Applications of Extended Kummer’s Summation Theorem

Abstract

In the theory of hypergeometric series and generalized hypergeometric series, classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series ${}_{2}F_1$; those of Watson, Dixon, Whipple, and Saalschutz for the series ${}_{3}F_2$; and others, play a key role. Applications of these classical summation theorems are well known. Berndt pointed out that a large number of interesting summations (including Ramanujan’s summations and the Gregory–Leibniz pi summation) can be obtained very quickly by employing the above-mentioned classical summation theorems. Also, several interesting results involving products of generalized hypergeometric series have been obtained by Bailey by employing the above-mentioned classical summation theorems. Recently, the above-mentioned classical summation theorems have been generalized and extended. In our present investigations, our aim is to demonstrate the applications of the extended Kummer’s summation theorem in establishing (i) extensions of Gauss’s second summation theorem and Bailey’s summation theorem; (ii) extensions of several summations (including Ramanujan’s summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) an extension of classical Dixon’s summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Leibniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey.

1. Introduction

The generalized hypergeometric function with p numerator and q denominator parameters is defined by [1,2,3,4]:

$$\begin{array}{c}\hfill {}_{p}F_q\left[\begin{array}{c}{a}_1,\cdots ,a_p\\ b_1,\cdots ,b_q\end{array};z\right]={}_{p}F_q[a_1,\cdots ,a_p;b_1,\cdots ,b_q;z]=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(a_1\right)}_n\cdots {\left(a_p\right)}_n}{{\left(b_1\right)}_n\cdots {\left(b_q\right)}_n}}·{\displaystyle \frac{z^n}{n!}},\end{array}$$

(1)

where ${\left(a\right)}_n$ denotes the Pochhammer symbol (or the raised factorial, since ${\left(1\right)}_n=n!$) defined, for any complex number a, by

$${\left(a\right)}_n=\left\{\begin{array}{cc}a(a+1)\cdots (a+n-1),\hfill & n\in \mathbb{N};\hfill \\ 1,\hfill & n=0.\hfill \end{array}\right.$$

(2)

Using the property of the gamma function relation $\mathrm{\Gamma }(a+1)=a\mathrm{\Gamma }\left(a\right)$, ${\left(a\right)}_n$ can be written in the form

$$\begin{array}{c}\hfill {\left(a\right)}_n={\displaystyle \frac{\mathrm{\Gamma }(a+n)}{\mathrm{\Gamma }\left(a\right)}},(n\in \mathbb{N}\cup \left\{0\right\}).\end{array}$$

(3)

Here, as usual, p and q are non-negative integers and parameters $a_j(1\le j\le p)$ and $b_j(1\le j\le q)$ can have arbitrary complex values with zero or negative integer values of $b_j$ excluded. The generalized hypergeometric function ${}_{p}F_q\left(z\right)$ converges for $\left|z\right|<\infty ,(p\le q),\left|z\right|<1(p=q+1)$ and $\left|z\right|=1(p=q+1)$ and $\Re \left(s\right)>0)$, where s is the parametric excess defined by

$$s=\sum _{j=1}^q{b}_j-\sum _{j=1}^p{a}_j.$$

For $p=2$ and $q=1$, we recover Gauss’s well-known hypergeometric function, viz.,

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{c}a,b\\ c\end{array};z\right]=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_n{\left(b\right)}_n}{{\left(c\right)}_n}}·{\displaystyle \frac{z^n}{n!}}.\end{array}$$

(4)

It is not out of place to mention here that whenever hypergeometric functions ${}_{2}F_1$ or the generalized hypergeometric functions ${}_{p}F_q$ reduce to gamma functions, the results are very important from the application point of view. Thus, the classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series ${}_{2}F_1$; those of Watson, Dixon, Whipple, and Saalschütz for the series ${}_{3}F_2$; and others, play an important role in the theory of generalized hypergeometric series. Here, we would like to mention the following summation theorems for ready reference.

Gauss’s summation theorem [1,2,3,4]:

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{cc}a,& b\\ & c\end{array};1\right]={\displaystyle \frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }(c-a-b)}{\mathrm{\Gamma }(c-a)\mathrm{\Gamma }(c-b)}},\end{array}$$

(5)

provided $\mathfrak{R}(c-a-b)>0$.

Gauss’s second summation theorem [1,2,3,4]:

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{c}a,b\\ {\displaystyle \frac{1}{2}}(a+b+1)\end{array};{\displaystyle \frac{1}{2}}\right]={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(\frac{1}{2}a+\frac{1}{2}b+\frac{1}{2})}{\mathrm{\Gamma }(\frac{1}{2}a+\frac{1}{2})\mathrm{\Gamma }(\frac{1}{2}b+\frac{1}{2})}}.\end{array}$$

(6)

Bailey’s summation theorem [1,2,3,4]:

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{c}a,1-a\\ c\end{array};{\displaystyle \frac{1}{2}}\right]={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(\frac{1}{2}c+\frac{1}{2})}{\mathrm{\Gamma }(\frac{1}{2}a+\frac{1}{2}c)\mathrm{\Gamma }(\frac{1}{2}c-\frac{1}{2}a+\frac{1}{2})}}.\end{array}$$

(7)

Kummer’s summation theorem [1,2,3,4]:

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{c}a,b\\ 1+a-b\end{array};-1\right]={\displaystyle \frac{\mathrm{\Gamma }(1+\frac{1}{2}a)\mathrm{\Gamma }(1+a-b)}{\mathrm{\Gamma }(1+a)\mathrm{\Gamma }(1+\frac{1}{2}a-b)}}.\end{array}$$

(8)

The above-mentioned classical summation theorems have wide applications. By employing Gauss’s second summation theorem we can easily obtain the following two elegant results recorded in [2,3]:

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{c}-2n,\alpha \\ 2\alpha \end{array};2\right]={\displaystyle \frac{{(1/2)}_n}{{(\alpha +1/2)}_n}}.\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{c}-2n-1,\alpha \\ 2\alpha \end{array};2\right]=0,\end{array}$$

(10)

each for $n=0,1,2,\cdots$.

