Bisection Series Approach for Exotic ₃F₂(1)-Series

Abstract

By employing the bisection series approach, two classes of nonterminating ₃$F_2\left(1\right)$-series are examined. Several new summation formulae are established in closed form through the summation formulae of Gauss and Kummer for the ₂$F_1(\pm 1)$-series. They are expressed in terms of well-known functions such as $\pi$, Euler–Gamma, and logarithmic functions, which can be used in physics and applied sciences for numerical and theoretical analysis.

1. Introduction and Motivation

Following Bailey [1], the classical hypergeometric series reads as

$${}_{p+1}F_{\mathrm{p}}\left[\begin{array}{ccccc}{a}_0,& a_1,& a_2,& \cdots ,& a_p\\ & b_1,& b_2,& \cdots ,& b_p\end{array}|z\right]=\sum _{k=0}^{\infty }\frac{{\left(a_0\right)}_k{\left(a_1\right)}_k{\left(a_2\right)}_k\cdots {\left(a_p\right)}_k}{k!{\left(b_1\right)}_k{\left(b_2\right)}_k\cdots {\left(b_p\right)}_k}{z}^k,$$

where the shifted factorial is defined by ${\left(x\right)}_0=1$ and

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

The above series converges for $\left|z\right|<1$. When $z=1$, it is convergent provided that the real part of the sum of denominator parameters is greater than that of the numerator parameters.

For $n\in \mathbb{Z}$, we can also express ${\left(x\right)}_n$ in terms of the $\Gamma$-function quotient

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

Among the numerous important properties of the $\Gamma$-function (see Rainville [2] (§17)), we will make use of Euler’s reflection formula

$$\Gamma \left(x\right)\Gamma (1-x)=\frac{\pi }{sin\left(\pi x\right)},$$

and Legendre’s duplication formula

$$\Gamma \left(2x\right)=\frac{2^{2x-1}}{\sqrt{\pi }}\Gamma \left(x\right)\Gamma \left(\frac{1}{2}+x\right).$$

Recently, there has been growing interest in finding new summation formulae for hypergeometric series (cf. [3,4,5,6,7]). For the field $\mathbb{Q}$ of rational numbers, let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ (the collection of all the complex numbers that are roots of polynomials with rational coefficients) with ${\overline{\mathbb{Q}}}^{\times }$ = $\overline{\mathbb{Q}}\setminus \left\{0\right\}$. Asakura–Otsubo–Terasoma [8] examined the following nonterminating exotic series

$$\mathcal{F}(a,b,x):={}_{3}F_2\left[\begin{array}{c}a,b,x\\ a+b,1+x\end{array}|1\right],\mathrm{where}a,b,x\in \mathbb{Q},$$

and figured out linear conditions under which the value of the series $\mathcal{F}(a,b,x)$ times the beta function $B(a,b)$ belongs to the $\overline{\mathbb{Q}}$-linear subspace spanned by 1, $2\pi \mathbf{i}$ and $log\alpha$ for all $\alpha \in {\overline{\mathbb{Q}}}^{\times }$. This result characterizes the exact value of the series $\mathcal{F}(a,b,x)$ in quality, but not in quantity. Asakura–Yabu [9] provided, but without proofs, explicit formulae for $\mathcal{F}(\frac{1}{6},\frac{5}{6},x)$ when $x\in \left\{\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{6}\right\}$. Some of their contiguous series were recently evaluated by Chen [10]. The authors [11,12] recently further examined the series $\mathcal{F}(\alpha ,1-\alpha ,x)$ and their contiguous series for

$$\alpha \in \left\{\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{6},\frac{1}{5},\frac{2}{5},\frac{1}{8},\frac{3}{8},\frac{1}{10},\frac{3}{10},\frac{1}{12},\frac{5}{12}\right\},$$

and established, in particular for $x=\frac{1}{2}$, numerous closed summation formulae.

Motivated by these interesting results, the aim of this paper is to investigate the following variants of the series $\mathcal{F}(a,b,x)$ defined by

$${}_{3}F_2\left[\begin{array}{c}a,b,y\\ c,1+y\end{array}|1\right],\mathrm{where}a+b\ne c\mathrm{and}y\ne \frac{1}{2}.$$

Specifically, for $x<\frac{1}{4}$ and $y\in \mathbb{R}\setminus \mathbb{Z}$, we are going to examine the following two classes of infinite series:

$$\begin{array}{cc}\hfill \mathsf{\Phi }(x,y)& \hfill :={}_{3}F_2\left[\begin{array}{c}\hfill 2x,2x+\frac{1}{2},y\\ \hfill \frac{1}{2},1+y\end{array}|1\right],\\ \hfill \mathsf{\Psi }(x,y)& \hfill :={}_{3}F_2\left[\begin{array}{c}\hfill 2x,2x-\frac{1}{2},y\\ \hfill \frac{3}{2},1+y\end{array}|1\right].\end{array}$$

Consider the bisection series

$${}_{2}F_1\left[\begin{array}{c}\hfill 4x,2y\\ \hfill 1+2y\end{array}|1\right]\pm {}_{2}F_1\left[\begin{array}{c}\hfill 4x,2y\\ \hfill 1+2y\end{array}|-1\right]=\sum _{n=0}^{\infty }\frac{{\left(4x\right)}_n\left(2y\right)}{n!(2y+n)}\pm \sum _{n=0}^{\infty }{(-1)}^n\frac{{\left(4x\right)}_n\left(2y\right)}{n!(2y+n)}.$$

According to the parity of the summation index n, the two sums on the right can be unified as

$$\begin{array}{cc}\hfill “+”:& 2\times {}_{3}F_2\left[\begin{array}{c}\hfill 2x,2x+\frac{1}{2},y\\ \hfill \frac{1}{2},1+y\end{array}|1\right],\hfill \\ \hfill “-”:& \frac{16xy}{1+2y}\times {}_{3}F_2\left[\begin{array}{c}\hfill 1+2x,\frac{1}{2}+2x,\frac{1}{2}+y\\ \hfill \frac{3}{2},\frac{3}{2}+y\end{array}|1\right];\hfill \end{array}$$

where the last two ${}_{3}F_2$(1)-series result in $\mathsf{\Phi }(x,y)$ and $\mathsf{\Psi }(\frac{1}{2}+x,\frac{1}{2}+y)$, respectively.

