Evaluation of Certain Definite Integrals Involving Generalized Hypergeometric Functions

Abstract

In 2012, Chu investigated the generalization of classical Watson–Whipple–Dixon summation theorems in the form of analytical formulas. By employing four generalized Watson summation formulas, the objective of this paper is to evaluate a new class of several Eulerian-type integrals (single and double) and Laplace-type integrals involving a hypergeometric function. Several interesting special cases are also given. Symmetry arises spontaneously in the hypergeometric function.

1. Introduction

The generalized hypergeometric function is defined [1,2,3,4] by

$${}_{p}F_q\left[\begin{array}{c}{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_p\hfill \\ {\beta }_1,{\beta }_2,\dots ,{\beta }_q\hfill \end{array}\right|\zeta ]=\sum _{n=0}^{\infty }{\displaystyle \frac{{\prod }_{i=1}^p{\left({\alpha }_i\right)}_n}{{\prod }_{j=1}^q{\left({\beta }_j\right)}_n}}{\displaystyle \frac{{\zeta }^n}{n!}},$$

(1)

where ${\alpha }_i$ and ${\beta }_j$ are complex parameters such that the denominators of the summands on the right-hand side are not equal to zero or negative integers.

The symbol ${\left(\alpha \right)}_n$ is the well-known Pochhammer symbol [5] for any complex number $\alpha$ given by

$${\left(\alpha \right)}_n=\left\{\begin{array}{c}\alpha (\alpha +1)\dots (\alpha +n-1),n\in \mathbb{N}\hfill \\ 1,n=0\hfill \end{array}\right..$$

(2)

In terms of gamma function, it can be written as

$${\left(\alpha \right)}_n={\displaystyle \frac{\Gamma (\alpha +n)}{\Gamma \left(\alpha \right)}},\alpha \notin {\mathbb{Z}}_0^{-}.$$

The series given by (1) is convergent for all $p\le q$ by the ratio test [1]. For $p<q+1$, it converges everywhere and converges nowhere for $p>q+1$. Further, if $p=q+1$, it converges absolutely for $\zeta =1$ provided $\delta =\Re \left({\sum }_{j=1}^q{\beta }_j-{\sum }_{i=1}^p{\alpha }_i\right)>0$ holds and is conditionally convergent for $\left|\zeta \right|=1$ and $\zeta \ne -1$ if $-1<\delta \le 0$ and diverges for $\left|\zeta \right|=1$ and $\zeta \ne 1$ if $\delta \le -1$.

It is interesting to mention here that the generalized hypergeometric function exhibits symmetry in both its numerator and denominator parameters. Any reordering of the numerator parameters ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_p$ does not alter the function, and, similarly, any reordering of the denominator parameters ${\beta }_1,{\beta }_2,\dots ,{\beta }_q$ also yields the same function [6].

The result will be significant whenever a generalized hypergeometric series can be summed in terms of gamma functions because there are very few of these summation theorems in the literature. However, in this paper, we shall mention the following Watson summation theorem viz,

$${}_{3}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2,{\alpha }_3\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}},2{\alpha }_3\hfill \end{array}\right|1]={\displaystyle \frac{\sqrt{\pi }\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma (\frac{1}{2}+{\alpha }_3)\Gamma (\frac{1-{\alpha }_1-{\alpha }_2}{2}+{\alpha }_3)}{\Gamma \left(\frac{1+{\alpha }_1}{2}\right)\Gamma \left(\frac{1+{\alpha }_2}{2}\right)\Gamma (\frac{1-{\alpha }_1}{2}+{\alpha }_3)\Gamma (\frac{1-{\alpha }_2}{2}+{\alpha }_3)}}=W_{0,0}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

(3)

provided $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2)>-1$.

For the two integer parameters m and n, let us denote

$$W_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={}_{3}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2,{\alpha }_3\hfill \\ {\displaystyle \frac{1+{\alpha }_1+{\alpha }_2+m}{2}},2{\alpha }_3+n\hfill \end{array}\right|1].$$

Clearly, for $m=n=0$, we obtain (3).

In 1992, Lavoie, Grondin, and Rathie [3] obtained the generalization of Watson series (3) with two integer parameters m and n in the following form:

$$\begin{array}{c}{}_{3}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2,{\alpha }_3\hfill \\ {\displaystyle \frac{1+{\alpha }_1+{\alpha }_2+m}{2}},2{\alpha }_3+n\hfill \end{array}|1\right]\hfill \\ =G_{m,n}{\displaystyle \frac{2^{{\alpha }_1+{\alpha }_2+m-2}\Gamma \left(\frac{1}{2}({\alpha }_1+{\alpha }_2+m+1)\right)\Gamma ({\alpha }_3+⌊\frac{n}{2}⌋+\frac{1}{2})\Gamma ({\alpha }_3-\frac{{\alpha }_1}{2}-\frac{{\alpha }_2}{2}-\left|\frac{m+n}{2}\right|+\frac{n}{2}+\frac{1}{2})}{\Gamma \left(\frac{1}{2}\right)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)}}\hfill \\ \times \left[H_{m,n}{\displaystyle \frac{\Gamma \left(\frac{{\alpha }_1}{2}+\frac{1-{(-1)}^m}{4}\right)\Gamma \left(\frac{{\alpha }_2}{2}\right)}{\Gamma \left({\alpha }_3-\frac{{\alpha }_1}{2}+\frac{1}{2}+⌊\frac{n}{2}⌋-\frac{{(-1)}^n(1-{(-1)}^m)}{4}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_2}{2}+\frac{1}{2}+⌊\frac{n}{2}⌋\right)}}\right.\hfill \\ \left.+I_{m,n}{\displaystyle \frac{\Gamma \left(\frac{{\alpha }_1}{2}+\frac{1+{(-1)}^m}{4})\right)\Gamma (\frac{{\alpha }_2}{2}+\frac{1}{2})}{\Gamma \left({\alpha }_3-\frac{{\alpha }_1}{2}+⌊\frac{n+1}{2}⌋+\frac{{(-1)}^n(1-{(-1)}^m)}{4}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_2}{2}+⌊\frac{n+1}{2}⌋\right)}}\right]\hfill \end{array}$$

