }^3}{32}}-4\mathrm{Im}{Li}_3\left({\displaystyle \frac{1+i}{2}}\right)-{\displaystyle \frac{3\pi {log}^{2}2}{8}}\approx 0.0605,\hfill \end{array}$$

where G is Catalan’s constant defined by (3). By (49) and (50)

$$\begin{array}{cc}\hfill \sum _{n_1>n_2>0}{\displaystyle \frac{a_{n_1}}{(2n_1+1)\left(2n_2\right)}}=& \mathrm{\Xi }(\beta ,\alpha ;1,1)-\sum _{n>0}{\displaystyle \frac{a_n}{(2n+1)\left(2n\right)}}\hfill \\ \hfill =& \mathrm{\Xi }(\beta ,\alpha ;1,1)-log2+{\displaystyle \frac{\pi }{2}}-1\hfill \\ \hfill =& 2G-{\displaystyle \frac{1}{2}}\pi log2-log2+{\displaystyle \frac{1}{2}}\pi -1\approx 0.6207.\hfill \end{array}$$

Let $H_n=1+1/2+\dots +1/n$ be the nth harmonic number. By combining the above examples, we have

$$\begin{array}{c}\hfill \sum _{n\ge 0}{\displaystyle \frac{a_n{H}_{2n}}{2n+1}}=\mathrm{\Xi }(\beta ,\alpha ;1,1)+\mathrm{\Xi }(\beta ,\beta ;1,1)-\mathrm{\Xi }(\beta ;2)=2G\end{array}$$

which is consistent with a formula on ([31] p. 10). Similarly, all the four formulas on the top of ([31], p. 10) can be verified.

We have computed more examples with different types of blocks and put them in Appendix A. For the convenience of the readers, we have summarized all these examples in Table 2. Recall that the block type $\mathit{b}\mathit{t}\left(l\right)=\alpha ,\beta ,\gamma$ if $l\left(n\right)=2n,2n+1,2n-1$, respectively. Further, set $\mathit{b}\mathit{t}(l_1,\dots ,l_d)=(\mathit{b}\mathit{t}\left(l_1\right),\dots ,\mathit{b}\mathit{t}\left(l_d\right))$.

Table 2. Summary of Examples 3–5, A1–A8, where $\mathsf{V}={\mathsf{CMZV}}^4$ and ${\delta }_{c,1}$ is the Kronecker symbol.

$\mathit{b}\mathit{t}\left(\mathit{l}\right)$	s	$\mathbf{\Xi }(\mathit{l};\mathit{s})\in$	Example #
$(\alpha /\beta )$	s	${\mathsf{V}}_s$	3
$({\beta }^{★},{\beta }^{★})$	$(s,1)$	$i{\mathsf{V}}_{s+1}$	4
$({\beta }^{★},\alpha )$	$(s,1)$	$i{\mathsf{V}}_{s+1}$	5
$(\alpha ,{\beta }^{★})$	$(s,1)$	${\mathsf{V}}_{s+1}$	A1
$({\beta }^{★},\alpha ,\alpha )$	$(s,1,1)$	$i{\mathsf{V}}_{s+2}$	A2
$({\beta }^{★},\alpha ,{\beta }^{★})$	$(s,1,1)$	$i{\mathsf{V}}_{s+2}$	A3
$(\alpha ,\gamma )$	$(s,1)$	${\mathsf{V}}_{\le s+1}$	A4
$(\gamma ,\alpha )$	$(s,1)$	$i{\mathsf{V}}_{\le s}+{\mathsf{V}}_1$	A5
$(\gamma ,\alpha )$	$(s,b),s\ge 2,b\ge 2$	$i{\mathsf{V}}_{\le s+b-1}+{\mathsf{V}}_b$	A5
$(\gamma ,{\beta }^{★})$	$(s,1),b\ge 2$	$i{\mathsf{V}}_{\le s}+{\mathsf{V}}_1$	A6
$(\gamma ,{\beta }^{★})$	$(s,b),s\ge 2,b\ge 2$	$i{\mathsf{V}}_{\le s+b-1}$	A6
$(\gamma ,\gamma ,{\beta }^{★})$	$(s,b,c)$	$i{\mathsf{V}}_{\le s+b+c-1}+{\delta }_{c,1}{\mathsf{V}}_1$	A7
$(\gamma ,\alpha ,\gamma )$	$(s,b,c)$	${\mathsf{V}}_{\le s+b+c-1}\otimes \mathbb{Q}\left[i\right]$	A8

Table 2. Summary of Examples 3–5, A1–A8, where $\mathsf{V}={\mathsf{CMZV}}^4$ and ${\delta }_{c,1}$ is the Kronecker symbol.

