Fractional Hypergeometric Functions

Abstract

The fractional calculus of special functions has significant importance and applications in various fields of science and engineering. Here, we aim to find the fractional integral and differential formulas of the extended hypergeometric-type functions by using the Marichev–Saigo–Maeda operators. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases due to the high degree of symmetry of the involved functions.

1. Introduction

Fractional calculus is nowadays one of the most rapidly growing subjects of mathematical analysis, in spite of the fact that it is nearly 300 years old. The giants of mathematics, G.W. Leibnitz and L. Euler, thought about the possibility to perform differentiation of non-integer order. The real birth and far-reaching development of fractional calculus are due to the numerous efforts of mathematicians from the XIX century to the beginning of the XX century. A current trend in contemporary fractional calculus operators is the generalization of fractional calculus operators (GFCO). Due to the expansion of numerous and even unexpected recent applications of the operators of the classical fractional calculus operators, GFCO has become a very powerful tool stimulating the development of this field.

Since then, many known researchers and applied scientists have made valuable contributions in the modifications of the fractional calculus operators and their generalizations. The generalized Marichev–Saigo–Maeda fractional integral was introduced by Marichev [1] as Mellin-type convolution operators with the Appell function $F_3$ in their kernel. These operators were rediscovered and studied by Saigo [2] and, subsequently, by Saigo and Maeda [3] as generalizations of the Saigo fractional integral operators, which were first studied by Saigo [4] and then applied by Srivastava and Saigo [5], Srivastava and Agarwal [6] and Choi and Agarwal [7] in their systematic investigations. On the other hand, Virchenko et al. [8] defined some fractional integral operators involving the generalized function ${}_r\tilde{\tilde{F}}\left(z\right)$ and their inverses in terms of the Mellin transformation. At the same time, some researchers studied the fractional operators associated with Mittag-Leffler-type functions [9,10] and some studied extended Caputo-type and Erdlyi–Kober-type fractional derivative operators [11,12,13]. We can find more details in this direction in the survey papers by Kiryakova [14,15] and associated books [16,17,18].

2. Preliminaries

In recent years, many extensions and generalizations of special functions have witnessed a significant evolution. This modification in the theory of special functions offers an analytic foundation base for the many problems in mathematical physics and engineering sciences, which have been solved absolutely and which have various practical applications. The theory of special functions revolves around the two most important basic special functions, i.e., the beta function and the Gamma function, because most of the special functions are expressed either in terms of these functions.

Classical Euler beta function defined as follows [19].

$$B(x_1,x_2)={\int }_0^1{t}^{x_1-1}{(1-t)}^{x_2-1}dt,\Re \left(x_1\right),\Re \left(x_2\right)>0.$$

(1)

Gamma function defined as follows [19].

$$\Gamma \left(x_1\right)={\int }_0^{\infty }{t}^{x_1-1}{e}^{-t}dt,\Re \left(x_1\right)>0.$$

(2)

Furthermore, the mathematical and physical applications of hypergeometric functions can be found in various areas of applied mathematics, mathematical physics, and engineering. The Gauss hypergeometric function is a solution of a homogenous second-order differential equation, which is called the hypergeometric differential equation, and it is given by

$$z(1-z)\frac{d^{2}w}{d{z}^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw=0$$

(3)

The Gauss hypergeometric function ${}_2{F}_1$ is defined as [20]

$${}_2{F}_1(a,b,c;z)=F(a,b,c;z)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k{\left(b\right)}_k}{{\left(c\right)}_k}\frac{z^k}{k!},$$

(4)

where ${\left(u\right)}_k$ represents the Pochhammer symbol defined below:

$${\left(u\right)}_k:=\frac{\Gamma \left(u+k\right)}{\Gamma \left(u\right)}=\left\{\begin{array}{cc}1\hfill & k=0;u\in \mathbb{C}\backslash \left\{0\right\},\hfill \\ u\left(u+1\right)\cdots \left(u+k-1\right)\hfill & k\in \mathbb{N};u\in \mathbb{C}.\hfill \end{array}\right.$$

Later, Kummer replaced the parameter z by $\frac{z}{b}$ and took the limit $b\to \infty$ in the (3); then, the hypergeometric differential equation becomes a confluent hypergeometric differential equation or Kummer’s equation.

$$z\frac{d^{2}w}{d{z}^2}+(c-z)\frac{dw}{dz}-aw=0$$

(5)

The confluent hypergeometric function is the solution of the above differential Equation (5).