Therefore, we have established two algebraic identities as in the following lemma.

Lemma 1

($x<\frac{1}{4}$ and $y\in \mathbb{R}$ with $2y\notin \mathbb{Z}\setminus \mathbb{N}$).

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& \mathsf{\Phi }(x,y)=\frac{1}{2}\left\{{}_{2}F_1\left[\begin{array}{c}\hfill 4x,2y\\ \hfill 1+2y\end{array}|1\right]+{}_{2}F_1\left[\begin{array}{c}\hfill 4x,2y\\ \hfill 1+2y\end{array}|-1\right]\right\},\hfill \\ \hfill \left(\mathrm{b}\right)& \mathsf{\Psi }(x,y)=\frac{y}{2(1-2x)(1-2y)}\left\{{}_{2}F_1\left[\begin{array}{c}\hfill 4x-2,2y-1\\ \hfill 2y\end{array}|1\right]-{}_{2}F_1\left[\begin{array}{c}\hfill 4x-2,2y-1\\ \hfill 2y\end{array}|-1\right]\right\}.\hfill \end{array}$$

By applying this lemma, two classes of exotic ${}_{3}F_2\left(1\right)$-series will be evaluated in closed form through the Gauss and Kummer’s summation formulae, as well as a general theorem due to the second author [13] (Theorem 9) for the ${}_{2}F_1(-1)$-series. Several remarkable identities will be established. To the best of our knowledge, none of them have appeared previously in the literature (cf. [14,15]).

2. Evaluations When $\mathit{y}\to \mathit{x}$

Making the replacement $y\to x$ in Lemma 1, we can evaluate the ${}_{2}F_1$-series using the following two summation formulae due to Gauss and Kummer (cf. Bailey [1] (§1.3 & §2.3)):

$$\begin{array}{c}{}_{2}F_1\left[\begin{array}{cc}a,& b\\ & c\end{array}|1\right]=\frac{\Gamma \left(c\right)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)},\Re (c-a-b)>0;\hfill \\ {}_{2}F_1\left[\begin{array}{c}a,b\\ 1+a-b\end{array}|-1\right]=\frac{\Gamma (1+a-b)\Gamma (1+\frac{a}{2})}{\Gamma (1+a)\Gamma (1+\frac{a}{2}-b)},\Re \left(b\right)<1.\hfill \end{array}$$

This leads us to the following remarkable nonterminating series identities.

Theorem 1

($x<\frac{1}{4}$).

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill 2x,2x+\frac{1}{2},x\\ \hfill \frac{1}{2},1+x\end{array}|1\right]=\frac{\Gamma (1+2x)\Gamma (1-4x)}{2\Gamma (1-2x)}+\frac{{\Gamma }^2(1+2x)}{2\Gamma (1+4x)}\hfill \\ & =\frac{\sqrt{\pi }{\Gamma }^2(1+x)\Gamma (\frac{1}{2}-2x)}{\Gamma (1+2x){\Gamma }^2(\frac{1}{2}-x)}=\frac{\sqrt{\pi }{\cos }^2\left(\pi x\right)\Gamma (1+2x)}{2^{4x}\cos \left(2\pi x\right)\Gamma (\frac{1}{2}+2x)},\\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill 2x,2x-\frac{1}{2},x\\ \hfill \frac{3}{2},1+x\end{array}|1\right]=\frac{x}{2{(1-2x)}^2}\times \left\{\frac{\Gamma \left(2x\right)\Gamma (3-4x)}{\Gamma (2-2x)}-\frac{{\Gamma }^2\left(2x\right)}{\Gamma (4x-1)}\right\}\hfill \\ & =\frac{\sqrt{\pi }{\Gamma }^2(1+x)\Gamma (\frac{3}{2}-2x)}{2\Gamma (1+2x){\Gamma }^2(\frac{3}{2}-x)}=\frac{1-4x}{{(1-2x)}^2}\frac{\sqrt{\pi }{\cos }^2\left(\pi x\right)\Gamma (1+2x)}{2^{4x}\cos \left(2\pi x\right)\Gamma (\frac{1}{2}+2x)}.\end{array}$$

According to this theorem, we deduce the following curious relation

$${}_{3}F_2\left[\begin{array}{c}\hfill 2x,2x-\frac{1}{2},x\\ \hfill \frac{3}{2},1+x\end{array}|1\right]=\frac{1-4x}{{(1-2x)}^2}\times {}_{3}F_2\left[\begin{array}{c}\hfill 2x,2x+\frac{1}{2},x\\ \hfill \frac{1}{2},1+x\end{array}|1\right],$$

that we failed to locate in the literature (e.g., [1,14]). By assigning the variable $x<\frac{1}{4}$ to particular values, we derive from Theorem 1 the following interesting identities, that don’t seem to have appeared previously in the literature.

2.1. $x=\frac{1}{5}$

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{2}{5},& \frac{9}{10},& \hfill \frac{1}{5}\\ & \frac{1}{2},& \hfill \frac{6}{5}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{6}{5}\right)\Gamma \left(\frac{1}{10}\right)}{\Gamma \left(\frac{7}{5}\right){\Gamma }^2\left(\frac{3}{10}\right)}=\frac{\sqrt{\pi }(2+\sqrt{5})\Gamma \left(\frac{7}{5}\right)}{2\sqrt[5]{16}\Gamma \left(\frac{9}{10}\right)},\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{2}{5},& \hfill -\frac{1}{10},& \hfill \frac{1}{5}\\ & \hfill \frac{3}{2},& \hfill \frac{6}{5}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{6}{5}\right)\Gamma \left(\frac{11}{10}\right)}{2\Gamma \left(\frac{7}{5}\right){\Gamma }^2\left(\frac{13}{10}\right)}=\frac{5\sqrt{\pi }(2+\sqrt{5})\Gamma \left(\frac{7}{5}\right)}{18\sqrt[5]{16}\Gamma \left(\frac{9}{10}\right)}.\hfill \end{array}$$