$\mathit{b}\mathit{t}\left(\mathit{l}\right)$	s	$\mathbf{\Xi }(\mathit{l};\mathit{s})\in$	Example #
$(\alpha /\beta )$	s	${\mathsf{V}}_s$	3
$({\beta }^{★},{\beta }^{★})$	$(s,1)$	$i{\mathsf{V}}_{s+1}$	4
$({\beta }^{★},\alpha )$	$(s,1)$	$i{\mathsf{V}}_{s+1}$	5
$(\alpha ,{\beta }^{★})$	$(s,1)$	${\mathsf{V}}_{s+1}$	A1
$({\beta }^{★},\alpha ,\alpha )$	$(s,1,1)$	$i{\mathsf{V}}_{s+2}$	A2
$({\beta }^{★},\alpha ,{\beta }^{★})$	$(s,1,1)$	$i{\mathsf{V}}_{s+2}$	A3
$(\alpha ,\gamma )$	$(s,1)$	${\mathsf{V}}_{\le s+1}$	A4
$(\gamma ,\alpha )$	$(s,1)$	$i{\mathsf{V}}_{\le s}+{\mathsf{V}}_1$	A5
$(\gamma ,\alpha )$	$(s,b),s\ge 2,b\ge 2$	$i{\mathsf{V}}_{\le s+b-1}+{\mathsf{V}}_b$	A5
$(\gamma ,{\beta }^{★})$	$(s,1),b\ge 2$	$i{\mathsf{V}}_{\le s}+{\mathsf{V}}_1$	A6
$(\gamma ,{\beta }^{★})$	$(s,b),s\ge 2,b\ge 2$	$i{\mathsf{V}}_{\le s+b-1}$	A6
$(\gamma ,\gamma ,{\beta }^{★})$	$(s,b,c)$	$i{\mathsf{V}}_{\le s+b+c-1}+{\delta }_{c,1}{\mathsf{V}}_1$	A7
$(\gamma ,\alpha ,\gamma )$	$(s,b,c)$	${\mathsf{V}}_{\le s+b+c-1}\otimes \mathbb{Q}\left[i\right]$	A8

In general, we can apply Theorem 8 to compute even every possible variation of $\mathrm{\Xi }(\mathit{l};\mathit{s})$ in which the index ordering can be change to either strict or non-strict inequalities, in terms of the CMZVs at level 4. However, the general case needs the complicated process of using partial fractions, which is demonstrated in Example A8. Also, from Table 2, we can see clearly that the weight drops only when the leading block is a $\gamma$-block.

6. Apéry-Type Binomial Series Involving Squares of Central Binomial Coefficients

In this section, we will consider another class of Apéry-type binomial series by replacing $a_n$ by $a_n^2$ in all the series appearing in the previous sections.

Theorem 9. 

Let $d\in \mathbb{N}$, $s=(s_1,\dots ,s_d)\in {\mathbb{N}}^d$. Let $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n\pm 1$.

(a)

Let $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n+1$. Then, we have

$$\begin{array}{c}\hfill \mathrm{\Psi }(\mathit{l};\mathit{s}):=\sum _{n_1\underset{1}{\succ }\dots \underset{d-1}{\succ }{n}_d\underset{d}{\succ }0}{\displaystyle \frac{a_{n_1}^2}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}\in {\displaystyle \frac{i}{\pi }}{\mathsf{CMZV}}_{\left|s\right|+1}^4,\end{array}$$

where $\underset{j}{\succ }$ is “≥” if if $l_j\left(n\right)=2n+1$, and is “>” otherwise.

(b)

More generally, if $l_1\left(n\right),\dots ,l_d\left(n\right)=2n,2n\pm 1$, then we have

$$\begin{array}{c}\hfill \sum _{n_1\underset{1}{\succ }{n}_2\succ \dots \succ n_d\succ 0}{\displaystyle \frac{a_{n_1}^2}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}\in {\displaystyle \frac{1}{\pi }}{\mathsf{CMZV}}_{\le \left|s\right|+\eta (s_1,l)}^4\otimes \mathbb{Q}\left[i\right],\end{array}$$

where “≻” is either “≥” or “>” provided the series is defined, and

$$\eta (s_1,l)=\left\{\begin{array}{cc}0,\hfill & \mathit{if}\mathit{b}\mathit{t}\left(l_1\right)=\gamma ,s_1=1\mathit{and}d=1;\hfill \\ 0,\hfill & \mathit{if}\mathit{b}\mathit{t}\left(l_1\right)=\gamma ,s_1=1\mathit{and}\mathit{b}\mathit{t}\left(l_2\right)=\alpha ;\hfill \\ -1,\hfill & \mathit{if}\mathit{b}\mathit{t}\left(l_1\right)=\gamma ,s_1=1\mathit{and}\mathit{b}\mathit{t}\left(l_2\right)\ne \alpha ;\hfill \\ -1,\hfill & \mathit{if}\mathit{b}\mathit{t}\left(l_1\right)=\gamma ,s_1\ge 2;\hfill \\ 1,\hfill & \mathit{if}\mathit{b}\mathit{t}\left(l_1\right)\ne \gamma .\hfill \end{array}\right.$$

(51)

Proof. 

The key observation is to apply the Wallis integral

$$\begin{array}{c}\hfill {\int }_0^1{\displaystyle \frac{x^{2n}}{\sqrt{1-x^2}}}dx={\int }_0^{\pi /2}{sin}^{2n}tdt={\displaystyle \frac{1}{2}}B\left(n+{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{\pi }{2}}{a}_n.\end{array}$$

In the following proof, we drop the restriction on the summation indices to save space.

(i) When $l_1\left(n\right)=2n+1$ by (16), we see that the sum

$$\begin{array}{c}\hfill \sum {\displaystyle \frac{a_{n_1}(siny)}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}=cscy{\int }_0^y{F}_{s_1}(sinttantdt-dtdsect)\circ \prod _{j=2}^d\circ {\lambda }_{l_j,s_j}\circ \overleftarrow{a_n(sint)}.\end{array}$$

Thus, integrating over $(0,\pi /2)$ and dividing by $\pi /2$, we obtain

$$\begin{array}{c}\hfill \sum {\displaystyle \frac{a_{n_1}^2}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}={\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}csctdt{F}_{s_1}(sinttantdt-dtdsect)\prod _{j=2}^d\circ {\lambda }_{l_j,s_j}\circ \overleftarrow{a_n(sint)}.\end{array}$$

(52)

The claim follows immediately since there are odd number of $dt$’s in this case, as shown in the proof of Theorem 8(a).