Also, by employing the Bailey summation theorem (7), we may obtain the following interesting result recorded in [5]:

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{c}-n,\alpha \\ -2n\end{array};2\right]={\displaystyle \frac{2^{2n}n!}{\left(2n\right)!}}{({\displaystyle \frac{1}{2}}\alpha +{\displaystyle \frac{1}{2}})}_n.\end{array}$$

(11)

In addition to this, we mention the following very interesting summation formulas (including Ramanujan’s summations):

$$1+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{1}{2}}\right)}^2+{\displaystyle \frac{1}{2^2}}{\left({\displaystyle \frac{1.3}{2.4}}\right)}^2+\cdots ={\displaystyle \frac{\sqrt{\pi }}{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}},$$

(12)

$$1-{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{5}}-{\displaystyle \frac{1}{7}}+\cdots ={\displaystyle \frac{\pi }{4}},$$

(13)

$$1-{\left({\displaystyle \frac{1}{2}}\right)}^2+{\left({\displaystyle \frac{1.3}{2.4}}\right)}^2-\cdots ={\displaystyle \frac{\sqrt{\pi }}{\sqrt{2}{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}},$$

(14)

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{x-1}{x+1}}+{\displaystyle \frac{(x-1)(x-2)}{(x+1)(x+2)}}+\cdots ={\displaystyle \frac{2^{2x-1}{\mathrm{\Gamma }}^2(x+1)}{\mathrm{\Gamma }(2x+1)}}(\Re \left(x\right)>0),\end{array}$$

(15)

and

$$\begin{array}{c}\hfill 1-{\left({\displaystyle \frac{1}{4}}\right)}^2+{\left({\displaystyle \frac{1.5}{4.8}}\right)}^2-\cdots ={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{8}\right)\mathrm{\Gamma }\left(\frac{3}{8}\right)}{2^{\frac{7}{4}}{\pi }^{\frac{3}{2}}}}.\end{array}$$

(16)

As pointed out by Berndt [6], the above-mentioned Ramanujan summations (excluding (13) and (16)) can be obtained very quickly by employing hypergeometric identities. For this, the summation (12) can be obtained by Gauss’s second summation theorem (6) by letting $a=b={\displaystyle \frac{1}{2}}$ or by employing Bailey’s summation theorem (7) by letting $a={\displaystyle \frac{1}{2}}$ and $c=1$. The summation (13) is the well-known Leibniz–Gregory pi series, and can be obtained by employing Kummer’s summation theorem (8) by letting $a=1$ and $b={\displaystyle \frac{1}{2}}$. The summations (14) and (15) can be obtained from Kummer’s summation theorem (8) by letting $a=b={\displaystyle \frac{1}{2}}$ and $a=1$ and $b=1-x$, respectively. Summation (16), which is due to Brychkov [5], can also be obtained by employing Kummer’s summation theorem by letting $a=b={\displaystyle \frac{1}{4}}$.

Further, in terms of hypergeometric function, the summations (12)–(16) can be written in the following manner:

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\\ \\ 1\end{array};{\displaystyle \frac{1}{2}}\right]={\displaystyle \frac{\sqrt{\pi }}{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}},$$

(17)

$${}_{2}F_1\left[\begin{array}{c}1,{\displaystyle \frac{1}{2}}\\ \\ {\displaystyle \frac{3}{2}}\end{array};-1\right]={\displaystyle \frac{\pi }{4}},$$

(18)

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\\ \\ 1\end{array};-1\right]={\displaystyle \frac{\sqrt{\pi }}{\sqrt{2}{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}},$$

(19)

$${}_{2}F_1\left[\begin{array}{c}1,1-x\\ \\ 1+x\end{array};-1\right]={\displaystyle \frac{2^{2x-1}{\mathrm{\Gamma }}^2(x+1)}{\mathrm{\Gamma }(2x+1)}};\left(Re\left(x\right)>0\right),$$

(20)

and

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\\ \\ 1\end{array};-1\right]={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{8}\right)\mathrm{\Gamma }\left(\frac{3}{8}\right)}{2^{\frac{7}{4}}{\pi }^{\frac{3}{2}}}}.$$

(21)

Moreover, in a very interesting, popular, and useful research paper (in addition to other known and new results), Bailey [7] obtained the following results by employing the classical Kummer summation theorem (8):

$${}_{0}F_1\left[\begin{array}{c}-\\ e\end{array};x\right]\times {}_{0}F_1\left[\begin{array}{c}-\\ e\end{array};-x\right]={}_{0}F_3\left[\begin{array}{c}-\\ e,{\displaystyle \frac{1}{2}}e,{\displaystyle \frac{1}{2}}e+{\displaystyle \frac{1}{2}}\end{array};-{\displaystyle \frac{x^2}{4}}\right],$$

(22)

$$\begin{array}{cc}{}_{0}F_1\left[\begin{array}{c}-\\ e\end{array};x\right]\times {}_{0}F_1\left[\begin{array}{c}-\\ 2-e\end{array};-x\right]& ={}_{0}F_3\left[\begin{array}{c}-\\ {\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}(e+1),{\displaystyle \frac{1}{2}}(3-e)\end{array};-{\displaystyle \frac{x^2}{4}}\right]\\ & \hspace{1em}\hspace{1em}+{\displaystyle \frac{2(1-e)x}{e(2-e)}}{}_{0}F_3\left[\begin{array}{c}-\\ {\displaystyle \frac{3}{2}},1+{\displaystyle \frac{1}{2}}e,2-{\displaystyle \frac{1}{2}}e\end{array};-{\displaystyle \frac{x^2}{4}}\right],\end{array}$$

(23)

$$\begin{array}{c}\hfill e^x{}_{0}F_1\left[\begin{array}{c}-\\ {\displaystyle \frac{1}{2}}\end{array};-{\displaystyle \frac{x^2}{4}}\right]=\sum _{m=0}^{\infty }{\displaystyle \frac{x^m{2}^{\frac{m}{2}}}{m!}}\cos {\displaystyle \frac{m\pi }{4}},\end{array}$$

(24)

$$\begin{array}{c}\hfill e^x{}_{0}F_1\left[\begin{array}{c}-\\ {\displaystyle \frac{3}{2}}\end{array};-{\displaystyle \frac{x^2}{4}}\right]=\sum _{m=1}^{\infty }{\displaystyle \frac{x^{m-1}{2}^{\frac{m}{2}}}{m!}}\sin {\displaystyle \frac{m\pi }{4}},\end{array}$$

(25)

$$\begin{array}{c}\hfill {(1-x)}^{-2\alpha }{}_{2}F_1\left[\begin{array}{c}\alpha ,\alpha +{\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{1}{2}}\end{array};-{\displaystyle \frac{x^2}{{(1-x)}^2}}\right]=\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(2\alpha \right)}_m}{\left(m\right)!}}{2}^{{\displaystyle \frac{m}{2}}}{x}^m\cos {\displaystyle \frac{m\pi }{4}},\end{array}$$