2.2. $x=\frac{1}{6}$

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{3},& \hfill \frac{5}{6},& \hfill \frac{1}{6}\\ & \hfill \frac{1}{2},& \hfill \frac{7}{6}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{7}{6}\right)\Gamma \left(\frac{1}{6}\right)}{\Gamma \left(\frac{4}{3}\right){\Gamma }^2\left(\frac{1}{3}\right)}=\frac{3\sqrt{\pi }\Gamma \left(\frac{4}{3}\right)}{2\sqrt[3]{4}\Gamma \left(\frac{5}{6}\right)},\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{3},& \hfill -\frac{1}{6},& \hfill \frac{1}{6}\\ & \hfill \frac{3}{2},& \hfill \frac{7}{6}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^3\left(\frac{7}{6}\right)}{2{\Gamma }^3\left(\frac{4}{3}\right)}=\frac{9\sqrt{\pi }\Gamma \left(\frac{4}{3}\right)}{8\sqrt[3]{4}\Gamma \left(\frac{5}{6}\right)}.\hfill \end{array}$$

2.3. $x=\frac{1}{7}$

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{2}{7},& \frac{11}{14},& \hfill \frac{1}{7}\\ & \frac{1}{2},& \hfill \frac{8}{7}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{8}{7}\right)\Gamma \left(\frac{3}{14}\right)}{\Gamma \left(\frac{9}{7}\right){\Gamma }^2\left(\frac{5}{14}\right)},\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{2}{7},& \hfill -\frac{3}{14},& \hfill \frac{1}{7}\\ & \hfill \frac{3}{2},& \hfill \frac{8}{7}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{8}{7}\right)\Gamma \left(\frac{17}{14}\right)}{2\Gamma \left(\frac{9}{7}\right){\Gamma }^2\left(\frac{19}{14}\right)}.\hfill \end{array}$$

2.4. $x=\frac{1}{8}$

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{4},& \hfill \frac{3}{4},& \hfill \frac{1}{8}\\ & \hfill \frac{1}{2},& \hfill \frac{9}{8}\end{array}|1\right]=\frac{4\sqrt{\pi }{\Gamma }^2\left(\frac{9}{8}\right)}{{\Gamma }^2\left(\frac{3}{8}\right)}=\frac{\sqrt{\pi }(1+\sqrt{2})\Gamma \left(\frac{5}{4}\right)}{2\sqrt{2}\Gamma \left(\frac{3}{4}\right)},\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{4},& \hfill -\frac{1}{4},& \hfill \frac{1}{8}\\ & \hfill \frac{3}{2},& \hfill \frac{9}{8}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{9}{8}\right)}{2{\Gamma }^2\left(\frac{11}{8}\right)}=\frac{2\sqrt{\pi }(2+\sqrt{2})\Gamma \left(\frac{5}{4}\right)}{9\Gamma \left(\frac{3}{4}\right)}.\hfill \end{array}$$

2.5. $x=\frac{1}{9}$

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{2}{9},& \frac{13}{18},& \frac{1}{9}\\ & \frac{1}{2},& \frac{10}{9}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{10}{9}\right)\Gamma \left(\frac{5}{18}\right)}{\Gamma \left(\frac{11}{9}\right){\Gamma }^2\left(\frac{7}{18}\right)},\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{2}{9},& \hfill -\frac{5}{18},& \frac{1}{9}\\ & \hfill \frac{3}{2},& \frac{10}{9}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{10}{9}\right)\Gamma \left(\frac{23}{18}\right)}{2\Gamma \left(\frac{11}{9}\right){\Gamma }^2\left(\frac{25}{18}\right)}.\hfill \end{array}$$

2.6. $x=\frac{1}{10}$

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{5},& \frac{7}{10},& \hfill \frac{1}{10}\\ & \frac{1}{2},& \hfill \frac{11}{10}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{11}{10}\right)\Gamma \left(\frac{3}{10}\right)}{\Gamma \left(\frac{6}{5}\right){\Gamma }^2\left(\frac{2}{5}\right)}=\frac{\sqrt{5\pi }\Gamma \left(\frac{6}{5}\right)}{2\sqrt[5]{4}\Gamma \left(\frac{7}{10}\right)},\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{5},& \hfill -\frac{3}{10},& \hfill \frac{1}{10}\\ & \hfill \frac{3}{2},& \hfill \frac{11}{10}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{11}{10}\right)\Gamma \left(\frac{13}{10}\right)}{2\Gamma \left(\frac{6}{5}\right){\Gamma }^2\left(\frac{7}{5}\right)}=\frac{15\sqrt{5\pi }\Gamma \left(\frac{6}{5}\right)}{32\sqrt[5]{4}\Gamma \left(\frac{7}{10}\right)}.\hfill \end{array}$$

2.7. $x=\frac{1}{11}$

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{2}{11},& \frac{15}{22},& \hfill \frac{1}{11}\\ & \frac{1}{2},& \hfill \frac{12}{11}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{12}{11}\right)\Gamma \left(\frac{7}{22}\right)}{\Gamma \left(\frac{13}{11}\right){\Gamma }^2\left(\frac{9}{22}\right)},\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{2}{11},& \hfill -\frac{7}{22},& \hfill \frac{1}{11}\\ & \hfill \frac{3}{2},& \hfill \frac{12}{11}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{12}{11}\right)\Gamma \left(\frac{29}{22}\right)}{2\Gamma \left(\frac{13}{11}\right){\Gamma }^2\left(\frac{31}{22}\right)}.\hfill \end{array}$$