(ii) If $l_1\left(n\right)=2n$, then we see that

$$\begin{array}{c}\hfill \sum {\displaystyle \frac{a_{n_1}(siny)}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}={\int }_0^y{F}_{s_1}(tantdt-csctdtdsect)\circ \prod _{j=2}^d\circ {\lambda }_{l_j,s_j}\circ \overleftarrow{a_n(sint)}.\end{array}$$

Thus, integrating over $(0,\pi /2)$ and dividing by $\pi /2$, we obtain

$$\begin{array}{c}\hfill \sum {\displaystyle \frac{a_{n_1}^2}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}={\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}dt{F}_{s_1}(tantdt-csctdtdsect)\circ \prod _{j=2}^d\circ {\lambda }_{l_j,s_j}\circ \overleftarrow{a_n(sint)}.\end{array}$$

(53)

The corollary holds as well in this case as the number of $dt$’s is changed to odd because of the leading $dt$, since originally, as shown in the proof of Theorem 8(a), the number of $dt$’s was even. The appearance of $\eta (s_1,\mathit{l})=0$ is due to the special behavior of $\alpha$-block as manifested by Example A12.

(iii) If $l_1\left(n\right)=2n-1$, then there are two cases. By (16), if $s_1=1$, then we have

$$\begin{array}{c}\hfill \sum {\displaystyle \frac{a_{n_1}(siny)}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}=cosy{\int }_0^y(dsect)\circ \prod _{j=2}^d\circ {\lambda }_{l_j,s_j}\circ \overleftarrow{a_n(sint)}.\end{array}$$

Hence, integrating over $(0,\pi /2)$ and dividing by $\pi /2$, we obtain

$$\begin{array}{cc}\hfill \sum {\displaystyle \frac{a_{n_1}^2}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}=& {\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}costdt(dsect)\circ \prod _{j=2}^d\circ {\lambda }_{l_j,s_j}\circ \overleftarrow{a_n(sint)}.\hfill \end{array}$$

If $s_1\ge 2$, then

$$\begin{array}{c}\hfill \sum {\displaystyle \frac{a_{n_1}(siny)}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}=siny{\int }_0^y{F}_{s-1}{\displaystyle \frac{dt}{{tan}^{2}t}}(dsect)\circ \prod _{j=2}^d\circ {\lambda }_{l_j,s_j}\circ \overleftarrow{a_n(sint)}.\end{array}$$

(54)

Thus, integrating over $(0,\pi /2)$ and dividing by $\pi /2$, we obtain

$$\begin{array}{c}\hfill \sum {\displaystyle \frac{a_{n_1}^2}{l_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}={\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}sintdt{F}_{s-1}{\displaystyle \frac{dt}{{tan}^{2}t}}(dsect)\circ \prod _{j=2}^d\circ {\lambda }_{l_j,s_j}\circ \overleftarrow{a_n(sint)}.\end{array}$$

(55)

To complete the proof of the theorem, we only need to pay attention to the CMZV weight increasing in (a) and weight drop phenomenon in (b) and (c) associated with a leading $\gamma$-block (i.e., $\eta \left(l_1\right)=-1$ when $\mathit{b}\mathit{t}\left(l_1\right)=\gamma$). The first phenomenon in (a) is obvious from (i) and (ii) above. By (iii), it is also easy to see the weight cannot increase. We can show that, in fact, the weight may drop by 1, and we can carry out a case-by-case study using the examples given in the next section and the Appendix. The two cases with $\eta (s_1,\mathit{l})=0$ are given by Example 8 ($\mathit{b}\mathit{t}\left(l_1\right)=\gamma$, $s_1=1$ and $d=1$), Example A13 ($\mathit{b}\mathit{t}(l_1,l_2)=(\gamma ,\alpha )$, $(s_1,s_2)=(1,s),s\ge 2$), and Example A14 ($bt(l_1,l_2)=(\gamma ,\alpha )$, $(s_1,s_2)=(1,1)$). One can quickly find these in Table 4. This concludes the proof of the theorem. □

We summarize the results of Theorem 9 in Table 3. Set

$$\begin{array}{cc}\hfill \mathrm{\Psi }(\mathit{l};\mathit{s};\mathrm{Rel}):=& \sum _{n_1\succ \dots \succ n_d\succ 0}{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2n_1}{n_1}\right)}^2}{{16}^{n_1}{l}_1{\left(n_1\right)}^{s_1}\dots l_d{\left(n_d\right)}^{s_d}}}.\hfill \end{array}$$

Table 3. Summary of Theorem 9, where $\mathsf{V}={\mathsf{CMZV}}^4$ and $\alpha /\beta$ means either $\alpha$ or $\beta$.

$\mathit{b}\mathit{t}\left(\mathit{l}\right)$	$\mathbf{Rel}$	$\mathbf{\Psi }(\mathit{l};\mathit{s};\mathbf{Rel})\in$
$\left({\{\alpha /\beta \}}^d\right)$	$\{\underset{1}{\succ },\dots ,\underset{d}{\succ }\}$	${\displaystyle \frac{i}{\pi }}{\mathsf{V}}_{\left|s\right|+1}\mathsf{V}$
$(\alpha /\beta ,{\{\alpha /\beta /\gamma \}}^{d-1})$	$\{\underset{1}{\succ },\succ ,\dots ,\succ \}$	${\displaystyle \frac{1}{\pi }}{\mathsf{V}}_{\le \left|s\right|+1}\mathsf{V}\otimes \mathbb{Q}\left[i\right]$
$(\gamma ,\beta ,{\{\alpha /\beta /\gamma \}}^{d-2})$	$\{\underset{1}{\succ },\succ ,\dots ,\succ \}$	${\displaystyle \frac{1}{\pi }}{\mathsf{V}}_{\le \left|s\right|-1}\mathsf{V}\otimes \mathbb{Q}\left[i\right]$
$(\gamma ,\varnothing ,\dots ,\varnothing )$	$\{>\}$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{\le \left|s\right|-1}+i{\mathsf{V}}_1)$
$(\gamma ,\alpha ,{\{\alpha /\beta /\gamma \}}^{d-2})$	$\{\underset{1}{\succ },\succ ,\dots ,\succ \}$	${\displaystyle \frac{1}{\pi }}{\mathsf{V}}_{\le max\left\{\right|s|-1,2\}}\mathsf{V}\otimes \mathbb{Q}\left[i\right]$

Table 3. Summary of Theorem 9, where $\mathsf{V}={\mathsf{CMZV}}^4$ and $\alpha /\beta$ means either $\alpha$ or $\beta$.

