Further Closed Formulae of Exotic ₃F₂-Series

Abstract

By making use of the linearization method, we examine a class of nonterminating ${}_3{F}_2$-series with five free integer parameters that yields twenty summation formulae. Under the Kummer and Thomae transformations, six classes of exotic ${}_3{F}_2$-series are consequently evaluated in closed forms. There are overall 100 identities recorded in the present paper

1. Introduction and Outline

Denote by $\mathbb{N}$ and $\mathbb{Z}$, respectively, the sets of natural numbers and integers with ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$. The shifted factorials are given by ${\left(x\right)}_0={⟨x⟩}_0=\equiv 1$ and

$$\left. \begin{array}{c}{\left(x\right)}_n=x(x+1)\cdots (x+n-1)\\ {⟨x⟩}_n=x(x-1)\cdots (x-n+1)\end{array}\right\}\mathrm{for}n\in \mathbb{N}.$$

We can express them, even when $n\in \mathbb{Z}$, as the quotients

$${\left(x\right)}_n=\frac{\Gamma (x+n)}{\Gamma \left(x\right)}\mathrm{and}{⟨x⟩}_n=\frac{\Gamma (1+x)}{\Gamma (1+x-n)},$$

where the $\Gamma$-function is defined by the Euler integral

$$\Gamma \left(x\right)={\int }_0^{\infty }{u}^{x-1}{e}^{-u}\mathrm{d}u\mathrm{for}\Re \left(x\right)>0.$$

For brevity, their fractional forms are concisely shortened as

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

According to Bailey [1], the generalized hypergeometric series is defined by

$${}_{1+p}{F}_p\left[\begin{array}{cccc}{a}_0,& a_1,& \cdots ,& a_p\\ & b_1,& \cdots ,& b_p\end{array}|z\right]=\sum _{n=0}^{\infty }\frac{{\left(a_0\right)}_n{\left(a_1\right)}_n\cdots {\left(a_p\right)}_n}{n!{\left(b_1\right)}_n\cdots {\left(b_p\right)}_n}{z}^n.$$

When $z=1$, this series is convergent only if the “parameter excess” (i.e., the difference between the sum of the denominator parameters and that of the numerator ones) has a positive real part.

There exist many strange evaluations of hypergeometric series (cf. [2,3,4,5,6,7,8] for example). Recently, Campbell, D’Aurizio and Sondow [9,10] discovered two mysterious-looking formulae (see D1 and D12)

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}\frac{1}{4},& \frac{3}{4},& \frac{1}{2}\\ \\ & 1,& \frac{3}{2}\end{array}|1\right]& =\frac{4ln(1+\sqrt{2})}{\pi },\hfill \\ \hfill {}_3{F}_2\left[\begin{array}{ccc}\frac{1}{4},& \frac{3}{4},& \hfill -\frac{1}{2}\\ \\ & 1,& \hfill \frac{1}{2}\end{array}|1\right]& =\frac{\sqrt{2}+ln(1+\sqrt{2})}{\pi }.\hfill \end{array}$$

Campbell and Abrarov [11] found, among the others, the following two further ones (see F10 and G8)

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}\frac{3}{2},& \frac{3}{4},& \hfill -\frac{1}{4}\\ \\ & 1,& \hfill \frac{7}{4}\end{array}|1\right]& =\frac{3\sqrt{\pi }\left\{3\sqrt{2}-log(1+\sqrt{2})\right\}}{2\Gamma {\left(\frac{1}{4}\right)}^2},\hfill \\ \hfill {}_3{F}_2\left[\begin{array}{ccc}\frac{3}{2},& \frac{1}{4},& \frac{5}{4}\\ \\ & 1,& \frac{9}{4}\end{array}|1\right]& =\frac{5\sqrt{\pi }\left\{3\sqrt{2}-log(1+\sqrt{2})\right\}}{8\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \end{array}$$

These series are said “exotic” because one numerator parameter minus a denominator parameter results in a negative integer. By examining carefully these seemingly unrelated series, we find that they are connected, under the Thomae and Kummer transformation (cf. Bailey [1] §3.2 and Page 98), to the following ${}_3{F}_2$-series

$$\mathcal{F}(a,c,e;b,d):={}_3{F}_2\left[\begin{array}{ccc}1+a,& c,& \frac{1}{2}+e\\ \\ & \frac{3}{4}+b,& \frac{5}{4}+d\end{array}|1\right],\left\{\genfrac{}{}{0.0pt}{}{\Delta :=\frac{1}{2}+b+d-a-c-e>0}{\sigma :=b+d-a-c-e\ge 0}\right\},$$

where $a,b,c,d,e\in \mathbb{Z}$ satisfying the conditions $a\ge 0$ and $c>0$ so that the both series involved are nonterminating. When $\sigma =b+d-a-c-e\ge 0$, the series is convergent, because in this case the parameter excess $\Delta =\sigma +\frac{1}{2}>0$ (i.e., the sum of the denominator parameters minus that of the numerator ones).

Classically, there are three typical summation theorems (for the ${}_3{F}_2$-series) discovered by Dixon, Watson and Whipple (cf. Bailey [1] §3.1, §3.3 and §3.4). However, neither of them can evaluate the afore-displayed series in closed form. In particular, the formulae for the ${}_3{F}_2$-series presented in this paper are not present in the recent paper by the author [12], and two useful compendiums: ([13] §8.1.2 and [14] §7.4.4), where numerous closed formulae are collected for the ${}_3{F}_2\left(1\right)$ series with numerical parameters.

By applying the linearization method (cf. [15,16,17,18]), we shall transform, in the next section, the evaluation of $\mathcal{F}$-series into the ${\Omega }_{m,n}$-series treated recently by the author [19]. The main results are summarized in the conclusive theorem as well as twenty closed formulae for the $\mathcal{F}$-series. Finally in Section 3, analytic formulae for six further classes of exotic ${}_3{F}_2$-series will be provided by employing the Thomae and Kummer transformations (cf. Bailey [1] §3.2 and Page 98) to the $\mathcal{F}$-series.

In order to ensure the accuracy, all the formulae appearing in this paper have been checked numerically by appropriately devised Mathematica commands.

2. Linearization Procedure for the $\mathcal{F}$-Series

In this section, we shall reduce, by means of the linearization method (cf. [15,16,17,18]), the $\mathcal{F}$-series to specific instances of a known ${\Omega }_{m,n}(x,y)$ function, that has recently been examined by the author [19].

2.1. $a=0$

According to the Chu–Vandermonde convolution identity on binomial coefficients, it is routine to establish the following lemma.

Lemma 1

(Linear relation: $m\in {\mathbb{N}}_0$).

$${(A+n)}_m=\sum _{k=0}^m{(B+n)}_k{X}_{k}where{X}_k=\left(\genfrac{}{}{0pt}{}{m}{k}\right){(A-B)}_{m-k}.$$

Specifying the above relation to the equality

$${(1+n)}_a=\sum _{k=0}^a{(c+n)}_k{X}_k\left(a\right)\mathrm{where}{X}_k\left(a\right)=\left(\genfrac{}{}{0pt}{}{a}{k}\right){(1-c)}_{a-k}$$

and then substituting it into the $\mathcal{F}$-series, we have the double series

$$\begin{array}{cc}\hfill \mathcal{F}(a,c,e;b,d)& =\sum _{n=0}^{\infty }{\left[\begin{array}{ccc}1+a,& c,& \frac{1}{2}+e\\ \\ 1,& \frac{3}{4}+b,& \frac{5}{4}+d\end{array}\right]}_n\sum _{k=0}^a\frac{{(c+n)}_k}{{(1+n)}_a}{X}_k\left(a\right)\hfill \\ & =\sum _{k=0}^a\frac{{\left(c\right)}_k}{{\left(1\right)}_a}{X}_k\left(a\right)\sum _{n=0}^{\infty }{\left[\begin{array}{cc}c+k,& \frac{1}{2}+e\\ \\ \frac{3}{4}+b,& \frac{5}{4}+d\end{array}\right]}_n.\hfill \end{array}$$

This results in the reduction formula as below.

Proposition 1

(Reduction formula from $a>0$ to $a=0$).

$$\mathcal{F}(a,c,e;b,d)=\sum _{k=0}^a{(-1)}^a\left(\genfrac{}{}{0pt}{}{-c}{k}\right)\left(\genfrac{}{}{0pt}{}{c-1}{a-k}\right)\mathcal{F}(0,c+k,e;b,d).$$

2.2. $b=d$

The $\mathcal{F}$-series can further be reduced to the case $b=d$.