2.8. $x=\frac{1}{12}$

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{6},& \hfill \frac{2}{3},& \hfill \frac{1}{12}\\ & \hfill \frac{1}{2},& \hfill \frac{13}{12}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{13}{12}\right)\Gamma \left(\frac{1}{3}\right)}{\Gamma \left(\frac{7}{6}\right){\Gamma }^2\left(\frac{5}{12}\right)}=\frac{\sqrt{\pi }(3+2\sqrt{3})\Gamma \left(\frac{7}{6}\right)}{6\sqrt[3]{2}\Gamma \left(\frac{2}{3}\right)},\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{6},& \hfill -\frac{1}{3},& \hfill \frac{1}{12}\\ & \hfill \frac{3}{2},& \hfill \frac{13}{12}\end{array}|1\right]=\frac{\sqrt{\pi }{\Gamma }^2\left(\frac{13}{12}\right)\Gamma \left(\frac{4}{3}\right)}{2\Gamma \left(\frac{7}{6}\right){\Gamma }^2\left(\frac{17}{12}\right)}=\frac{2\sqrt{\pi }\sqrt[3]{4}(3+2\sqrt{3})\Gamma \left(\frac{7}{6}\right)}{25\Gamma \left(\frac{2}{3}\right)}.\hfill \end{array}$$

3. Evaluations When $\mathbf{4}\mathit{x}+\mathbf{2}\mathit{y}\in \mathbb{Z}$ and $\mathbf{4}\mathit{y}$ Is Odd

In this case, write

$$\left\{\begin{array}{c}4y\to 2n-1\\ 4x+2y\to n-m\end{array}\right\}⟹\left\{\begin{array}{c}y=\frac{n}{2}-\frac{1}{4}\\ x=\frac{1}{8}-\frac{m}{4}\end{array}\right\}\mathrm{where}\left\{\begin{array}{c}m\in {\mathbb{N}}_0\\ n\in \mathbb{Z}\end{array}\right\}.$$

Under these substitutions, the alternating series becomes

$${}_{2}F_1\left[\begin{array}{c}\hfill 4x,2y\\ \hfill 1+2y\end{array}|-1\right]⟹{}_{2}F_1\left[\begin{array}{c}\hfill \frac{1}{2}-m,n-\frac{1}{2}\\ \hfill n+\frac{1}{2}\end{array}|-1\right].$$

Then the two transformations in Lemma 1 can be reformulated as follows.

Proposition 1

($m\in {\mathbb{N}}_0$ and $n\in \mathbb{Z}$).

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4}-\frac{m}{2},\frac{3}{4}-\frac{m}{2},\frac{n}{2}-\frac{1}{4}\\ \hfill \frac{1}{2},\frac{n}{2}+\frac{3}{4}\end{array}|1\right]=\frac{1}{2}{}_{2}F_1\left[\begin{array}{c}\hfill \frac{1}{2}-m,n-\frac{1}{2}\\ \hfill n+\frac{1}{2}\end{array}|1\right]\hfill \\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +\frac{1}{2}{}_{2}F_1\left[\begin{array}{c}\hfill \frac{1}{2}-m,n-\frac{1}{2}\\ \hfill n+\frac{1}{2}\end{array}|-1\right],\hfill \\ \hfill \left(b\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4}-\frac{m}{2},-\frac{m}{2}-\frac{1}{4},\frac{n}{2}-\frac{1}{4}\\ \hfill \frac{3}{2},\frac{n}{2}+\frac{3}{4}\end{array}|1\right]=\frac{2n-1}{(3+2m)(3-2n)}\hfill \\ & \times \left\{{}_{2}F_1\left[\begin{array}{c}\hfill -\frac{3}{2}-m,n-\frac{3}{2}\\ \hfill n-\frac{1}{2}\end{array}|1\right]-{}_{2}F_1\left[\begin{array}{c}\hfill -\frac{3}{2}-m,n-\frac{3}{2}\\ \hfill n-\frac{1}{2}\end{array}|-1\right]\right\}.\hfill \end{array}$$

The two formulae in the above theorem can be further reformulated. First by means of the Gauss summation theorem, we can evaluate

$$\begin{array}{c}{}_{2}F_1\left[\begin{array}{c}\hfill \frac{1}{2}-m,n-\frac{1}{2}\\ \hfill n+\frac{1}{2}\end{array}|1\right]=\frac{\pi {\left(\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_n}{\Gamma (m+n)},\hfill \\ {}_{2}F_1\left[\begin{array}{c}\hfill -\frac{3}{2}-m,n-\frac{3}{2}\\ \hfill n-\frac{1}{2}\end{array}|1\right]=\frac{\pi {\left(\frac{1}{2}\right)}_{m+2}{\left(\frac{1}{2}\right)}_{n-1}}{\Gamma (m+n+1)}.\hfill \end{array}$$

Then recall the ${\mathsf{\Omega }}_{m,n}(x,y)$ examined in [13]

$${\mathsf{\Omega }}_{m,n}(x,y)={}_{2}F_1\left[\begin{array}{cc}x,& m-x\\ & n+\frac{1}{2}\end{array}|y^2\right],\mathrm{where}m,n\in \mathbb{Z}.$$

The two formulae in Proposition 1 can accordingly be simplified as shown below.

Theorem 2

($m\in {\mathbb{N}}_0$ and $n\in \mathbb{Z}$).

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4}-\frac{m}{2},\frac{3}{4}-\frac{m}{2},\frac{n}{2}-\frac{1}{4}\\ \hfill \frac{1}{2},\frac{n}{2}+\frac{3}{4}\end{array}|1\right]=\frac{\pi {\left(\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_n}{2\Gamma (m+n)}+\frac{1}{2}{\mathsf{\Omega }}_{n-m,n}\left(n-\frac{1}{2},\mathbf{i}\right),\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4}-\frac{m}{2},-\frac{m}{2}-\frac{1}{4},\frac{n}{2}-\frac{1}{4}\\ \hfill \frac{3}{2},\frac{n}{2}+\frac{3}{4}\end{array}|1\right]=\frac{2n-1}{(3+2m)(3-2n)}\hfill \\ & \hfill \times \left\{\frac{\pi {\left(\frac{1}{2}\right)}_{m+2}{\left(\frac{1}{2}\right)}_{n-1}}{\Gamma (m+n+1)}-{\mathsf{\Omega }}_{n-m-3,n-1}\left(n-\frac{3}{2},\mathbf{i}\right)\right\}.\hfill \end{array}$$

Recall that the two rightmost ${}_{2}F_1(-1)$-series can explicitly be evaluated in closed form by a general theorem due to the second author [13] (Theorem 9), and the corollaries wherein. By specifying $m\in {\mathbb{N}}_0$ and $n\in \mathbb{Z}$ to specific small values, we can establish numerous infinite series identities. Nine pairs of sample identities for nonterminating ${}_{3}F_2\left(1\right)$-series are highlighted below as examples.