(26)

$$\begin{array}{c}\hfill {(1-x)}^{-2\alpha }{}_{2}F_1\left[\begin{array}{c}\alpha ,\alpha +{\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{3}{2}}\end{array};-{\displaystyle \frac{x^2}{{(1-x)}^2}}\right]=\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(2\alpha \right)}_m}{(m+1)!}}{2}^{{\displaystyle \frac{m}{2}}+{\displaystyle \frac{1}{2}}}{x}^m\sin {\displaystyle \frac{(m+1)\pi }{4}}.\end{array}$$

(27)

Recently, a good deal of progress has been made in the direction of generalizing the above-mentioned classical summation theorems. For this, we refer to a paper by Rakha and Rathie [8]. Also, the above-mentioned classical summation theorems have been extended by Kim et al. [9]. In our present investigation, we use the following extension of Kummer’s summation theorem [9]. For $\Re \left(d\right)\notin Z_0^{-}$, we have

$$\begin{array}{c}\hfill {}_{3}F_2\left[\begin{array}{c}a,b,d+1\\ 2+a-b,d\end{array};-1\right]={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(2+a-b)}{2^a(1-b)}}\left[{\displaystyle \frac{\left(\frac{1+a-b}{d}-1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}a\right)\mathrm{\Gamma }(\frac{1}{2}a-b+\frac{3}{2})}}+{\displaystyle \frac{\left(1-\frac{a}{d}\right)}{\mathrm{\Gamma }(\frac{1}{2}a+\frac{1}{2})\mathrm{\Gamma }(\frac{1}{2}a-b+1)}}\right].\end{array}$$

(28)

When $d=1+a-b$, we recover the Kummer summation theorem (8).

The paper is organized as follows. In Section 2, extensions of Gauss’s second summation theorem and Bailey’s summation theorem are given by a different method to that given by Kim et al. [9]. In Section 3, extensions of the results (17)–(21) are given by employing extensions of Gauss’s second summation theorem (29), Bailey’s summation theorem (30), and Kummer’s summation theorem (28). A few interesting special cases (known as well as new) and the limiting cases are also provided. In Section 4, extensions of the results (22) and (23), involving products of generalized hypergeometric series, are established. In Section 5, extensions of Bailey’s results (24) and (25) are given. In Section 6, extensions of Bailey’s results (26) and (27) are given. Finally, in Section 7, we derive an extension of the well-known classical Dixon theorem obtained earlier by Kim et al. [9] by following a different method. The results established in this paper are simple, interesting, easily established, and may be useful.

2. Extensions of Gauss’s Second Summation Theorem and Bailey’s Summation Theorem

In this section, we establish the extensions of Gauss’s second summation theorem (6) and Bailey’s summation theorem (7) asserted in the following theorem.

Theorem 1.

The following results hold true for $\Re \left(d\right)\notin Z_0^{-}$.

Extension of Gauss’s second summation theorem:

$$\begin{array}{c}\hfill {}_{3}F_2\left[\begin{array}{c}a,b,d+1\\ {\displaystyle \frac{a+b+3}{2}},d\end{array};{\displaystyle \frac{1}{2}}\right]={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{a+b+3}{2}\right)\mathrm{\Gamma }\left(\frac{a-b-1}{2}\right)}{\mathrm{\Gamma }(\frac{1}{2}a-\frac{1}{2}b+\frac{3}{2})}}\left[{\displaystyle \frac{\frac{1}{2}(a+b-1)-\frac{ab}{d}}{\mathrm{\Gamma }(\frac{1}{2}a+\frac{1}{2})\mathrm{\Gamma }(\frac{1}{2}b+\frac{1}{2})}}+{\displaystyle \frac{\frac{a+b+1}{d}-2}{\mathrm{\Gamma }\left(\frac{1}{2}a\right)\mathrm{\Gamma }\left(\frac{1}{2}b\right)}}\right].\end{array}$$

(29)

Extension of Bailey’s summation theorem:

$$\begin{array}{cc}{}_{3}F_2\left[\begin{array}{c}a,1-a,d+1\\ c+1,d\end{array};{\displaystyle \frac{1}{2}}\right]& =2^{-c}\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}\right)\mathrm{\Gamma }(c+1)\\ & \times \left[{\displaystyle \frac{\left(\frac{2}{d}\right)}{\mathrm{\Gamma }(\frac{1}{2}a+\frac{1}{2}c)\mathrm{\Gamma }(\frac{1}{2}c-\frac{1}{2}a+\frac{1}{2})}}+{\displaystyle \frac{(1-\frac{c}{d})}{\mathrm{\Gamma }(\frac{1}{2}a+\frac{1}{2}c+\frac{1}{2})\mathrm{\Gamma }(\frac{1}{2}c-\frac{1}{2}a+1)}}\right].\end{array}$$

(30)

Proof. 

In order to derive (29), we proceed as follows. Consider the extension of Euler’s first transformation obtained earlier by Rathie and Paris [10] in the form

$${(1-x)}^{-A}{}_{3}F_2\left[\begin{array}{c}A,B,D+1\\ C+1,D\end{array};-{\displaystyle \frac{x}{1-x}}\right]={}_{3}F_2\left[\begin{array}{c}A,C-B,f+1\\ C+1,f\end{array};x\right],$$

(31)

where f is given by

$$\begin{array}{c}\hfill f={\displaystyle \frac{D(B-C)}{B-D}}.\end{array}$$

(32)

Further, in (31), if we take $x\to -1$, we obtain the following result:

$${}_{3}F_2\left[\begin{array}{c}A,B,D+1\\ C+1,D\end{array};{\displaystyle \frac{1}{2}}\right]=2^A{}_{3}F_2\left[\begin{array}{c}A,C-B,f+1\\ C+1,f\end{array};-1\right].$$

(33)

Now, taking $A=a,B=b,D=d$, and $C={\displaystyle \frac{1}{2}}(a+b+1)$ in (33), we obtain the following form:

$${}_{3}F_2\left[\begin{array}{c}a,b,d+1\\ {\displaystyle \frac{1}{2}}(a+b+3),d\end{array};{\displaystyle \frac{1}{2}}\right]=2^a{}_{3}F_2\left[\begin{array}{c}a,{\displaystyle \frac{1}{2}}(a-b+1),f+1\\ {\displaystyle \frac{1}{2}}(a+b+3),f\end{array};-1\right],$$

(34)

where

$$f={\displaystyle \frac{d(b-a-1)}{2(b-d)}}.$$

It is now easy to see that the ${}_{3}F_2$ appearing on the right-hand side of (34) can be evaluated with the help of the extension of Kummer’s summation theorem (28), and after some simplification, we easily arrive at the right-hand side of (29). This completes the proof of (29).

