Symmetric and Asymmetric Double Series via Expansions over Legendre Polynomials

Abstract

By examining Fourier–Legendre series pairs $\mathcal{F}\left(y\right)$ and $\mathcal{G}\left(y\right)$, fifteen classes of symmetric and asymmetric bivariate series will systematically be investigated. A compendium of 360 remarkable summation formulae will be established. They may serve as a complementary work to the paper by Chu and Campbell (The Ramanujan Journal (2023)).

1. Introduction and Outline

For $n\in {\mathbb{N}}_0$ and an indeterminate x, the shifted factorials are defined by

$${\left(x\right)}_0=1\mathrm{and}{\left(x\right)}_n=x(x+1)\cdots (x+n-1)\mathrm{for}n\in \mathbb{N}.$$

In general, for $n\in \mathbb{Z}$, we can express it as the $\Gamma$-function ratio

$${\left(x\right)}_n={\displaystyle \frac{\Gamma (x+n)}{\Gamma \left(x\right)}},\mathrm{where}\Gamma \left(x\right)={\int }_0^{\infty }{\tau }^{x-1}{e}^{-\tau }\mathrm{d}\tau \mathrm{for}\Re \left(x\right)>0.$$

According to Bailey [1], the generalized hypergeometric series reads as

$${}_{p}F_q\left[\begin{array}{c}{a}_1,\cdots ,a_p\\ c_1,\cdots ,c_q\end{array}|z\right]=\sum _{n=0}^{\infty }{\left[\begin{array}{c}{a}_1,\cdots ,a_p\\ c_1,\cdots ,c_q\end{array}\right]}_n{\displaystyle \frac{z^n}{n!}},$$

where for the sake of brevity, the factorial quotient is abbreviated to

$${\left[\begin{array}{cccc}\alpha ,& \beta ,& \cdots ,& \gamma \\ A,& B,& \cdots ,& C\end{array}\right]}_n={\displaystyle \frac{{\left(\alpha \right)}_n{\left(\beta \right)}_n\cdots {\left(\gamma \right)}_n}{{\left(A\right)}_n{\left(B\right)}_n\cdots {\left(C\right)}_n}}.$$

The Fourier–Legendre series plays important rules in mathematics, theoretical physics and applied sciences (see for example [2,3]). By integrating Fourier–Legendre series, Campbell [4] recently discovered the following elegant identity:

$${\displaystyle \frac{14\zeta \left(3\right)}{{\pi }^2}}=\sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}(1+i+j)}}.$$

Subsequently, this method was systematically developed by Chu and Campbell [5] to evaluate, in closed form, numerous double series containing binomial and multinomial coefficients. Denoting by G the usual Catalan constant (cf. [6,7]), the authors confine themselves to reproduce the following remarkable ones (where the series in the middle is symmetric with respect to i and j):

$$\begin{array}{cc}& \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}(1+j)(2+i+j)}}={\displaystyle \frac{8}{{\pi }^2}}.\hfill \\ & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}(1+i+j)}}={\displaystyle \frac{16G}{{\pi }^2}}.\hfill \\ & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}(1+i+j)}}={\displaystyle \frac{12\sqrt{3}ln2}{{\pi }^2}}.\hfill \end{array}$$

Observe that the common feature among these double series is that they are almost tensor products of two independent sums in i and j, weakly tied by linear functions of i and j in denominators. We shall investigate, in this paper, another large class of double series representations for ${\pi }^{\pm }$, consisting of two independent single sums in i and j, strongly tied by the binomial coefficient $\left({\displaystyle \genfrac{}{}{0pt}{}{i+j}{i}}\right)$ in denominators. Two typical examples are as follows:

$$\begin{array}{cc}& \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{\pi }{2}}.\hfill \\ & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{2}{\pi }}.\hfill \end{array}$$

They are two representatives of a totally new class of numerous double series identities (counting 360 exactly). In addition, it is worth highlighting that the former series possesses a perfect symmetry with respect to summation indices i and j, while the latter one is quasi symmetric. Numerous identities in the next section (see Section 2.1, Section 2.6, Section 2.10 and Section 2.13) have similar properties.

Suppose that $\mathcal{F}\left(y\right)$ and $\mathcal{G}\left(y\right)$ are two analytic functions in $[0,1]$ with power series expressions:

$$\mathcal{F}\left(y\right)=\sum _{k=0}^{\infty }{a}_k{y}^k\mathrm{and}\mathcal{G}\left(y\right)=\sum _{k=0}^{\infty }{b}_k{y}^k.$$

(1)

Then, it is not hard to check that $\mathcal{F}\left({\displaystyle \frac{1+x}{2}}\right)$ and $\mathcal{G}\left({\displaystyle \frac{1+x}{2}}\right)$ are also analytic in $[-1,1]$. Hence, we can expand these two composite functions in terms of Legendre polynomials (cf. [8,9] and Rainville [10], Chapter 10):

$$\begin{array}{cc}\mathcal{F}\left({\displaystyle \frac{1+x}{2}}\right)=\sum _{n=0}^{\infty }{A}_n{P}_n\left(x\right),\hfill & A_n=\left(n+{\displaystyle \frac{1}{2}}\right){\int }_{-1}^1\mathcal{F}\left({\displaystyle \frac{1+x}{2}}\right)P_n\left(x\right)dx,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\mathcal{G}\left({\displaystyle \frac{1+x}{2}}\right)=\sum _{n=0}^{\infty }{B}_n{P}_n\left(x\right),\hfill & B_n=\left(n+{\displaystyle \frac{1}{2}}\right){\int }_{-1}^1\mathcal{G}\left({\displaystyle \frac{1+x}{2}}\right)P_n\left(x\right)dx.\hfill \end{array}$$

(3)

Recently, Campbell and the second author discovered the following remarkable formula [5] (Theorem 11). Letting the sequences $\{a_i,b_i:i\in {\mathbb{N}}_0\}$ and $\{A_n,B_n:\in {\mathbb{N}}_0\}$ be as in (1), (2) and (3) under the given conditions of $\mathcal{F}$ and $\mathcal{G}$, we have “the universal identity” of infinite series:

$$\begin{array}{|c|}\hline {\displaystyle \sum _{i,j=0}^{\infty }}{\displaystyle \frac{a_i{b}_j}{i+j+1}}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{A_n{B}_n}{2n+1}}.\\ \hline\end{array}$$

(4)

By combining this theorem with the forty Legendre–Fourier series shown in [5] (Corollaries 6–10), numerous double infinite series were evaluated in closed form in the just-cited paper.

Now, taking into account Equation (3), there is a companion Fourier–Legendre series

$$\begin{array}{|c|}\hline \mathcal{G}\left({\displaystyle \frac{1-x}{2}}\right)={\displaystyle \sum _{n=0}^{\infty }}{(-1)}^n{B}_n{P}_n\left(x\right)\mathrm{over}-1<x<1.\\ \hline\end{array}$$

We can similarly examine the integral

$$\begin{array}{cc}\hfill {\int }_{-1}^1\mathcal{F}\left({\displaystyle \frac{1+x}{2}}\right)\mathcal{G}\left({\displaystyle \frac{1-x}{2}}\right)& =\sum _{i,j=0}^{\infty }{a}_i{b}_j{\int }_{-1}^1{\left({\displaystyle \frac{1+x}{2}}\right)}^i{\left({\displaystyle \frac{1-x}{2}}\right)}^{j}dx\hfill \\ \hfill \begin{array}{|c|}\hline x=1-2y\\ \hline\end{array}& =2\sum _{i,j=0}^{\infty }{a}_i{b}_j{\int }_0^1{y}^j{(1+y)}^{i}dy\hfill \\ & =2\sum _{i,j=0}^{\infty }{\displaystyle \frac{a_i{b}_j}{\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}.\hfill \end{array}$$

On the other hand, consider the integral of the corresponding Fourier–Legendre series

$$\begin{array}{cc}\hfill {\int }_{-1}^1\mathcal{F}\left({\displaystyle \frac{1+x}{2}}\right)\mathcal{G}\left({\displaystyle \frac{1-x}{2}}\right)& ={\int }_{-1}^{1}dx\sum _{m=0}^{\infty }{A}_m{P}_m\left(x\right)\sum _{n=0}^{\infty }{(-1)}^n{B}_n{P}_n\left(x\right)\hfill \\ & =\sum _{m,n=0}^{\infty }{(-1)}^n{A}_m{B}_n{\int }_{-1}^1{P}_m\left(x\right)P_n\left(x\right)dx\hfill \\ & =\sum _{m,n=0}^{\infty }{(-1)}^n{A}_m{B}_n{\displaystyle \frac{2\chi (m=n)}{2n+1}},\hfill \end{array}$$

where $\chi$ stands for the logical function with $\chi \left(\mathrm{true}\right)=1$ and $\chi \left(\mathrm{false}\right)=0$. Therefore, we have established another general double series identity which serves as a counterpart of (4):

$$\begin{array}{|c|}\hline {\displaystyle \sum _{i,j=0}^{\infty }}{\displaystyle \frac{a_i{b}_j}{\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \sum _{n=0}^{\infty }}{(-1)}^n{\displaystyle \frac{A_n{B}_n}{2n+1}}.\\ \hline\end{array}$$

(5)

As the reader will observe in the sequel, this “universal formula” serves as a framework for a very large class of new double series, as illustrated by the preceding representatives. In particular, when $\mathcal{F}\left(y\right)=\mathcal{G}\left(y\right)$, the above series becomes symmetric in i and j.