3.1. $m=0\&n=1$

The corresponding two ${}_{3}F_2$-series in Theorem 2 read as

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4},\frac{3}{4},\frac{1}{4}\\ \hfill \frac{1}{2},\frac{5}{4}\end{array}|1\right]=\frac{\pi }{4}+\frac{1}{2}{\mathsf{\Omega }}_{1,1}\left(\frac{1}{2},\mathbf{i}\right),\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4},-\frac{1}{4},\frac{1}{4}\\ \hfill \frac{3}{2},\frac{5}{4}\end{array}|1\right]=\frac{\pi }{4}-\frac{1}{3}{\mathsf{\Omega }}_{-2,0}\left(-\frac{1}{2},\mathbf{i}\right).\hfill \end{array}$$

According to [13] (Corollaries 11 & 19), the above two ${}_{2}F_1(-1)$-series result in

$$\begin{array}{c}{\mathsf{\Omega }}_{1,1}\left(\frac{1}{2},\mathbf{i}\right)=\mathrm{arcsinh}\left(1\right)=\ln (1+\sqrt{2}),\hfill \\ {\mathsf{\Omega }}_{-2,0}\left(-\frac{1}{2},\mathbf{i}\right)=\frac{1}{\sqrt{2}}-\frac{3}{2}\ln (1+\sqrt{2});\hfill \end{array}$$

which leads us to the following two interesting identities:

$$\begin{array}{cc}\hfill \left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4},\frac{3}{4},\frac{1}{4}\\ \hfill \frac{1}{2},\frac{5}{4}\end{array}|1\right]=\frac{\pi }{4}+\frac{\ln (1+\sqrt{2})}{2},\hfill \\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4},-\frac{1}{4},\frac{1}{4}\\ \hfill \frac{3}{2},\frac{5}{4}\end{array}|1\right]=\frac{\pi }{4}-\frac{\sqrt{2}}{6}+\frac{\ln (1+\sqrt{2})}{2}.\end{array}$$

3.2. $m=1\&n=0$

The two ${}_{3}F_2$-series in Theorem 2 can be stated as

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{1}{4},& \hfill \frac{1}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{1}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{\pi }{4}+\frac{1}{2}{\mathsf{\Omega }}_{-1,0}\left(-\frac{1}{2},\mathbf{i}\right),\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{1}{4},& \hfill -\frac{3}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{\pi }{4}+\frac{1}{15}{\mathsf{\Omega }}_{-4,-1}\left(-\frac{3}{2},\mathbf{i}\right).\hfill \end{array}$$

By employing [13] (Corollary 12 and Theorem 9), we can evaluate the last two ${}_{2}F_1(-1)$-series, respectively, as follows:

$$\begin{array}{c}{\mathsf{\Omega }}_{-1,0}\left(-\frac{1}{2},\mathbf{i}\right)=\sqrt{2}-\ln (1+\sqrt{2}),\hfill \\ {\mathsf{\Omega }}_{-4,-1}(-\frac{3}{2},\mathbf{i})=\frac{13}{\sqrt{2}}-\frac{15}{2}\ln (1+\sqrt{2}).\hfill \end{array}$$

Consequently, we have establish two further identities:

$$\begin{array}{cc}\hfill \left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{1}{4},& \hfill \frac{1}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{1}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{\pi }{4}+\frac{1}{\sqrt{2}}-\frac{\ln (1+\sqrt{2})}{2},\hfill \\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{1}{4},& \hfill -\frac{3}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{\pi }{4}+\frac{13\sqrt{2}}{30}-\frac{\ln (1+\sqrt{2})}{2}.\hfill \end{array}$$

3.3. $m=1\&n=1$

The two ${}_{3}F_2$-series displayed in Theorem 2 become

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{1}{4},\frac{1}{4},\frac{1}{4}\\ \hfill \frac{1}{2},\frac{5}{4}\end{array}|1\right]=\frac{\pi }{8}+\frac{1}{2}{\mathsf{\Omega }}_{0,1}\left(\frac{1}{2},\mathbf{i}\right),\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{1}{4},-\frac{3}{4},\frac{1}{4}\\ \hfill \frac{3}{2},\frac{5}{4}\end{array}|1\right]=\frac{3\pi }{16}-\frac{1}{5}{\mathsf{\Omega }}_{-3,0}\left(-\frac{1}{2},\mathbf{i}\right).\hfill \end{array}$$

Evaluating the last two ${}_{2}F_1(-1)$-series by [13] (Corollary 14 and Theorem 9)

$$\begin{array}{c}{\mathsf{\Omega }}_{0,1}\left(\frac{1}{2},\mathbf{i}\right)=\frac{1}{\sqrt{2}}+\frac{\ln (1+\sqrt{2})}{2},\hfill \\ {\mathsf{\Omega }}_{-3,0}\left(-\frac{1}{2},\mathbf{i}\right)=-\frac{3\sqrt{2}}{8}-\frac{15}{8}\ln (1+\sqrt{2}),\hfill \end{array}$$

we find the following two infinite series identities:

$$\begin{array}{cc}\hfill \left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{1}{4},\frac{1}{4},\frac{1}{4}\\ \hfill \frac{1}{2},\frac{5}{4}\end{array}|1\right]=\frac{\pi }{8}+\frac{\sqrt{2}}{4}+\frac{\ln (1+\sqrt{2})}{4},\hfill \\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{1}{4},-\frac{3}{4},\frac{1}{4}\\ \hfill \frac{3}{2},\frac{5}{4}\end{array}|1\right]=\frac{3\pi }{16}+\frac{3\sqrt{2}}{40}+\frac{3}{8}\ln (1+\sqrt{2}).\hfill \end{array}$$