(4)

for $m,n=-2,-1,0,1,2$. Here, as usual, $⌊x⌋$ is the greatest integer less than or equal to x and its modulus is denoted by $\left|x\right|$. For the values of the coefficients $G_{m,n}$, $H_{m,n}$, and $I_{m,n}$ one can refer to [3]. Later, in 2012, Chu [7] established analytical formulas explicitly for Watson series (3) with two integer parameters m ∈${\mathbb{N}}_0$ and n∈${\mathbb{N}}_0$ where ${\mathbb{N}}_0=\{0,1,2,\dots \}$ in the following form:

$$\begin{array}{c}{W}_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)\hfill \\ ={\displaystyle \sum _{i=0}^m}{\displaystyle \sum _{j=0}^n}{(-1)}^{i+j}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\displaystyle \frac{{\alpha }_1-{\alpha }_2-m+2i}{{\alpha }_1-{\alpha }_2-m+i}}{\displaystyle \frac{{\left({\alpha }_1\right)}_{i+j}{\left({\alpha }_2\right)}_{m-i+j}}{{({\alpha }_2-{\alpha }_1-i)}_m{(2{\alpha }_3-1)}_{2j}}}\hfill \\ \times {\displaystyle \frac{{\left({\alpha }_3\right)}_j{(2{\alpha }_3-1)}_j}{{\left(\frac{1+{\alpha }_1+{\alpha }_2+m}{2}\right)}_j{(2{\alpha }_3+n)}_j}}{W}_{0,0}({\alpha }_1+i+j,{\alpha }_2+m-i+j,{\alpha }_3+j)\hfill \end{array}$$

(5)

$$\begin{array}{c}{W}_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)\hfill \\ ={\displaystyle \sum _{i=0}^m}{\displaystyle \sum _{j=0}^n}{(-1)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{m}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\displaystyle \frac{{\alpha }_1-{\alpha }_2-m+2i}{{\alpha }_1-{\alpha }_2-m+i}}{\displaystyle \frac{{\left({\alpha }_1\right)}_{i+j}{\left({\alpha }_2\right)}_{m-i+j}}{{({\alpha }_2-{\alpha }_1-i)}_m{(2{\alpha }_3-2n-1)}_{2j}}}\hfill \\ \times {\displaystyle \frac{{({\alpha }_3-n)}_j{(2{\alpha }_3-2n-1)}_j}{{\left(\frac{1+{\alpha }_1+{\alpha }_2+m}{2}\right)}_j{(2{\alpha }_3-n)}_j}}{W}_{0,0}({\alpha }_1+i+j,{\alpha }_2+m-i+j,{\alpha }_3+j-n)\hfill \end{array}$$

(6)

$$\begin{array}{c}{W}_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)\hfill \\ ={\displaystyle \frac{{\left(\frac{1-{\alpha }_1+{\alpha }_2-m}{2}\right)}_m}{{\left(\frac{1+{\alpha }_1+{\alpha }_2-m}{2}\right)}_m}}{\displaystyle \sum _{i=0}^m}{\displaystyle \sum _{j=0}^n}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{m}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\displaystyle \frac{{\alpha }_1-{\alpha }_2-m+2i}{{\alpha }_1-{\alpha }_2-m+i}}{\displaystyle \frac{{\left({\alpha }_1\right)}_{i+j}{\left({\alpha }_2\right)}_{m-i+j}}{{({\alpha }_2-{\alpha }_1-i)}_m{(2{\alpha }_3-1)}_{2j}}}\hfill \\ \times {\displaystyle \frac{{\left({\alpha }_3\right)}_j{(2{\alpha }_3-1)}_j}{{\left(\frac{1+{\alpha }_1+{\alpha }_2+m}{2}\right)}_j{(2{\alpha }_3+n)}_j}}{W}_{0,0}({\alpha }_1+i+j,{\alpha }_2+m-i+j,{\alpha }_3+j)\hfill \end{array}$$

(7)

$$\begin{array}{c}{W}_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)\hfill \\ ={\displaystyle \frac{{\left(\frac{1-{\alpha }_1+{\alpha }_2-m}{2}\right)}_m}{{\left(\frac{1+{\alpha }_1+{\alpha }_2-m}{2}\right)}_m}}{\displaystyle \sum _{i=0}^m}{\displaystyle \sum _{j=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\displaystyle \frac{{\alpha }_1-{\alpha }_2-m+2i}{{\alpha }_1-{\alpha }_2-m+i}}{\displaystyle \frac{{\left({\alpha }_1\right)}_{i+j}{\left({\alpha }_2\right)}_{m-i+j}}{{({\alpha }_2-{\alpha }_1-i)}_m{(2{\alpha }_3-2n-1)}_{2j}}}\hfill \\ \times {\displaystyle \frac{{({\alpha }_3-n)}_j{(2{\alpha }_3-2n-1)}_j}{{\left(\frac{1+{\alpha }_1+{\alpha }_2+m}{2}\right)}_j{(2{\alpha }_3-n)}_j}}{W}_{0,0}({\alpha }_1+i+j,{\alpha }_2+m-i+j,{\alpha }_3+j-n)\hfill \end{array}$$

(8)

On the other hand, the Eulerian integrals are an important class of finite integrals. The integrals of this type can be written in the form

$$I={\int }_0^1{x}^{c-1}{(1-x)}^{d-1}f\left(x\right)dx$$

(9)

provided $\Re \left(c\right)>0$ and $\Re \left(d\right)>0.$

Next, if the function f(x) in the integrand is such that it can be expanded in a power series such as

$$f\left(x\right)=\sum _{n=0}^{\infty }{c}_n{x}^n$$

(10)

provided that the radius of convergence of (10) is not greater than unity, then, from (9), we have

$$\begin{array}{c}I={\int }_0^1{x}^{c-1}{(1-x)}^{d-1}{\displaystyle \sum _{n=0}^{\infty }}{c}_n{x}^{n}dx\hfill \\ ={\displaystyle \sum _{n=0}^{\infty }}{c}_n\left\{{\int }_0^1{x}^{c+n-1}{(1-x)}^{d-1}\right\}dx\hfill \end{array}$$

(11)

Evaluating the beta integral, and after a little simplification, we have

$$I={\displaystyle \frac{\Gamma \left(c\right)\Gamma \left(d\right)}{\Gamma (c+d)}}\sum _{n=0}^{\infty }{c}_n{\displaystyle \frac{{\left(c\right)}_n}{{(c+d)}_n}}$$

(12)

This is a general result.