Analogous to what has been done in [5], by appropriately examining the Fourier–Legendre series pairs $\mathcal{F}\left(y\right)$ and $\mathcal{G}\left(y\right)$ (recorded in the Appendix A), numerous double series formulae will be established in Section 2 by employing (5). Finally, this paper will conclude with Section 3, where we shall present exceptional double series whose values involve the golden ratio.

2. Fifteen Classes of Double Series

By combining appropriately the “A, B, C, D and E” series displayed in Appendix A, we are going to derive 360 double infinite series identities in accordance with relation (5). They will be divided into fifteen classes according to the Fourier–Legendre series pairs.

2.1. Combined Products from Series in Class A

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{A}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{\pi }{2}},\hfill \end{array}$$

(6)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{A}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2j)}}={\displaystyle \frac{\pi }{4}}+{\displaystyle \frac{1}{\pi }},\hfill \end{array}$$

(7)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{A}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j){(1-2j)}^2}}={\displaystyle \frac{\pi }{4}}+{\displaystyle \frac{2}{\pi }},\hfill \end{array}$$

(8)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{A}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{2}{\pi }},\hfill \end{array}$$

(9)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{A}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right)\left(\genfrac{}{}{0pt}{}{1+2j}{j}\right)}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2j)}}={\displaystyle \frac{\pi }{8}},\hfill \end{array}$$

(10)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{A}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right)\left(\genfrac{}{}{0pt}{}{1+2j}{j}\right)(3+2j)}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2j)(3-2j)}}={\displaystyle \frac{2}{9\pi }},\hfill \end{array}$$

(11)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{A}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{\pi }{16}},\hfill \end{array}$$

(12)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{A}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)(1-2j)}}={\displaystyle \frac{\pi }{8}}+{\displaystyle \frac{1}{\pi }},\hfill \end{array}$$

(13)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{A}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i){(1-2j)}^2}}={\displaystyle \frac{5\pi }{32}}+{\displaystyle \frac{3}{2\pi }},\hfill \end{array}$$

(14)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{A}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2i)(1+j)}}={\displaystyle \frac{\pi }{16}}+{\displaystyle \frac{1}{\pi }},\hfill \end{array}$$

(15)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{A}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right)\left(\genfrac{}{}{0pt}{}{1+2j}{j}\right)}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2i)(1-2j)}}={\displaystyle \frac{\pi }{16}}+{\displaystyle \frac{1}{3\pi }},\hfill \end{array}$$

(16)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{A}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right)\left(\genfrac{}{}{0pt}{}{1+2j}{j}\right)(3+2j)}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2i)(1-2j)(3-2j)}}={\displaystyle \frac{1}{9\pi }},\hfill \end{array}$$

(17)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{A}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1-2i)(1+j)(2+j)}}={\displaystyle \frac{\pi }{32}}+{\displaystyle \frac{2}{9\pi }},\hfill \end{array}$$

(18)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{A}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j){(1-2i)}^2{(1-2j)}^2}}={\displaystyle \frac{13\pi }{64}}+{\displaystyle \frac{2}{\pi }},\hfill \end{array}$$

(19)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{A}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j){(1-2i)}^2(1+j)}}={\displaystyle \frac{3\pi }{32}}+{\displaystyle \frac{1}{\pi }},\hfill \end{array}$$

(20)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{A}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)}}={\displaystyle \frac{\pi }{16}},\hfill \end{array}$$

(21)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{A}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right)\left(\genfrac{}{}{0pt}{}{1+2j}{j}\right)}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1-2j)}}={\displaystyle \frac{4}{9\pi }},\hfill \end{array}$$

(22)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{A}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(1+j)(2+j)}}={\displaystyle \frac{4}{27\pi }},\hfill \end{array}$$

(23)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{A}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{2j}{j}\right)\left(\genfrac{}{}{0pt}{}{1+2j}{j}\right)}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1-2j)}}={\displaystyle \frac{\pi }{32}},\hfill \end{array}$$

(24)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{A}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{(3+2i)\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{2j}{j}\right)\left(\genfrac{}{}{0pt}{}{1+2j}{j}\right)}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1-2j)(3-2j)}}={\displaystyle \frac{4}{45\pi }},\hfill \end{array}$$

(25)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{A}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right){\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-2i)(1+j)(2+j)}}={\displaystyle \frac{3\pi }{256}},\hfill \end{array}$$

(26)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{A}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{(3+2i)\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{2j}{j}\right)\left(\genfrac{}{}{0pt}{}{1+2j}{j}\right)(3+2j)}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1-2j)(3-2j)}}=0,\hfill \end{array}$$

(27)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{A}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{(3+2i)\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right){\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-2i)(3-2i)(1+j)(2+j)}}={\displaystyle \frac{4}{75\pi }},\hfill \end{array}$$

(28)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{A}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}^2}{4^{2i+2j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j)(1+i)(2+i)(1+j)(2+j)}}={\displaystyle \frac{3\pi }{1024}}.\hfill \end{array}$$

(29)

2.2. Crossing Products from Series in Classes A and B

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{B}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{18(2-\sqrt{3})}{\pi }},\hfill \end{array}$$

(30)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{B}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-3j)}}={\displaystyle \frac{36(13-6\sqrt{3})}{25\pi }},\hfill \end{array}$$

(31)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{B}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(2-3j)}}={\displaystyle \frac{18(25-12\sqrt{3})}{49\pi }},\hfill \end{array}$$

(32)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{B}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{648(2+\sqrt{3})}{1225\pi }},\hfill \end{array}$$

(33)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-3j)}}={\displaystyle \frac{648(2-\sqrt{3})}{121\pi }},\hfill \end{array}$$

(34)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{B}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)(2+3j)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)(2-3j)}}={\displaystyle \frac{648(2-\sqrt{3})}{169\pi }},\hfill \end{array}$$

(35)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{B}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{46656(2-\sqrt{3})}{20449\pi }},\hfill \end{array}$$

(36)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{B}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)}}={\displaystyle \frac{18(36-17\sqrt{3})}{35\pi }},\hfill \end{array}$$

(37)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{B}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)(1-3j)}}={\displaystyle \frac{216(93-41\sqrt{3})}{1925\pi }},\hfill \end{array}$$

(38)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{B}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)(2-3j)}}={\displaystyle \frac{108(129-58\sqrt{3})}{3185\pi }},\hfill \end{array}$$

(39)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{B}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right){(1-2i)}^2(1+i+j)}}={\displaystyle \frac{18(1296-577\sqrt{3})}{1225\pi }},\hfill \end{array}$$

(40)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{B}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+i)}}={\displaystyle \frac{648(2+\sqrt{3})}{1225\pi }},\hfill \end{array}$$

(41)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{B}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)}}={\displaystyle \frac{46656(2-\sqrt{3})}{20449\pi }},\hfill \end{array}$$

(42)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1-3j)}}={\displaystyle \frac{46656(2+\sqrt{3})}{354025\pi }},\hfill \end{array}$$

(43)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{B}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)(2+3j)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)(2-3j)}}={\displaystyle \frac{46656(2+\sqrt{3})}{442225\pi }},\hfill \end{array}$$

(44)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{B}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2i)}}={\displaystyle \frac{648(2-\sqrt{3})}{143\pi }},\hfill \end{array}$$

(45)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{B}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1+j)}}={\displaystyle \frac{46656(2+\sqrt{3})}{395675\pi }},\hfill \end{array}$$

(46)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1-3j)}}={\displaystyle \frac{46656(2-\sqrt{3})}{36179\pi }},\hfill \end{array}$$

(47)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{B}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)(2+3j)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1+j)(2-3j)}}={\displaystyle \frac{46656(2-\sqrt{3})}{46475\pi }},\hfill \end{array}$$

(48)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{B}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)(3+2i)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2i)(3-2i)}}={\displaystyle \frac{648(2+\sqrt{3})}{11305\pi }},\hfill \end{array}$$

(49)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{B}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)(3+2i)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1+j)}}={\displaystyle \frac{46656(2-\sqrt{3})}{82225\pi }},\hfill \end{array}$$

(50)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)(3+2i)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1-3j)}}={\displaystyle \frac{46656(2+\sqrt{3})}{1639225\pi }},\hfill \end{array}$$

(51)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{B}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)(3+2i)(2+3j)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1+j)(2-3j)}}={\displaystyle \frac{46656(2+\sqrt{3})}{2453185\pi }},\hfill \end{array}$$

(52)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{B}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(2+i+j)(1+i)(2+i)}}={\displaystyle \frac{46656(2-\sqrt{3})}{20449\pi }},\hfill \end{array}$$

(53)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{B}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)}}={\displaystyle \frac{5038848(2+\sqrt{3})}{127803025\pi }},\hfill \end{array}$$

(54)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1-3j)}}={\displaystyle \frac{5038848(2-\sqrt{3})}{10817521\pi }},\hfill \end{array}$$

(55)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{B}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)(2+3j)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)(2-3j)}}={\displaystyle \frac{5038848(2-\sqrt{3})}{12780625\pi }},\hfill \end{array}$$

(56)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{B}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3j}{4}^{2i}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j)(1+i)(2+i)(1+j)(2+j)}}={\displaystyle \frac{725594112(2-\sqrt{3})}{6760950625\pi }}.\hfill \end{array}$$

(57)

2.3. Crossing Products from Series in Classes A and C

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{8(2-\sqrt{2})}{\pi }},\hfill \end{array}$$

(58)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{C}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-4j)}}={\displaystyle \frac{16(5-2\sqrt{2})}{9\pi }},\hfill \end{array}$$

(59)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(3-4j)}}={\displaystyle \frac{16(13-6\sqrt{2})}{75\pi }},\hfill \end{array}$$