On the other hand, in order to derive (30), if we take $A=a,B=1-a,C=c$, and $D=d$ in (33), then we have

$${}_{3}F_2\left[\begin{array}{c}a,1-a,d+1\\ c+1,d\end{array};{\displaystyle \frac{1}{2}}\right]=2^a{}_{3}F_2\left[\begin{array}{c}a,a+c-1,f+1\\ c+1,f\end{array};-1\right],$$

(35)

where

$$\begin{array}{c}\hfill f={\displaystyle \frac{d(1-a-c)}{1-a-d}}.\end{array}$$

Again, it is easy to see that the ${}_{3}F_2$ appearing on the right-hand side of (35) can now be evaluated with the help of the extension of Kummer’s summation theorem (28) and after some simplification, we easily arrive at the right-hand side of (30). This completes the proof of (30). □

Corollary 1.

In results (29) and (30), if we take $d={\displaystyle \frac{1}{2}}(a+b+1)$ and $d=c$, we, respectively, recover Gauss’s second summation theorem (6) and Bailey’s summation theorem (7). Thus, results (29) and (30) can be regarded as extensions of results (6) and (7), respectively.

3. Extensions of Summations (17) to (21)

In this section, we establish extensions of summations (17)–(21) asserted in the following theorem.

Theorem 2.

For $\Re \left(d\right)\notin Z_0^{-}$, the following summation formulas hold true.

$${}_{3}F_2\left[\begin{array}{c}{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}},d+1\\ \\ 2,d\end{array};{\displaystyle \frac{1}{2}}\right]={\displaystyle \frac{\sqrt{\pi }}{2}}\left\{{\displaystyle \frac{\left(\frac{2}{d}\right)}{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}}+{\displaystyle \frac{8(1-\frac{1}{d})}{{\pi }^2}}{\mathrm{\Gamma }}^2\left({\displaystyle \frac{3}{4}}\right)\right\},$$

(36)

$${}_{3}F_2\left[\begin{array}{c}1,{\displaystyle \frac{1}{2}},d+1\\ \\ {\displaystyle \frac{5}{2}},d\end{array};-1\right]={\displaystyle \frac{3\pi }{4}}\left\{{\displaystyle \frac{2}{\pi }}\left({\displaystyle \frac{3}{2d}}-1\right)+\left(1-{\displaystyle \frac{1}{d}}\right)\right\},$$

(37)

$${}_{3}F_2\left[\begin{array}{c}{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}},d+1\\ \\ 2,d\end{array};-1\right]=\sqrt{2\pi }\left\{{\displaystyle \frac{2\left(\frac{1}{d}-1\right)}{{\pi }^2}}{\mathrm{\Gamma }}^2\left({\displaystyle \frac{3}{4}}\right)+{\displaystyle \frac{\left(1-\frac{1}{2d}\right)}{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}}\right\},$$

(38)

$${}_{3}F_2\left[\begin{array}{c}1,1-x,d+1\\ \\ 2+x,d\end{array};-1\right]={\displaystyle \frac{(x+1)\sqrt{\pi }\mathrm{\Gamma }\left(x\right)}{2}}\left\{{\displaystyle \frac{\left(\frac{1+x}{d}-1\right)}{\sqrt{\pi }\mathrm{\Gamma }(x+1)}}+{\displaystyle \frac{\left(1-\frac{1}{d}\right)}{\mathrm{\Gamma }(x+\frac{1}{2})}}\right\},$$

(39)

for $\Re \left(x\right)>0$

• and

$${}_{3}F_2\left[\begin{array}{c}{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},d+1\\ \\ 2,d\end{array};-1\right]={\displaystyle \frac{2^{\frac{7}{4}}\sqrt{\pi }}{3}}\left\{{\displaystyle \frac{\left(\frac{1}{d}-1\right)}{\mathrm{\Gamma }\left(\frac{1}{8}\right)\mathrm{\Gamma }\left(\frac{11}{8}\right)}})+{\displaystyle \frac{\left(1-\frac{1}{4d}\right)}{\mathrm{\Gamma }\left(\frac{5}{8}\right)\mathrm{\Gamma }\left(\frac{7}{8}\right)}}\right\}.$$

(40)

Proof. 

The derivations of summations (36)–(40) asserted in the theorem are quite straightforward. For this, it can be easily seen that:

(a)

Summation (36) follows from result (29) by letting $a=b={\displaystyle \frac{1}{2}}$ or from result (30) by letting $a={\displaystyle \frac{1}{2}}$ and $c=1$.

(b)

Summation (37) follows at once from result (28) by letting $a=1$ and $b={\displaystyle \frac{1}{2}}$.

(c)

Summation (38) follows at once from result (28) by letting $a=b={\displaystyle \frac{1}{2}}$.

(d)

Summation (39) follows at once from result (28) by letting $a=1$ and $b=1-x$.

(e)

Summation (40) follows at once from result (28) by letting $a=b={\displaystyle \frac{1}{4}}$.

□

Corollary 2.

(a) 

(i) In (36), if we take $d=1$, we at once obtain result (17).

(ii) In (36), if we take $d={\displaystyle \frac{1}{2}}$, we obtain the following result:

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}\\ 2\end{array};{\displaystyle \frac{1}{2}}\right]=2\sqrt{\pi }\left\{{\displaystyle \frac{1}{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}}-{\displaystyle \frac{2}{{\pi }^2}}{\mathrm{\Gamma }}^2\left({\displaystyle \frac{3}{4}}\right)\right\}.$$

(iii) In (36), if we set $d\to \infty$, we obtain

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\\ 1\end{array};{\displaystyle \frac{1}{2}}\right]={\displaystyle \frac{4}{{\pi }^{\frac{3}{2}}}}{\mathrm{\Gamma }}^2\left({\displaystyle \frac{3}{4}}\right).$$

Corollary 3.

(b) 

(i) In (37), if we take $d=1$, we at once obtain result (18).

(ii) In (37), if we take $d={\displaystyle \frac{1}{2}}$ and $d=1$, we, respectively, obtain the following results:

$${}_{2}F_1\left[\begin{array}{c}1,{\displaystyle \frac{3}{2}}\\ \\ {\displaystyle \frac{5}{2}}\end{array};-1\right]={\displaystyle \frac{3\pi }{4}}\left({\displaystyle \frac{4}{\pi }}-1\right),$$

and

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{2}},2\\ \\ {\displaystyle \frac{5}{2}}\end{array};-1\right]={\displaystyle \frac{3}{4}}.$$

(iii) In (37), if we set $d\to \infty$, we obtain

$${}_{2}F_1\left[\begin{array}{c}1,{\displaystyle \frac{1}{2}}\\ \\ {\displaystyle \frac{5}{2}}\end{array};-1\right]={\displaystyle \frac{3}{4}}(\pi -2).$$

Corollary 4.