Further, f(x) is in the form of a Gauss hypergeometric function ₂F₁ as follows:

$$f\left(x\right)={}_{2}F_1\left[\begin{array}{c}a,b\hfill \\ e\hfill \end{array}|sx\right].$$

(13)

Hence,

$$I={\displaystyle \frac{\Gamma \left(c\right)\Gamma \left(d\right)}{\Gamma (c+d)}}\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_n{\left(b\right)}_n{\left(c\right)}_n}{{\left(e\right)}_n{(c+d)}_n}}{\displaystyle \frac{{\left(sx\right)}^n}{n!}}={\displaystyle \frac{\Gamma \left(c\right)\Gamma \left(d\right)}{\Gamma (c+d)}}{}_{3}F_2\left[\begin{array}{c}a,b,c\hfill \\ e,c+d\hfill \end{array}|s\right].$$

(14)

Many summations theorems may be employed to sum the ₃F₂ function that appeared on the right-hand side of (14). For example, if, in (14), we take $s=1$, $d=c$, and $e={\displaystyle \frac{1}{2}}(a+b+1)$, then (14) takes the following form:

$$\begin{array}{c}{\int }_0^1{x}^{c-1}{(1-x)}^{c-1}{}_{2}F_1\left[\begin{array}{c}a,b\hfill \\ {\displaystyle \frac{1}{2}}(a+b+1)\hfill \end{array}|x\right]dx\hfill \\ ={\displaystyle \frac{\Gamma \left(c\right)\Gamma \left(c\right)}{\Gamma \left(2c\right)}}{}_{3}F_2\left[\begin{array}{c}a,b,c\hfill \\ {\displaystyle \frac{1}{2}}(a+b+1),2c\hfill \end{array}|1\right]\hfill \end{array}$$

(15)

We now observe that the ₃F₂ appearing on the right-hand side of (15) can be evaluated with the help of the classical Watson summation theorem (3), and, finally, we obtain the following Eulerian integral:

$${\int }_0^1{x}^{c-1}{(1-x)}^{c-1}{}_{2}F_1\left[\begin{array}{c}a,b\hfill \\ {\displaystyle \frac{1}{2}}(a+b+1)\hfill \end{array}|x\right]dx={\displaystyle \frac{\Gamma \left(c\right)\Gamma \left(c\right)}{\Gamma \left(2c\right)}}{W}_{0,0}(a,b,c)$$

(16)

provided $\Re \left(c\right)>0$, $\Re (2c-a-b)>-1$, and $W_{0,0}(a,b,c)$ is the same as given in (3).

Similar results can easily be obtained by employing the following Edwards double integral [8]:

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_1}{(1-\mu )}^{{\alpha }_1-1}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{1-{\alpha }_1-{\alpha }_2}d\mu d\nu ={\displaystyle \frac{\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)}{\Gamma ({\alpha }_1+{\alpha }_2)}}$$

(17)

provided $\Re \left({\alpha }_1\right)>0$ and $\Re \left({\alpha }_2\right)>0$ and the following well-known gamma integral

$${\int }_0^{\infty }{e}^{-st}{t}^{\nu -1}dt={\displaystyle \frac{\Gamma \left(\nu \right)}{s^{\nu }}}$$

provided $\Re \left(s\right)>0$ and $\Re \left(\nu \right)>0.$

The details are given in the following sections.

The rest of the paper is organized as follows.

In Section 2, eight Eulerian-type single integrals involving a hypergeometric function have been evaluated in terms of a gamma function by using results (5) to (8). Section 3 deals with sixteen Eulerian-type double integrals involving a hypergeometric function and eight Laplace-type integrals using a generalized hypergeometric function are obtained in Section 4. We discuss several interesting special cases of our main results in each section.

We conclude this section by remarking that the integrals (single and multiple) involving hypergeometric functions play an important role in the area of applied mathematics, statistics, engineering, physics, and several other branches. A large number of interesting applications can be seen in the seventh chapter of the standard text of Exton [9].

2. Evaluation of Eulerian-Type Integrals Involving Hypergeometric Function

The eight new classes of Eulerian-type integrals involving generalized hypergeometric functions to be established in this section are asserted in the following theorems.

Theorem 1.

For $m,n\in {\mathbb{N}}_0$, $\Re \left({\alpha }_3\right)>0$, and $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m+2n+1)>0$, the following result holds true:

$${\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3+n-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+m+1}{2}}\hfill \end{array}|\mu \right]d\mu ={\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma ({\alpha }_3+n)}{\Gamma (2{\alpha }_3+n)}}{W}_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\Psi }_1$$

(18)

where $W_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as in (5).

Proof. 

To prove the result (18), denoting the left-hand side of (18) by I, expressing ₂F₁ as a series, changing the order of integration and summation, which is easily seen to be justified due to the uniform convergence of the series in the interval $[0,1]$, and evaluating the beta integral, we have

$$I=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left({\alpha }_1\right)}_k{\left({\alpha }_2\right)}_k}{{\left[\frac{1}{2}({\alpha }_1+{\alpha }_2+m+1)\right]}_{k}k!}}{\displaystyle \frac{\Gamma ({\alpha }_3+k)\Gamma ({\alpha }_3+n)}{\Gamma (2{\alpha }_3+k+n)}}$$

Using (2) and after some simplification, we have

$$I={\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma ({\alpha }_3+n)}{\Gamma (2{\alpha }_3+n)}}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left({\alpha }_1\right)}_k{\left({\alpha }_2\right)}_k{\left({\alpha }_3\right)}_k}{{\left[\frac{1}{2}({\alpha }_1+{\alpha }_2+m+1)\right]}_k{(2{\alpha }_3+n)}_{k}k!}}$$

Summing up the series, we finally obtain

$$I={{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma ({\alpha }_3+n)}{\Gamma (2{\alpha }_3+n)}}}_3{F}_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2,{\alpha }_3\hfill \\ {\displaystyle \frac{1+{\alpha }_1+{\alpha }_2+m}{2}},2{\alpha }_3+n\hfill \end{array}\right|1]$$

Now, we observe that the ₃F₂’s appearance can be evaluated with the help of the result (5) and we easily arrive at the right side of (18). This completes the proof of the result (18) asserted in Theorem 1. □

Corollary 1.

In Theorem 1, if we take m = 0, n = 1; m = 1, n = 0; and m = n = 1, we respectively obtain the following integrals:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-2}{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2-1}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+1\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]={\delta }_1.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+2}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-2}{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi ({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]={\delta }_2.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+2}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-2}{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi ({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (1+2{\alpha }_3-{\alpha }_1)\Gamma (1+2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[(2{\alpha }_3-{\alpha }_1+{\alpha }_2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma ({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+1)\right.\hfill \\ -\left.(2{\alpha }_3+{\alpha }_1-{\alpha }_2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]={\delta }_3.\hfill \end{array}$$

In the same manner, the results and the special cases given in the following theorems and corollaries can be easily established by applying the corresponding summation Formulas (5) to (8). Hence, they are given here without proof.

Theorem 2.

For $m,n\in {\mathbb{N}}_0$, $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2+m)>0$, and $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m+2n+1)>0$, the following result holds true:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+m-1)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3+n\hfill \end{array}|\mu \right]d\mu \hfill \\ ={\displaystyle \frac{\Gamma \left({\alpha }_2\right)\Gamma \left[\frac{1}{2}({\alpha }_1-{\alpha }_2+m+1)\right]}{\Gamma \left[\frac{1}{2}({\alpha }_1+{\alpha }_2+m+1)\right]}}{W}_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\Psi }_2\hfill \end{array}$$

(19)

where $W_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as in (5).

Corollary 2.