3.4. $m=2\&n=0$

The corresponding two ${}_{3}F_2$-series in Theorem 2 read as

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{1}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{1}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{3\pi }{8}+\frac{1}{2}{\mathsf{\Omega }}_{-2,0}\left(-\frac{1}{2},\mathbf{i}\right),\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{5}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{5\pi }{16}+\frac{1}{21}{\mathsf{\Omega }}_{-5,-1}\left(-\frac{3}{2},\mathbf{i}\right).\end{array}$$

According to [13] (Corollary 19 and Theorem 9), the above two ${}_{2}F_1(-1)$-series can be evaluated in closed form, respectively, as

$$\begin{array}{cc}& \hfill {\mathsf{\Omega }}_{-2,0}\left(-\frac{1}{2},\mathbf{i}\right)=\frac{1}{\sqrt{2}}-\frac{3}{2}\ln (1+\sqrt{2}),\\ & \hfill {\mathsf{\Omega }}_{-5,-1}\left(-\frac{3}{2},\mathbf{i}\right)=\frac{43\sqrt{2}}{8}-\frac{105}{8}\ln (1+\sqrt{2}).\end{array}$$

This gives rise to the following two hypergeometric series identities:

$$\begin{array}{cc}\hfill \left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{1}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{1}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{3\pi }{8}+\frac{\sqrt{2}}{4}-\frac{3}{4}\ln (1+\sqrt{2}),\hfill \\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{5}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{5\pi }{16}+\frac{43\sqrt{2}}{168}-\frac{5}{8}\ln (1+\sqrt{2}).\hfill \end{array}$$

3.5. $m=0\&n=2$

The two ${}_{3}F_2$-series in Theorem 2 can be stated as

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4},\frac{3}{4},\frac{3}{4}\\ \hfill \frac{1}{2},\frac{7}{4}\end{array}|1\right]=\frac{3\pi }{8}+\frac{1}{2}{\mathsf{\Omega }}_{2,2}\left(\frac{1}{2},\mathbf{i}\right),\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4},-\frac{1}{4},\frac{3}{4}\\ \hfill \frac{3}{2},\frac{7}{4}\end{array}|1\right]={\mathsf{\Omega }}_{-1,1}\left(\frac{1}{2},\mathbf{i}\right)-\frac{3\pi }{16}.\hfill \end{array}$$

Evaluating the above two ${}_{2}F_1(-1)$-series by [13] (Corollaries 15 & 20)

$$\begin{array}{c}{\mathsf{\Omega }}_{2,2}\left(\frac{1}{2},\mathbf{i}\right)=\frac{3}{\sqrt{2}}-\frac{3}{2}\ln (1+\sqrt{2}),\hfill \\ {\mathsf{\Omega }}_{-1,1}\left(\frac{1}{2},\mathbf{i}\right)=\frac{7\sqrt{2}}{8}+\frac{3}{8}\ln (1+\sqrt{2});\hfill \end{array}$$

we derive the following two infinite series identities:

$$\begin{array}{cc}\left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4},\frac{3}{4},\frac{3}{4}\\ \hfill \frac{1}{2},\frac{7}{4}\end{array}|1\right]=\frac{3\pi }{8}+\frac{3\sqrt{2}}{4}-\frac{3}{4}\ln (1+\sqrt{2}),\\ \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill \frac{1}{4},-\frac{1}{4},\frac{3}{4}\\ \hfill \frac{3}{2},\frac{7}{4}\end{array}|1\right]=\frac{7\sqrt{2}}{8}-\frac{3\pi }{16}+\frac{3}{8}\ln (1+\sqrt{2}).\end{array}$$

3.6. $m=2\&n=1$

The two ${}_{3}F_2$-series displayed in Theorem 2 become

$$\begin{array}{cc}\left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{3}{4},-\frac{1}{4},\frac{1}{4}\\ \hfill \frac{1}{2},\frac{5}{4}\end{array}|1\right]=\frac{3\pi }{32}+\frac{1}{2}{\mathsf{\Omega }}_{-1,1}\left(\frac{1}{2},\mathbf{i}\right),\\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{3}{4},-\frac{5}{4},\frac{1}{4}\\ \hfill \frac{3}{2},\frac{5}{4}\end{array}|1\right]=\frac{5\pi }{32}-\frac{1}{7}{\mathsf{\Omega }}_{-4,0}\left(-\frac{1}{2},\mathbf{i}\right).\hfill \end{array}$$

By utilizing [13] (Corollary 20 and Theorem 9), the last two ${}_{2}F_1(-1)$-series can be evaluated in closed form, respectively, as follows:

$$\begin{array}{c}{\mathsf{\Omega }}_{-1,1}\left(\frac{1}{2},\mathbf{i}\right)=\frac{7\sqrt{2}}{8}+\frac{3}{8}\ln (1+\sqrt{2}),\hfill \\ {\mathsf{\Omega }}_{-4,0}\left(-\frac{1}{2},\mathbf{i}\right)=-\frac{85\sqrt{2}}{48}-\frac{35}{16}\ln (1+\sqrt{2}).\hfill \end{array}$$

Therefore, we obtain the following two new identities:

$$\begin{array}{cc}\left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{3}{4},-\frac{1}{4},\frac{1}{4}\\ \hfill \frac{1}{2},\frac{5}{4}\end{array}|1\right]=\frac{3\pi }{32}+\frac{7\sqrt{2}}{16}+\frac{3}{16}\ln (1+\sqrt{2}),\\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{3}{4},-\frac{5}{4},\frac{1}{4}\\ \hfill \frac{3}{2},\frac{5}{4}\end{array}|1\right]=\frac{5\pi }{32}+\frac{85\sqrt{2}}{336}+\frac{5}{16}\ln (1+\sqrt{2}).\hfill \end{array}$$