(c) 

(i) In (38), if we take $d=1$, we at once obtain result (19).

(ii) In (38), if we take $d={\displaystyle \frac{1}{2}}$, we obtain the following result:

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}\\ \\ 2\end{array};-1\right]={\displaystyle \frac{2\sqrt{2}}{\pi \sqrt{\pi }}}{\mathrm{\Gamma }}^2\left({\displaystyle \frac{3}{4}}\right).$$

(iii) In (38), if we set $d\to \infty$, we obtain

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\\ \\ 2\end{array};-1\right]=\sqrt{2\pi }\left\{{\displaystyle \frac{1}{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}}-{\displaystyle \frac{2}{{\pi }^2}}{\mathrm{\Gamma }}^2\left({\displaystyle \frac{3}{4}}\right)\right\}.$$

Corollary 5.

(d) 

(i) In (39), if we take $d=1+x$, we at once obtain result (20).

(ii) In (39), if we take $d=1$, we obtain the following result:

$${}_{2}F_1\left[\begin{array}{c}2,1-x\\ \\ 1+x\end{array};-1\right]={\displaystyle \frac{1+x}{2}}.$$

(iii) In (39), if we set $d\to \infty$, we obtain

$${}_{2}F_1\left[\begin{array}{c}1,1-x\\ \\ 2+x\end{array};-1\right]={\displaystyle \frac{(x+1)\sqrt{\pi }\mathrm{\Gamma }\left(x\right)}{2}}\left\{{\displaystyle \frac{1}{\mathrm{\Gamma }(x+\frac{1}{2})}}-{\displaystyle \frac{1}{\sqrt{\pi }\mathrm{\Gamma }(x+1)}}\right\}.$$

Corollary 6.

(e) 

(i) In (40), if we take $d=1$, we obtain at once result (21).

(ii) In (40), if we take $d={\displaystyle \frac{1}{4}}$, we obtain the following result:

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{4}},{\displaystyle \frac{5}{4}}\\ \\ 2\end{array};-1\right]={\displaystyle \frac{2^{\frac{7}{4}}\sqrt{\pi }}{\mathrm{\Gamma }\left(\frac{1}{8}\right)\mathrm{\Gamma }\left(\frac{11}{8}\right)}}.$$

(iii) In (40), if we set $d\to \infty$, we obtain

$${}_{2}F_1\left[\begin{array}{c}{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\\ \\ 2\end{array};-1\right]={\displaystyle \frac{2^{\frac{7}{4}}\sqrt{\pi }}{3}}\left\{{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\frac{5}{8}\right)\mathrm{\Gamma }\left(\frac{7}{8}\right)}}-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\frac{1}{8}\right)\mathrm{\Gamma }\left(\frac{11}{8}\right)}}\right\}.$$

It is not out of place to mention here that the results provided in this section have been verified with the help of Mathematica, a general system of performing mathematics by computer.

We conclude this section by remarking that most of the special cases mentioned above are believed to be new.

4. Extension of Results (22) and (23)

In this section, we establish the extensions of Bailey’s results (22) and (23) asserted in the following theorem.

Theorem 3.

The following results hold true for $\Re \left(d\right)\notin Z_0^{-}$.

$$\begin{array}{c}\hfill {}_{0}F_1\left[\begin{array}{c}-\\ e\end{array};x\right]\times {}_{1}F_2\left[\begin{array}{c}d+1\\ e+1,d\end{array};-x\right]={}_{1}F_4\left[\begin{array}{c}1+{\displaystyle \frac{1}{2}}d\\ e,{\displaystyle \frac{1}{2}}e+{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}e+1,{\displaystyle \frac{1}{2}}d\end{array};-{\displaystyle \frac{x^2}{4}}\right]\\ \hfill -{\displaystyle \frac{x(e-d)}{de(e+1)}}{}_{0}F_3\left[\begin{array}{c}-\\ e+1,{\displaystyle \frac{1}{2}}e+1,{\displaystyle \frac{1}{2}}e+{\displaystyle \frac{3}{2}}\end{array};-{\displaystyle \frac{x^2}{4}}\right],\end{array}$$

(41)

and

$$\begin{array}{c}{}_{0}F_1\left[\begin{array}{c}-\\ e\end{array};x\right]\times {}_{1}F_2\left[\begin{array}{c}d+1\\ 3-e,d\end{array};-x\right]\hfill \\ ={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(3-e)}{2^{1-e}}}\{{\displaystyle \frac{\frac{2-e}{d}-1}{\mathrm{\Gamma }(\frac{1}{2}-\frac{1}{2}e)\mathrm{\Gamma }(2-\frac{1}{2}e)}}{}_{0}F_3\left[\begin{array}{c}-\\ {\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}}e,2-{\displaystyle \frac{1}{2}}e\end{array};-{\displaystyle \frac{x^2}{4}}\right]\hfill \\ +{\displaystyle \frac{\frac{e-1}{d}+1}{\mathrm{\Gamma }(1-\frac{1}{2}e)\mathrm{\Gamma }(\frac{3}{2}-\frac{1}{2}e)}}{}_{1}F_4\left[\begin{array}{c}{\displaystyle \frac{1}{2}}(d+e+1)\\ {\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}}e+{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{2}}e,{\displaystyle \frac{1}{2}}(d+e-1)\end{array};-{\displaystyle \frac{x^2}{4}}\right]\}\hfill \\ +{\displaystyle \frac{x\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(3-e)}{e{2}^{1-e}}}\{{\displaystyle \frac{\frac{2-e}{d}-1}{\mathrm{\Gamma }(-\frac{1}{2}e)\mathrm{\Gamma }(\frac{5}{2}-\frac{1}{2}e)}}{}_{1}F_4\left[\begin{array}{c}1\\ 2,{\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}e,{\displaystyle \frac{5}{2}}-{\displaystyle \frac{1}{2}}e\end{array};-{\displaystyle \frac{x^2}{4}}\right]\hfill \\ +{\displaystyle \frac{\frac{e}{d}+1}{\mathrm{\Gamma }(\frac{1}{2}-\frac{1}{2}e)\mathrm{\Gamma }(2-\frac{1}{2}e)}}{}_{2}F_5\left[\begin{array}{c}1,{\displaystyle \frac{1}{2}}(d+e)+1\\ 2,{\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}}e+1,2-{\displaystyle \frac{1}{2}}e,{\displaystyle \frac{1}{2}}(d+e)\end{array};-{\displaystyle \frac{x^2}{4}}\right].\}.\hfill \end{array}$$

(42)

Proof. 