(60)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{128(2+\sqrt{2})}{225\pi }},\hfill \end{array}$$

(61)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-4j)}}={\displaystyle \frac{128(2-\sqrt{2})}{49\pi }},\hfill \end{array}$$

(62)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)(3-4j)}}={\displaystyle \frac{128(2-\sqrt{2})}{81\pi }},\hfill \end{array}$$

(63)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{4096(2-\sqrt{2})}{3969\pi }},\hfill \end{array}$$

(64)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)}}={\displaystyle \frac{8(16-7\sqrt{2})}{15\pi }},\hfill \end{array}$$

(65)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{C}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)(1-4j)}}={\displaystyle \frac{32(52-19\sqrt{2})}{315\pi }},\hfill \end{array}$$

(66)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)(3-4j)}}={\displaystyle \frac{32(28-11\sqrt{2})}{675\pi }},\hfill \end{array}$$

(67)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2i)(1+j)}}={\displaystyle \frac{128(32-9\sqrt{2})}{1575\pi }},\hfill \end{array}$$

(68)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j){(1-2i)}^2(3-4j)}}={\displaystyle \frac{32(64-23\sqrt{2})}{1215\pi }},\hfill \end{array}$$

(69)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)}}={\displaystyle \frac{128(2+\sqrt{2})}{225\pi }},\hfill \end{array}$$

(70)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{C}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(1-4j)}}={\displaystyle \frac{256(137-44\sqrt{2})}{11025\pi }},\hfill \end{array}$$

(71)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(3-4j)}}={\displaystyle \frac{256(17-4\sqrt{2})}{6075\pi }},\hfill \end{array}$$

(72)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)}}={\displaystyle \frac{4096(2-\sqrt{2})}{3969\pi }},\hfill \end{array}$$

(73)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)}}={\displaystyle \frac{128(2-\sqrt{2})}{63\pi }},\hfill \end{array}$$

(74)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{C}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)(1-4j)}}={\displaystyle \frac{256(31-12\sqrt{2})}{3465\pi }},\hfill \end{array}$$

(75)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)(3-4j)}}={\displaystyle \frac{256(173-76\sqrt{2})}{61425\pi }},\hfill \end{array}$$

(76)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1+j)}}={\displaystyle \frac{4096(2+\sqrt{2})}{32175\pi }},\hfill \end{array}$$

(77)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+2i)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)(3-2i)}}={\displaystyle \frac{128(2+\sqrt{2})}{2145\pi }},\hfill \end{array}$$

(78)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+2i)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1+j)}}={\displaystyle \frac{4096(2-\sqrt{2})}{16065\pi }},\hfill \end{array}$$

(79)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)(3+2i)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1-4j)}}={\displaystyle \frac{4096(2+\sqrt{2})}{122265\pi }},\hfill \end{array}$$

(80)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+2i)(3+4j)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1+j)(3-4j)}}={\displaystyle \frac{4096(2+\sqrt{2})}{225225\pi }},\hfill \end{array}$$

(81)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+2i)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-2i)(3-2i)(1+j)(2+j)}}={\displaystyle \frac{65536(2+\sqrt{2})}{4279275\pi }},\hfill \end{array}$$

(82)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(3+i+j)(1+i)(2+i)}}={\displaystyle \frac{4096(2-\sqrt{2})}{3969\pi }},\hfill \end{array}$$

(83)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)}}={\displaystyle \frac{65536(2+\sqrt{2})}{1533675\pi }},\hfill \end{array}$$

(84)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1-4j)}}={\displaystyle \frac{65536(2-\sqrt{2})}{297675\pi }},\hfill \end{array}$$

(85)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)(3-4j)}}={\displaystyle \frac{65536(2-\sqrt{2})}{382347\pi }},\hfill \end{array}$$

(86)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{2i+3j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j)(1+i)(2+i)(1+j)(2+j)}}={\displaystyle \frac{4194304(2-\sqrt{2})}{86028075\pi }}.\hfill \end{array}$$

(87)

2.4. Crossing Products from Series in Classes A and D

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{9}{2\pi }},\hfill \end{array}$$

(88)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-6j)}}={\displaystyle \frac{63}{16\pi }},\hfill \end{array}$$

(89)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(5-6j)}}={\displaystyle \frac{171}{320\pi }},\hfill \end{array}$$

(90)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{243}{128\pi }},\hfill \end{array}$$

(91)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-6j)}}={\displaystyle \frac{81}{50\pi }},\hfill \end{array}$$

(92)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)(5-6j)}}={\displaystyle \frac{81}{98\pi }},\hfill \end{array}$$

(93)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{729}{1225\pi }},\hfill \end{array}$$

(94)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)}}={\displaystyle \frac{99}{32\pi }},\hfill \end{array}$$

(95)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)(1-6j)}}={\displaystyle \frac{837}{320\pi }},\hfill \end{array}$$

(96)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-2i)(5-6j)}}={\displaystyle \frac{1431}{4480\pi }},\hfill \end{array}$$

(97)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2i)(1+j)}}={\displaystyle \frac{13689}{8960\pi }},\hfill \end{array}$$

(98)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-2i)(1-6j)}}={\displaystyle \frac{16443}{12800\pi }},\hfill \end{array}$$

(99)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)(1-2i)(5-6j)}}={\displaystyle \frac{37827}{62720\pi }},\hfill \end{array}$$

(100)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}2\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1-2i)(1+j)(2+j)}}={\displaystyle \frac{1282311}{2508800\pi }},\hfill \end{array}$$

(101)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j){(1-2i)}^2}}={\displaystyle \frac{1035}{256\pi }},\hfill \end{array}$$

(102)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j){(1-2i)}^2(1-6j)}}={\displaystyle \frac{44739}{12800\pi }},\hfill \end{array}$$

(103)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j){(1-2i)}^2(5-6j)}}={\displaystyle \frac{57159}{125440\pi }},\hfill \end{array}$$

(104)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j){(1-2i)}^2(1+j)}}={\displaystyle \frac{572427}{313600\pi }},\hfill \end{array}$$

(105)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j){(1-2i)}^2(1-6j)}}={\displaystyle \frac{317763}{204800\pi }},\hfill \end{array}$$

(106)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j){(1-2i)}^2(1+j)(5-6j)}}={\displaystyle \frac{485433}{627200\pi }},\hfill \end{array}$$

(107)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}3\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j){(1-2i)}^2(1+j)(2+j)}}={\displaystyle \frac{23410377}{40140800\pi }},\hfill \end{array}$$

(108)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)}}={\displaystyle \frac{243}{128\pi }},\hfill \end{array}$$

(109)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(1-6j)}}={\displaystyle \frac{11259}{6400\pi }},\hfill \end{array}$$

(110)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(5-6j)}}={\displaystyle \frac{17091}{62720\pi }},\hfill \end{array}$$

(111)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)}}={\displaystyle \frac{729}{1225\pi }},\hfill \end{array}$$

(112)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1-6j)}}={\displaystyle \frac{2187}{4096\pi }},\hfill \end{array}$$

(113)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)(5-6j)}}={\displaystyle \frac{2187}{6400\pi }},\hfill \end{array}$$

(114)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(1+j)(2+j)}}={\displaystyle \frac{59049}{409600\pi }},\hfill \end{array}$$

(115)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)}}={\displaystyle \frac{81}{70\pi }},\hfill \end{array}$$

(116)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)(1-6j)}}={\displaystyle \frac{18873}{17920\pi }},\hfill \end{array}$$

(117)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)(5-6j)}}={\displaystyle \frac{1377}{8960\pi }},\hfill \end{array}$$

(118)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1+j)}}={\displaystyle \frac{2187}{5120\pi }},\hfill \end{array}$$

(119)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1-6j)}}={\displaystyle \frac{729}{1925\pi }},\hfill \end{array}$$

(120)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(1+j)(5-6j)}}={\displaystyle \frac{729}{3185\pi }},\hfill \end{array}$$

(121)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-2i)(1+j)(2+j)}}={\displaystyle \frac{19683}{175175\pi }},\hfill \end{array}$$

(122)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+2i)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)(3-2i)}}={\displaystyle \frac{243}{1280\pi }},\hfill \end{array}$$

(123)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+2i)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)(3-2i)(1-6j)}}={\displaystyle \frac{29079}{197120\pi }},\hfill \end{array}$$

(124)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+2i)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-2i)(3-2i)(5-6j)}}={\displaystyle \frac{2349}{232960\pi }},\hfill \end{array}$$

(125)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+2i)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1+j)}}={\displaystyle \frac{729}{5005\pi }},\hfill \end{array}$$

(126)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)(3+2i)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1-6j)}}={\displaystyle \frac{2187}{17920\pi }},\hfill \end{array}$$

(127)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+2i)(5+6j)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-2i)(3-2i)(1+j)(5-6j)}}={\displaystyle \frac{2187}{40960\pi }},\hfill \end{array}$$

(128)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+2i)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-2i)(3-2i)(1+j)(2+j)}}={\displaystyle \frac{59049}{1146880\pi }},\hfill \end{array}$$

(129)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(3+i+j)(1+i)(2+i)}}={\displaystyle \frac{729}{1225\pi }},\hfill \end{array}$$

(130)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(3+i+j)(1+i)(2+i)(1-6j)}}={\displaystyle \frac{5665059}{10035200\pi }},\hfill \end{array}$$

(131)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(3+i+j)(1+i)(2+i)(5-6j)}}={\displaystyle \frac{293787}{3136000\pi }},\hfill \end{array}$$

(132)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)}}={\displaystyle \frac{59049}{409600\pi }},\hfill \end{array}$$