In Theorem 2, if we take m = 0, n = 1; m = 1, n = 0; and m = n = 1, we respectively obtain the following integrals:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3+1\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-2}{\displaystyle \frac{\Gamma \left(\frac{1+{\alpha }_1-{\alpha }_2}{2}\right)\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left({\alpha }_3-\left(\frac{{\alpha }_1+{\alpha }_2-1}{2}\right)\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+1\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]={\delta }_4.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-3}{\displaystyle \frac{\Gamma \left({\alpha }_3+\frac{1}{2}\right)\Gamma \left(1+\frac{{\alpha }_1-{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_3-{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_3-{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]={\delta }_5.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3+1\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-2}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma (1+\frac{{\alpha }_1-{\alpha }_2}{2})\Gamma \left({\alpha }_3-\left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\right)}{{\pi }^{\frac{3}{2}}({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma (1+2{\alpha }_3-{\alpha }_1)\Gamma (1+2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[(2{\alpha }_3-{\alpha }_1+{\alpha }_2)\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.(2{\alpha }_3+{\alpha }_1-{\alpha }_2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_6.\hfill \end{array}$$

Theorem 3.

For $m,n\in {\mathbb{N}}_0$, $\Re ({\alpha }_3-n)>0$, and $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m-2n+1)>0$, the following result holds true:

$${\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-n-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+m+1}{2}}\hfill \end{array}|\mu \right]d\mu ={\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma ({\alpha }_3-n)}{\Gamma (2{\alpha }_3-n)}}{W}_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\Psi }_3$$

(20)

where $W_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as in (6).

Corollary 3.

In Theorem 3, if we take m = 0, n = 1; m = 1, n = 0; and m = n = 1, we respectively obtain the following integrals:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-2}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-4}{\displaystyle \frac{\Gamma ({\alpha }_3-1)\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]={\delta }_7.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+2}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-2}{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi ({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]={\delta }_8.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-2}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+2}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-3}{\displaystyle \frac{\Gamma ({\alpha }_3-1)\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi ({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}-{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}-{\displaystyle \frac{1}{2}}\right)\right]={\delta }_9.\hfill \end{array}$$

Theorem 4.

For $m,n\in {\mathbb{N}}_0$, $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2+m+1)>0$, and $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m-2n+1)>0$, the following result holds true:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+m-1)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3-n\hfill \end{array}|\mu \right]d\mu \hfill \\ ={\displaystyle \frac{\Gamma \left({\alpha }_2\right)\Gamma \left[\frac{1}{2}({\alpha }_1-{\alpha }_2+m+1)\right]}{\Gamma \left[\frac{1}{2}({\alpha }_1+{\alpha }_2+m+1)\right]}}{W}_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\Psi }_4\hfill \end{array}$$

(21)

where $W_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as in (6).

Corollary 4.

In Theorem 4, if we take m = 0, n = 1; m = 1, n = 0; and m = n = 1, we respectively obtain the following integrals:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3-1\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-6}{\displaystyle \frac{\Gamma ({\alpha }_3-\frac{1}{2})\Gamma \left(\frac{1+{\alpha }_1-{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]={\delta }_{10}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-3}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma (1+\frac{{\alpha }_1-{\alpha }_2}{2})\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{11}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3-1\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-5}{\displaystyle \frac{\Gamma (1+\frac{{\alpha }_1-{\alpha }_2}{2})\Gamma ({\alpha }_3-\frac{1}{2})\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]={\delta }_{12}.\hfill \end{array}$$

Theorem 5.

For $m,n\in {\mathbb{N}}_0$, $\Re \left({\alpha }_3\right)>0$, and $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m+2n+1)>0$, the following result holds true:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3+n-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2-m+1}{2}}\hfill \end{array}|\mu \right]d\mu ={\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma ({\alpha }_3+n)}{\Gamma (2{\alpha }_3+n)}}{W}_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\Psi }_5\hfill \end{array}$$

(22)

where $W_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as in (7).

Corollar 5.

In Theorem 5, if we take m = 0, n = 1; m = 1, n = 0; and m = n = 1, we respectively obtain the following integrals:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-2}{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3+\frac{1-{\alpha }_1-{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+1-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+1-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{13}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-3}{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{14}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{{\alpha }_2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-2}{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3+1-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+1-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left(1+{\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+1-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{15}.\hfill \end{array}$$

Theorem 6.

For $m,n\in {\mathbb{N}}_0$, $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2-m+1)>0$, and $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m+2n+1)>0$, the following result holds true:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-m-1)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3+n\hfill \end{array}|\mu \right]d\mu \hfill \\ ={\displaystyle \frac{\Gamma \left({\alpha }_2\right)\Gamma \left[\frac{1}{2}({\alpha }_1-{\alpha }_2-m+1)\right]}{\Gamma \left[\frac{1}{2}({\alpha }_1+{\alpha }_2-m+1)\right]}}{W}_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\Psi }_6\hfill \end{array}$$

(23)

where $W_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as in (7).

Corollary 6.

In Theorem 6, if we take m = 0, n = 1; m = 1, n = 0; and m = n = 1, we respectively obtain the following integrals:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3+1\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-2}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma (1+\frac{{\alpha }_1-{\alpha }_2}{2})\Gamma \left({\alpha }_3+\frac{1-{\alpha }_1-{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left(1+{\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{16}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-4}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(\frac{{\alpha }_1-{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{17}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3+1\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-2}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(\frac{{\alpha }_1-{\alpha }_2}{2}\right)\Gamma \left(1+{\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{18}.\hfill \end{array}$$

Theorem 7.

For $m,n\in {\mathbb{N}}_0$, $\Re ({\alpha }_3-n)>0$, and $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m-2n+1)>0$, the following result holds true:

$${\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-n-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2-m+1}{2}}\hfill \end{array}|\mu \right]d\mu ={\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma ({\alpha }_3-n)}{\Gamma (2{\alpha }_3-n)}}{W}_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\Psi }_7$$

(24)

where $W_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as in (8).

Corollary 7.

In Theorem 7, if we take m = 0, n = 1; m = 1, n = 0; and m = n = 1, we respectively obtain the following integrals:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-2}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-4}{\displaystyle \frac{\Gamma ({\alpha }_3-1)\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]={\delta }_{19}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-3}{\displaystyle \frac{\Gamma \left({\alpha }_3\right)\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{20}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-2}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{2{\alpha }_3-5}{\displaystyle \frac{\Gamma ({\alpha }_3-1)\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-1-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[(2{\alpha }_3-{\alpha }_1+{\alpha }_2-2)\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.(2{\alpha }_3+{\alpha }_1-{\alpha }_2-2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]={\delta }_{21}.\hfill \end{array}$$

Theorem 8.