When $b>d$, we can specify Lemma 1 to the equality

$${(\frac{5}{4}+d+n)}_{b-d}=\sum _{k=0}^{b-d}{(c+n)}_k{Y}_k(b,d)\mathrm{where}{Y}_k(b,d)=\left(\genfrac{}{}{0pt}{}{b-d}{k}\right){(\frac{5}{4}-c+d)}_{b-d-k}.$$

Putting this inside the $\mathcal{F}$-series, we have the double series

$$\begin{array}{cc}\hfill \mathcal{F}(0,c,e;b,d)& =\sum _{n=0}^{\infty }{\left[\begin{array}{c}c,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+d\end{array}\right]}_n\sum _{k=0}^{b-d}\frac{{(c+n)}_k}{{(\frac{5}{4}+d+n)}_{b-d}}{Y}_k(b,d)\hfill \\ & =\sum _{k=0}^{b-d}\frac{{\left(c\right)}_k}{{(\frac{5}{4}+d)}_{b-d}}{Y}_k(b,d)\sum _{n=0}^{\infty }{\left[\begin{array}{c}c+k,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+b\end{array}\right]}_n.\hfill \end{array}$$

This yields the following reduction formula.

Proposition 2

(Reduction formula from $b>d$ to $b=d$).

$$\mathcal{F}(0,c,e;b,d)=\sum _{k=0}^{b-d}\left(\genfrac{}{}{0pt}{}{b-d}{k}\right)\frac{{\left(c\right)}_k{(\frac{5}{4}-c+d)}_{b-d-k}}{{(\frac{5}{4}+d)}_{b-d}}\mathcal{F}(0,c+k,e;b,b).$$

Alternatively, for $b<d$, we can specify Lemma 1 to the equality

$${(\frac{3}{4}+b+n)}_{d-b}=\sum _{k=0}^{d-b}{(c+n)}_k{\mathcal{Y}}_k(b,d)\mathrm{where}{\mathcal{Y}}_k(b,d)=\left(\genfrac{}{}{0pt}{}{d-b}{k}\right){(\frac{3}{4}+b-c)}_{d-b-k}.$$

Substituting this into the $\mathcal{F}$-series, we have the double series

$$\begin{array}{cc}\hfill \mathcal{F}(0,c,e;b,d)& =\sum _{n=0}^{\infty }{\left[\begin{array}{c}c,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+d\end{array}\right]}_n\sum _{k=0}^{d-b}\frac{{(c+n)}_k}{{(\frac{3}{4}+b+n)}_{d-b}}{\mathcal{Y}}_k(b,d)\hfill \\ & =\sum _{k=0}^{d-b}\frac{{\left(c\right)}_k}{{(\frac{3}{4}+b)}_{d-b}}{\mathcal{Y}}_k(b,d)\sum _{n=0}^{\infty }{\left[\begin{array}{c}c+k,\frac{1}{2}+e\\ \\ \frac{3}{4}+d,\frac{5}{4}+d\end{array}\right]}_n.\hfill \end{array}$$

This gives rise to another reduction formula.

Proposition 3

(Reduction formula from $b<d$ to $b=d$).

$$\mathcal{F}(0,c,e;b,d)=\sum _{k=0}^{d-b}\left(\genfrac{}{}{0pt}{}{d-b}{k}\right)\frac{{\left(c\right)}_k{(\frac{3}{4}+b-c)}_{d-b-k}}{{(\frac{3}{4}+b)}_{d-b}}\mathcal{F}(0,c+k,e;d,d).$$

2.3. $c=e$

The $\mathcal{F}$-series can further be reduced to the case $c=e$. For this purpose, we have to show the following linearization lemma.

Lemma 2

(Linear relation: $m\in {\mathbb{N}}_0$).

$${(A+n)}_m=\sum _{k=0}^m{⟨B+2n⟩}_k{Z}_{k}where{Z}_k=\sum _{i=0}^k\frac{{(-1)}^{k-i}}{k!}\left(\genfrac{}{}{0pt}{}{k}{i}\right){(A-\frac{B-i}{2})}_m.$$

Proof. 

By substitution, it suffices to evaluate the double sum

$$\mathcal{S}:=\sum _{k=0}^m{⟨B+2n⟩}_k\sum _{i=0}^k\frac{{(-1)}^{k-i}}{k!}\left(\genfrac{}{}{0pt}{}{k}{i}\right){(A-\frac{B-i}{2})}_m={(A+n)}_m.$$

By exchanging the order of summations, we can reformulate it as

$$\begin{array}{cc}\hfill \mathcal{S}& =\sum _{i=0}^m\frac{{⟨B+2n⟩}_i}{i!}{(A-\frac{B-i}{2})}_m\sum _{k=i}^m{(-1)}^{k-i}\left(\genfrac{}{}{0pt}{}{B+2n-i}{k-i}\right)\hfill \\ & =\sum _{i=0}^m{(-1)}^{m-i}\frac{{⟨B+2n⟩}_i}{i!}{(A-\frac{B-i}{2})}_m\left(\genfrac{}{}{0pt}{}{B+2n-i-1}{m-i}\right)\hfill \\ & =\frac{{⟨B+2n⟩}_{m+1}}{m!}\sum _{i=0}^m{(-1)}^{m-i}\left(\genfrac{}{}{0pt}{}{m}{i}\right)\frac{{(A-\frac{B-i}{2})}_m}{B+2n-i}\hfill \\ & =\frac{{(B+2n)}_{m+1}}{m!}\times \frac{m!{(A+n)}_m}{{⟨B+2n⟩}_{m+1}}={(A+n)}_m,\hfill \end{array}$$

where the last line is justified by finite difference calculus (cf. [20,21]). □

First for $c<e$, we have from Lemma 2 the equality

$$\begin{array}{cc}& {(\frac{1}{2}+c+n)}_{e-c}=\sum _{k=0}^{e-c}{⟨2b+2n+\frac{1}{2}⟩}_k{\mathcal{Z}}_k(b,c,e),\hfill \\ \hfill \mathrm{where}& {\mathcal{Z}}_k(b,c,e)=\sum _{i=0}^k\frac{{(-1)}^{k-i}}{k!}\left(\genfrac{}{}{0pt}{}{k}{i}\right){(c-b+\frac{1+2i}{4})}_{e-c}.\hfill \end{array}$$

By inserting this into the $\mathcal{F}$-series, we obtain the double series below

$$\begin{array}{cc}\hfill \mathcal{F}(0,c,e;b,b)& =\sum _{n=0}^{\infty }{\left[\begin{array}{c}c,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+b\end{array}\right]}_n\sum _{k=0}^{e-c}\frac{{⟨2b+2n+\frac{1}{2}⟩}_k}{{(\frac{1}{2}+c+n)}_{e-c}}{\mathcal{Z}}_k(b,c,e)\hfill \\ & =\sum _{k=0}^{e-c}{(-1)}^k\frac{{(-\frac{1}{2}-2b)}_k}{{(\frac{1}{2}+c)}_{e-c}}{\mathcal{Z}}_k(b,c,e)\sum _{n=0}^{\infty }{\left[\begin{array}{c}c,\frac{1}{2}+c\\ \\ \frac{3-2k}{4}+b,\frac{5-2k}{4}+b\end{array}\right]}_n.\hfill \end{array}$$

Writing the inner sum concerning n in terms of the $\mathcal{F}$-series, we immediately establish the reduction formula as in the following proposition.

Proposition 4

(Reduction formula from $c<e$ to $c=e$).

$$\mathcal{F}(0,c,e;b,b)=\sum _{k=0}^{e-c}{(-1)}^k\frac{{(-\frac{1}{2}-2b)}_k}{{(\frac{1}{2}+c)}_{e-c}}{\mathcal{Z}}_k(b,c,e)\mathcal{F}(0,c,c;b-\frac{k}{2},b-\frac{k}{2}).$$

When $c>e$ and $e>0$, we infer from Lemma 2 that

$${(e+n)}_{c-e}=\sum _{k=0}^{c-e}{⟨2b+2n+\frac{1}{2}⟩}_k{\mathcal{Z}}_k(b,c,e),$$

$$\mathrm{where}\hspace{1em}{\mathcal{Z}}_k(b,c,e)=\sum _{i=0}^k\frac{{(-1)}^{k-i}}{k!}\left(\genfrac{}{}{0pt}{}{k}{i}\right){(e-b+\frac{2i-1}{4})}_{c-e}.$$

(1)

Putting this inside the $\mathcal{F}$-series, we can analogously treat the double series

$$\begin{array}{cc}\hfill \mathcal{F}(0,c,e;b,b)& =\sum _{n=0}^{\infty }{\left[\begin{array}{c}c,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+b\end{array}\right]}_n\sum _{k=0}^{c-e}\frac{{⟨2b+2n+\frac{1}{2}⟩}_k}{{(e+n)}_{c-e}}{\mathcal{Z}}_k(b,c,e)\hfill \\ & =\sum _{k=0}^{c-e}{(-1)}^k\frac{{(-\frac{1}{2}-2b)}_k}{{\left(e\right)}_{c-e}}{\mathcal{Z}}_k(b,c,e)\sum _{n=0}^{\infty }{\left[\begin{array}{c}e,\frac{1}{2}+e\\ \\ \frac{3-2k}{4}+b,\frac{5-2k}{4}+b\end{array}\right]}_n.\hfill \end{array}$$