In order to derive (41), we proceed as follows. Let

$${}_{0}F_1\left[\begin{array}{c}-\\ e\end{array};x\right]\times {}_{1}F_2\left[\begin{array}{c}d+1\\ e+1,d\end{array};-x\right]=\sum _{n=0}^{\infty }{a}_n{x}^n.$$

(43)

Then, it is not difficult to see the coefficient of $x^n$ in the product, after some simplification, takes the following form:

$$a_n={\displaystyle \frac{1}{n!{\left(e\right)}_n}}{}_{3}F_2\left[\begin{array}{c}-n,1-e-n,d+1\\ e+1,d\end{array};-1\right].$$

(44)

Now, ${}_{3}F_2$ can be evaluated with the help of an extension of Kummer’s summation theorem (28), and after some algebra, we have

$$a_n={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(1+e)2^n}{n!{\left(e\right)}_n(e+n)}}\left[{\displaystyle \frac{(\frac{e}{d}-1)}{\mathrm{\Gamma }(-\frac{1}{2}n)\mathrm{\Gamma }(\frac{1}{2}n+e+\frac{1}{2})}}+{\displaystyle \frac{\left(1+\frac{n}{d}\right)}{\mathrm{\Gamma }(\frac{1}{2}-\frac{1}{2}n)\mathrm{\Gamma }(\frac{1}{2}n+e)}}\right].$$

(45)

Finally, changing n to $2n$ and $2n+1$, respectively, and making use of several elementary identities, we have

$$\begin{array}{c}\hfill a_{2n}={\displaystyle \frac{{(\frac{1}{2}d+1)}_n{(-1)}^n}{{\left(e\right)}_n{(\frac{1}{2}e+\frac{1}{2})}_n{(\frac{1}{2}e+1)}_n{\left(\frac{1}{2}d\right)}_n{2}^{2n}n!}}\end{array}$$

(46)

and

$$\begin{array}{c}\hfill a_{2n+1}=-{\displaystyle \frac{(\frac{e}{d}-1)}{e(e+1)}}{\displaystyle \frac{{(-1)}^n}{{(e+1)}_n{(\frac{1}{2}e+1)}_n{(\frac{1}{2}e+\frac{3}{2})}_n{2}^{2n}n!}}\end{array}$$

(47)

Upon substituting the values of $a_{2n}$ and $a_{2n+1}$ into the right-hand side of (43), and then, summing up the series, we arrive at (41). This completes the proof of (41).

In exactly the same manner, result (42) can be established. □

Corollary 7.

In results (41) and (42), if we take $d=e$ and $d=2-e$, we immediately recover Bailey’s results (22) and (23). Thus, results (41) and (42) can be regarded as extensions of results (22) and (23).

5. Extensions of Bailey’s Results (24) and (25)

In this section, we establish extensions of Bailey’s results (24) and (25), asserted in the following theorem.

Theorem 4.

The following results hold true for $\Re \left(d\right)\notin Z_0^{-}$.

$$\begin{array}{c}\hfill e^x{}_{1}F_2\left[\begin{array}{c}d+1\\ {\displaystyle \frac{3}{2}},d\end{array};-{\displaystyle \frac{x^2}{4}}\right]=\sum _{m=0}^{\infty }{\displaystyle \frac{x^m{2}^{m/2}}{(m+1)!}}\left\{(1+{\displaystyle \frac{m}{2d}})cos{\displaystyle \frac{m\pi }{4}}-({\displaystyle \frac{1}{2d}}-1)sin{\displaystyle \frac{m\pi }{4}}\right\},\end{array}$$

(48)

and

$$\begin{array}{c}\hfill e^x{}_{1}F_2\left[\begin{array}{c}d+1\\ {\displaystyle \frac{5}{2}},d\end{array};-{\displaystyle \frac{x^2}{4}}\right]=\sum _{m=1}^{\infty }{\displaystyle \frac{3x^{m-1}{2}^{m/2}}{(m-1)!(m+1)}}\left\{{\displaystyle \frac{(1-\frac{2-m}{2d})}{m}}sin{\displaystyle \frac{m\pi }{4}}+{\displaystyle \frac{(\frac{3}{2d}-1)}{m+2}}cos{\displaystyle \frac{m\pi }{4}}\right\}.\end{array}$$

(49)

Proof. 

In order to derive result (48), we proceed as follows. Denoting the left-hand side of (48) by $\mathbf{S}$, expressing the two functions as series, we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n{(d+1)}_n{x}^{m+2n}}{{\left(\frac{3}{2}\right)}_n{\left(d\right)}_n{2}^{2n}m!n!}}.\end{array}$$

Replacing m by $m-2n$ and using result [3],

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\sum _{k=0}^{\infty }A(k,n)=\sum _{n=0}^{\infty }\sum _{k=0}^{\left[{\displaystyle \frac{n}{2}}\right]}A(k,n-2k),\end{array}$$

(50)

we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }\sum _{n=0}^{\left[{\displaystyle \frac{m}{2}}\right]}{\displaystyle \frac{{(-1)}^n{(d+1)}_n{x}^m}{{\left(\frac{3}{2}\right)}_n{\left(d\right)}_n{2}^{2n}(m-2n)!n!}}.\end{array}$$

Using the identity $(m-2n)!={\displaystyle \frac{m!}{{(-m)}_{2n}}}={\displaystyle \frac{m!}{2^{2n}{(-\frac{m}{2})}_n{(-\frac{m}{2}+\frac{1}{2})}_n}}$, we have, after some simplification,

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }{\displaystyle \frac{x^m}{m!}}\sum _{n=0}^{\left[{\displaystyle \frac{m}{2}}\right]}{\displaystyle \frac{{(-1)}^n{(-\frac{1}{2}m)}_n{(\frac{1}{2}-\frac{1}{2}m)}_n{(d+1)}_n}{{\left(\frac{3}{2}\right)}_n{\left(d\right)}_{n}n!}}.\end{array}$$

Summing up the inner series, we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }{\displaystyle \frac{x^m}{m!}}{}_{3}F_2\left[\begin{array}{c}-{\displaystyle \frac{1}{2}}m,{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}m,d+1\\ {\displaystyle \frac{3}{2}},d\end{array};-1\right].\end{array}$$

Finally, applying the extended Kummer’s summation theorem (28) and using the results

$$\begin{array}{c}\hfill \mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }(1-z)={\displaystyle \frac{\pi }{sin\pi z}}\mathrm{and}\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}+z\right)\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}-z\right)={\displaystyle \frac{\pi }{cos\pi z}},\end{array}$$

(51)

and after some algebra, we easily arrive at the right-hand side of (48). This completes the proof of (48).