The confluent hypergeometric function is defined as [20]

$${}_1{F}_1(a,c;z)=\Phi (a,c;z)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k}{{\left(c\right)}_k}\frac{z^k}{k!}.$$

(6)

Very recently, Goyal et al. [21] introduced an extension of the beta function using the Wiman function, thus studying the various properties and relationships of that function:

$$B_{(u_1,u_2)}^{\left(u\right)}(y_1,y_2)={\int }_0^1{t}^{y_1-1}{(1-t)}^{y_2-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt,$$

(7)

where $min\left\{\Re \left(y_1\right),\Re \left(y_2\right)\right\}>0$, $\Re \left(u_1\right)>0,\Re \left(u_2\right)>0$, $u\ge 0$ and $E_{u_1,u_2}\left(z\right)$ is 2-parameter Mittag-Leffler function given by [22].

Motivated by the above work, Jain et al. [23] extended the Gauss hypergeometric function and confluent hypergeometric function by using the above extended beta function and studied the various properties of these extended functions. They also studied the increasing or decreasing nature (monotonicity), log-concavity, and log-convexity of the extended beta function defined in [21].

The extended Gauss hypergeometric function is defined as [23]:

$$\begin{array}{c}\hfill F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;z)=\sum _{k=0}^{\infty }\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+k,q_2-p_1)}{B(q_1,q_2-q_1)}{\left(q_0\right)}_k\frac{z^k}{k!},\end{array}$$

(8)

where $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0,\Re \left(s_2\right)>0$, $s\ge 0$, $\left|z\right|<1$ and $B_{(s_1,s_2)}^{\left(s\right)}(w_1,w_2)$ is the extended beta function.

The extended confluent hypergeometric function is defined as [23]:

$$\begin{array}{c}\hfill {\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;z)=\sum _{k=0}^{\infty }\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+k,q_2-q_1)}{B(q_1,q_2-q_1)}\frac{z^k}{k!},\end{array}$$

(9)

where $(\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0,\Re \left(s_2\right)>0$ and $s\ge 0).$

Remark 1.

If we set $s_1=s_2=1$ and $s=0$, then the extended Gauss hypergeometric function $F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;z)$ (8), and the extended confluent hypergeometric function ${\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;z)$ (9), reduced to the Gauss hypergeometric function ${}_2{F}_1(q_0,q_1,q_2;z)$ (4) and the confluent hypergeometric function $\Phi (q_1,q_2;z)$ (6).

The concept of the Hadamard product (convolution) of the functions f and g is very important for our results.

The Hadamard product of f and g can be defined as follows [24]:

$$(f\ast g)\left(z\right)=\sum _{n=0}^{\infty }{a}_n{b}_n{z}^n=(g\ast f)\left(z\right),$$

(10)

where $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ and $g\left(z\right)={\sum }_{n=0}^{\infty }{b}_n{z}^n$.

The third Appell function $F_3$ (also known as one of the functions in Horn’s list) is defined as follows (see [25]):

$$\begin{array}{c}\hfill F_3(a,a^{\prime },b,b^{\prime };c;x;y)=\sum _{m,n=0}^{\infty }\frac{{\left(a\right)}_m{\left(a\right)}_n{\left(b\right)}_m{\left(b^{\prime }\right)}_n}{{\left(c\right)}_{m+n}}\frac{x^m}{m!}\frac{y^n}{n!}\end{array}$$

(11)

here, $(max\mid x\mid ,\mid y\mid <1)$

Now, we recall here the generalized Marichev–Saigo–Maeda fractional integral and fractional derivative operators, introduced by Marichev [1], as Mellin-type convolution operators with an Appell function $F_3$ in their kernel, which are defined as follows:

Definition 1.