Instead, for $c>e$ and $e\le 0$, reformulate first the $\mathcal{F}$-series by reindexing

$$\begin{array}{cc}\hfill \mathcal{F}(0,c,e;b,b)=\mathcal{F}(0,1+c-e,1;1+b-e,1+b-e)& \\ \hfill \times {\left[\begin{array}{c}c,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+b\end{array}\right]}_{1-e}+\sum _{n=0}^{-e}{\left[\begin{array}{c}c,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+b\end{array}\right]}_n.\end{array}$$

Then according to Lemma 2, we have another equality

$${(1+n)}_{c-e}=\sum _{k=0}^{c-e}{⟨\frac{5}{2}+2b-2e+2n⟩}_k{\mathcal{Z}}_k(b,c,e),$$

where the connection coefficients ${\mathcal{Z}}_k(b,c,e)$ coincide with those given by (1). Now, by substitution, we have another double series

$$\begin{array}{cc}& \mathcal{F}(0,1+c-e,1;1+b-e,1+b-e)\hfill \\ & =\sum _{n=0}^{\infty }{\left[\begin{array}{c}1+c-e,\frac{3}{2}\\ \\ \frac{7}{4}+b-e,\frac{9}{4}+b-e\end{array}\right]}_n\sum _{k=0}^{c-e}\frac{{⟨\frac{5}{2}+2b-2e+2n⟩}_k}{{(1+n)}_{c-e}}{\mathcal{Z}}_k(b,c,e)\hfill \\ & =\sum _{k=0}^{c-e}{(-1)}^k\frac{{(2e-2b-\frac{5}{2})}_k}{(c-e)!}{\mathcal{Z}}_k(b,c,e)\sum _{n=0}^{\infty }{\left[\begin{array}{c}1,\frac{3}{2}\\ \\ \frac{7-2k}{4}+b-e,\frac{9-2k}{4}+b-e\end{array}\right]}_n.\hfill \end{array}$$

Summing up, we have established the reduction formula to the case $c=e$.

Proposition 5

(Reduction formula from $c>e$ to $c=e$).

$$\begin{array}{cc}\hfill e>0:\mathcal{F}(0,c,e;b,b)=& \sum _{k=0}^{c-e}{(-1)}^k\frac{{(-\frac{1}{2}-2b)}_k}{{\left(e\right)}_{c-e}}{\mathcal{Z}}_k(b,c,e)\mathcal{F}(0,e,e;b-\frac{k}{2},b-\frac{k}{2}),\hfill \\ \hfill e\le 0:\mathcal{F}(0,c,e;b,b)=& \sum _{n=0}^{-e}{\left[\begin{array}{c}c,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+b\end{array}\right]}_n+\sum _{k=0}^{c-e}{(-1)}^k\frac{{(2e-2b-\frac{5}{2})}_k}{(c-e)!}{\mathcal{Z}}_k(b,c,e)\hfill \\ \hfill \times & {\left[\begin{array}{c}c,\frac{1}{2}+e\\ \\ \frac{3}{4}+b,\frac{5}{4}+b\end{array}\right]}_{1-e}\mathcal{F}(0,1,1;1+b-e-\frac{k}{2},1+b-e-\frac{k}{2}).\hfill \end{array}$$

Observe that the parameter excess $\Delta \ge \frac{1}{2}$ for the $\mathcal{F}$-series is not diminished hitherto by the established reduction formulae. Consequently, all the $\mathcal{F}$-series displayed on the right hand sides of Propositions 4 and 5 have the parameter excess $\Delta \ge \frac{1}{2}$, and can be expressed as the following bisection series

$$\begin{array}{cc}\hfill \mathcal{F}(0,c,c;b,b)=\sum _{n=0}^{\infty }\frac{{\left(2c\right)}_{2n}}{{(2b+\frac{3}{2})}_{2n}}=\frac{1}{2}\times {}_2{F}_1\left[\begin{array}{c}1,2c\\ \frac{3}{2}+2b\end{array}|1\right]& \\ \hfill +\frac{1}{2}\times {}_2{F}_1\left[\begin{array}{c}1,2c\\ \frac{3}{2}+2b\end{array}|-1\right]& ,\hfill \end{array}$$

where $b,c\in \mathbb{N}$ subject to the condition $b\ge c$. Therefore, to evaluate the $\mathcal{F}$-series explicitly, it suffices to do that for the above bisection series.

2.4. ${\Omega }_{m,n}$-Series

In a recent paper [19], the author examined a more general series

$${\Omega }_{m,n}(x,y):={}_2{F}_1\left[\begin{array}{cc}x,& m-x\\ & n+\frac{1}{2}\end{array}|y^2\right]\mathrm{where}m,n\in \mathbb{Z}$$

(2)

and proved the following evaluation formula.

Theorem 1

(Chu [19] Theorems 4 and 8: Recurrence formula). For the two natural numbers m and n satisfying $m<n$, there holds the following formula

$$\begin{array}{cc}\hfill {\Omega }_{m,n}(x,y)& =\frac{{\left(\frac{1}{2}\right)}_n}{y^{2n}}\sum _{i=0}^{n-m}\left(\genfrac{}{}{0pt}{}{n-m}{i}\right)\frac{{\left(x\right)}_i{(m-x)}_{n-m-i}}{{(2x-n+i)}_i{(m-2x-i)}_{n-m-i}}\hfill \\ & \times \sum _{k=0}^n{(-1)}^{n-k}\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{2x+2i-2k}{{(2x+2i-n-k)}_{n+1}}{\Omega }_{0,0}(x+i-k,y),\hfill \end{array}$$

where the series ${\Omega }_{0,0}$ is evaluated by

$${\Omega }_{0,0}(x,y)={}_2{F}_1\left[\begin{array}{cc}x,& -x\\ & \frac{1}{2}\end{array}|y^2\right]=cos(2xarcsiny).$$

Hence, the $\mathcal{F}$-series can be evaluated in terms of the $\Omega$-series by the theorem below.

Theorem 2

($b\ge c:b,c\in \mathbb{N}$).

$$\begin{array}{cccc}& \mathcal{F}(0,c,c;b,b)=\frac{1}{2}\underset{x\to 1}{lim}{\Omega }_{2c+1,2b+1}(x,1)+\frac{1}{2}\underset{x\to 1}{lim}{\Omega }_{2c+1,2b+1}(x,\sqrt{-1})\hfill & & \\ \hfill with& {\Omega }_{0,0}(x,1)=cos\left(\pi x\right)and{\Omega }_{0,0}(x,\sqrt{-1})=cosh\left(2xln(1+\sqrt{2})\right).\hfill & \end{array}$$

2.5. Conclusive Theorem and Examples (Class-A)

Based on the preceding reduction formulae, we may evaluate, for any quintuple integers $a,b,c,d,e\in \mathbb{Z}$ subject to $a\ge 0$, $c>0$ and $\sigma =b+d-a-c-e>0$, the $\mathcal{F}$-series by carrying out the following procedure:

• Step-A: If $a=0$, go directly to Step-B. Otherwise for $a>0$, according to Proposition 1, express $\mathcal{F}(a,c,e;b,d)$ in terms of $\mathcal{F}(0,c,e;b,d)$, and then go to Step-B.
• Step-B: By means of Propositions 2 and 3, express $\mathcal{F}(0,c,e;b,d)$ in terms of $\mathcal{F}(0,c,e;b,b)$, and then go to Step-C.
• Step-C: In virtu of Propositions 4 and 5, express $\mathcal{F}(0,c,e;b,b)$ in terms of $\mathcal{F}(0,c,c;b,b)$, and then go to Step-D.
• Step-D: Finally by applying Theorems 1 and 2, evaluate $\mathcal{F}(0,c,c;b,b)$ explicitly in terms of the $\Omega$-series.

Therefore, we have validated the conclusive theorem as below.

Theorem 3

(Conclusion). For any quintuple integers

$$a,b,c,d,e\in \mathbb{Z}subjecttoa\ge 0,c>0and\sigma =b+d-a-c-e>0,$$

the nonterminating $\mathcal{F}(a,c,e;b,d)$ series can always be evaluated by finitely linear sums of trigonometric function $cos\left(\pi x\right)$ and hyperbolic function $cosh\left(2xln(1+\sqrt{2})\right)$, where $x\in \mathbb{Z}$ and the coefficients are rational numbers.