For $m,n\in {\mathbb{N}}_0$, $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2-m+1)>0$, and $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m-2n+1)>0$, the following result holds true:

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-m-1)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3-n\hfill \end{array}|\mu \right]d\mu \hfill \\ ={\displaystyle \frac{\Gamma \left({\alpha }_2\right)\Gamma \left[\frac{1}{2}({\alpha }_1-{\alpha }_2-m+1)\right]}{\Gamma \left[\frac{1}{2}({\alpha }_1+{\alpha }_2-m+1)\right]}}{W}_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\Psi }_8\hfill \end{array}$$

(25)

where $W_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as in (8).

Corollary 8.

In Theorem 8, if we take m = 0, n = 1; m = 1, n = 0; and m = n = 1, we respectively obtain the following integrals:

$$\begin{array}{c}{\int }_0^1{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3-1\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-6}{\displaystyle \frac{\Gamma ({\alpha }_3-\frac{1}{2})\Gamma \left(\frac{1+{\alpha }_1-{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]={\delta }_{22}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-4}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(\frac{{\alpha }_1-{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]={\delta }_{23}.\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3-1\hfill \end{array}|\mu \right]d\mu \hfill \\ =2^{4{\alpha }_3-7}{\displaystyle \frac{\Gamma ({\alpha }_3-\frac{1}{2})\Gamma \left(\frac{{\alpha }_1-{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-1-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_1\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[(2{\alpha }_3-{\alpha }_1+{\alpha }_2-2)\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.(2{\alpha }_3+{\alpha }_1-{\alpha }_2-2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]={\delta }_{24}\hfill \end{array}$$

Remark 1.

It is interesting to mention here that in Theorem 1, 3, 5, or 7 and in Theorem 2, 4, 6, or 8, if we set $m=n=0$, then we respectively obtain the following interesting integrals:

$${\int }_0^1{{\mu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-1}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\hfill \\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\hfill \end{array}|\mu \right]d\mu ={\displaystyle \frac{{[\Gamma \left({\alpha }_3\right)]}^2}{\Gamma \left(2{\alpha }_3\right)}}{W}_{0,0}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\delta }_{25}.$$

$${\int }_0^1{{\mu }^{{\alpha }_2-1}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\hfill \\ 2{\alpha }_3\hfill \end{array}|\mu \right]d\mu ={\displaystyle \frac{\Gamma \left({\alpha }_2\right)\Gamma \left[\frac{1}{2}({\alpha }_1-{\alpha }_2+1)\right]}{\Gamma \left[\frac{1}{2}({\alpha }_1+{\alpha }_2+1)\right]}}{W}_{0,0}({\alpha }_1,{\alpha }_2,{\alpha }_3)={\delta }_{26}.$$

3. Evaluation of Double Integrals Involving Hypergeometric Function

The sixteen new classes of double integrals involving a hypergeometric function to be established in this section are asserted in the following theorems.

Theorem 9.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3+n-1}{{(1-\mu \nu )}^{1-2{\alpha }_3-n}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+m+1}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\Psi }_1\hfill \end{array}$$

(26)

provided $\Re \left({\alpha }_3\right)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m+2n+1)>0$, and ${\Psi }_1$ is the same as given in (18).

Proof. 

To prove (26), we express the hypergeometric function as a series, change the order of summation and integration, which can be easily justified due to the uniform convergence of the series in the interval $[0,1]$, and evaluate the double integral with the help of (17); we finally arrive at the right-hand side of (26). □

Corollary 9.

In Theorem 9, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$${\int }_0^1{\int }_0^1{\left[\nu (1-\nu )\right]}^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_1$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+2}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_2$$

$${\int }_0^1{\int }_0^1{\left[\nu (1-\nu )\right]}^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+2}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_3$$

where ${\delta }_1$, ${\delta }_2$, and ${\delta }_3$ are the same as given in Corollary 1.

Theorem 10.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3+n}{(1-\mu )}^{{\alpha }_3+n-1}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3-n}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+m+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\Psi }_1\hfill \end{array}$$

(27)

provided $\Re \left({\alpha }_3\right)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m+2n+1)>0$, and ${\Psi }_1$ is the same as given in (18).

Corollary 10.

In Theorem 2, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3+1}{(1-\mu )}^{{\alpha }_3}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_1$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{\left[\right(1-\mu \left)\right(1-\nu \left)\right]}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2\mu }}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+2}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_2$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3+1}{(1-\mu )}^{{\alpha }_3}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+2}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_3$$

where ${\delta }_1$, ${\delta }_2$, and ${\delta }_3$ are the same as given in Corollary 1.

Theorem 11.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+m-1)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2-m)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+n\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\Psi }_2\hfill \end{array}$$

(28)

provided $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2+m)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m+2n+1)>0$, and ${\Psi }_2$ is the same as given in (19).

Corollary 11.

In Theorem 11, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+1\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_4\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{{(1-\mu \nu )}^{-{\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_5\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{{(1-\mu \nu )}^{-{\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+1\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_6\hfill \end{array}$$

where ${\delta }_4$, ${\delta }_5$, and ${\delta }_6$ are the same as given in Corollary 2.

Theorem 12.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+m+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+m-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2-m)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+n\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_2\hfill \end{array}$$

(29)

provided $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2+m)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m+2n+1)>0$, and ${\Psi }_2$ is the same as given in (19).

Corollary 12.

In Theorem 12, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+1\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_4\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+2)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{-{\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_5\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+2)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{-{\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+1\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_6\hfill \end{array}$$

where ${\delta }_4$, ${\delta }_5$, and ${\delta }_6$ are the same as in Corollary 2.

Theorem 13.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-n-1}{(1-\mu \nu )}^{1-2{\alpha }_3+n}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+m+1)\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_3\hfill \end{array}$$

(30)

provided $\Re ({\alpha }_3-n)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m-2n+1)>0$, and ${\Psi }_3$ is the same as given in (20).

Corollary 13.

In Theorem 13, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-2}{{(1-\mu \nu )}^{2-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1)\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_7\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2)\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_8\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-2}{{(1-\mu \nu )}^{2-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2)\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_9\hfill \end{array}$$

where ${\delta }_7$, ${\delta }_8$, and ${\delta }_9$ are the same as given in Corollary 3.

Theorem 14.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3-n}{(1-\mu )}^{{\alpha }_3-n-1}{(1-\nu )}^{{\alpha }_3-1}{(1-\mu \nu )}^{1-2{\alpha }_3+n}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+m+1)\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_3\hfill \end{array}$$

(31)

provided $\Re ({\alpha }_3-n)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m-2n+1)>0$, and ${\Psi }_3$ is the same as given in (20).

Corollary 14.

In Theorem 14, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-2}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{2-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1)\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_7\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2)\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_8\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-2}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{2-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2)\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_9\hfill \end{array}$$

where ${\delta }_7$, ${\delta }_8$, and ${\delta }_9$ are the same as given in Corollary 3.

Theorem 15.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+m-1)}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2-m)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-n\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_4\hfill \end{array}$$

(32)

provided $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2+m+1)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m-2n+1)>0$, and ${\Psi }_4$ is the same as given in (21).