$\mathit{b}\mathit{t}\left(\mathit{l}\right)$	$\mathbf{Rel}$	$\mathbf{\Psi }(\mathit{l};\mathit{s};\mathbf{Rel})\in$
$\left({\{\alpha /\beta \}}^d\right)$	$\{\underset{1}{\succ },\dots ,\underset{d}{\succ }\}$	${\displaystyle \frac{i}{\pi }}{\mathsf{V}}_{\left|s\right|+1}\mathsf{V}$
$(\alpha /\beta ,{\{\alpha /\beta /\gamma \}}^{d-1})$	$\{\underset{1}{\succ },\succ ,\dots ,\succ \}$	${\displaystyle \frac{1}{\pi }}{\mathsf{V}}_{\le \left|s\right|+1}\mathsf{V}\otimes \mathbb{Q}\left[i\right]$
$(\gamma ,\beta ,{\{\alpha /\beta /\gamma \}}^{d-2})$	$\{\underset{1}{\succ },\succ ,\dots ,\succ \}$	${\displaystyle \frac{1}{\pi }}{\mathsf{V}}_{\le \left|s\right|-1}\mathsf{V}\otimes \mathbb{Q}\left[i\right]$
$(\gamma ,\varnothing ,\dots ,\varnothing )$	$\{>\}$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{\le \left|s\right|-1}+i{\mathsf{V}}_1)$
$(\gamma ,\alpha ,{\{\alpha /\beta /\gamma \}}^{d-2})$	$\{\underset{1}{\succ },\succ ,\dots ,\succ \}$	${\displaystyle \frac{1}{\pi }}{\mathsf{V}}_{\le max\left\{\right|s|-1,2\}}\mathsf{V}\otimes \mathbb{Q}\left[i\right]$

In the following, we present a few examples illustrating the ideas in Theorem 9. We will adopt the convention $\mathrm{\Psi }\left(\mathit{b}\mathit{t}\right(\mathit{l});\mathit{s})=\mathrm{\Psi }(\mathit{l};\mathit{s})$.

Example 6. 

We consider the series given by an α-block. From (39), (41), and (53), we have

$$\begin{array}{cc}\hfill \mathrm{\Psi }(\alpha ;s)=\sum _{n>0}{\displaystyle \frac{a_n^2}{{\left(2n\right)}^s}}=& {\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}dt{F}_s(csct-cott)dt\hfill \\ \hfill =& {(-1)}^s{\displaystyle \frac{2i}{\pi }}{\int }_0^1(2{\mathit{x}}_{-1}-{\mathit{x}}_{-i}-{\mathit{x}}_i){\mathit{y}}^{s-1}{\mathit{d}}_{-i,i}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \sum _{n>0}{\displaystyle \frac{a_n^2}{2n}}=-{\displaystyle \frac{2i}{\pi }}{\int }_0^1(2{\mathit{x}}_{-1}-{\mathit{x}}_{-i}-{\mathit{x}}_i){\mathit{d}}_{-i,i}={\displaystyle \frac{2}{\pi }}(\pi log2-2G)\approx 0.22,\end{array}$$

(56)

$$\begin{array}{cc}\hfill \sum _{n>0}{\displaystyle \frac{a_n^2}{{\left(2n\right)}^2}}=& {\displaystyle \frac{2i}{\pi }}{\int }_0^1(2{\mathit{x}}_{-1}-{\mathit{x}}_{-i}-{\mathit{x}}_i)\mathit{y}{\mathit{d}}_{-i,i}\hfill \\ \hfill =& {\displaystyle \frac{2}{\pi }}\left({\displaystyle \frac{3}{16}}{\pi }^3-8\mathrm{Im}{Li}_3\left({\displaystyle \frac{1+i}{2}}\right)-{\displaystyle \frac{3}{4}}\pi {log}^{2}2\right)\approx 0.077,\hfill \\ \hfill \sum _{n>0}{\displaystyle \frac{a_n^2}{{\left(2n\right)}^3}}=& -{\displaystyle \frac{2i}{\pi }}{\int }_0^1(2{\mathit{x}}_{-1}-{\mathit{x}}_{-i}-{\mathit{x}}_i){\mathit{y}}^2{\mathit{d}}_{-i,i}\hfill \\ \hfill =& {\displaystyle \frac{2}{\pi }}\left(24\beta \left(4\right)-32\mathrm{Im}{Li}_4\left({\displaystyle \frac{1+i}{2}}\right)-{\displaystyle \frac{3}{8}}{\pi }^{3}log2+{\displaystyle \frac{\pi }{4}}(2{log}^{3}2+\zeta \left(3\right))\right)\approx 0.034.\hfill \end{array}$$

(57)

Example 7. 