According to the afore-described procedure, we have written appropriate Mathematica commands to determine explicitly closed form expressions for $\mathcal{F}(a,c,e;b,d)$ series. Twenty summation formulae are displayed below, where the argument “1” will be suppressed from the notation of ${}_3{F}_2$-series for the sake of brevity. We shall call these series “Class-A”. Among them, an equivalent form of A5 has been obtained by Campbell and Abrarov ([11] Equation (18)).

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{A}\mathbf{1}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill \frac{1}{2};& \hfill \frac{5}{4},& \hfill \frac{7}{4}]& =\frac{3}{\sqrt{2}}log(1+\sqrt{2}).\hfill \\ \mathbf{A}\mathbf{2}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill \frac{1}{2};& \hfill \frac{7}{4},& \hfill \frac{9}{4}]& =-5\left\{1-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{3}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill \frac{1}{2};& \hfill \frac{5}{4},& \hfill \frac{11}{4}]& =\frac{-7}{15}\left\{1-3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{4}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill \frac{3}{2};& \hfill \frac{5}{4},& \hfill \frac{11}{4}]& =\frac{7}{12}\left\{2+3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{5}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill \frac{3}{2};& \hfill \frac{7}{4},& \hfill \frac{9}{4}]& =\frac{15}{4}\left\{2-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{6}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill \frac{3}{2};& \hfill \frac{9}{4},& \hfill \frac{11}{4}]& =\frac{35}{6}\left\{4-3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{7}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill -\frac{1}{2};& \hfill \frac{3}{4},& \hfill \frac{5}{4}]& =\frac{1}{3}\left\{1-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{8}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill -\frac{1}{2};& \hfill \frac{1}{4},& \hfill \frac{7}{4}]& =\frac{3}{5}\left\{1-3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{9}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill -\frac{1}{2};& \hfill \frac{7}{4},& \hfill \frac{9}{4}]& =\frac{-3}{7}\left\{3-4\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{10}.\hfill & {}_3{F}_2[1,\hfill & \hfill 1,& \hfill -\frac{3}{2};& \hfill \frac{3}{4},& \hfill \frac{5}{4}]& =\frac{1}{35}\left\{3-4\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \end{array}\hfill & \end{array}$$

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{A}\mathbf{11}.\hfill & {}_3{F}_2[1,\hfill & \hfill 2,& \hfill \frac{1}{2};& \hfill \frac{7}{4},& \hfill \frac{9}{4}]& =\frac{5}{8}\left\{2+\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{12}.\hfill & {}_3{F}_2[1,\hfill & \hfill 2,& \hfill -\frac{1}{2};& \hfill \frac{5}{4},& \hfill \frac{7}{4}]& =\frac{3}{20}\left\{2-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{13}.\hfill & {}_3{F}_2[1,\hfill & \hfill 2,& \hfill -\frac{1}{2};& \hfill \frac{3}{4},& \hfill \frac{9}{4}]& =\frac{5}{84}\left\{2-5\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{14}.\hfill & {}_3{F}_2[1,\hfill & \hfill 2,& \hfill -\frac{1}{2};& \hfill \frac{7}{4},& \hfill \frac{9}{4}]& =\frac{1}{14}\left\{8+\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{15}.\hfill & {}_3{F}_2[1,\hfill & \hfill 2,& \hfill -\frac{3}{2};& \hfill \frac{3}{4},& \hfill \frac{9}{4}]& =\frac{-5}{77}\left\{1+\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{16}.\hfill & {}_3{F}_2[2,\hfill & \hfill 2,& \hfill \frac{1}{2};& \hfill \frac{7}{4},& \hfill \frac{13}{4}]& =\frac{135}{224}\left\{6-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{17}.\hfill & {}_3{F}_2[2,\hfill & \hfill 2,& \hfill -\frac{1}{2};& \hfill \frac{5}{4},& \hfill \frac{11}{4}]& =\frac{7}{48}\left\{2-3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{18}.\hfill & {}_3{F}_2[2,\hfill & \hfill 2,& \hfill -\frac{1}{2};& \hfill \frac{9}{4},& \hfill \frac{11}{4}]& =\frac{1}{24}\left\{2+9\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{19}.\hfill & {}_3{F}_2[2,\hfill & \hfill 2,& \hfill -\frac{3}{2};& \hfill \frac{7}{4},& \hfill \frac{13}{4}]& =\frac{1}{22}\left\{8-3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{A}\mathbf{20}.\hfill & {}_3{F}_2[2,\hfill & \hfill 2,& \hfill -\frac{3}{2};& \hfill \frac{11}{4},& \hfill \frac{13}{4}]& =\frac{-1}{13}\left\{13-15\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \end{array}\hfill & \end{array}$$

3. The Thomae and Kummer Transformations

In the classical theory of hypergeometric series, the Thomae and Kummer transformations are fundamental (cf. Bailey [1] §3.2 and Page 98, where $\sigma =b+d-a-c-e$):

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{c}\hfill a,c,e\\ \hfill b,d\end{array}|1\right]=& {}_3{F}_2\left[\begin{array}{c}\hfill \sigma ,b-a,d-a\\ \hfill c+\sigma ,e+\sigma \end{array}|1\right]\Gamma \left[\begin{array}{c}\sigma ,b,d\\ a,c+\sigma ,e+\sigma \end{array}\right]\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{c}\hfill a,c,e\\ \hfill b,d\end{array}|1\right]=& {}_3{F}_2\left[\begin{array}{c}\hfill a,b-c,b-e\\ \hfill \sigma +a,b\end{array}|1\right]\Gamma \left[\begin{array}{c}\sigma ,d\\ \sigma +a,d-a\end{array}\right].\hfill \end{array}$$

(4)

They will be applied to the $\mathcal{F}$-series to evaluete six classes of exotic ${}_3{F}_2$-series.

3.1. Class B

Applying the Kummer transformation (4), we can express the following “Class-B” series in terms of the $\mathcal{F}$-series (where $\sigma =b+d-a-c-e$):

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}1+a,& c+\frac{1}{4},& e+\frac{3}{4}\\ \\ & b+\frac{3}{2},& d+\frac{5}{4}\end{array}|1\right]=\Gamma \left[\begin{array}{c}b+\frac{3}{2},\sigma +\frac{3}{4}\\ \\ b-a+\frac{1}{2},\sigma +a+\frac{7}{4}\end{array}\right]& \\ \hfill \times {}_3{F}_2\left[\begin{array}{ccc}1+a,& d-c+1,& d-e+\frac{1}{2}\\ \\ & d+\frac{5}{4},& \sigma +a+\frac{7}{4}\end{array}|1\right]& .\hfill \end{array}$$

Then we can derive the following closed formulae for these series (except for divergent series) from those displayed in “Class A”.

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{B}\mathbf{1}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{1}{4},& \hfill \frac{3}{4};& \hfill \frac{3}{2},& \hfill \frac{5}{4}]& =\sqrt{2}log(1+\sqrt{2}).\hfill \\ \mathbf{B}\mathbf{2}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{1}{4},& \hfill \frac{7}{4};& \hfill \frac{5}{2},& \hfill \frac{5}{4}]& =\frac{2}{5}\left\{1+2\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{B}\mathbf{3}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{1}{4},& \hfill \frac{7}{4};& \hfill \frac{5}{2},& \hfill \frac{9}{4}]& =\frac{3}{2}\left\{2-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{B}\mathbf{4}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{3}{2},& \hfill \frac{9}{4}]& =\frac{5}{2}\left\{2-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{B}\mathbf{5}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{5}{2},& \hfill \frac{9}{4}]& =5\left\{4-3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{B}\mathbf{6}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{7}{4},& \hfill \frac{9}{4};& \hfill \frac{5}{2},& \hfill \frac{13}{4}]& =\frac{9}{5}\left\{8-5\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{B}\mathbf{7}.\hfill & {}_3{F}_2[2,\hfill & \hfill \frac{1}{4},& \hfill \frac{7}{4};& \hfill \frac{7}{2},& \hfill \frac{5}{4}]& =\frac{2}{9}\left\{4+3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{B}\mathbf{8}.\hfill & {}_3{F}_2[2,\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{5}{2},& \hfill \frac{9}{4}]& =\frac{5}{4}\left\{-2+3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{B}\mathbf{9}.\hfill & {}_3{F}_2[2,\hfill & \hfill \frac{5}{4},& \hfill \frac{7}{4};& \hfill \frac{7}{2},& \hfill \frac{9}{4}]& =5\left\{2-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{B}\mathbf{10}.\hfill & {}_3{F}_2[2,\hfill & \hfill \frac{9}{4},& \hfill \frac{11}{4};& \hfill \frac{9}{2},& \hfill \frac{13}{4}]& =30\left\{4-3\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \end{array}\hfill & \end{array}$$

3.2. Class C

By means of the Kummer transformation (4), we can express the “Class-C” series below in terms of the $\mathcal{F}$-series (where $\sigma =b+d-a-c-e$):

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}1+a,& c+\frac{1}{4},& e+\frac{3}{4}\\ \\ & b+\frac{3}{2},& d+\frac{3}{4}\end{array}|1\right]=\Gamma \left[\begin{array}{c}\sigma +\frac{1}{4},b+\frac{3}{2}\\ \\ \sigma +a+\frac{5}{4},b-a+\frac{1}{2}\end{array}\right]& \\ \hfill \times {}_3{F}_2\left[\begin{array}{ccc}1+a,& d-e,& d-c+\frac{1}{2}\\ \\ & d+\frac{3}{4},& \sigma +a+\frac{5}{4}\end{array}|1\right]& .\hfill \end{array}$$

Then the closed formulae below for these series ( except for divergent series) follow directly from those recorded in “Class A”.