(133)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1-6j)}}={\displaystyle \frac{19683}{148225\pi }},\hfill \end{array}$$

(134)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)(5-6j)}}={\displaystyle \frac{19683}{207025\pi }},\hfill \end{array}$$

(135)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j)(1+i)(2+i)(1+j)(2+j)}}={\displaystyle \frac{708588}{25050025\pi }}.\hfill \end{array}$$

(136)

2.5. Crossing Products from Series in Classes A and E

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{4^{2i}{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{25(5-\sqrt{5})}{16\pi }},\hfill \end{array}$$

(137)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}1\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{4^{2i}{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{25(3-\sqrt{5})}{4\pi }},\hfill \end{array}$$

(138)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{4^{2i}{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{625(3+\sqrt{5})}{1764\pi }},\hfill \end{array}$$

(139)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}4\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{4^{2i}{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{625(5+\sqrt{5})}{2304\pi }},\hfill \end{array}$$

(140)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right){\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{4^{2i}{(j!)}^2(1-2i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{625(5-\sqrt{5})}{1526\pi }},\hfill \end{array}$$

(141)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}5\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right){\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{4^{2i}{(j!)}^2(1-2i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{625(3-\sqrt{5})}{396\pi }},\hfill \end{array}$$

(142)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)(3+2i){\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{4^{2i}{(j!)}^2(1-2i)(3-2i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{625(3+\sqrt{5})}{18564\pi }},\hfill \end{array}$$

(143)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}6\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)\left(\genfrac{}{}{0pt}{}{1+2i}{i}\right)(3+2i){\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{4^{2i}{(j!)}^2(1-2i)(3-2i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{625(5+\sqrt{5})}{21504\pi }},\hfill \end{array}$$

(144)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{4^{2i}{(j!)}^2(1+i)(2+i)(3+i+j)\left(\genfrac{}{}{0pt}{}{2+i+j}{j}\right)}}={\displaystyle \frac{15625(5-\sqrt{5})}{73728\pi }},\hfill \end{array}$$

(145)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{A}7\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2{\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{4^{2i}{(j!)}^2(1+i)(2+i)(3+i+j)\left(\genfrac{}{}{0pt}{}{2+i+j}{j}\right)}}={\displaystyle \frac{15625(3-\sqrt{5})}{19602\pi }}.\hfill \end{array}$$

(146)

2.6. Combined Products from Series in Class B

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{B}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{3}{2}},\hfill \end{array}$$

(147)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{B}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-3j)}}={\displaystyle \frac{3}{4}}+{\displaystyle \frac{3\sqrt{3}}{4\pi }},\hfill \end{array}$$

(148)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{B}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(2-3j)}}={\displaystyle \frac{3}{8}}+{\displaystyle \frac{3\sqrt{3}}{16\pi }},\hfill \end{array}$$

(149)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{B}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{9\sqrt{3}}{8\pi }},\hfill \end{array}$$

(150)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{B}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{27}{140}},\hfill \end{array}$$

(151)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{B}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-3i)(1-3j)}}={\displaystyle \frac{9}{20}}+{\displaystyle \frac{3\sqrt{3}}{4\pi }},\hfill \end{array}$$

(152)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{B}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-3i)(2-3j)}}={\displaystyle \frac{3}{16}}+{\displaystyle \frac{9\sqrt{3}}{32\pi }},\hfill \end{array}$$

(153)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{B}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-3i)(1+j)}}={\displaystyle \frac{9}{40}}+{\displaystyle \frac{9\sqrt{3}}{16\pi }},\hfill \end{array}$$

(154)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-3i)(1-3j)}}={\displaystyle \frac{9}{40}}+{\displaystyle \frac{9\sqrt{3}}{32\pi }},\hfill \end{array}$$

(155)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{B}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1-3i)(1+j)(2+j)}}={\displaystyle \frac{27}{280}}+{\displaystyle \frac{9\sqrt{3}}{64\pi }},\hfill \end{array}$$

(156)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{B}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(2-3i)(2-3j)}}={\displaystyle \frac{9}{112}}+{\displaystyle \frac{3\sqrt{3}}{32\pi }},\hfill \end{array}$$

(157)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{B}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(2-3i)(1+j)}}={\displaystyle \frac{9}{112}}+{\displaystyle \frac{9\sqrt{3}}{32\pi }},\hfill \end{array}$$

(158)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(2-3i)(1-3j)}}={\displaystyle \frac{9}{80}}+{\displaystyle \frac{3\sqrt{3}}{32\pi }},\hfill \end{array}$$

(159)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{B}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(2-3i)(1+j)(2+j)}}={\displaystyle \frac{27}{560}}+{\displaystyle \frac{9\sqrt{3}}{160\pi }},\hfill \end{array}$$

(160)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1-3j)}}={\displaystyle \frac{9\sqrt{3}}{32\pi }},\hfill \end{array}$$

(161)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{B}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)(2+3j)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)(2-3j)}}={\displaystyle \frac{9\sqrt{3}}{40\pi }},\hfill \end{array}$$

(162)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{B}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(1+j)(2+j)}}={\displaystyle \frac{27\sqrt{3}}{320\pi }},\hfill \end{array}$$

(163)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{B}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+3j}{1+j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-3i)(1-3j)}}={\displaystyle \frac{27}{220}},\hfill \end{array}$$

(164)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{B}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-3i)(1+j)(2+j)}}={\displaystyle \frac{243}{6160}},\hfill \end{array}$$

(165)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{B}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{3j}{j,j,j}\right)}{3^{3i+3j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j)(1+i)(2+i)(1+j)(2+j)}}={\displaystyle \frac{739}{80080}}.\hfill \end{array}$$

(166)

2.7. Crossing Products from Series in Classes B and C

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{72(\sqrt{3}-\sqrt{2})}{5\pi }},\hfill \end{array}$$

(167)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{C}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-4j)}}={\displaystyle \frac{72(41\sqrt{3}-36\sqrt{2})}{385\pi }},\hfill \end{array}$$

(168)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(3-4j)}}={\displaystyle \frac{24(113\sqrt{3}-108\sqrt{2})}{1105\pi }},\hfill \end{array}$$

(169)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{10368(\sqrt{2}+\sqrt{3})}{17017\pi }},\hfill \end{array}$$

(170)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-4j)}}={\displaystyle \frac{10368(\sqrt{3}-\sqrt{2})}{2185\pi }},\hfill \end{array}$$

(171)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)(3-4j)}}={\displaystyle \frac{10368(\sqrt{3}-\sqrt{2})}{3625\pi }},\hfill \end{array}$$

(172)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{2985984(\sqrt{3}-\sqrt{2})}{1584125\pi }},\hfill \end{array}$$

(173)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-3i)}}={\displaystyle \frac{72(48\sqrt{3}-43\sqrt{2})}{455\pi }},\hfill \end{array}$$

(174)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(2-3i)}}={\displaystyle \frac{36(96\sqrt{3}-91\sqrt{2})}{935\pi }},\hfill \end{array}$$

(175)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{C}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(2-3i)(1-4j)}}={\displaystyle \frac{432(712\sqrt{3}-597\sqrt{2})}{150535\pi }},\hfill \end{array}$$

(176)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)}}={\displaystyle \frac{10368(\sqrt{2}+\sqrt{3})}{17017\pi }},\hfill \end{array}$$

(177)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)}}={\displaystyle \frac{2985984(\sqrt{3}-\sqrt{2})}{1584125\pi }},\hfill \end{array}$$

(178)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1-4j)}}={\displaystyle \frac{2985984(\sqrt{2}+\sqrt{3})}{18463445\pi }},\hfill \end{array}$$

(179)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)(3-4j)}}={\displaystyle \frac{2985984(\sqrt{2}+\sqrt{3})}{25814789\pi }},\hfill \end{array}$$

(180)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(1+j)(2+j)}}={\displaystyle \frac{1289945088(\sqrt{2}+\sqrt{3})}{28009046065\pi }},\hfill \end{array}$$

(181)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-3i)}}={\displaystyle \frac{10368(\sqrt{3}-\sqrt{2})}{2375\pi }},\hfill \end{array}$$

(182)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-3i)(1+j)}}={\displaystyle \frac{2985984(\sqrt{2}+\sqrt{3})}{19518499\pi }},\hfill \end{array}$$

(183)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-3i)(1-4j)}}={\displaystyle \frac{2985984(\sqrt{3}-\sqrt{2})}{2348875\pi }},\hfill \end{array}$$

(184)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-3i)(1+j)(3-4j)}}={\displaystyle \frac{2985984(\sqrt{3}-\sqrt{2})}{3374875\pi }},\hfill \end{array}$$

(185)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-3i)(1+j)(2+j)}}={\displaystyle \frac{1289945088(\sqrt{3}-\sqrt{2})}{3337751375\pi }},\hfill \end{array}$$

(186)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(2+3i)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(2-3i)}}={\displaystyle \frac{10368(\sqrt{3}-\sqrt{2})}{3335\pi }},\hfill \end{array}$$

(187)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(2+3i)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(2-3i)(1+j)}}={\displaystyle \frac{2985984(\sqrt{2}+\sqrt{3})}{24419395\pi }},\hfill \end{array}$$

(188)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)(2+3i)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(2-3i)(1-4j)}}={\displaystyle \frac{2985984(\sqrt{3}-\sqrt{2})}{2978155\pi }},\hfill \end{array}$$

(189)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(2+3i)(3+4j)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(2-3i)(1+j)(3-4j)}}={\displaystyle \frac{2985984(\sqrt{3}-\sqrt{2})}{4418875\pi }},\hfill \end{array}$$