We consider the series given by a ${\beta }^{★}$-block. From (40), (42), and (52), we have

$$\begin{array}{cc}\hfill \mathrm{\Psi }(\beta ;s)=\sum _{n\ge 0}{\displaystyle \frac{a_n^2}{{(2n+1)}^s}}={\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}csctdt{F}_{s}dt=& {(-1)}^{s+1}{\displaystyle \frac{2i}{\pi }}{\int }_0^1{\mathit{d}}_{-i,i}{\mathit{y}}^{s-1}{\mathit{d}}_{-1,1}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \sum _{n\ge 0}{\displaystyle \frac{a_n^2}{2n+1}}=& {\displaystyle \frac{2i}{\pi }}{\int }_0^1{\mathit{d}}_{-i,i}{\mathit{d}}_{-1,1}={\displaystyle \frac{4G}{\pi }}\approx 1.1662,\hfill \\ \hfill \sum _{n\ge 0}{\displaystyle \frac{a_n^2}{{(2n+1)}^2}}=& -{\displaystyle \frac{2i}{\pi }}{\int }_0^1{\mathit{d}}_{-i,i}\mathit{y}{\mathit{d}}_{-1,1}={\displaystyle \frac{2}{\pi }}\left({\displaystyle \frac{3}{16}}{\pi }^3-8\mathrm{Im}{Li}_3\left({\displaystyle \frac{1+i}{2}}\right)+{\displaystyle \frac{\pi }{4}}{log}^{2}2\right)\approx 1.0379,\hfill \\ \hfill \sum _{n\ge 0}{\displaystyle \frac{a_n^2}{{(2n+1)}^3}}=& {\displaystyle \frac{2i}{\pi }}{\int }_0^1{\mathit{d}}_{-i,i}{\mathit{y}}^2{\mathit{d}}_{-1,1}\hfill \\ \hfill =& {\displaystyle \frac{2}{\pi }}\left(\mathrm{Im}{Li}_4\left({\displaystyle \frac{1+i}{2}}\right)-24\beta \left(4\right)+{\displaystyle \frac{3}{8}}{\pi }^{3}log2+{\displaystyle \frac{\pi }{6}}{log}^{3}2\right)\approx 1.01.\hfill \end{array}$$

Example 8. 

We consider the series given by a single γ-block. From (9), we have

$$\begin{array}{cc}\hfill \mathrm{\Psi }(\gamma ;1)=\sum _{n>0}{\displaystyle \frac{a_n^2}{2n-1}}=& {\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}costdtdsect\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill =& {\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}(1-cost)dt={\displaystyle \frac{2}{\pi }}\left({\displaystyle \frac{\pi }{2}}-1\right)\approx 0.3633.\hfill \end{array}$$

(59)

Note that there is no weight drop here so it is one of the two reasons we need to introduce $\eta (s_1,l)=0$ in Theorem 9. The other reason is given in Examples A13 and A14. For all $s\ge 2$

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_s:=\mathrm{\Psi }(\gamma ;s)=& \sum _{n>0}{\displaystyle \frac{a_n^2}{{(2n-1)}^s}}={\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}sintdt{F}_{s-1}{\displaystyle \frac{dt}{{tan}^{2}t}}dsect\hfill \\ \hfill =& {\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}sintdt{F}_{s-1}({csc}^{2}t-1)dtdsect\hfill \\ \hfill =& {\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}sintdt{F}_{s-1}(d(-cott)(dsect)-dtdsect)\hfill \\ \hfill =& -{\mathrm{\Gamma }}_{s-1}+{\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}sintdt{F}_{s-1}(sectdt-dtdsect)\hfill \\ \hfill =& -{\mathrm{\Gamma }}_{s-1}+{\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}sintdt{F}_{s-1}dt=-{\mathrm{\Gamma }}_{s-1}+I_{s-1}\hfill \\ \hfill =& {\displaystyle \frac{2}{\pi }}\left({(-1)}^s{\int }_0^{\pi /2}sintdt{\displaystyle \frac{dt}{{tan}^{2}t}}dsect+\sum _{j=2}^{s-1}{(-1)}^{s-1-j}{I}_j\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill I_j=& {\int }_0^{\pi /2}sintdt{F}_{j}dt={\int }_0^{\pi /2}costcottdt{F}_{j-1}dt={\int }_0^{\pi /2}csctdt{F}_{j-1}dt-I_{j-1}\hfill \\ \hfill =& {(-1)}^{j-1}{\int }_0^{\pi /2}sintdtdt+\sum _{k=1}^{j-1}{(-1)}^{j-1-k}{\int }_0^{\pi /2}csctdt{F}_{k}dt,\hfill \\ \hfill =& {(-1)}^{j-1}+\sum _{k=1}^{j-1}{(-1)}^{j}i{\int }_0^1{\mathit{d}}_{-i,i}{\mathit{y}}^{k-1}{\mathit{d}}_{-1,1}.\hfill \end{array}$$

Hence, for all $s\ge 2$, we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_s=\sum _{n>0}{\displaystyle \frac{a_n^2}{{(2n-1)}^s}}=& {(-1)}^s{\displaystyle \frac{2}{\pi }}\left(s-{\displaystyle \frac{\pi }{2}}-i\sum _{k=2}^{s-1}(s-k){\int }_0^1{\mathit{d}}_{-i,i}{\mathit{y}}^{k-2}{\mathit{d}}_{-1,1}\right).\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill \sum _{n>0}{\displaystyle \frac{a_n^2}{{(2n-1)}^2}}=& {\displaystyle \frac{2}{\pi }}\left(2-{\displaystyle \frac{\pi }{2}}\right)\approx 0.2732,\hfill \\ \hfill \sum _{n>0}{\displaystyle \frac{a_n^2}{{(2n-1)}^3}}=& {\displaystyle \frac{2}{\pi }}\left({\displaystyle \frac{\pi }{2}}+2G-3\right)\approx 0.2563.\hfill \end{array}$$

We have computed more examples with different types of blocks and put them in Appendix B. For the convenience of the readers, we have summarized all these examples in Table 4.