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{C}\mathbf{1}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{1}{4},& \hfill \frac{3}{4};& \hfill \frac{3}{2},& \hfill \frac{7}{4}]& =\frac{3}{2}\left\{2-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{C}\mathbf{2}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{1}{4},& \hfill \frac{3}{4};& \hfill \frac{5}{2},& \hfill \frac{7}{4}]& =\frac{3}{5}\left\{8-5\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{C}\mathbf{3}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{3}{2},& \hfill \frac{7}{4}]& =3\sqrt{2}log(1+\sqrt{2}).\hfill \\ \mathbf{C}\mathbf{4}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{3}{4},& \hfill \frac{9}{4};& \hfill \frac{5}{2},& \hfill \frac{7}{4}]& =\frac{6}{5}\left\{1+2\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{C}\mathbf{5}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{5}{4},& \hfill -\frac{1}{4};& \hfill \frac{3}{2},& \hfill \frac{3}{4}]& =\frac{2}{3}\left\{1-\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{C}\mathbf{6}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{5}{4},& \hfill -\frac{1}{4};& \hfill \frac{3}{2},& \hfill \frac{7}{4}]& =\frac{1}{4}\left\{2+\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{C}\mathbf{7}.\hfill & {}_3{F}_2[1,\hfill & \hfill \frac{5}{4},& \hfill -\frac{1}{4};& \hfill \frac{5}{2},& \hfill \frac{3}{4}]& =\frac{2}{7}\left\{5-2\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{C}\mathbf{8}.\hfill & {}_3{F}_2[2,\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{5}{2},& \hfill \frac{7}{4}]& =\frac{3}{2}\left\{2+\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{C}\mathbf{9}.\hfill & {}_3{F}_2[2,\hfill & \hfill \frac{3}{4},& \hfill \frac{9}{4};& \hfill \frac{7}{2},& \hfill \frac{7}{4}]& =\frac{6}{7}\left\{8+\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{C}\mathbf{10}.\hfill & {}_3{F}_2[2,\hfill & \hfill \frac{3}{4},& \hfill \frac{13}{4};& \hfill \frac{7}{2},& \hfill \frac{11}{4}]& =\frac{1}{2}\left\{2+9\sqrt{2}log(1+\sqrt{2})\right\}.\hfill \end{array}\hfill & \end{array}$$

3.3. Class D

By virtue of the Thomae transformation (3), we can express the following “Class-D” series in terms of the $\mathcal{F}$-series (where $\sigma =b+d-a-c-e$):

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}a+\frac{1}{2},& c+\frac{1}{4},& e+\frac{3}{4}\\ \\ & b+1,& d+\frac{1}{2}\end{array}|1\right]=\Gamma \left[\begin{array}{c}\sigma ,b+1,d+\frac{1}{2}\\ \\ a+\frac{1}{2},\sigma +c+\frac{1}{4}\sigma +e+\frac{3}{4}\end{array}\right]& \\ \hfill \times {}_3{F}_2\left[\begin{array}{ccc}\sigma ,& d-a,& b-a+\frac{1}{2}\\ \\ & \sigma +c+\frac{1}{4},& \sigma +e+\frac{3}{4}\end{array}|1\right]& .\hfill \end{array}$$

(5)

Then we find the closed formulae below for these series ( except for divergent series) as consequences of those produced in “Class A”.

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{D}\mathbf{1}.\hfill & {}_3{F}_2[\frac{1}{4},\hfill & \hfill \frac{3}{4},& \hfill \frac{1}{2};& \hfill 1,& \hfill \frac{3}{2}]& =\frac{4log(1+\sqrt{2})}{\pi }.\hfill \\ \mathbf{D}\mathbf{2}.\hfill & {}_3{F}_2[\frac{1}{4},\hfill & \hfill \frac{7}{4},& \hfill \frac{1}{2};& \hfill 2,& \hfill \frac{3}{2}]& =\frac{8\left\{\sqrt{2}+3log(1+\sqrt{2})\right\}}{9\pi }.\hfill \\ \mathbf{D}\mathbf{3}.\hfill & {}_3{F}_2[\frac{1}{4},\hfill & \hfill \frac{7}{4},& \hfill \frac{1}{2};& \hfill 1,& \hfill \frac{5}{2}]& =\frac{2\left\{\sqrt{2}+9log(1+\sqrt{2})\right\}}{5\pi }.\hfill \\ \mathbf{D}\mathbf{4}.\hfill & {}_3{F}_2[\frac{1}{4},\hfill & \hfill \frac{7}{4},& \hfill \frac{3}{2};& \hfill 1,& \hfill \frac{7}{2}]& =\frac{8\left\{2\sqrt{2}+3log(1+\sqrt{2})\right\}}{9\pi }.\hfill \\ \mathbf{D}\mathbf{5}.\hfill & {}_3{F}_2[\frac{3}{4},\hfill & \hfill \frac{5}{4},& \hfill \frac{1}{2};& \hfill 1,& \hfill \frac{5}{2}]& =\frac{2\left\{\sqrt{2}+log(1+\sqrt{2})\right\}}{\pi }.\hfill \end{array}\hfill & \end{array}$$

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{D}\mathbf{6}.\hfill & {}_3{F}_2[\frac{3}{4},\hfill & \hfill \frac{5}{4},& \hfill \frac{1}{2};& \hfill 2,& \hfill \frac{3}{2}]& =\frac{8\left\{\sqrt{2}-log(1+\sqrt{2})\right\}}{\pi }.\hfill \\ \mathbf{D}\mathbf{7}.\hfill & {}_3{F}_2[\frac{3}{4},\hfill & \hfill \frac{5}{4},& \hfill \frac{3}{2};& \hfill 1,& \hfill \frac{7}{2}]& =\frac{8\left\{4\sqrt{2}+log(1+\sqrt{2})\right\}}{7\pi }.\hfill \\ \mathbf{D}\mathbf{8}.\hfill & {}_3{F}_2[\frac{5}{4},\hfill & \hfill -\frac{1}{4},& \hfill \frac{1}{2};& \hfill 1,& \hfill \frac{3}{2}]& =\frac{4\left\{2\sqrt{2}-log(1+\sqrt{2})\right\}}{3\pi }.\hfill \\ \mathbf{D}\mathbf{9}.\hfill & {}_3{F}_2[\frac{5}{4},\hfill & \hfill -\frac{1}{4},& \hfill \frac{3}{2};& \hfill 1,& \hfill \frac{5}{2}]& =\frac{4\left\{5\sqrt{2}-4log(1+\sqrt{2})\right\}}{7\pi }.\hfill \\ \mathbf{D}\mathbf{10}.\hfill & {}_3{F}_2[\frac{5}{4},\hfill & \hfill \frac{7}{4},& \hfill \frac{3}{2};& \hfill 2,& \hfill \frac{7}{2}]& =\frac{16\left\{\sqrt{2}-log(1+\sqrt{2})\right\}}{\pi }.\hfill \end{array}\hfill & \end{array}$$

Observing that the parameter excess of the ${}_3{F}_2$-series displayed on the right hand side of (5) equals $\Delta =\frac{1}{2}+a$, the equality (5) valid only when $a\ge 0$ and $\sigma \ge 0$. It remains a problem to evaluate, for $a<0$, the ${}_3{F}_2$-series on the left of (5). This can also be resolved by the linearization method.

According to the Pfaff–Saalschütz summation theorem (cf. Bailey [1] §2.2), it is not hard to confirm the linear relation in the following lemma.