(190)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(2+3i)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(2-3i)(1+j)(2+j)}}={\displaystyle \frac{1289945088(\sqrt{3}-\sqrt{2})}{3946055375\pi }},\hfill \end{array}$$

(191)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(3+i+j)(1+i)(2+i)}}={\displaystyle \frac{2985984(\sqrt{3}-\sqrt{2})}{1584125\pi }},\hfill \end{array}$$

(192)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)}}={\displaystyle \frac{1289945088(\sqrt{2}+\sqrt{3})}{28009046065\pi }},\hfill \end{array}$$

(193)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1-4j)}}={\displaystyle \frac{1289945088(\sqrt{3}-\sqrt{2})}{3201516625\pi }},\hfill \end{array}$$

(194)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)(3-4j)}}={\displaystyle \frac{1289945088(\sqrt{3}-\sqrt{2})}{4113972625\pi }},\hfill \end{array}$$

(195)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{3^{3i}{4}^{3j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j){(1+i)}_2{(1+j)}_2}}={\displaystyle \frac{743008370688(\sqrt{3}-\sqrt{2})}{8314338675125\pi }}.\hfill \end{array}$$

(196)

2.8. Crossing Products from Series in Classes B and D

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{6(\sqrt{3}-1)}{\pi }},\hfill \end{array}$$

(197)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-6j)}}={\displaystyle \frac{6(3\sqrt{3}-2)}{5\pi }},\hfill \end{array}$$

(198)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(5-6j)}}={\displaystyle \frac{2(11\sqrt{3}-10)}{35\pi }},\hfill \end{array}$$

(199)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{24(1+\sqrt{3})}{35\pi }},\hfill \end{array}$$

(200)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-6j)}}={\displaystyle \frac{24(\sqrt{3}-1)}{11\pi }},\hfill \end{array}$$

(201)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)(5-6j)}}={\displaystyle \frac{72(\sqrt{3}-1)}{65\pi }},\hfill \end{array}$$

(202)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{576(\sqrt{3}-1)}{715\pi }},\hfill \end{array}$$

(203)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-3i)}}={\displaystyle \frac{6(4\sqrt{3}-3)}{7\pi }},\hfill \end{array}$$

(204)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-3i)(1-6j)}}={\displaystyle \frac{4(22\sqrt{3}-13)}{35\pi }},\hfill \end{array}$$

(205)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-3i)(5-6j)}}={\displaystyle \frac{4(38\sqrt{3}-25)}{455\pi }},\hfill \end{array}$$

(206)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-3i)(1+j)}}={\displaystyle \frac{24(24\sqrt{3}-11)}{455\pi }},\hfill \end{array}$$

(207)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-3i)(1-6j)}}={\displaystyle \frac{24(104\sqrt{3}-71)}{1925\pi }},\hfill \end{array}$$

(208)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-3i)(1+j)(5-6j)}}={\displaystyle \frac{24(232\sqrt{3}-167)}{8645\pi }},\hfill \end{array}$$

(209)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}2\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1-3i)(1+j)(2+j)}}={\displaystyle \frac{576(404\sqrt{3}-261)}{475475\pi }},\hfill \end{array}$$

(210)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(2-3i)}}={\displaystyle \frac{8\sqrt{3}-7}{5\pi }},\hfill \end{array}$$

(211)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(2-3i)(1-6j)}}={\displaystyle \frac{2(28\sqrt{3}-17)}{55\pi }},\hfill \end{array}$$

(212)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(2-3i)(5-6j)}}={\displaystyle \frac{6(4\sqrt{3}-3)}{175\pi }},\hfill \end{array}$$

(213)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(2-3i)(1+j)}}={\displaystyle \frac{12(16\sqrt{3}-5)}{385\pi }},\hfill \end{array}$$

(214)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(2-3i)(1-6j)}}={\displaystyle \frac{12(48\sqrt{3}-37)}{935\pi }},\hfill \end{array}$$

(215)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(2-3i)(1+j)(5-6j)}}={\displaystyle \frac{12(80\sqrt{3}-67)}{3185\pi }},\hfill \end{array}$$

(216)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}3\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(2-3i)(1+j)(2+j)}}={\displaystyle \frac{288(488\sqrt{3}-345)}{595595\pi }},\hfill \end{array}$$

(217)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)}}={\displaystyle \frac{24(1+\sqrt{3})}{35\pi }},\hfill \end{array}$$

(218)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(1-6j)}}={\displaystyle \frac{24(23\sqrt{3}-12)}{385\pi }},\hfill \end{array}$$

(219)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(5-6j)}}={\displaystyle \frac{24(17\sqrt{3}-4)}{2275\pi }},\hfill \end{array}$$

(220)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)}}={\displaystyle \frac{576(\sqrt{3}-1)}{715\pi }},\hfill \end{array}$$

(221)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1-6j)}}={\displaystyle \frac{576(1+\sqrt{3})}{2975\pi }},\hfill \end{array}$$

(222)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)(5-6j)}}={\displaystyle \frac{576(1+\sqrt{3})}{4655\pi }},\hfill \end{array}$$

(223)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(1+j)(2+j)}}={\displaystyle \frac{20736(1+\sqrt{3})}{395675\pi }},\hfill \end{array}$$

(224)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-3i)}}={\displaystyle \frac{24(\sqrt{3}-1)}{13\pi }},\hfill \end{array}$$

(225)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-3i)(1-6j)}}={\displaystyle \frac{24(107\sqrt{3}-68)}{2275\pi }},\hfill \end{array}$$

(226)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-3i)(5-6j)}}={\displaystyle \frac{24(73\sqrt{3}-60)}{8645\pi }},\hfill \end{array}$$

(227)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-3i)(1+j)}}={\displaystyle \frac{576(1+\sqrt{3})}{3325\pi }},\hfill \end{array}$$

(228)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-3i)(1-6j)}}={\displaystyle \frac{576(\sqrt{3}-1)}{1001\pi }},\hfill \end{array}$$

(229)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-3i)(1+j)(5-6j)}}={\displaystyle \frac{576(\sqrt{3}-1)}{1625\pi }},\hfill \end{array}$$

(230)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-3i)(1+j)(2+j)}}={\displaystyle \frac{20736(\sqrt{3}-1)}{125125\pi }},\hfill \end{array}$$

(231)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(2+3i)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(2-3i)}}={\displaystyle \frac{72(\sqrt{3}-1)}{55\pi }},\hfill \end{array}$$

(232)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(2+3i)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(2-3i)(1-6j)}}={\displaystyle \frac{24(31\sqrt{3}-20)}{935\pi }},\hfill \end{array}$$

(233)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(2+3i)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(2-3i)(5-6j)}}={\displaystyle \frac{24(79\sqrt{3}-68)}{13475\pi }},\hfill \end{array}$$

(234)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(2+3i)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(2-3i)(1+j)}}={\displaystyle \frac{576(1+\sqrt{3})}{4165\pi }},\hfill \end{array}$$

(235)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)(2+3i)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(2-3i)(1-6j)}}={\displaystyle \frac{576(\sqrt{3}-1)}{1265\pi }},\hfill \end{array}$$

(236)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(2+3i)(5+6j)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(2-3i)(1+j)(5-6j)}}={\displaystyle \frac{192(\sqrt{3}-1)}{715\pi }},\hfill \end{array}$$

(237)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(2+3i)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(2-3i)(1+j)(2+j)}}={\displaystyle \frac{2304(\sqrt{3}-1)}{16445\pi }},\hfill \end{array}$$

(238)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(3+i+j)(1+i)(2+i)}}={\displaystyle \frac{576(\sqrt{3}-1)}{715\pi }},\hfill \end{array}$$

(239)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(3+i+j)(1+i)(2+i)(1-6j)}}={\displaystyle \frac{576(369\sqrt{3}-226)}{425425\pi }},\hfill \end{array}$$

(240)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{2+i}\right)(3+i+j)(1+i)(2+i)(5-6j)}}={\displaystyle \frac{576(537\sqrt{3}-394)}{3328325\pi }},\hfill \end{array}$$

(241)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)}}={\displaystyle \frac{20736(1+\sqrt{3})}{395675\pi }},\hfill \end{array}$$

(242)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1-6j)}}={\displaystyle \frac{20736(\sqrt{3}-1)}{115115\pi }},\hfill \end{array}$$

(243)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{2+i}\right)(4+i+j)(1+i)(2+i)(1+j)(5-6j)}}={\displaystyle \frac{2304(\sqrt{3}-1)}{17875\pi }},\hfill \end{array}$$

(244)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}7\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(7+i+j){(1+i)}_2{(1+j)}_2}}={\displaystyle \frac{110592(\sqrt{3}-1)}{2877875\pi }}.\hfill \end{array}$$

(245)

2.9. Crossing Products from Series in Classes B and E

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right){\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{3^{3i}{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{225(1+2\sqrt{3}-\sqrt{5})}{119\pi }},\hfill \end{array}$$

(246)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right){\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{3^{3i}{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{225(2\sqrt{3}-1-\sqrt{5})}{11\pi }},\hfill \end{array}$$

(247)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{3^{3i}{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{4\sqrt{6}(3\sqrt{2}+\sqrt{3}-3)}{7\pi }},\hfill \end{array}$$

(248)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}1\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{3^{3i}{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{4\sqrt{6}(3-3\sqrt{2}+\sqrt{3})}{\pi }},\hfill \end{array}$$

(249)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{3^{3i}{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{384(\sqrt{3}+\sqrt{2-\sqrt{3}})}{475\pi }},\hfill \end{array}$$