In exactly the same manner, result (49) can also be established. □

Corollary 8.

In results (48) and (49), if we take $d=1/2$ and $d=3/2$, we immediately recover Bailey’s results (24) and (25), respectively. Thus, results (48) and (49) can be regarded as extensions of Bailey’s results (24) and (25).

6. Extensions of Results (26) and (27)

In this section, we establish extensions of Bailey’s results (26) and (27), asserted in the following theorem.

Theorem 5.

The following results hold true for $\Re \left(d\right)\notin Z_0^{-}$

$$\begin{array}{c}{(1-x)}^{-2\alpha }{}_{3}F_2\left[\begin{array}{c}\alpha ,\alpha +{\displaystyle \frac{1}{2}},d+1\\ {\displaystyle \frac{3}{2}},d\end{array};-{\displaystyle \frac{x^2}{{(1-x)}^2}}\right]\hfill \\ =\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(2\alpha \right)}_m{2}^{m/2}}{(m+1)!}}{x}^m\left[(1+{\displaystyle \frac{m}{2d}})cos{\displaystyle \frac{m\pi }{4}}-({\displaystyle \frac{1}{2d}}-1)sin{\displaystyle \frac{m\pi }{4}}\right],\hfill \end{array}$$

(52)

and

$$\begin{array}{c}{(1-x)}^{-2\alpha }{}_{3}F_2\left[\begin{array}{c}\alpha ,\alpha +{\displaystyle \frac{1}{2}},d+1\hfill \\ {\displaystyle \frac{5}{2}},d\hfill \end{array};-{\displaystyle \frac{x^2}{{(1-x)}^2}}\right]\hfill \\ =\sum _{m=0}^{\infty }{\displaystyle \frac{3{\left(2\alpha \right)}_m{2}^{m/2+3/2}}{\left(m\right)!(m+2)}}{x}^m\left[{\displaystyle \frac{(\frac{3}{2d}-1)}{m+3}}sin{\displaystyle \frac{\pi }{4}}(1-m)+{\displaystyle \frac{(\frac{m-1}{2d}+1)}{m+1}}sin{\displaystyle \frac{\pi }{4}}(1+m)\right].\hfill \end{array}$$

(53)

Proof. 

In order to derive result (52), we proceed as follows. Denoting the left-hand side of (52) by $\mathbf{S}$, expressing ${}_{3}F_2$ as a series, and after some simplification, we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }{\displaystyle \frac{{(-1)}^n{\left(\alpha \right)}_n{(\alpha +\frac{1}{2})}_n{(d+1)}_n{x}^{2n}}{{\left(\frac{3}{2}\right)}_n{\left(d\right)}_{n}n!}}{(1-x)}^{-(2\alpha +2n)}.\end{array}$$

Using the binomial theorem [3] in the above identity, we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n{\left(\alpha \right)}_n{(\alpha +\frac{1}{2})}_n{(d+1)}_n{(2\alpha +2n)}_m}{{\left(\frac{3}{2}\right)}_n{\left(d\right)}_{n}n!m!}}{x}^{m+2n}.\end{array}$$

Replacing m by $m-2n$ and using (50), we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }\sum _{n=0}^{\left[{\displaystyle \frac{m}{2}}\right]}{\displaystyle \frac{{(-1)}^n{\left(\alpha \right)}_n{(\alpha +\frac{1}{2})}_n{(d+1)}_n{(2\alpha +2n)}_{m-2n}}{{\left(\frac{3}{2}\right)}_n{\left(d\right)}_{n}n!(m-2n)!}}{x}^m.\end{array}$$

Using ${(2\alpha +2n)}_{m-2n}={\displaystyle \frac{{\left(2\alpha \right)}_m}{{\left(2\alpha \right)}_n}}$ and the results given in the previous section, we have, after some simplification,

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(2\alpha \right)}_m}{m!}}{x}^m\sum _{n=0}^{\left[{\displaystyle \frac{m}{2}}\right]}{\displaystyle \frac{{(-1)}^n{(-\frac{1}{2}m)}_n{(\frac{1}{2}-\frac{1}{2}m)}_n{(d+1)}_n}{{\left(\frac{3}{2}\right)}_n{\left(d\right)}_{n}n!}}.\end{array}$$

Summing up the inner series, we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(2\alpha \right)}_m}{m!}}{x}^m{}_{3}F_2\left[\begin{array}{c}-{\displaystyle \frac{1}{2}}m,{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}m,d+1\\ {\displaystyle \frac{3}{2}},d\end{array};-1\right].\end{array}$$

Finally, applying the extended Kummer’s summation theorem (28) and making use of result (51) and after some simplification, we easily arrive at the right-hand side of (52). This completes the proof of (52).

In exactly the same manner, result (53) can also be established. □

Corollary 9.

In results (52) and (53), if we take $d=1/2$ and $d=3/2$, we immediately recover Bailey’s results (26) and (27), respectively. Thus, results (52) and (53) can be regarded as extensions of Bailey’s results (26) and (27), respectively.

Further, in (52) and (53), if we replace x by ${\displaystyle \frac{x}{2\alpha }}$ and take $\alpha \to \infty$, we recover, respectively, results (48) and (49).

7. Extension of Classical Dixon’s Summation Theorem

In this section, we establish the following extension of the classical Dixon’s summation theorem for the series ${}_{3}F_2$ asserted in the following theorem.

Theorem 6.

The following results hold true for $\Re \left(d\right)\notin Z_0^{-}$.

$$\begin{array}{cc}& {}_{4}F_3\left[\begin{array}{c}a,b,c,d+1\\ 2+a-b,1+a-c,d\end{array};1\right]\hfill \\ =& {\displaystyle \frac{\alpha }{2^a(b-1)}}{\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(2+a-b)\mathrm{\Gamma }(1+a-c)\mathrm{\Gamma }(\frac{3}{2}+\frac{1}{2}a-b-c)}{\mathrm{\Gamma }\left(\frac{1}{2}a\right)\mathrm{\Gamma }(\frac{1}{2}a-b+\frac{3}{2})\mathrm{\Gamma }(\frac{1}{2}a-c+\frac{1}{2})\mathrm{\Gamma }(2+a-b-c)}}\hfill \\ +& {\displaystyle \frac{\beta }{2^{a+1}(b-1)}}{\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(1+a-b)\mathrm{\Gamma }(1+a-c)\mathrm{\Gamma }(1+\frac{1}{2}a-b-c)}{\mathrm{\Gamma }(\frac{1}{2}a+\frac{1}{2})\mathrm{\Gamma }(\frac{1}{2}a-b+1)\mathrm{\Gamma }(\frac{1}{2}a-c+1)\mathrm{\Gamma }(1+a-b-c)}},\hfill \end{array}$$

(54)

provided $\mathfrak{R}(a-2b-2c)>-2,$ with

$$\begin{array}{c}\alpha =1-{\displaystyle \frac{1}{d}}(1+a-b),\hfill \\ and\beta ={\displaystyle \frac{1+a-b}{1+a-b-c}}\left[{\displaystyle \frac{a}{d}}(1+a-b-2c)-2({\displaystyle \frac{1}{2}}a-b-c+1)\right].\hfill \end{array}$$

Proof. 