Table 4. Summary of Examples 6–8, A9–A17, where $\mathsf{V}={\mathsf{CMZV}}^4$.

$\mathit{b}\mathit{t}(\mathit{l})$	s	$\mathbf{\Psi }(\mathit{l};\mathit{s})\in$	Example #
$(\alpha )$	$(s)$	${\displaystyle \frac{i}{\pi }}{\mathsf{V}}_{s+1}$	6
$({\beta }^{★})$	$(s)$	${\displaystyle \frac{i}{\pi }}{\mathsf{V}}_{s+1}$	7
$(\gamma )$	$(s)$, $s\ge 2$	${\displaystyle \frac{1}{\pi }}(\mathbb{Q}+i{\mathsf{V}}_{<s})$	8
$({\left\{\gamma \right\}}^d)$	$({\left\{1\right\}}^d)$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{<d}+i{\mathsf{V}}_1)$	8 ($d=1$), A9
$(\gamma ,\gamma )$	$(s,1)$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{\le 1}+i{\mathsf{V}}_{\le s})$	A10
$({\left\{\gamma \right\}}^d)$	$(2,{\left\{1\right\}}^{d-1})$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{<d}+i{\mathsf{V}}_1)$	A11
$(\gamma ,\alpha )$	$(2,1)$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_1+i{\mathsf{V}}_{\le 2})$	A12
$({\left\{\gamma \right\}}^d,\alpha )$	$({\left\{1\right\}}^d,s)$, $s\ge 2$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{<s+d}+i{\mathsf{V}}_{s,s+1})$	A13
$({\left\{\gamma \right\}}^d,\alpha )$	$({\left\{1\right\}}^{d+1})$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{\le d}+i{\mathsf{V}}_{\le 2})$	A14
$(\gamma ,\alpha )$	$(s,2)$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_2+i{\mathsf{V}}_{\le s+1})$	A15
$(\gamma ,\alpha ,\alpha )$	$(s,1,1)$, $s\ge 2$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_2+i{\mathsf{V}}_{\le s+1})$	A16
$({\left\{\gamma \right\}}^d,\beta )$	$({\left\{1\right\}}^{d+1})$	${\displaystyle \frac{1}{\pi }}{\mathsf{V}}_{\le d}$	A17

Table 4. Summary of Examples 6–8, A9–A17, where $\mathsf{V}={\mathsf{CMZV}}^4$.

$\mathit{b}\mathit{t}(\mathit{l})$	s	$\mathbf{\Psi }(\mathit{l};\mathit{s})\in$	Example #
$(\alpha )$	$(s)$	${\displaystyle \frac{i}{\pi }}{\mathsf{V}}_{s+1}$	6
$({\beta }^{★})$	$(s)$	${\displaystyle \frac{i}{\pi }}{\mathsf{V}}_{s+1}$	7
$(\gamma )$	$(s)$, $s\ge 2$	${\displaystyle \frac{1}{\pi }}(\mathbb{Q}+i{\mathsf{V}}_{<s})$	8
$({\left\{\gamma \right\}}^d)$	$({\left\{1\right\}}^d)$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{<d}+i{\mathsf{V}}_1)$	8 ($d=1$), A9
$(\gamma ,\gamma )$	$(s,1)$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{\le 1}+i{\mathsf{V}}_{\le s})$	A10
$({\left\{\gamma \right\}}^d)$	$(2,{\left\{1\right\}}^{d-1})$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{<d}+i{\mathsf{V}}_1)$	A11
$(\gamma ,\alpha )$	$(2,1)$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_1+i{\mathsf{V}}_{\le 2})$	A12
$({\left\{\gamma \right\}}^d,\alpha )$	$({\left\{1\right\}}^d,s)$, $s\ge 2$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{<s+d}+i{\mathsf{V}}_{s,s+1})$	A13
$({\left\{\gamma \right\}}^d,\alpha )$	$({\left\{1\right\}}^{d+1})$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_{\le d}+i{\mathsf{V}}_{\le 2})$	A14
$(\gamma ,\alpha )$	$(s,2)$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_2+i{\mathsf{V}}_{\le s+1})$	A15
$(\gamma ,\alpha ,\alpha )$	$(s,1,1)$, $s\ge 2$	${\displaystyle \frac{1}{\pi }}({\mathsf{V}}_2+i{\mathsf{V}}_{\le s+1})$	A16
$({\left\{\gamma \right\}}^d,\beta )$	$({\left\{1\right\}}^{d+1})$	${\displaystyle \frac{1}{\pi }}{\mathsf{V}}_{\le d}$	A17

Corollary 3. 

We have

$$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\displaystyle \frac{a_n^2{H}_n^{\left(3\right)}}{2n-1}}={\displaystyle \frac{2}{\pi }}\left(16G+4\pi -8\pi log2-\zeta \left(3\right)-8\right).\end{array}$$

Proof. 