Lemma 3

(Linear relation: $m\in {\mathbb{N}}_0$).

$${(A+n)}_m=\sum _{k=0}^m{⟨n⟩}_k{(B+n)}_{m-k}{X}_k,where{X}_k={(-1)}^k\left(\genfrac{}{}{0pt}{}{m}{k}\right)\frac{{\left(A\right)}_m{(A-B)}_k}{{\left(B\right)}_m{\left(A\right)}_k}.$$

By specializing this to the equality

$$\begin{array}{cc}& {(1+b+n)}_{-a}=\sum _{k=0}^{-a}{⟨n⟩}_k{(\frac{1}{2}+a+n)}_{-a-k}{X}_a^k,\hfill \\ & \mathrm{where}{X}_k\left(a\right)=\frac{(-a)!}{{⟨-\frac{1}{2}⟩}_{-a}}\left(\genfrac{}{}{0pt}{}{a-b-\frac{1}{2}}{k}\right)\left(\genfrac{}{}{0pt}{}{b-a}{-a-k}\right)\hfill \end{array}$$

and then substituting it into the ${}_3{F}_2$-series, we may manipulate the double sum

$$\begin{array}{cc}& {}_3{F}_2\left[\begin{array}{ccc}\frac{1}{2}+a,& \frac{1}{4}+c,& \frac{3}{4}+e\\ \\ & 1+b,& \frac{1}{2}+d\end{array}|1\right]\hfill \\ & =\sum _{n=0}^{\infty }{\left[\begin{array}{c}\frac{1}{2}+a,\frac{1}{4}+c,\frac{3}{4}+e\\ \\ 1,1+b,\frac{1}{2}+d\end{array}\right]}_n\sum _{k=0}^{-a}\frac{{⟨n⟩}_k{(\frac{1}{2}+a+n)}_{-a-k}}{{(1+b+n)}_{-a}}{X}_k\left(a\right)\hfill \\ & =\sum _{k=0}^{-a}\frac{{(\frac{1}{2}+a)}_{-a-k}}{{(1+b)}_{-a}}{X}_k\left(a\right)\sum _{n=0}^{\infty }\frac{{⟨n⟩}_k}{n!}{\left[\begin{array}{c}\frac{1}{2}-k,\frac{1}{4}+c,\frac{3}{4}+e\\ \\ 1-a+b,\frac{1}{2}+d\end{array}\right]}_n.\hfill \end{array}$$

Performing the replacement $n\to n+k$, we can express the last sum with respect to n as

$${\left[\begin{array}{c}\frac{1}{2}-k,\frac{1}{4}+c,\frac{3}{4}+e\\ \\ 1-a+b,\frac{1}{2}+d\end{array}\right]}_k{}_3{F}_2\left[\begin{array}{c}\frac{1}{2},\frac{1}{4}+c+k,\frac{3}{4}+e+k\\ \\ 1-a+b+k,\frac{1}{2}+d+k\end{array}|1\right].$$

Therefore, we have established, after some simplifications, the following transformation formula.

Theorem 4

(Reduction formula from $a<0$ to $a=0$).

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}\frac{1}{2}+a,& \frac{1}{4}+c,& \frac{3}{4}+e\\ \\ & 1+b,& \frac{1}{2}+d\end{array}|1\right]& =\sum _{k=0}^{-a}{\left[\begin{array}{c}a,\frac{1}{2}-a+b,\frac{1}{4}+c,\frac{3}{4}+e\\ \\ 1,1-a+b,1+b,\frac{1}{2}+d\end{array}\right]}_k\hfill \\ & \times {}_3{F}_2\left[\begin{array}{c}\frac{1}{2},\frac{1}{4}+c+k,\frac{3}{4}+e+k\\ \\ 1-a+b+k,\frac{1}{2}+d+k\end{array}|1\right].\hfill \end{array}$$

It should be emphasized that under this transformation, the parameter excess $\Delta =\sigma =b+d-a-c-e$ remains invariant for all the ${}_3{F}_2$-series. However the ${}_3{F}_2$-series on the right belongs to Class-D and can therefore be evaluated by (5). Ten more formulae are recorded below.

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{D}\mathbf{11}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill -\frac{1}{4};& \hfill 1,& \hfill \frac{3}{2}]& =\frac{13\sqrt{2}+log(1+\sqrt{2})}{6\pi }.\hfill \\ \mathbf{D}\mathbf{12}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{3}{4};& \hfill 1,& \hfill \frac{1}{2}]& =\frac{\sqrt{2}+log(1+\sqrt{2})}{\pi }.\hfill \\ \mathbf{D}\mathbf{13}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{3}{4};& \hfill 1,& \hfill \frac{3}{2}]& =\frac{\sqrt{2}+5log(1+\sqrt{2})}{2\pi }.\hfill \\ \mathbf{D}\mathbf{14}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{7}{4};& \hfill 1,& \hfill \frac{3}{2}]& =\frac{5\sqrt{2}+9log(1+\sqrt{2})}{6\pi }.\hfill \\ \mathbf{D}\mathbf{15}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill 1,& \hfill \frac{3}{2}]& =\frac{3\sqrt{2}-log(1+\sqrt{2})}{2\pi }.\hfill \\ \mathbf{D}\mathbf{16}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill -\frac{3}{4};& \hfill 1,& \hfill \frac{1}{2}]& =\frac{5\sqrt{2}+9log(1+\sqrt{2})}{3\pi }.\hfill \\ \mathbf{D}\mathbf{17}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill -\frac{3}{4};& \hfill 2,& \hfill \frac{1}{2}]& =\frac{34\sqrt{2}+42log(1+\sqrt{2})}{21\pi }.\hfill \\ \mathbf{D}\mathbf{18}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{5}{4},& \hfill \frac{7}{4};& \hfill 2,& \hfill \frac{3}{2}]& =\frac{7\sqrt{2}+3ln(1+\sqrt{2})}{9\pi }.\hfill \\ \mathbf{D}\mathbf{19}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill -\frac{3}{4},& \hfill \frac{7}{4};& \hfill 1& \hfill \frac{3}{2}]& =\frac{43\sqrt{2}+87log(1+\sqrt{2})}{30\pi }.\hfill \\ \mathbf{D}\mathbf{20}.\hfill & {}_3{F}_2[-\frac{3}{2},\hfill & \hfill -\frac{1}{4},& \hfill -\frac{3}{4};& \hfill 1,& \hfill \frac{1}{2}]& =\frac{31\sqrt{2}-37log(1+\sqrt{2})}{8\pi }.\hfill \end{array}\hfill & \end{array}$$

Campbell, D’Aurizio and Sondow [9,10,22] discovered some formulae in Class-D.

• The formula D1 has been found by them in ([9] Equation (10)), where they also conjectured D12. For this last evaluation, five different proofs have been provided by the same authors [10].
• By making use of beta integrals, Campbell recoded in ([22] Theorems 2,3,7 and Example 12) four formulae. The first one ([22] Theorem 2) is corrected by D18. The second one ([22] Theorem 3) is incorrect. The third one ([22] Theorem 7) is simplified by D2. The fourth one ([22] Example 12) is too complicated to reproduce here.

3.4. Class E

Again in view of the Thomae transformation (3), we can express the “Class-E” series below in terms of the $\mathcal{F}$-series (where $\sigma =b+d-a-c-e$):

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}a+\frac{1}{2},& c+\frac{1}{4},& e+\frac{3}{4}\\ \\ & b+\frac{1}{2},& d+\frac{3}{2}\end{array}|1\right]=\Gamma \left[\begin{array}{c}\sigma +\frac{1}{2},b+\frac{1}{2},d+\frac{3}{2}\\ \\ a+\frac{1}{2},\sigma +c+\frac{3}{4},\sigma +e+\frac{5}{4}\end{array}\right]& \\ \hfill \times {}_3{F}_2\left[\begin{array}{ccc}1+d-a,& b-a,& \sigma +\frac{1}{2}\\ \\ & \sigma +c+\frac{3}{4},& \sigma +e+\frac{5}{4}\end{array}|1\right]& .\hfill \end{array}$$

(6)

Consequently, the closed formulae below for these series ( except for divergent series) can be deduced from those exhibited in “Class A”. Among them, E2 simplifies a formula of Campbell ([22] Example 5).