(250)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}4\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{3^{3i}{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{384(\sqrt{3}+\sqrt{2+\sqrt{3}})}{715\pi }},\hfill \end{array}$$

(251)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{3^{3i}{(j!)}^2(1-3i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{64(\sqrt{6}+6-3\sqrt{2})}{119\sqrt{3}\pi }},\hfill \end{array}$$

(252)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}5\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+3i}{1+i,i,i}\right){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{3^{3i}{(j!)}^2(1-3i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{576(\sqrt{6}-6+3\sqrt{2})}{161\sqrt{3}\pi }},\hfill \end{array}$$

(253)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)(3i+2){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{3^{3i}{(j!)}^2(1+i)(2-3i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{576(\sqrt{2}-\sqrt{6}+2\sqrt{3})}{1519\sqrt{3}\pi }},\hfill \end{array}$$

(254)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{B}6\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{3i}{i,i,i}\right)(3i+2){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{3^{3i}{(j!)}^2(1+i)(2-3i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{64(\sqrt{2}+\sqrt{6}-2\sqrt{3})}{25\pi }}.\hfill \end{array}$$

(255)

2.10. Combined Products from Series in Class C

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{C}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}=\sqrt{2},\hfill \end{array}$$

(256)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{C}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-4j)}}={\displaystyle \frac{2+\pi }{\sqrt{2}\pi }},\hfill \end{array}$$

(257)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(3-4j)}}={\displaystyle \frac{2+3\pi }{9\sqrt{2}\pi }},\hfill \end{array}$$

(258)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{4\sqrt{2}}{3\pi }},\hfill \end{array}$$

(259)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j+1}{i}\right)(2+i+j)(1-4j)}}={\displaystyle \frac{\sqrt{2}}{3}},\hfill \end{array}$$

(260)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j+1}{i}\right)(2+i+j)(1+j)(3-4j)}}={\displaystyle \frac{\sqrt{2}}{5}},\hfill \end{array}$$

(261)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j+2}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{2\sqrt{2}}{15}},\hfill \end{array}$$

(262)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{C}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-4i)(1-4j)}}={\displaystyle \frac{\sqrt{2}(3+\pi )}{3\pi }},\hfill \end{array}$$

(263)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-4i)(3-4j)}}={\displaystyle \frac{8+3\pi }{18\sqrt{2}\pi }},\hfill \end{array}$$

(264)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-4i)(1+j)}}={\displaystyle \frac{4+\pi }{3\sqrt{2}\pi }},\hfill \end{array}$$

(265)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-4i)(1-4j)}}={\displaystyle \frac{12+5\pi }{15\sqrt{2}\pi }},\hfill \end{array}$$

(266)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-4i)(1+j)(3-4j)}}={\displaystyle \frac{20+9\pi }{45\sqrt{2}\pi }},\hfill \end{array}$$

(267)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1-4i)(1+j)(2+j)}}={\displaystyle \frac{\sqrt{2}(8+3\pi )}{45\pi }},\hfill \end{array}$$

(268)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}3\right)\times \left(\mathrm{C}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(3-4i)(3-4j)}}={\displaystyle \frac{\sqrt{2}(5+3\pi )}{135\pi }},\hfill \end{array}$$

(269)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}3\right)\times \left(\mathrm{C}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(3-4i)(1+j)}}={\displaystyle \frac{20+3\pi }{45\sqrt{2}\pi }},\hfill \end{array}$$

(270)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}3\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(3-4i)(1-4j)}}={\displaystyle \frac{4+3\pi }{27\sqrt{2}\pi }},\hfill \end{array}$$

(271)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}3\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(3-4i)(3-4j)(1+j)}}={\displaystyle \frac{20+21\pi }{315\sqrt{2}\pi }},\hfill \end{array}$$

(272)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}3\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(3-4i)(1+j)(2+j)}}={\displaystyle \frac{\sqrt{2}(40+21\pi )}{945\pi }},\hfill \end{array}$$

(273)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1-4j)}}={\displaystyle \frac{16\sqrt{2}}{45\pi }},\hfill \end{array}$$

(274)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(3-4j)(1+j)}}={\displaystyle \frac{16\sqrt{2}}{63\pi }},\hfill \end{array}$$

(275)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(1+j)(2+j)}}={\displaystyle \frac{32\sqrt{2}}{315\pi }},\hfill \end{array}$$

(276)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}5\right)\times \left(\mathrm{C}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+4i}{1+i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{1+4j}{1+j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-4i)(1-4j)}}={\displaystyle \frac{2\sqrt{2}}{21}},\hfill \end{array}$$

(277)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}5\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+4i}{1+i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4j)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-4i)(3-4j)(1+j)}}={\displaystyle \frac{\sqrt{2}}{15}},\hfill \end{array}$$

(278)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}5\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+4i}{1+i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-4i)(1+j)(2+j)}}={\displaystyle \frac{\sqrt{2}}{35}},\hfill \end{array}$$

(279)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}6\right)\times \left(\mathrm{C}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4i)(3+4j)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(3-4i)(3-4j)(1+i)(1+j)}}={\displaystyle \frac{2\sqrt{2}}{45}},\hfill \end{array}$$

(280)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}6\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)(3+4i)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(3-4i)(1+j)(2+j)}}={\displaystyle \frac{\sqrt{2}}{45}},\hfill \end{array}$$

(281)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}7\right)\times \left(\mathrm{C}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{4j}{j,j,2j}\right)}{4^{3i+3j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j)(1+i)(2+i)(1+j)(2+j)}}={\displaystyle \frac{2\sqrt{2}}{315}}.\hfill \end{array}$$

(282)

2.11. Crossing Products from Series in Classes C and D

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{72(\sqrt{2}-1)}{7\pi }},\hfill \end{array}$$

(283)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-6j)}}={\displaystyle \frac{72(31\sqrt{2}-24)}{385\pi }},\hfill \end{array}$$

(284)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(5-6j)}}={\displaystyle \frac{72(127\sqrt{2}-120)}{8645\pi }},\hfill \end{array}$$

(285)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{10368(1+\sqrt{2})}{13585\pi }},\hfill \end{array}$$

(286)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-6j)}}={\displaystyle \frac{10368(\sqrt{2}-1)}{2737\pi }},\hfill \end{array}$$

(287)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)(5-6j)}}={\displaystyle \frac{10368(\sqrt{2}-1)}{5425\pi }},\hfill \end{array}$$

(288)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{2985984(\sqrt{2}-1)}{2121175\pi }},\hfill \end{array}$$

(289)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-4i)}}={\displaystyle \frac{72(36\sqrt{2}-29)}{455\pi }},\hfill \end{array}$$

(290)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-4i)(1-6j)}}={\displaystyle \frac{864(417\sqrt{2}-298)}{85085\pi }},\hfill \end{array}$$

(291)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}3\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(3-4i)}}={\displaystyle \frac{24(108\sqrt{2}-101)}{1463\pi }},\hfill \end{array}$$

(292)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}3\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(3-4i)(1-6j)}}={\displaystyle \frac{288(603\sqrt{2}-442)}{168245\pi }},\hfill \end{array}$$

(293)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}3\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(3-4i)(5-6j)}}={\displaystyle \frac{288(1467\sqrt{2}-1250)}{2947945\pi }},\hfill \end{array}$$

(294)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)}}={\displaystyle \frac{10368(1+\sqrt{2})}{13585\pi }},\hfill \end{array}$$

(295)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)}}={\displaystyle \frac{2985984(\sqrt{2}-1)}{2121175\pi }},\hfill \end{array}$$

(296)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}5\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+4i}{1+i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1-4i)}}={\displaystyle \frac{10368(\sqrt{2}-1)}{2975\pi }},\hfill \end{array}$$

(297)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}5\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+4i}{1+i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-4i)(1+j)}}={\displaystyle \frac{2985984(1+\sqrt{2})}{14576705\pi }},\hfill \end{array}$$

(298)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}5\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+4i}{1+i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-4i)(1-6j)}}={\displaystyle \frac{2985984(\sqrt{2}-1)}{2805425\pi }},\hfill \end{array}$$

(299)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}6\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+4i)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{1+i}\right)(2+i+j)(1+i)(3-4i)}}={\displaystyle \frac{10368(\sqrt{2}-1)}{4991\pi }},\hfill \end{array}$$

(300)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}6\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+4i)(5+6j)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)(3-4i)(5-6j)}}={\displaystyle \frac{2985984(\sqrt{2}-1)}{6862625\pi }},\hfill \end{array}$$

(301)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}6\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(3+4i)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(3-4i)(1+j)(2+j)}}={\displaystyle \frac{1289945088(\sqrt{2}-1)}{5483237375\pi }},\hfill \end{array}$$

(302)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}7\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3j}{4}^{3i+2j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j)(1+i)(2+i)(1+j)(2+j)}}={\displaystyle \frac{743008370688(\sqrt{2}-1)}{11015823886375\pi }}.\hfill \end{array}$$

(303)

2.12. Crossing Products from Series in Classes C and E

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{4^{3i}{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{9\sqrt{6-3\sqrt{3}}}{2\pi }},\hfill \end{array}$$

(304)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}1\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{4^{3i}{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{9\sqrt{2-\sqrt{3}}}{\pi }},\hfill \end{array}$$

(305)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{4^{3i}{(j!)}^2(1-4i)(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{9(23-5\sqrt{3})}{28\sqrt{2}\pi }},\hfill \end{array}$$

(306)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}2\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{4^{3i}{(j!)}^2(1-4i)(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{9(11\sqrt{3}-7)}{20\sqrt{2}\pi }},\hfill \end{array}$$