By partial fractions, we have

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

Thus, by (58) and (A3)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{8}}\sum _{n=1}^{\infty }{\displaystyle \frac{a_n^2{H}_n^{\left(3\right)}}{2n-1}}=& \sum _{n=1}^{\infty }{\displaystyle \frac{a_n^2}{(2n-1){\left(2n\right)}^3}}+\sum _{n>m>0}{\displaystyle \frac{a_n^2}{(2n-1){\left(2m\right)}^3}}\hfill \\ \hfill =& {\displaystyle \frac{2}{\pi }}\left({\displaystyle \frac{\pi }{2}}-1\right)+V_1\left(3\right)-\sum _{n=1}^{\infty }{\displaystyle \frac{a_n^2}{{\left(2n\right)}^3}}-\sum _{n=1}^{\infty }{\displaystyle \frac{a_n^2}{{\left(2n\right)}^2}}-\sum _{n=1}^{\infty }{\displaystyle \frac{a_n^2}{2n}}.\hfill \end{array}$$

Now, the last three sums are given in Example 6. The corollary follows immediately. □

7. A Corollary and Some Related Sums from Other Works

In this last section, we first answer affirmatively a few questions we posted at the end of [29]. For $\mathit{k}\in {\mathbb{N}}^d$ and $\mathit{l}\in {\mathbb{N}}^e$, we define

$$\begin{array}{cc}\hfill {\zeta }_n\left(\mathit{k}\right):=& \sum _{n\ge m_1>\dots >m_d>0}{\displaystyle \frac{1}{m_1^{k_1}\dots m_d^{k_d}}},\hfill \\ \hfill t_n\left(\mathit{l}\right):=& \sum _{n\ge r_1>\dots >r_e>0}{\displaystyle \frac{1}{{(2r_1-1)}^{l_1}\dots {(2r_e-1)}^{l_e}}}.\hfill \end{array}$$

Corollary 4. 

For all $m\in \mathbb{N}$, $p\in {\mathbb{N}}_{\ge 2}$, $q\in {\mathbb{N}}_{\ge 3}$, and all compositions of positive integers k and l (including the cases $\mathit{k}=\varnothing$ or $\mathit{l}=\varnothing$), we have

$$\begin{array}{cccc}\hfill \left(\mathrm{a}\right)& \sum _{n=1}^{\infty }{a}_n{\displaystyle \frac{{\zeta }_n\left(\mathit{k}\right)t_n\left(\mathit{l}\right)}{n^p}}\in {\mathsf{CMZV}}_{\left|\mathit{k}\right|+\left|\mathit{l}\right|+p}^4,\hfill & \hfill \left(\mathrm{b}\right)& \sum _{n=1}^{\infty }{a}_n^2{\displaystyle \frac{{\zeta }_n\left(\mathit{k}\right)t_n\left(\mathit{l}\right)}{n^q}}\in {\displaystyle \frac{i}{\pi }}{\mathsf{CMZV}}_{\left|\mathit{k}\right|+\left|\mathit{l}\right|+q+1}^4,\hfill \\ \hfill \left(\mathrm{c}\right)& \sum _{n=0}^{\infty }{a}_n{\displaystyle \frac{{\zeta }_n\left(\mathit{k}\right)t_n\left(\mathit{l}\right)}{{(2n+1)}^p}}\in i{\mathsf{CMZV}}_{\left|\mathit{k}\right|+\left|\mathit{l}\right|+p}^4,\hfill & \hfill \left(\mathrm{d}\right)& \sum _{n=0}^{\infty }{a}_n^2{\displaystyle \frac{{\zeta }_n\left(\mathit{k}\right)t_n\left(\mathit{l}\right)}{{(2n+1)}^q}}\in {\displaystyle \frac{i}{\pi }}{\mathsf{CMZV}}_{\left|\mathit{k}\right|+\left|\mathit{l}\right|+q+1}^4.\hfill \end{array}$$

Proof. 

Write

$$\begin{array}{c}\hfill {\zeta }_n\left(\mathit{k}\right)=\sum _{n\ge m_1>\dots >m_d>0}{\displaystyle \frac{1}{m_1^{k_1}\dots m_d^{k_d}}},t_n\left(\mathit{l}\right)=\sum _{n>r_1>\dots >r_e\ge 0}{\displaystyle \frac{1}{{(2r_1+1)}^{l_1}\dots {(2r_e+1)}^{l_e}}}.\end{array}$$

We only need to note the following facts: (i) for any summation index m for ${\zeta }_n\left(\mathit{k}\right)$ and summation index r for $t_n\left(\mathit{l}\right)$, there are only two possibilities: $m>r$ or $r\ge m$; (ii) we can re-write

$$\begin{array}{c}\hfill \sum _{n>r_1}{\displaystyle \frac{1}{{(2n+1)}^q{(2r_1+1)}^{l_1}}}=\sum _{n\ge r_1}{\displaystyle \frac{1}{{(2n+1)}^q{(2r_1+1)}^{l_1}}}-{\displaystyle \frac{1}{{(2n+1)}^{q+l_1}}}\end{array}$$

and obtain similar identities when n and $r_1$ are replaced by $r_j$ and $r_{j+1}$. Therefore, we see that (a) and (c) (resp. (b) and (d)) are special cases of Theorem 8(a) (resp. Theorem 9(a)). □

We now turn to some identities we found in the literature. For $n,k\in \mathbb{N}$, it is conventional to define the generalized harmonic numbers by

$$\begin{array}{c}\hfill H_n^{\left(k\right)}:={\zeta }_n\left(k\right)=1+{\displaystyle \frac{1}{2^k}}+{\displaystyle \frac{1}{3}}+\dots +{\displaystyle \frac{1}{n^k}}.\end{array}$$