3.7. $m=2\&n=-1$

The corresponding two ${}_{3}F_2$-series in Theorem 2 read as

$$\begin{array}{cc}\left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{1}{4},& \hfill -\frac{3}{4}\\ & \hfill \frac{1}{2},& \hfill \frac{1}{4}\end{array}|1\right]=-\frac{3\pi }{4}+\frac{1}{2}{\mathsf{\Omega }}_{-3,-1}\left(-\frac{3}{2},\mathbf{i}\right),\\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{5}{4},& \hfill -\frac{3}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{1}{4}\end{array}|1\right]=\frac{3}{35}{\mathsf{\Omega }}_{-6,-2}\left(-\frac{5}{2},\mathbf{i}\right)-\frac{3\pi }{4}.\hfill \end{array}$$

For the above two ${}_{2}F_1(-1)$-series, we can evaluate them by [13] (Theorem 9)

$$\begin{array}{c}{\mathsf{\Omega }}_{-3,-1}\left(-\frac{3}{2},\mathbf{i}\right)=5\sqrt{2}-3\ln (1+\sqrt{2}),\hfill \\ {\mathsf{\Omega }}_{-6,-2}\left(-\frac{5}{2},\mathbf{i}\right)=\frac{139\sqrt{2}}{6}-\frac{35}{2}\ln (1+\sqrt{2}).\hfill \end{array}$$

As a consequences, we obtain the following two interesting identities:

$$\begin{array}{cc}\hfill \left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{1}{4},& \hfill -\frac{3}{4}\\ & \hfill \frac{1}{2},& \hfill \frac{1}{4}\end{array}|1\right]=\frac{5}{\sqrt{2}}-\frac{3\pi }{4}-\frac{3}{2}\ln (1+\sqrt{2}),\hfill \\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{5}{4},& \hfill -\frac{3}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{1}{4}\end{array}|1\right]=\frac{139\sqrt{2}}{70}-\frac{3\pi }{4}-\frac{3}{2}\ln (1+\sqrt{2}).\hfill \end{array}$$

3.8. $m=1\&n=2$

The two ${}_{3}F_2$-series in Theorem 2 can be stated as

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{1}{4},\frac{1}{4},\frac{3}{4}\\ \hfill \frac{1}{2},\frac{7}{4}\end{array}|1\right]=\frac{3\pi }{32}+\frac{1}{2}{\mathsf{\Omega }}_{1,2}\left(\frac{3}{2},\mathbf{i}\right),\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{1}{4},-\frac{3}{4},\frac{3}{4}\\ \hfill \frac{3}{2},\frac{7}{4}\end{array}|1\right]=\frac{3}{5}{\mathsf{\Omega }}_{-2,1}\left(\frac{1}{2},\mathbf{i}\right)-\frac{3\pi }{32}.\hfill \end{array}$$

By employing [13] (Corollary 16 and Theorem 9), the last two ${}_{2}F_1(-1)$-series can be evaluated in closed form as

$$\begin{array}{c}{\mathsf{\Omega }}_{1,2}\left(\frac{3}{2},\mathbf{i}\right)=\frac{9\sqrt{2}}{8}-\frac{3}{8}\ln (1+\sqrt{2}),\hfill \\ {\mathsf{\Omega }}_{-2,1}\left(\frac{1}{2},\mathbf{i}\right)=\frac{67\sqrt{2}}{48}+\frac{5}{16}\ln (1+\sqrt{2}).\hfill \end{array}$$

Accordingly, this leads us to the two new identities:

$$\begin{array}{cc}\hfill \left(\mathrm{A}\right)& \hfill {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{1}{4},\frac{1}{4},\frac{3}{4}\\ \hfill \frac{1}{2},\frac{7}{4}\end{array}|1\right]=\frac{3\pi }{32}+\frac{9\sqrt{2}}{16}-\frac{3}{16}\ln (1+\sqrt{2}),\\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{c}\hfill -\frac{1}{4},-\frac{3}{4},\frac{3}{4}\\ \hfill \frac{3}{2},\frac{7}{4}\end{array}|1\right]=\frac{67\sqrt{2}}{80}-\frac{3\pi }{32}+\frac{3}{16}\ln (1+\sqrt{2}).\hfill \end{array}$$

3.9. $m=2\&n=2$

The two ${}_{3}F_2$-series displayed in Theorem 2 become

$$\begin{array}{cc}\hfill \left(\mathrm{a}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{1}{4},& \hfill \frac{3}{4}\\ & \hfill \frac{1}{2},& \hfill \frac{7}{4}\end{array}|1\right]=\frac{3\pi }{64}+\frac{1}{2}{\mathsf{\Omega }}_{0,2}\left(\frac{3}{2},\mathbf{i}\right),\hfill \\ \hfill \left(\mathrm{b}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{5}{4},& \hfill \frac{3}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{7}{4}\end{array}|1\right]=\frac{3}{7}{\mathsf{\Omega }}_{-3,1}\left(\frac{1}{2},\mathbf{i}\right)-\frac{15\pi }{256}.\hfill \end{array}$$

Evaluating the above two ${}_{2}F_1(-1)$-series by [13] (Theorem 9)

$$\begin{array}{c}{\mathsf{\Omega }}_{0,2}\left(\frac{3}{2},\mathbf{i}\right)=\frac{25\sqrt{2}}{16}-\frac{3}{16}\ln (1+\sqrt{2}),\hfill \\ {\mathsf{\Omega }}_{-3,1}\left(\frac{1}{2},\mathbf{i}\right)=\frac{853\sqrt{2}}{384}+\frac{105}{384}\ln (1+\sqrt{2});\hfill \end{array}$$

we finally arrive at the following two infinite series identities:

$$\begin{array}{cc}\hfill \left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{1}{4},& \hfill \frac{3}{4}\\ & \hfill \frac{1}{2},& \hfill \frac{7}{4}\end{array}|1\right]=\frac{3\pi }{64}+\frac{25\sqrt{2}}{32}-\frac{3}{32}\ln (1+\sqrt{2}),\hfill \\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{5}{4},& \hfill \frac{3}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{7}{4}\end{array}|1\right]=\frac{853\sqrt{2}}{896}-\frac{15\pi }{256}+\frac{15}{128}\ln (1+\sqrt{2}).\hfill \end{array}$$

4. Concluding Comments and Further Observation

By means of the bisection series approach, we succeeded in finding several novel values for the hypergeometric ${}_{3}F_2\left(1\right)$-series. More identities regarding the ${}_{3}F_2\left(1\right)$-series can be found in [16,17,18,19,20]. These closed formulae suggest that the following two related topics may further be explored.