(307)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{E}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{1}{8}\right)}_j{\left(\frac{7}{8}\right)}_j}{4^{3i}{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{2048(\sqrt{2}+\sqrt{2-\sqrt{2}})}{2457\pi }},\hfill \end{array}$$

(308)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{E}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{3}{8}\right)}_j{\left(\frac{5}{8}\right)}_j}{4^{3i}{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{2048(\sqrt{2}+\sqrt{2+\sqrt{2}})}{3465\pi }},\hfill \end{array}$$

(309)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{4^{3i}{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{162\sqrt{2+\sqrt{3}}}{175\pi }},\hfill \end{array}$$

(310)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}4\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{4^{3i}{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{81\sqrt{6+3\sqrt{3}}}{140\pi }},\hfill \end{array}$$

(311)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}5\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+4i}{1+i,i,2i}\right){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{4^{3i}{(j!)}^2(1-4i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{81(3-\sqrt{3})}{52\sqrt{2}\pi }},\hfill \end{array}$$

(312)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}5\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+4i}{1+i,i,2i}\right){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{4^{3i}{(j!)}^2(1-4i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{162(\sqrt{3}-1)}{55\sqrt{2}\pi }},\hfill \end{array}$$

(313)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}6\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)(4i+3){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{4^{3i}{(j!)}^2(1+i)(3-4i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{81(3-\sqrt{3})}{88\sqrt{2}\pi }},\hfill \end{array}$$

(314)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}6\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right)(4i+3){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{4^{3i}{(j!)}^2(1+i)(3-4i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{162(\sqrt{3}-1)}{91\sqrt{2}\pi }},\hfill \end{array}$$

(315)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}7\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{4^{3i}{(j!)}^2(1+i)(2+i)(3+i+j)\left(\genfrac{}{}{0pt}{}{2+i+j}{j}\right)}}={\displaystyle \frac{729\sqrt{6-3\sqrt{3}}}{1144\pi }},\hfill \end{array}$$

(316)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{C}7\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{4i}{i,i,2i}\right){\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{4^{3i}{(j!)}^2(1+i)(2+i)(3+i+j)\left(\genfrac{}{}{0pt}{}{2+i+j}{j}\right)}}={\displaystyle \frac{5832(\sqrt{3}-1)}{5005\sqrt{2}\pi }}.\hfill \end{array}$$

(317)

2.13. Combined Products from Series in Class D

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{D}1\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)}}={\displaystyle \frac{3\sqrt{3}}{4}},\hfill \end{array}$$

(318)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-6j)}}={\displaystyle \frac{3(4+\sqrt{3}\pi )}{8\pi }},\hfill \end{array}$$

(319)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(5-6j)}}={\displaystyle \frac{3(4+5\sqrt{3}\pi )}{200\pi }},\hfill \end{array}$$

(320)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)}}={\displaystyle \frac{9}{5\pi }},\hfill \end{array}$$

(321)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-6j)}}={\displaystyle \frac{9\sqrt{3}}{32}},\hfill \end{array}$$

(322)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1+j)(5-6j)}}={\displaystyle \frac{9\sqrt{3}}{64}},\hfill \end{array}$$

(323)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1+j)(2+j)}}={\displaystyle \frac{27\sqrt{3}}{256}},\hfill \end{array}$$

(324)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}2\right)\times \left(\mathrm{D}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-6i)(1-6j)}}={\displaystyle \frac{3}{2\pi }}+{\displaystyle \frac{9\sqrt{3}}{32}},\hfill \end{array}$$

(325)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}2\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(1-6i)(5-6j)}}={\displaystyle \frac{9}{50\pi }}+{\displaystyle \frac{3\sqrt{3}}{80}},\hfill \end{array}$$

(326)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}2\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-6i)(1+j)}}={\displaystyle \frac{9}{10\pi }}+{\displaystyle \frac{9\sqrt{3}}{64}},\hfill \end{array}$$

(327)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}2\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-6i)(1-6j)}}={\displaystyle \frac{9}{14\pi }}+{\displaystyle \frac{9\sqrt{3}}{64}},\hfill \end{array}$$

(328)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}2\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(1-6i)(5-6j)(1+j)}}={\displaystyle \frac{3}{10\pi }}+{\displaystyle \frac{9\sqrt{3}}{128}},\hfill \end{array}$$

(329)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}2\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(1-6i)(1+j)(2+j)}}={\displaystyle \frac{9}{35\pi }}+{\displaystyle \frac{27\sqrt{3}}{512}},\hfill \end{array}$$

(330)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}3\right)\times \left(\mathrm{D}3\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(1+i+j)(5-6i)(5-6j)}}={\displaystyle \frac{3}{250\pi }}+{\displaystyle \frac{9\sqrt{3}}{1600}},\hfill \end{array}$$

(331)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}3\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(5-6i)(1+j)}}={\displaystyle \frac{9}{50\pi }}+{\displaystyle \frac{9\sqrt{3}}{640}},\hfill \end{array}$$

(332)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}3\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(5-6i)(1-6j)}}={\displaystyle \frac{3}{50\pi }}+{\displaystyle \frac{9\sqrt{3}}{320}},\hfill \end{array}$$

(333)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}3\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{1+i+j}{i}\right)(2+i+j)(5-6i)(5-6j)(1+j)}}={\displaystyle \frac{9}{550\pi }}+{\displaystyle \frac{9\sqrt{3}}{640}},\hfill \end{array}$$

(334)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}3\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{i}\right)(3+i+j)(5-6i)(1+j)(2+j)}}={\displaystyle \frac{9}{275\pi }}+{\displaystyle \frac{27\sqrt{3}}{2560}},\hfill \end{array}$$

(335)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}4\right)\times \left(\mathrm{D}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)}}={\displaystyle \frac{27\sqrt{3}}{256}},\hfill \end{array}$$

(336)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}4\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1-6j)}}={\displaystyle \frac{18}{35\pi }},\hfill \end{array}$$

(337)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}4\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1+i)(1+j)(5-6j)}}={\displaystyle \frac{18}{55\pi }},\hfill \end{array}$$

(338)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}4\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1+i)(1+j)(2+j)}}={\displaystyle \frac{54}{385\pi }},\hfill \end{array}$$

(339)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}5\right)\times \left(\mathrm{D}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+6i}{1+i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{1+6j}{1+j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-6i)(1-6j)}}={\displaystyle \frac{27\sqrt{3}}{320}},\hfill \end{array}$$

(340)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}5\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+6i}{1+i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6j)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(1-6i)(5-6j)(1+j)}}={\displaystyle \frac{27\sqrt{3}}{512}},\hfill \end{array}$$

(341)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}5\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+6i}{1+i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(1-6i)(1+j)(2+j)}}={\displaystyle \frac{243\sqrt{3}}{10240}},\hfill \end{array}$$

(342)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}6\right)\times \left(\mathrm{D}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6i)(5+6j)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{2+i+j}{1+i}\right)(3+i+j)(5-6i)(5-6j)(1+i)(1+j)}}={\displaystyle \frac{27\sqrt{3}}{896}},\hfill \end{array}$$

(343)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}6\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)(5+6i)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{3+i+j}{1+i}\right)(4+i+j)(5-6i)(1+i)(1+j)(2+j)}}={\displaystyle \frac{243\sqrt{3}}{14336}},\hfill \end{array}$$

(344)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}7\right)\times \left(\mathrm{D}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{6i}{i,2i,3i}\right)\left(\genfrac{}{}{0pt}{}{6j}{j,2j,3j}\right)}{3^{3i+3j}{4}^{2i+2j}\left(\genfrac{}{}{0pt}{}{4+i+j}{2+i}\right)(5+i+j){(1+i)}_2{(1+j)}_2}}={\displaystyle \frac{729\sqrt{3}}{143360}}.\hfill \end{array}$$

(345)

2.14. Crossing Products from Series in Classes D and E

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{6}\right)}_i{\left(\frac{5}{6}\right)}_i{\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{225(3-\sqrt{5})}{44\pi }},\hfill \end{array}$$

(346)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}1\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{6}\right)}_i{\left(\frac{5}{6}\right)}_i{\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{225(\sqrt{5}-1)}{64\pi }},\hfill \end{array}$$

(347)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}4\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{6}\right)}_i{\left(\frac{5}{6}\right)}_i{\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{50625(1+\sqrt{5})}{93184\pi }},\hfill \end{array}$$

(348)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}4\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{6}\right)}_i{\left(\frac{5}{6}\right)}_i{\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{50625(3+\sqrt{5})}{142324\pi }},\hfill \end{array}$$

(349)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}4\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{6}\right)}_i{\left(\frac{5}{6}\right)}_i{\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{{(i!)}^2{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{576(2-\sqrt{2}+\sqrt{6})}{1001\pi }},\hfill \end{array}$$

(350)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}4\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{6}\right)}_i{\left(\frac{5}{6}\right)}_i{\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{{(i!)}^2{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{192(2+\sqrt{2}+\sqrt{6})}{595\pi }},\hfill \end{array}$$

(351)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}5\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{-1}{6}\right)}_i{\left(\frac{7}{6}\right)}_i{\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{50625(3-\sqrt{5})}{25916\pi }},\hfill \end{array}$$

(352)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}5\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{-1}{6}\right)}_i{\left(\frac{7}{6}\right)}_i{\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{50625(\sqrt{5}-1)}{39424\pi }},\hfill \end{array}$$

(353)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}6\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{-5}{6}\right)}_i{\left(\frac{11}{6}\right)}_i{\left(\frac{1}{10}\right)}_j{\left(\frac{9}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{50625(3-\sqrt{5})}{52316\pi }},\hfill \end{array}$$