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{E}\mathbf{1}.\hfill & {}_3{F}_2[\frac{1}{4},\hfill & \hfill \frac{3}{4},& \hfill \frac{1}{2};& \hfill \frac{3}{2},& \hfill \frac{3}{2}]& =2\sqrt{2}-2log(1+\sqrt{2}).\hfill \\ \mathbf{E}\mathbf{2}.\hfill & {}_3{F}_2[\frac{1}{4},\hfill & \hfill \frac{7}{4},& \hfill \frac{1}{2};& \hfill \frac{3}{2},& \hfill \frac{3}{2}]& =\frac{1}{3}\left\{4\sqrt{2}-2log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{E}\mathbf{3}.\hfill & {}_3{F}_2[\frac{1}{4},\hfill & \hfill \frac{7}{4},& \hfill \frac{3}{2};& \hfill \frac{5}{2},& \hfill \frac{5}{2}]& =\frac{12}{25}\left\{8\sqrt{2}-10log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{E}\mathbf{4}.\hfill & {}_3{F}_2[\frac{3}{4},\hfill & \hfill \frac{5}{4},& \hfill \frac{1}{2};& \hfill \frac{3}{2},& \hfill \frac{3}{2}]& =2log(1+\sqrt{2}).\hfill \\ \mathbf{E}\mathbf{5}.\hfill & {}_3{F}_2[\frac{3}{4},\hfill & \hfill \frac{5}{4},& \hfill \frac{3}{2};& \hfill \frac{5}{2},& \hfill \frac{5}{2}]& =4\left\{4\sqrt{2}-6log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{E}\mathbf{6}.\hfill & {}_3{F}_2[\frac{3}{4},\hfill & \hfill \frac{9}{4},& \hfill \frac{1}{2};& \hfill \frac{3}{2},& \hfill \frac{5}{2}]& =\frac{1}{5}\left\{\sqrt{2}+9log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{E}\mathbf{7}.\hfill & {}_3{F}_2[\frac{5}{4},\hfill & \hfill \frac{7}{4},& \hfill \frac{1}{2};& \hfill \frac{3}{2},& \hfill \frac{5}{2}]& =\sqrt{2}+log(1+\sqrt{2}).\hfill \\ \mathbf{E}\mathbf{8}.\hfill & {}_3{F}_2[\frac{5}{4},\hfill & \hfill \frac{11}{4},& \hfill \frac{1}{2};& \hfill \frac{5}{2},& \hfill \frac{5}{2}]& =\frac{9}{14}\left\{3\sqrt{2}-log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{E}\mathbf{9}.\hfill & {}_3{F}_2[\frac{7}{4},\hfill & \hfill \frac{9}{4},& \hfill \frac{3}{2};& \hfill \frac{5}{2},& \hfill \frac{7}{2}]& =12\left\{\sqrt{2}-log(1+\sqrt{2})\right\}.\hfill \\ \mathbf{E}\mathbf{10}.\hfill & {}_3{F}_2[\frac{9}{4},\hfill & \hfill \frac{11}{4},& \hfill \frac{5}{2};& \hfill \frac{7}{2},& \hfill \frac{9}{2}]& =40\left\{4\sqrt{2}-6log(1+\sqrt{2})\right\}.\hfill \end{array}\hfill & \end{array}$$

Analogous to the series in Class-D, the parameter excess of the ${}_3{F}_2$-series displayed on the right hand side of (6) equals $\Delta =\frac{1}{2}+a$, which converges only when $a\ge 0$. We can also evaluate that ${}_3{F}_2$-series by reducing the case $a<0$ to $a=0$.

By means of Lemma 3, we have the equality

$$\begin{array}{cc}& {(\frac{1}{2}+b+n)}_{-a}=\sum _{k=0}^{-a}{⟨n⟩}_k{(\frac{1}{2}+a+n)}_{-a-k}{X}_a^k,\hfill \\ & \mathrm{where}{X}_k\left(a\right)=\frac{(-a)!}{{⟨-\frac{1}{2}⟩}_{-a}}\left(\genfrac{}{}{0pt}{}{a-b}{k}\right)\left(\genfrac{}{}{0pt}{}{b-a-\frac{1}{2}}{-a-k}\right)\hfill \end{array}$$

and then insert it in the ${}_3{F}_2$-series, we can handle the double sum

$$\begin{array}{cc}& {}_3{F}_2\left[\begin{array}{ccc}\frac{1}{2}+a,& \frac{1}{4}+c,& \frac{3}{4}+e\\ \\ & \frac{1}{2}+b,& \frac{3}{2}+d\end{array}|1\right]\hfill \\ & =\sum _{n=0}^{\infty }{\left[\begin{array}{c}\frac{1}{2}+a,\frac{1}{4}+c,\frac{3}{4}+e\\ \\ 1,\frac{1}{2}+b,\frac{3}{2}+d\end{array}\right]}_n\sum _{k=0}^{-a}\frac{{⟨n⟩}_k{(\frac{1}{2}+a+n)}_{-a-k}}{{(\frac{1}{2}+b+n)}_{-a}}{X}_k\left(a\right)\hfill \\ & =\sum _{k=0}^{-a}\frac{{(\frac{1}{2}+a)}_{-a-k}}{{(\frac{1}{2}+b)}_{-a}}{X}_k\left(a\right)\sum _{n=0}^{\infty }\frac{{⟨n⟩}_k}{n!}{\left[\begin{array}{c}\frac{1}{2}-k,\frac{1}{4}+c,\frac{3}{4}+e\\ \\ \frac{1}{2}-a+b,\frac{3}{2}+d\end{array}\right]}_n.\hfill \end{array}$$

Making the replacement $n\to n+k$, we can express the last sum as

$${\left[\begin{array}{c}\frac{1}{2}-k,\frac{1}{4}+c,\frac{3}{4}+e\\ \\ \frac{1}{2}-a+b,\frac{3}{2}+d\end{array}\right]}_k{}_3{F}_2\left[\begin{array}{c}\frac{1}{2},\frac{1}{4}+c+k,\frac{3}{4}+e+k\\ \\ \frac{1}{2}-a+b+k,\frac{3}{2}+d+k\end{array}|1\right].$$

After some simplifications, we establish the transformation below.

Theorem 5

(Reduction formula from $a<0$ to $a=0$).

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}\frac{1}{2}+a,& \frac{1}{4}+c,& \frac{3}{4}+e\\ \\ & \frac{1}{2}+b,& \frac{3}{2}+d\end{array}|1\right]& =\sum _{k=0}^{-a}{\left[\begin{array}{c}a,b-a,\frac{1}{4}+c,\frac{3}{4}+e\\ \\ 1,\frac{1}{2}-a+b,\frac{1}{2}+b,\frac{3}{2}+d\end{array}\right]}_k\hfill \\ & \times {}_3{F}_2\left[\begin{array}{c}\frac{1}{2},\frac{1}{4}+c+k,\frac{3}{4}+e+k\\ \\ \frac{1}{2}-a+b+k,\frac{3}{2}+d+k\end{array}|1\right].\hfill \end{array}$$

Under this transformation, the parameter excess $\Delta =\sigma =b+d-a-c-e$ remains invariant for all the ${}_3{F}_2$-series involved. However the ${}_3{F}_2$-series on the right belongs to Class-E and can therefore be evaluated by (6). We record ten more examples.

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{E}\mathbf{11}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{3}{4};& \hfill \frac{1}{2},& \hfill \frac{3}{2}]& =\frac{6-\sqrt{2}log(1+\sqrt{2})}{4\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{12}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{7}{4};& \hfill \frac{1}{2},& \hfill \frac{3}{2}]& =\frac{14-5\sqrt{2}log(1+\sqrt{2})}{12\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{13}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{1}{2},& \hfill \frac{3}{2}]& =\frac{2-3\sqrt{2}log(1+\sqrt{2})}{4\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{14}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{3}{2},& \hfill \frac{3}{2}]& =\frac{2+5\sqrt{2}log(1+\sqrt{2})}{8\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{15}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{5}{4},& \hfill \frac{7}{4};& \hfill \frac{1}{2},& \hfill \frac{5}{2}]& =\frac{3\left\{2+5\sqrt{2}log(1+\sqrt{2})\right\}}{-16\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{16}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{5}{4},& \hfill \frac{7}{4};& \hfill \frac{3}{2},& \hfill \frac{3}{2}]& =\frac{10-7\sqrt{2}log(1+\sqrt{2})}{24\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{17}.\hfill & {}_3{F}_2[-\frac{3}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{1}{2},& \hfill \frac{3}{2}]& =\frac{10-39\sqrt{2}log(1+\sqrt{2})}{128\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{18}.\hfill & {}_3{F}_2[-\frac{3}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{5}{4};& \hfill \frac{1}{2},& \hfill \frac{5}{2}]& =\frac{3\left\{62-37\sqrt{2}log(1+\sqrt{2})\right\}}{256\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{19}.\hfill & {}_3{F}_2[-\frac{3}{2},\hfill & \hfill \frac{5}{4},& \hfill \frac{7}{4};& \hfill \frac{3}{2},& \hfill \frac{3}{2}]& =\frac{62-37\sqrt{2}log(1+\sqrt{2})}{512\sqrt{2}}.\hfill \\ \mathbf{E}\mathbf{20}.\hfill & {}_3{F}_2[-\frac{3}{2},\hfill & \hfill \frac{5}{4},& \hfill -\frac{1}{4};& \hfill \frac{1}{2},& \hfill \frac{3}{2}]& =\frac{3\left\{42+41\sqrt{2}log(1+\sqrt{2})\right\}}{128\sqrt{2}}.\hfill \end{array}\hfill & \end{array}$$

3.5. Class F

By invoking the Kummer transformation (4), we can express the “Class-F” series below in terms of the $\mathcal{F}$-series (where $\sigma =b+d-a-c-e$):

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}a+\frac{1}{2},& c+\frac{3}{4},& e+\frac{3}{4}\\ \\ & b+1,& d+\frac{7}{4}\end{array}|1\right]=\Gamma \left[\begin{array}{c}b+1,\sigma +\frac{3}{4}\\ \\ b-a+\frac{1}{2},\sigma +a+\frac{5}{4}\end{array}\right]& \\ \hfill \times {}_3{F}_2\left[\begin{array}{ccc}1+d-c,& 1+d-e,& a+\frac{1}{2}\\ \\ & \sigma +a+\frac{5}{4},& d+\frac{7}{4}\end{array}|1\right]& .\hfill \end{array}$$

Then the closed formulae below for these series ( except for divergent series) can be established from those shown in “Class A”. Among them, the formula F10 is due to Campbell and Abrarov ([11] Corollary 5).