From the examples above and those contained in Appendix B, we can derive immediately the following identity which also appeared in ([32] Thm. 2.7) and [33]:

$$\begin{array}{cc}\hfill \sum _{n>0}{\displaystyle \frac{a_n^2{H}_{2n}}{{(2n-1)}^2}}=& \sum _{n>0}{\displaystyle \frac{a_n^2}{{(2n-1)}^3}}+\sum _{n>m>0}{\displaystyle \frac{a_n^2}{{(2n-1)}^2(2m-1)}}+\sum _{n>m>0}{\displaystyle \frac{a_n^2}{{(2n-1)}^2\left(2m\right)}}\hfill \\ \hfill & +\sum _{n>0}{\displaystyle \frac{a_n^2}{{(2n-1)}^2}}-\sum _{n>0}{\displaystyle \frac{a_n^2}{(2n-1)}}+\sum _{n>0}{\displaystyle \frac{a_n^2}{\left(2n\right)}}\hfill \\ \hfill =& {\displaystyle \frac{2}{\pi }}(\left({\displaystyle \frac{\pi }{2}}+2G-3\right)+\left(3-{\displaystyle \frac{\pi }{2}}-2log2\right)+\left(2G-\pi log2+\pi -4log2\right)\hfill \\ \hfill & +\left(2-{\displaystyle \frac{\pi }{2}}\right)-\left({\displaystyle \frac{\pi }{2}}-1\right)+\left(\pi log2-2G\right))\hfill \\ \hfill =& {\displaystyle \frac{4G-12log2+6}{\pi }}.\hfill \end{array}$$

Similarly, we can also verify

$$\sum _{n>0}{\displaystyle \frac{a_n^2{H}_{2n}}{2n-1}}={\displaystyle \frac{2}{\pi }}(3log2-1)$$

([32] Thm. 2.5) and ([34] Thm. 5.15),

$$\sum _{n>0}{\displaystyle \frac{a_n^2{H}_n}{2n-1}}={\displaystyle \frac{8log2-4}{\pi }}$$

([35] Thm. 1),

$$\sum _{n>0}{\displaystyle \frac{a_n^2{H}_n}{{(2n-1)}^2}}={\displaystyle \frac{12-16log2}{\pi }}$$

([35] Thm. 2) and ([31], p. 10),

$$\sum _{n>0}{\displaystyle \frac{a_n^2(H_n^2+H_n^{\left(2\right)})}{2n-1}}={\displaystyle \frac{4\pi }{3}}-{\displaystyle \frac{32{log}^{2}2-32log2+16}{\pi }}$$

([35] Thm. 4).

For the last equation, we may use the stuffle relation

$$\begin{array}{c}\hfill H_n^2=2{\zeta }_n(1,1)+H_n^{\left(2\right)}.\end{array}$$

(60)

As a further application, we can derive ([34] Thm. 5.12) as follows. Noting that $(2n-1)a_{n-1}=2n{a}_n$ for all $n\in \mathbb{N}$, we obtain by shifting index $n\to n-1$

$$\begin{array}{cc}\hfill \sum _{n>0}{\displaystyle \frac{a_n^2}{(n+1)}}{H}_n^{\left(2\right)}=\sum _{n>k>0}{\displaystyle \frac{4n{a}_n^2}{{(2n-1)}^2{k}^2}}=& \sum _{n>k>0}{\displaystyle \frac{2a_n^2}{(2n-1)k^2}}+\sum _{n>k>0}{\displaystyle \frac{2a_n^2}{{(2n-1)}^2{k}^2}}\hfill \\ \hfill =& 8V_1\left(2\right)+8W_2={\displaystyle \frac{2}{\pi }}\left(16G+{\displaystyle \frac{{\pi }^2}{3}}-8\pi log2\right),\hfill \end{array}$$

where $V_1\left(2\right)$ and $W_2$ are given by Examples A13 and A15, respectively. Similarly, using the stuffle relation (60) and the identity

$$\begin{array}{cc}\hfill \sum _{n>0}{\displaystyle \frac{a_n^2}{(n+1)}}{\zeta }_n(1,1)=& \sum _{n>k>m>0}{\displaystyle \frac{4n{a}_n^2}{{(2n-1)}^{2}km}}\hfill \\ \hfill =& \sum _{n>k>m>0}{\displaystyle \frac{2a_n^2}{(2n-1)km}}+\sum _{n>k>m>0}{\displaystyle \frac{2a_n^2}{{(2n-1)}^{2}km}}\hfill \\ \hfill =& 8Y_1+8Y_2={\displaystyle \frac{2}{\pi }}\left(16\pi {log}^{2}2-16G-{\pi }^2+8\pi log2\right),\hfill \end{array}$$

where $Y_1$ and $Y_2$ are given by Example A16, we can confirm ([34] Thm. 5.12) immediately.

8. Conclusions and Future Plan

In this paper, by using iterated integrals, we have demonstrated that the Apéry-type binomial series

$$\begin{array}{c}\hfill \sum _{n_1>\dots >n_d>0}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2n_1}{n_1}\right)}{4^{n_1}{\left(2n_1\right)}^{s_1}\dots {\left(2n_d\right)}^{s_d}}}\mathrm{and}\sum _{n_1>\dots >n_d>0}{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2n_1}{n_1}\right)}^2}{4^{2n_1}{\left(2n_1\right)}^{s_1}\dots {\left(2n_d\right)}^{s_d}}}\end{array}$$

can be expressed as $\mathbb{Q}$-linear combinations of the real and/or imaginary parts of the colored multiple zeta values at level 4, with an extra factor of $1/\pi$ for the squared version. The same claim still holds if some or all the indices $2n_j$ are replaced by $2n_j\pm 1$ and “>” replaced by “≥” as long as the series converge.

From numerical evidence, it seems that similar results can be obtained if $4^{n_1}$ is replaced by $8^{n_1}$, ${12}^{n_1}$, or ${16}^{n_1}$; however, the level must be increased significantly. Currently, our method only shows that these series can be expressed in terms of multiple polylogarithms at suitable algebraic points. In the related work [36], we have studied the alternating versions of the series treated in this paper and those series in which the binomial coefficients appear on the denominators.

On the other hand, one can introduce more binomial factors besides the central ones as was conducted in [37,38,39,40,41,42]. It would be very interesting to find a unified way to deal with all the different types of binomial series including the alternating ones. We plan to return to this in the future.