(354)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{D}6\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{-5}{6}\right)}_i{\left(\frac{11}{6}\right)}_i{\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i)(2+i+j)\left(\genfrac{}{}{0pt}{}{1+i+j}{j}\right)}}={\displaystyle \frac{50625(\sqrt{5}-1)}{77824\pi }}.\hfill \end{array}$$

(355)

2.15. Combined Products from Series in Class E

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}1\right)\times \left(\mathrm{E}2\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{5}\right)}_i{\left(\frac{4}{5}\right)}_i\frac{2}{5}{)}_j{\left(\frac{3}{5}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{25\sqrt{5-2\sqrt{5}}}{4\pi }},\hfill \end{array}$$

(356)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}1\right)\times \left(\mathrm{E}5\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{5}\right)}_i{\left(\frac{4}{5}\right)}_i\frac{1}{10}{)}_j{\left(\frac{9}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{25(1-\sqrt{5}+\sqrt{10-2\sqrt{5}})}{7\pi }},\hfill \end{array}$$

(357)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}2\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{2}{5}\right)}_i{\left(\frac{3}{5}\right)}_i\frac{3}{10}{)}_j{\left(\frac{7}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{25(\sqrt{10+2\sqrt{5}}-1-\sqrt{5})}{3\pi }},\hfill \end{array}$$

(358)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}3\right)\times \left(\mathrm{E}4\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{8}\right)}_i{\left(\frac{7}{8}\right)}_i{\left(\frac{3}{8}\right)}_j{\left(\frac{5}{8}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{4\sqrt{4-2\sqrt{2}}}{\pi }},\hfill \end{array}$$

(359)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}5\right)\times \left(\mathrm{E}6\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{10}\right)}_i{\left(\frac{9}{10}\right)}_i{\left(\frac{3}{10}\right)}_j{\left(\frac{7}{10}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{25}{6\pi }},\hfill \end{array}$$

(360)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}5\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{10}\right)}_i{\left(\frac{9}{10}\right)}_i{\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{900(\sqrt{2}-1+\sqrt{5}-\sqrt{6})}{49\pi }},\hfill \end{array}$$

(361)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}5\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{10}\right)}_i{\left(\frac{9}{10}\right)}_i{\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{900(1+\sqrt{2}-\sqrt{5}+\sqrt{6})}{551\pi }},\hfill \end{array}$$

(362)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}6\right)\times \left(\mathrm{E}7\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{3}{10}\right)}_i{\left(\frac{7}{10}\right)}_i{\left(\frac{1}{12}\right)}_j{\left(\frac{11}{12}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{900(1+\sqrt{2}+\sqrt{5}-\sqrt{6})}{481\pi }},\hfill \end{array}$$

(363)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}6\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{3}{10}\right)}_i{\left(\frac{5}{10}\right)}_i{\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{900(\sqrt{2}-1-\sqrt{5}+\sqrt{6})}{119\pi }},\hfill \end{array}$$

(364)

$$\begin{array}{cccc}& \boxed{\left(\mathrm{E}7\right)\times \left(\mathrm{E}8\right)}\hfill & & \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{12}\right)}_i{\left(\frac{11}{12}\right)}_i{\left(\frac{5}{12}\right)}_j{\left(\frac{7}{12}\right)}_j}{{(i!)}^2{(j!)}^2(1+i+j)\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)}}={\displaystyle \frac{3\sqrt{2}}{\pi }}.\hfill \end{array}$$

(365)

3. Concluding Comments

Besides squares of central binomial coefficients as appearing in the double series recorded in the last section, it is possible to examine other binomial expressions, as long as their generating functions can explicitly be expressed in terms of Legendre polynomials. The reader is encouraged to make further exploration.

As an example, we are going to examine some asymmetric series with their values involving golden ratio. For $\left|x\right|<1$ and $0<y<1$, consider the binomial series

$${\displaystyle \frac{1}{\sqrt{1-x(1-y^2)}}}=\sum _{j=0}^{\infty }{x}^j\left({\displaystyle \genfrac{}{}{0pt}{}{2j}{j}}\right){\displaystyle \frac{{(1-y^2)}^j}{4^j}}.$$

This function also admits the expansion in terms of Legendre polynomials

$${\displaystyle \frac{1}{\sqrt{1-\frac{1+x}{2}(1-y^2)}}}={\displaystyle \frac{2}{1+y}}\sum _{n=0}^{\infty }{\left({\displaystyle \frac{1-y}{1+y}}\right)}^n{P}_n\left(x\right).$$

(366)

By multiplying the above series with (A1) displayed in Appendix A and then applying equality (4), we find two interesting double series. The first one reads as

$$\begin{array}{cc}& \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right){(1-y^2)}^j}{4^{2i+j}(i+j+1)}}={\displaystyle \frac{8}{\pi (1+y)}}\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\frac{1-y}{1+y}\right)}^n}{{(2n+1)}^2}}\hfill \\ & ={\displaystyle \frac{4}{\pi \sqrt{1-y^2}}}\left\{{\mathrm{Li}}_2\left(\sqrt{{\displaystyle \frac{1-y}{1+y}}}\right)-{\mathrm{Li}}_2\left(-\sqrt{{\displaystyle \frac{1-y}{1+y}}}\right)\right\}.\hfill \end{array}$$

Here and forth, ${\mathrm{Li}}_2\left(x\right)$ stands for the dilogarithm function (cf. [11]) explicitly given by

$${\mathrm{Li}}_2\left(x\right):=\sum _{n=1}^{\infty }{\displaystyle \frac{x^n}{n^2}},\mathrm{where}\left|x\right|\le 1.$$

When $y=0$ and $y=1/\sqrt{5}$, we obtain two infinite series identities:

$$\begin{array}{cc}\hfill \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}{4^{2i+j}(i+j+1)}}& ={\displaystyle \frac{4}{\pi }}\left\{{\mathrm{Li}}_2\left(1\right)-{\mathrm{Li}}_2\left(-1\right)\right\}\hfill \\ & ={\displaystyle \frac{4}{\pi }}\left\{{\displaystyle \frac{{\pi }^2}{6}}+{\displaystyle \frac{{\pi }^2}{12}}\right\}=\pi ,\hfill \\ \hfill \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}{4^{2i}{5}^j(i+j+1)}}& ={\displaystyle \frac{2\sqrt{5}}{\pi }}\left\{{\mathrm{Li}}_2\left({\displaystyle \frac{\sqrt{5}-1}{2}}\right)-{\mathrm{Li}}_2\left({\displaystyle \frac{1-\sqrt{5}}{2}}\right)\right\}\hfill \\ & ={\displaystyle \frac{2\sqrt{5}}{\pi }}\left\{{\displaystyle \frac{{\pi }^2}{6}}-{\displaystyle \frac{3}{2}}{ln}^2{\displaystyle \frac{1+\sqrt{5}}{2}}\right\}\hfill \\ & ={\displaystyle \frac{\sqrt{5}}{3\pi }}\left\{{\pi }^2-9{ln}^2{\displaystyle \frac{1+\sqrt{5}}{2}}\right\};\hfill \end{array}$$

where for the last one we have invoked two known values (cf. [11])

$$\begin{array}{cc}\hfill {\mathrm{Li}}_2\left({\displaystyle \frac{\sqrt{5}-1}{2}}\right)& ={\displaystyle \frac{{\pi }^2}{10}}-{ln}^2{\displaystyle \frac{1+\sqrt{5}}{2}},\hfill \\ \hfill {\mathrm{Li}}_2\left({\displaystyle \frac{1-\sqrt{5}}{2}}\right)& ={\displaystyle \frac{1}{2}}{ln}^2{\displaystyle \frac{1+\sqrt{5}}{2}}-{\displaystyle \frac{{\pi }^2}{15}}.\hfill \end{array}$$

Alternatively, we have another double series

$$\begin{array}{cc}& \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right){(1-y^2)}^j}{4^{2i+j}\left(\genfrac{}{}{0pt}{}{i+j}{j}\right)(i+j+1)}}={\displaystyle \frac{8}{\pi (1+y)}}\sum _{n=0}^{\infty }{(-1)}^n{\displaystyle \frac{{\left(\frac{1-y}{1+y}\right)}^n}{{(2n+1)}^2}}\hfill \\ & ={\displaystyle \frac{4}{\pi \mathbf{i}\sqrt{1-y^2}}}\left\{{\mathrm{Li}}_2\left(\sqrt{{\displaystyle \frac{y-1}{1+y}}}\right)-{\mathrm{Li}}_2\left(-\sqrt{{\displaystyle \frac{y-1}{1+y}}}\right)\right\}.\hfill \end{array}$$

When $y=0$, it reduces to the identity below

$$\begin{array}{cc}& \sum _{i,j=0}^{\infty }{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{2i}{i}\right)}^2\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}{4^{2i+j}\left(\genfrac{}{}{0pt}{}{i+j}{i}\right)(i+j+1)}}={\displaystyle \frac{4}{\pi \mathbf{i}}}\left\{{\mathrm{Li}}_2\left(\mathbf{i}\right)-{\mathrm{Li}}_2\left(-\mathbf{i}\right)\right\}\hfill \\ & ={\displaystyle \frac{4}{\pi \mathbf{i}}}\left\{\left(\mathbf{i}G-{\displaystyle \frac{{\pi }^2}{48}}\right)-\left(-\mathbf{i}G-{\displaystyle \frac{{\pi }^2}{48}}\right)\right\}={\displaystyle \frac{8G}{\pi }}.\hfill \end{array}$$