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{F}\mathbf{1}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill -\frac{1}{4},& \hfill -\frac{1}{4};& \hfill 1,& \hfill \frac{3}{4}]& =\frac{2\sqrt{\pi }\left\{5\sqrt{2}-4log(1+\sqrt{2})\right\}}{\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{2}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill -\frac{1}{4};& \hfill 1,& \hfill \frac{7}{4}]& =\frac{3\sqrt{\pi }\left\{8\sqrt{2}+2log(1+\sqrt{2})\right\}}{5\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{3}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{3}{4};& \hfill 1,& \hfill \frac{7}{4}]& =\frac{6\sqrt{\pi }\left\{\sqrt{2}+4log(1+\sqrt{2})\right\}}{5\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{4}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{7}{4};& \hfill 1,& \hfill \frac{11}{4}]& =\frac{7\sqrt{\pi }\left\{4\sqrt{2}+6log(1+\sqrt{2})\right\}}{15\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{5}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill -\frac{1}{4},& \hfill -\frac{1}{4};& \hfill 1,& \hfill \frac{3}{4}]& =\frac{2\sqrt{\pi }\left\{4\sqrt{2}-2log(1+\sqrt{2})\right\}}{\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{6}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill -\frac{1}{4},& \hfill -\frac{1}{4};& \hfill 1,& \hfill \frac{7}{4}]& =\frac{9\sqrt{\pi }\left\{3\sqrt{2}-log(1+\sqrt{2})\right\}}{4\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{7}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill -\frac{1}{4};& \hfill 1,& \hfill \frac{7}{4}]& =\frac{3\sqrt{\pi }\left\{\sqrt{2}+log(1+\sqrt{2})\right\}}{\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{8}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{3}{4};& \hfill 1,& \hfill \frac{7}{4}]& =\frac{12\sqrt{\pi }log(1+\sqrt{2})}{\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{9}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill \frac{3}{4},& \hfill \frac{7}{4};& \hfill 1,& \hfill \frac{11}{4}]& =\frac{7\sqrt{\pi }\left\{\sqrt{2}+9log(1+\sqrt{2})\right\}}{5\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \\ \mathbf{F}\mathbf{10}\hfill & {}_3{F}_2[\frac{3}{2},\hfill & \hfill \frac{3}{4},& \hfill -\frac{1}{4};& \hfill 1,& \hfill \frac{7}{4}]& =\frac{3\sqrt{\pi }\left\{3\sqrt{2}-log(1+\sqrt{2})\right\}}{2\Gamma {\left(\frac{1}{4}\right)}^2}.\hfill \end{array}\hfill & \end{array}$$

3.6. Class G

Finally, by employing the Kummer transformation (4), we can express the “Class-G” series below in terms of the $\mathcal{F}$-series (where $\sigma =b+d-a-c-e$):

$$\begin{array}{cc}\hfill {}_3{F}_2\left[\begin{array}{ccc}a+\frac{1}{2},& c+\frac{1}{4},& e+\frac{1}{4}\\ \\ & b+1,& d+\frac{1}{4}\end{array}|1\right]=\Gamma \left[\begin{array}{c}b+1,\sigma +\frac{1}{4}\\ \\ b-a+\frac{1}{2},\sigma +a+\frac{3}{4}\end{array}\right]& \\ \hfill \times {}_3{F}_2\left[\begin{array}{ccc}d-c,& d-e,& a+\frac{1}{2}\\ \\ & d+\frac{1}{4},& \sigma +a+\frac{3}{4}\end{array}|1\right]& .\hfill \end{array}$$

Then the closed formulae below for these series ( except for divergent series) can be shown from those displayed in “Class A”. Among them, the formula G8 is due to Campbell and Abrarov ([11] Corollary 4), who evaluated also another similar series ([11] Corollary 6).

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{G}\mathbf{1}.\hfill & {}_3{F}_2[-\frac{3}{2},\hfill & \hfill \frac{5}{4},& \hfill \frac{5}{4};& \hfill 1,& \hfill \frac{13}{4}]& =\frac{5\sqrt{\pi }\left\{4\sqrt{2}-3log(1+\sqrt{2})\right\}}{44\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \\ \mathbf{G}\mathbf{2}.\hfill & {}_3{F}_2[-\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill -\frac{3}{4};& \hfill 1,& \hfill \frac{5}{4}]& =\frac{\sqrt{\pi }\left\{2\sqrt{2}+3log(1+\sqrt{2})\right\}}{6\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \\ \mathbf{G}\mathbf{3}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill -\frac{3}{4};& \hfill 1,& \hfill \frac{5}{4}]& =\frac{\sqrt{\pi }\left\{\sqrt{2}+9log(1+\sqrt{2})\right\}}{12\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \\ \mathbf{G}\mathbf{4}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{1}{4};& \hfill 1,& \hfill \frac{5}{4}]& =\frac{\sqrt{\pi }log(1+\sqrt{2})}{\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \\ \mathbf{G}\mathbf{5}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{1}{4};& \hfill 2,& \hfill \frac{5}{4}]& =\frac{2\sqrt{\pi }\left\{-\sqrt{2}+6log(1+\sqrt{2})\right\}}{9\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \end{array}\hfill & \end{array}$$

$$\begin{array}{ccc}& \begin{array}{ccccccc}\mathbf{G}\mathbf{6}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{5}{4};& \hfill 2,& \hfill \frac{9}{4}]& =\frac{5\sqrt{\pi }\left\{\sqrt{2}-log(1+\sqrt{2})\right\}}{3\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \\ \mathbf{G}\mathbf{7}.\hfill & {}_3{F}_2[\frac{1}{2},\hfill & \hfill \frac{5}{4},& \hfill \frac{9}{4};& \hfill 2,& \hfill \frac{13}{4}]& =\frac{3\sqrt{\pi }\left\{-\sqrt{2}+5log(1+\sqrt{2})\right\}}{7\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \\ \mathbf{G}\mathbf{8}.\hfill & {}_3{F}_2[\frac{3}{2},\hfill & \hfill \frac{1}{4},& \hfill \frac{5}{4};& \hfill 1,& \hfill \frac{9}{4}]& =\frac{5\sqrt{\pi }\left\{3\sqrt{2}-log(1+\sqrt{2})\right\}}{8\Gamma {\left(\frac{3}{4}\right)}^2}\hfill \\ \mathbf{G}\mathbf{9}.\hfill & {}_3{F}_2[\frac{3}{2},\hfill & \hfill \frac{5}{4},& \hfill \frac{5}{4};& \hfill 2,& \hfill \frac{9}{4}]& =\frac{5\sqrt{\pi }\left\{2\sqrt{2}-2log(1+\sqrt{2})\right\}}{\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \\ \mathbf{G}\mathbf{10}.\hfill & {}_3{F}_2[\frac{3}{2},\hfill & \hfill \frac{5}{4},& \hfill \frac{9}{4};& \hfill 3,& \hfill \frac{13}{4}]& =\frac{6\sqrt{\pi }\left\{-6\sqrt{2}+10log(1+\sqrt{2})\right\}}{\Gamma {\left(\frac{3}{4}\right)}^2}.\hfill \end{array}\hfill & \end{array}$$

Concluding Comments

By combining the linearization method with the Kummer and Thomae transformations, we present 100 explicit formulae for 7 classes of nonterminating ${}_3{F}_2\left(1\right)$-series. They may potentially find applications in mathematics and physics as other mathematical formulae. Further explorations are encouraged to enrich this bank database of hypergeometric series identities.