One topic is about similar series with parameters having bigger discrepancies. Besides the sample series exhibited in Section 3, further cases with $n\le 0$ of Theorem 2 can be examined. For instance, using “$m=0\&n=0$”, “$m=1\&n=-1$” and “$m=2\&n=-2$”, all three series corresponding to Theorem 2(a) reduce to the ${}_{2}F_1\left(1\right)$-series. Instead, the remaining three series corresponding to (b) result in the three identities below:

$$\begin{array}{c}{}_{3}F_2\left[\begin{array}{ccc}\hfill \frac{1}{4},& \hfill -\frac{1}{4},& \hfill -\frac{1}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{\pi }{6}+\frac{5\sqrt{2}}{9}-\frac{\ln (1+\sqrt{2})}{3},\hfill \\ {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{1}{4},& \hfill -\frac{3}{4},& \hfill -\frac{3}{4}\\ & \hfill \frac{3}{2},& \hfill \frac{1}{4}\end{array}|1\right]=\frac{37\sqrt{2}}{25}-\frac{3\pi }{10}-\frac{3}{5}\ln (1+\sqrt{2}),\hfill \\ {}_{3}F_2\left[\begin{array}{ccc}\hfill -\frac{3}{4},& \hfill -\frac{5}{4},& \hfill -\frac{5}{4}\\ & \hfill \frac{3}{2},& \hfill -\frac{1}{4}\end{array}|1\right]=\frac{5\pi }{14}+\frac{379\sqrt{2}}{147}-\frac{5}{7}\ln (1+\sqrt{2}).\hfill \end{array}$$

To validate them, we have to invoke the following three series

$$\begin{array}{cc}& \hfill {\mathsf{\Omega }}_{-3,-1}\left(-\frac{3}{2},\mathbf{i}\right)={}_{2}F_1\left[\begin{array}{c}\hfill -\frac{3}{2},-\frac{3}{2}\\ \hfill -\frac{1}{2}\end{array}|-1\right]=5\sqrt{2}-3\ln (1+\sqrt{2}),\\ & \hfill {\mathsf{\Omega }}_{-5,-2}\left(-\frac{5}{2},\mathbf{i}\right)={}_{2}F_1\left[\begin{array}{c}\hfill -\frac{5}{2},-\frac{5}{2}\\ \hfill -\frac{3}{2}\end{array}|-1\right]=\frac{37\sqrt{2}}{3}-5\ln (1+\sqrt{2}),\\ & \hfill {\mathsf{\Omega }}_{-7,-3}\left(-\frac{7}{2},\mathbf{i}\right)={}_{2}F_1\left[\begin{array}{c}\hfill -\frac{7}{2},-\frac{7}{2}\\ \hfill -\frac{5}{2}\end{array}|-1\right]=\frac{379\sqrt{2}}{15}-7\ln (1+\sqrt{2});\end{array}$$

that have been justified by applying Theorem 9 according to the second author [13].

Another topic concerns the variety of ${}_{3}F_2\left(1\right)$-series under known transformations. Recall the Kummer and Thomae transformation formulae (cf. Bailey [1] (p. 98 and §3.2))

$$\begin{array}{cc}{}_{3}F_2\left[\begin{array}{c}\hfill a,c,e\\ \hfill b,d\end{array}|1\right]\hfill & ={}_{3}F_2\left[\begin{array}{c}a,b-c,b-e\\ a+\Delta ,b\end{array}|1\right]\frac{\Gamma (\Delta )\Gamma \left(d\right)}{\Gamma (a+\Delta )\Gamma (d-a)},\hfill \\ {}_{3}F_2\left[\begin{array}{c}\hfill a,c,e\\ \hfill b,d\end{array}|1\right]\hfill & ={}_{3}F_2\left[\begin{array}{c}b-a,d-a,\Delta \\ c+\Delta ,e+\Delta \end{array}|1\right]\frac{\Gamma (\Delta )\Gamma \left(b\right)\Gamma \left(d\right)}{\Gamma \left(a\right)\Gamma (c+\Delta )\Gamma (e+\Delta )};\hfill \end{array}$$

where $\Delta =b+d-a-c-e$ denotes the parameter excess. As immediate applications, Formula (A) in Section 3.1 is transformed into the two numerical series below

$$\begin{array}{cc}\hfill \left(\mathrm{A}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{4},& \frac{1}{4},& \hfill -\frac{1}{4}\\ & \frac{1}{2},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{{\Gamma }^2\left(\frac{3}{4}\right)}{\sqrt{2{\pi }^3}}\left\{\pi +2\ln (1+\sqrt{2})\right\},\hfill \\ \hfill \left(\mathrm{B}\right)& {}_{3}F_2\left[\begin{array}{ccc}\frac{1}{2},& \frac{1}{2},& \hfill -\frac{1}{4}\\ & \frac{3}{4},& \hfill \frac{3}{4}\end{array}|1\right]=\frac{{\Gamma }^4\left(\frac{3}{4}\right)}{{\pi }^2\sqrt{2}}\left\{\pi +2\ln (1+\sqrt{2})\right\}.\hfill \end{array}$$

These are not in the list provided by Brychkov [14] (Chapter 8). Numerous values can be obtained for further ${}_{3}F_2\left(1\right)$-series by applying the Kummer and Thomae transformations to other series evaluated in Section 3.