Corollary 15.

In Theorem 15, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-1\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_{10}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{{(1-\mu \nu )}^{-{\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_{11}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{{(1-\mu \nu )}^{-{\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-1\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_{12}\hfill \end{array}$$

where ${\delta }_{10}$, ${\delta }_{11}$, and ${\delta }_{12}$ are the same as given in Corollary 4.

Theorem 16.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+m+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+m-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2-m)}\hfill \\ \times {}_{2}F_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-n\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_4\hfill \end{array}$$

(33)

provided $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2+m+1)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2+m-2n+1)>0$, and ${\Psi }_4$ is the same as given in (21).

Corollary 16.

In Theorem 16, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-1\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{10}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+2)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{-{\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{11}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+2)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{-{\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-1\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{12}\hfill \end{array}$$

where ${\delta }_{10}$, ${\delta }_{11}$, and ${\delta }_{12}$ are the same as given in Corollary 4.

Theorem 17.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3+n-1}{{(1-\mu \nu )}^{1-2{\alpha }_3-n}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2-m+1}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\Psi }_5\hfill \end{array}$$

(34)

provided $\Re \left({\alpha }_3\right)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m+2n+1)>0$, and ${\Psi }_5$ is the same as given in (22).

Corollary 17.

In Theorem 17, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$${\int }_0^1{\int }_0^1{\left[\nu (1-\nu )\right]}^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{13}$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{\left[\right(1-\mu \left)\right(1-\nu \left)\right]}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{14}$$

$${\int }_0^1{\int }_0^1{\left[\nu (1-\nu )\right]}^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{15}$$

where ${\delta }_{13}$, ${\delta }_{14}$, and ${\delta }_{15}$ are the same as given in Corollary 5.

Theorem 18.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3+n}{(1-\mu )}^{{\alpha }_3+n-1}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3-n}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2-m+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\Psi }_5\hfill \end{array}$$

(35)

provided $\Re \left({\alpha }_3\right)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m+2n+1)>0$, and ${\Psi }_5$ is the same as given in (22).

Corollary 18.

In Theorem 18, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3+1}{(1-\mu )}^{{\alpha }_3}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{13}$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{\left[\right(1-\mu \left)\right(1-\nu \left)\right]}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{14}$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3+1}{(1-\mu )}^{{\alpha }_3}{(1-\nu )}^{{\alpha }_3-1}{{(1-\mu \nu )}^{-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{15}$$

where ${\delta }_{13}$, ${\delta }_{14}$, and ${\delta }_{15}$ are the same as given in Corollary 5.

Theorem 19.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-m-1)}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2+m)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+n\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_6\hfill \end{array}$$

(36)

provided $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2-m+1)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m+2n+1)>0$, and ${\Psi }_6$ is the same as given in (23).

Corollary 19.

In Theorem 19, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+1\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{16}$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(2-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{17}$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(2-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+1\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{18}$$

where ${\delta }_{16}$, ${\delta }_{17}$, and ${\delta }_{19}$ are the same as given in Corollary 6.

Theorem 20.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-m+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-m-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2+m)}\hfill \\ \times {}_{2}F_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+n\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_6\hfill \end{array}$$

(37)

provided $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2-m+1)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m+2n+1)>0$, and ${\Psi }_6$ is the same as given in (23).

Corollary 20.

In Theorem 20, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+1\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{16}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(2-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{17}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(2-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3+1\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{18}\hfill \end{array}$$

where ${\delta }_{16}$, ${\delta }_{17}$, and ${\delta }_{19}$ are the same as given in Corollary 6.

Theorem 21.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-n-1}{{(1-\mu \nu )}^{1-2{\alpha }_3+n}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2-m+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\Psi }_7\hfill \end{array}$$

(38)

provided $\Re ({\alpha }_3-n)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m-2n+1)>0$, and ${\Psi }_7$ is the same as given in (24).

Corollary 21.

In Theorem 21, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-2}{{(1-\mu \nu )}^{2-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{19}$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{\left[\right(1-\mu \left)\right(1-\nu \left)\right]}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{20}$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{(1-\mu )}^{{\alpha }_3-1}{(1-\nu )}^{{\alpha }_3-2}{{(1-\mu \nu )}^{2-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{21}$$

where ${\delta }_{19}$, ${\delta }_{20}$, and ${\delta }_{21}$ are the same as given in Corollary 7.

Theorem 22.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3-n}{(1-\mu )}^{{\alpha }_3-n-1}{(1-\nu )}^{{\alpha }_3-1}{(1-\mu \nu )}^{1-2{\alpha }_3+n}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2-m+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_7\hfill \end{array}$$

(39)

provided $\Re ({\alpha }_3-n)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m-2n+1)>0$, and ${\Psi }_7$ is the same as given in (24).

Corollary 22.

In Theorem 22, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$${\int }_0^1{\int }_0^1{\left[\nu (1-\nu )\right]}^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-2}{{(1-\mu \nu )}^{2-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{19}$$

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{\left[\right(1-\mu \left)\right(1-\nu \left)\right]}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{20}$$

$${\int }_0^1{\int }_0^1{\left[\nu (1-\nu )\right]}^{{\alpha }_3-1}{(1-\mu )}^{{\alpha }_3-2}{{(1-\mu \nu )}^{2-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{21}$$

where ${\delta }_{19}$, ${\delta }_{20}$, and ${\delta }_{21}$ are the same as given in Corollary 7.

Theorem 23.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-m-1)}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2+m)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-n\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_8\hfill \end{array}$$

(40)

provided $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2-m+1)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m-2n+1)>0$, and ${\Psi }_8$ is the same as given in (25).

Corollary 23.

In Theorem 23, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-1\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_{22}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(2-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_{23}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(2-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-1\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_{24}\hfill \end{array}$$

where ${\delta }_{22}$, ${\delta }_{23}$, and ${\delta }_{24}$ are the same as given in Corollary 8.

Theorem 24.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-m+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-m-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2+m)}\hfill \\ \times {}_{2}F_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-n\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\Psi }_8\hfill \end{array}$$

(41)

provided $\Re \left({\alpha }_2\right)>0$, $\Re ({\alpha }_1-{\alpha }_2-m+1)>0$, $\Re (2{\alpha }_3-{\alpha }_1-{\alpha }_2-m-2n+1)>0$, and ${\Psi }_8$ is the same as given in (25).

Corollary 24.

In Theorem 24, if we take $m=0,n=1;m=1,n=0$; and $m=n=1$, then we respectively obtain the following double integrals:

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-1\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{22}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(2-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{23}\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-2)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(2-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3-1\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{24}\hfill \end{array}$$

where ${\delta }_{22}$, ${\delta }_{23}$, and ${\delta }_{24}$ are the same as given in Corollary 8.

Remark 2.