In order to derive (54), we proceed as follows. Denoting the left-hand side of (54) by $\mathbf{S}$, and then, expressing ${}_{4}F_3$ as a series, we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_k{\left(b\right)}_k{(d+1)}_k{(-1)}^k}{{(2+a-b)}_k{\left(d\right)}_{k}k!}}\left\{{\displaystyle \frac{{(-1)}^k{\left(c\right)}_k}{{(1+a-c)}_k}}\right\}.\end{array}$$

Using a known result [3] (Exercise 5, p. 69),

$$\begin{array}{c}\hfill {}_{2}F_1\left[\begin{array}{c}-k,a+k\\ 1+a-c\end{array};1\right]={\displaystyle \frac{{(-1)}^k{\left(c\right)}_k}{{(1+a-c)}_k}},\end{array}$$

we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_k{\left(b\right)}_k{(d+1)}_k{(-1)}^k}{{(2+a-b)}_k{\left(d\right)}_{k}k!}}{}_{2}F_1\left[\begin{array}{c}-k,a+k\\ 1+a-c\end{array};1\right].\end{array}$$

Expressing ${}_{2}F_1$ as a series, we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{k=0}^{\infty }\sum _{m=0}^k{\displaystyle \frac{{\left(a\right)}_k{(a+k)}_m{\left(b\right)}_k{(d+1)}_k{(-k)}_m{(-1)}^k}{{(2+a-b)}_k{\left(d\right)}_k{(1+a-c)}_{m}k!m!}}.\end{array}$$

Using the known identities

$$\begin{array}{c}\hfill {\left(a\right)}_k{(a+k)}_m={\left(a\right)}_{k+m}\mathrm{and}{(-k)}_m={\displaystyle \frac{{(-1)}^{m}k!}{(k-m)!}},\end{array}$$

we have, after some simplification,

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{k=0}^{\infty }\sum _{m=0}^k{\displaystyle \frac{{\left(a\right)}_{k+m}{\left(b\right)}_k{(d+1)}_k{(-1)}^{k+m}}{{(2+a-b)}_k{\left(d\right)}_k{(1+a-c)}_m(k-m)!m!}}.\end{array}$$

Now, replacing k by $k+m$ and using the result [3] (Equation (2), p. 57)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\sum _{k=0}^{n}B(k,n)=\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }B(k,n+k),\end{array}$$

we have, after some algebra,

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_{2m}{\left(b\right)}_m{(d+1)}_m}{{(2+a-b)}_m{\left(d\right)}_m{(1+a-c)}_{m}m!}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(a+2m)}_k{(b+m)}_k{(d+1+m)}_k{(-1)}^k}{{(2+a-b+m)}_k{(d+m)}_{k}k!}}.\end{array}$$

Summing up the inner series, we have

$$\begin{array}{c}\hfill \mathbf{S}=\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_{2m}{\left(b\right)}_m{(d+1)}_m}{{(2+a-b)}_m{\left(d\right)}_m{(1+a-c)}_{m}m!}}{}_{3}F_2\left[\begin{array}{c}a+2m,b+m,d+1+m\\ 2+a-b+m,d+m\end{array};-1\right].\end{array}$$

Finally, ${}_{3}F_2$ can now be evaluated with the help of the extended Kummer’s summation theorem (28) and after much simplification, summing up the result and series in terms of ${}_{2}F_1$, which can be further evaluated with the help of the classical Gauss summation theorem (5), we arrive at the right-hand side of (54). This completes the proof of (54). □

Corollary 10.

In result (54), if we take $d=1+a-b$, we obtain

$$\begin{array}{cc}& {}_{3}F_2\left[\begin{array}{c}a,b,c\\ 1+a-b,1+a-c\end{array};1\right]\hfill \\ =& {\displaystyle \frac{\mathrm{\Gamma }(1+\frac{1}{2}a)\mathrm{\Gamma }(1+a-b)\mathrm{\Gamma }(1+a-c)\mathrm{\Gamma }(1+\frac{1}{2}a-b-c)}{\mathrm{\Gamma }(1+a)\mathrm{\Gamma }(1+\frac{1}{2}a-b)\mathrm{\Gamma }(1+\frac{1}{2}a-c)\mathrm{\Gamma }(1+a-b-c)}},\hfill \end{array}$$

(55)

provided $\mathfrak{R}(a-2b-2c)>-2$,

• which is the classical Dixon’s summation theorem [3,11]. Thus, result (54) is regarded as an extension of the result (55).

Remark: For some interesting summation formulas involving hypergeometric functions in two variables (i.e., in Kampé de Fériet), we refer the reader to research papers by Verma et al. [12] and Wang and Chen [13].

Concluding remarks: In this research paper, we have provided applications of the extension of Kummer’s summation theorem due to Kim et al. in establishing (i) extensions of Gauss’s second summation theorem and Bailey’s summation theorem; (ii) extensions of several summations (including Ramanujan’s summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) extension of classical Dixon’s summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Lebniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey. We believe that the results established in this research paper are simple, interesting, easily established, have not appeared in the literature previously, and represent a definite contribution in the theory of generalized hypergeometric series of one and several variables. It is hoped that the results could be of potential use in the area of applied mathematics, statistics, engineering, and mathematical physics. We conclude this research paper by remarking that the new results (29), (30), (36)–(40), (41), (42), (48), (49), (52), (53), and (54), in their most general form, for example, generalization of result (29) in the form

$${}_{3}F_2\left[\begin{array}{c}a,b,d+n\\ \\ {\displaystyle \frac{1}{2}}(a+b+3),d\end{array};{\displaystyle \frac{1}{2}}\right]$$

for $\Re \left(d\right)\ne {\mathbb{Z}}_0^{-}$ and $n\in {\mathbb{N}}_0$, is under investigation and will form a part of the subsequent paper in this direction.