It is interesting to mention here that in Theorem 9, 13, 17 or 21, in Theorem 10, 14, 18 or 22, in Theorem 11, 15, 19, or 23, and in Theorem 12, 16, 20, or 24, if we set $m=n=0$, then we respectively obtain the following interesting double integrals:

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{\left[\right(1-\mu \left)\right(1-\nu \left)\right]}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{25}$$

where ${\delta }_{25}$ is the same as given in Corollary 9.

$${\int }_0^1{\int }_0^1{\nu }^{{\alpha }_3}{\left[\right(1-\mu \left)\right(1-\nu \left)\right]}^{{\alpha }_3-1}{{(1-\mu \nu )}^{1-2{\alpha }_3}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{{\alpha }_1+{\alpha }_2+1}{2}}\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{25}$$

where ${\delta }_{25}$ is the same as given in Corollary 9.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\alpha }_2}{(1-\mu )}^{{\alpha }_2-1}{(1-\nu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}}_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{\nu (1-\mu )}{1-\mu \nu }}\right]d\mu d\nu \hfill \\ ={\delta }_{26}\hfill \end{array}$$

where ${\delta }_{26}$ is the same as given in Corollary 10.

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\nu }^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2+1)}{(1-\mu )}^{{\displaystyle \frac{1}{2}}({\alpha }_1-{\alpha }_2-1)}{(1-\nu )}^{{\alpha }_2-1}{(1-\mu \nu )}^{{\displaystyle \frac{1}{2}}(1-{\alpha }_1-{\alpha }_2)}\hfill \\ {\times }_2{F}_1\left[\begin{array}{c}{\alpha }_1,{\alpha }_3\\ 2{\alpha }_3\end{array}|{\displaystyle \frac{1-\nu }{1-\mu \nu }}\right]d\mu d\nu ={\delta }_{26}\hfill \end{array}$$

where ${\delta }_{26}$ is the same as given in Corollary 10.

The proof of Theorems 10 to 24 is similar to that of the Theorem 9 and, hence, it is omitted here.

4. Evaluation of Laplace-Type Integrals Involving Hypergeometric Function

The eight new classes of Laplace- type integrals involving a hypergeometric function to be established in this section are asserted in the following theorems.

Theorem 25.

For $m,n\in \mathbb{N}$, $\Re \left(s\right)>0$, and $\Re \left({\alpha }_1\right)>0$, the following result holds true:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+m+1),2{\alpha }_3+n\end{array}|st\right]dt=\Gamma \left({\alpha }_1\right)s^{-{\alpha }_1}{W}_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3).$$

(42)

where $W_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as given in (5).

Proof. 

To derive the result given in the Theorem 25, we express ${}_{2}F_2$ as a series, change the order of integration and summation, and then evaluate the integral. We finally arrive at the right-hand side of (42). □

Corollary 25.

In Theorem 25, if we take $m=0,n=1$; $m=1,n=0$; and $m=1,n=1$, we obtain the following integrals:

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3+1\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-2}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3+\frac{1-{\alpha }_1-{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+1\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2),2{\alpha }_3\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-3}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2),2{\alpha }_3+1\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-2}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_2\right)\Gamma (1+2{\alpha }_3-{\alpha }_1)\Gamma (1+2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[(2{\alpha }_3-{\alpha }_1+{\alpha }_2)\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma (1+{\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}})\right.\hfill \\ -\left.(2{\alpha }_3+{\alpha }_1-{\alpha }_2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

In exactly the same manner, the integrals given in the following Theorems 26, 27, 28, 29, 30, 31, and 32 can be evaluated; thus, they are given here without proof.

Theorem 26.

For $m,n\in \mathbb{N}$, $\Re \left(s\right)>0$, and $\Re \left({\alpha }_3\right)>0$, the following result holds true:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+m+1),2{\alpha }_3+n\end{array}|st\right]dt=\Gamma \left({\alpha }_3\right)s^{-{\alpha }_3}{W}_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3).$$

(43)

where $W_{m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as given in (5).

Corollary 26.

In Theorem 26, if we take $m=0,n=1$; $m=1,n=0$; and $m=1,n=1$, we obtain the following integrals:

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3+1\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-1}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma \left(2{\alpha }_3\right)\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3+\frac{1-{\alpha }_1-{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+1\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2),2{\alpha }_3\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-2}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma \left(2{\alpha }_3\right)\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi ({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2),2{\alpha }_3+1\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-1}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma \left(2{\alpha }_3\right)\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi ({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (1+2{\alpha }_3-{\alpha }_1)\Gamma (1+2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[(2{\alpha }_3-{\alpha }_1+{\alpha }_2)\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma (1+{\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}})\right.\hfill \\ -\left.(2{\alpha }_3+{\alpha }_1-{\alpha }_2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

Theorem 27.

For $m,n\in \mathbb{N}$, $\Re \left(s\right)>0$, and $\Re \left({\alpha }_1\right)>0$ the following result holds true:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+m+1),2{\alpha }_3-n\end{array}|st\right]dt=\Gamma \left({\alpha }_1\right)s^{-{\alpha }_1}{W}_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

(44)

where $W_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as given in (6).

Corollary 27.

In Theorem 27, if we take $m=0,n=1$; $m=1,n=0$; and $m=1,n=1$, we obtain the following integrals:

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3-1\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-6}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3-\frac{1}{2})\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2),2{\alpha }_3\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-3}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2),2{\alpha }_3-1\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-5}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3-\frac{1}{2})\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

Theorem 28.

For $m,n\in \mathbb{N}$, $\Re \left(s\right)>0$, and $\Re \left({\alpha }_3\right)>0$, the following result holds true:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+m+1),2{\alpha }_3-n\end{array}|st\right]dt=\Gamma \left({\alpha }_3\right)s^{-{\alpha }_3}{W}_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

(45)

where $W_{m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as given in (6).

Corollary 28.

In Theorem 28, if we take $m=0,n=1$; $m=1,n=0$; and $m=1,n=1$, we obtain the following integrals:

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3-1\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-4}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma (2{\alpha }_3-1)\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2),2{\alpha }_3\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-2}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma \left(2{\alpha }_3\right)\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi ({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+2),2{\alpha }_3-1\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-3}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma (2{\alpha }_3-1)\Gamma \left(1+\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi ({\alpha }_1-{\alpha }_2)\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

Theorem 29.

For $m,n\in \mathbb{N}$, $\Re \left(s\right)>0$, and $\Re \left({\alpha }_1\right)>0$ the following result holds true:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2-m+1),2{\alpha }_3+n\end{array}|st\right]dt=\Gamma \left({\alpha }_1\right)s^{-{\alpha }_1}{W}_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

(46)

where $W_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as given in (7).

Corollary 29.

In Theorem 29, if we take $m=0,n=1$; $m=1,n=0$; and $m=1,n=1$, we obtain the following integrals:

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3+1\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-2}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3+\frac{1-{\alpha }_1-{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+1\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2),2{\alpha }_3\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-4}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2),2{\alpha }_3+1\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-2}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3+1-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+1-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+1-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

Theorem 30.

For $m,n\in \mathbb{N}$, $\Re \left(s\right)>0$, and $\Re \left({\alpha }_3\right)>0$, the following result holds true:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2-m+1),2{\alpha }_3+n\end{array}|st\right]dt=\Gamma \left({\alpha }_3\right)s^{-{\alpha }_3}{W}_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

(47)

where $W_{-m,n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as given in (7).

Corollary 30.

In Theorem 30, if we take $m=0,n=1$; $m=1,n=0$; and $m=1,n=1$, we obtain the following integrals:

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3+1\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-1}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma \left(2{\alpha }_3\right)\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3+\frac{1-{\alpha }_1-{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+1\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+1\right)\right.\hfill \\ -\left.\Gamma \left({\displaystyle \frac{{\alpha }_1+1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2+1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{1}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2),2{\alpha }_3\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-3}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma \left(2{\alpha }_3\right)\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2),2{\alpha }_3+1\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-1}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma \left(2{\alpha }_3\right)\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left(1+{\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1+1)\Gamma (2{\alpha }_3-{\alpha }_2+1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left(1+{\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

Theorem 31.

For $m,n\in \mathbb{N}$, $\Re \left(s\right)>0$, and $\Re \left({\alpha }_1\right)>0$ the following result holds true:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2-m+1),2{\alpha }_3-n\end{array}|st\right]dt=\Gamma \left({\alpha }_1\right)s^{-{\alpha }_1}{W}_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

(48)

where $W_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as given in (8).

Corollary 31.

In Theorem 31, if we take $m=0,n=1$; $m=1,n=0$; and $m=1,n=1$, we obtain the following integrals:

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3-1\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-6}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3-\frac{1}{2})\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2),2{\alpha }_3\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-4}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3+\frac{1}{2})\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2),2{\alpha }_3-1\end{array}|st\right]dt\hfill \\ =2^{4{\alpha }_3-7}{s}^{-{\alpha }_1}{\displaystyle \frac{\Gamma ({\alpha }_3-\frac{1}{2})\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-1-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{{\pi }^{\frac{3}{2}}\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[(2{\alpha }_3-{\alpha }_1+{\alpha }_2-2)\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ -\left.(2{\alpha }_3+{\alpha }_1-{\alpha }_2-2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

Theorem 32.

For $m,n\in \mathbb{N}$, $\Re \left(s\right)>0$, and $\Re \left({\alpha }_3\right)>0$, the following result holds true:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2-m+1),2{\alpha }_3-n\end{array}|st\right]dt=\Gamma \left({\alpha }_3\right)s^{-{\alpha }_3}{W}_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

(49)

where $W_{-m,-n}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the same as given in (8).

Corollary 32.

In Theorem 32, if we take $m=0,n=1$; $m=1,n=0$; and $m=1,n=1$, we obtain the following integrals:

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3-1\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-4}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma (2{\alpha }_3-1)\Gamma \left(\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{1+{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2),2{\alpha }_3\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-3}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma \left(2{\alpha }_3\right)\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1)\Gamma (2{\alpha }_3-{\alpha }_2)}}\hfill \\ \times \left[\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3+{\displaystyle \frac{1-{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2),2{\alpha }_3-1\end{array}|st\right]dt\hfill \\ =2^{2{\alpha }_3-5}{s}^{-{\alpha }_3}{\displaystyle \frac{\Gamma (2{\alpha }_3-1)\Gamma \left(\frac{{\alpha }_1+{\alpha }_2}{2}\right)\Gamma \left({\alpha }_3-1-\frac{{\alpha }_1+{\alpha }_2}{2}\right)}{\pi \Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\Gamma (2{\alpha }_3-{\alpha }_1-1)\Gamma (2{\alpha }_3-{\alpha }_2-1)}}\hfill \\ \times \left[(2{\alpha }_3-{\alpha }_1+{\alpha }_2-2)\Gamma \left({\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_2}{2}}\right)\right.\hfill \\ +\left.(2{\alpha }_3+{\alpha }_1-{\alpha }_2-2)\Gamma \left({\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\displaystyle \frac{1+{\alpha }_2}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{{\alpha }_1}{2}}\right)\Gamma \left({\alpha }_3-{\displaystyle \frac{1+{\alpha }_2}{2}}\right)\right]\hfill \end{array}$$

Remark 3.

It is interesting to mention here that in Theorem 25, 27, 29, or 31 and in Theorem 26, 28, 30, or 32, if we set $m=n=0,$ we respectively obtain the following interesting integrals:

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_1-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_2,{\alpha }_3\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3\end{array}|st\right]dt=\Gamma \left({\alpha }_1\right)s^{-{\alpha }_1}{W}_{0,0}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

where $W_{0,0}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is given in (3).

$${\int }_0^{\infty }{e}^{-st}{t}^{{\alpha }_3-1}{}_{2}F_2\left[\begin{array}{c}{\alpha }_1,{\alpha }_2\\ {\displaystyle \frac{1}{2}}({\alpha }_1+{\alpha }_2+1),2{\alpha }_3\end{array}|st\right]dt=\Gamma \left({\alpha }_3\right)s^{-{\alpha }_3}{W}_{0,0}({\alpha }_1,{\alpha }_2,{\alpha }_3)$$

where $W_{0,0}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is given in (3).

5. Conclusions

In this paper, we have evaluated a new class of Eulerian-type integrals (single and double) and Laplace -type integrals involving a hypergeometric function by employing four generalizations of the classical Watson summation theorem with two integer parameters discovered by Chu. Several interesting special cases have also been given. Hypergeometric integrals are widely recognized for their significant role in various fields, including statistics, mathematical physics, and quantum chemistry. In statistics, they provide the foundation for representing probability distributions such as the beta and gamma distributions, along with density and cumulative distribution functions. In the realm of physics, these integrals appear in the study of elastic plate vibrations, heat conduction within cylinders, viscous fluid motion, and certain electrical networks. They also contribute to the analysis of heat flow in solids within conducting media and the exploration of specific non-linear oscillation phenomena. Quantum chemistry employs hypergeometric integrals in molecular calculations, while communications engineering utilizes them in contexts like Gaussian noise analysis. Additional applications include their use in electrical impedance theory and combinatorial analysis. Since all the results obtained in this paper are given in terms of the gamma function, they may be potentially useful in applied mathematics, engineering mathematics, and mathematical physics.

We conclude the paper by remarking that, following the same lines, Eulerian-type integrals (single and double) and Laplace -type integrals by employing generalizations of Dixon’s and Whipple’s summation theorems are under investigation and we will form two subsequent papers in this direction.