Let $a,a^{\prime },b,b^{\prime },c\in \mathbb{C}$ and $x>0$, then for $\Re \left(c\right)>0$

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}f\right)\left(x\right)\\ \hfill =\frac{x^{-a}}{\Gamma \left(c\right)}{\int }_0^x{(x-t)}^{c-1}{t}^{-a^{\prime }}{F}_3\left(a,a^{\prime },b,b^{\prime };c;1-\frac{t}{x},1-\frac{x}{t}\right)f\left(t\right)dt,\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \left(I_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}f\right)\left(x\right)\\ \hfill =\frac{x^{-a^{\prime }}}{\Gamma \left(c\right)}{\int }_x^{\infty }{(t-x)}^{c-1}{t}^{-a}{F}_3\left(a,a^{\prime },b,b^{\prime };c;1-\frac{x}{t},1-\frac{t}{x}\right)f\left(t\right)dt\end{array}$$

(13)

provided the function $f\left(t\right)$ is constrained such that the integrals in (12) and (13) exist.

The above fractional integral operators in Equations (12) and (13) can be written as

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}f\right)\left(x\right)={\left(\frac{d}{dx}\right)}^k\left(I_{0,x}^{a,a^{\prime },b+k,b^{\prime },c+k}f\right)\left(x\right)\\ \hfill (\Re (c)\le 0;k=[-\Re \left(c\right)+1\left]\right)\end{array}$$

(14)

and

$$\begin{array}{c}\hfill \left(I_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}f\right)\left(x\right)={\left(-\frac{d}{dx}\right)}^k\left(I_{x,\infty }^{a,a^{\prime },b,b^{\prime }+k,c+k}f\right)\left(x\right)\\ \hfill (\Re (c)\le 0;k=[-\Re \left(c\right)+1\left]\right)\end{array}$$

(15)

Furthermore, the corresponding Marichev–Saigo–Maeda fractional differential operators are given as:

Definition 2.

Let $a,a^{\prime },b,b^{\prime },c\in \mathbb{C}$ and $x>0$, then

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}f\right)\left(x\right)=\left(I_{0,x}^{-a^{\prime },-a,-b^{\prime },-b,-c}f\right)\left(x\right)\\ \hfill ={\left(\frac{d}{dx}\right)}^k\left(I_{0,x}^{-a^{\prime },-a,-b^{\prime }+k,-b,-c+k}f\right)\left(x\right)(\Re \left(c\right)>0;k=[\Re \left(c\right)]+1)\\ \hfill =\frac{1}{\Gamma (k-c)}{\left(\frac{d}{dx}\right)}^k{\left(x\right)}^{a^{\prime }}{\int }_0^x{(x-t)}^{k-c-1}{t}^a\\ \hfill \times F_3\left(-a^{\prime },-a,k-b^{\prime },-b;k-c;1-\frac{t}{x},1-\frac{x}{t}\right)f\left(t\right)dt\end{array}$$

(16)

and

$$\begin{array}{c}\hfill \left(D_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}f\right)\left(x\right)=\left(I_{x,\infty }^{-a^{\prime },-a,-b^{\prime },-b,-c}f\right)\left(x\right)\\ \hfill ={\left(-\frac{d}{dx}\right)}^k\left(I_{x,\infty }^{-a^{\prime },-a,-b^{\prime },-b+k,-c+k}f\right)\left(x\right)(\Re \left(c\right)>0;k=[\Re \left(c\right)]+1)\\ \hfill =\frac{1}{\Gamma (k-c)}{\left(-\frac{d}{dx}\right)}^k{\left(x\right)}^a{\int }_x^{\infty }{(t-x)}^{k-c-1}{t}^{a^{\prime }}\\ \hfill \times F_3\left(-a^{\prime },-a,-b^{\prime },k-b;k-c;1-\frac{x}{t},1-\frac{t}{x}\right)f\left(t\right)dt\end{array}$$

(17)

Remark 2.

Here, it is important to mention that the Appell function $F_3$ involved in Definitions (1) and (2) satisfies a system of two linear partial differential equations of the second order and reduces them to the Gauss hypergeometric function ${}_2{F}_1$ as follows (see ([26], p. 25, Equation (35)) and ([27], p. 301, Equation (9.4))):

$$\begin{array}{c}\hfill F_3(a,c-a,b,c-b;c;x;y){=}_2{F}_1\left[\begin{array}{cc}a,& b\\ c\end{array};x+y-xy\right]\end{array}$$

(18)

It is easily observed that

$$\begin{array}{c}\hfill F_3(a,0,b,b^{\prime };c;x;y)=F_3(a,a^{\prime },b,0;c;x;y){=}_2{F}_1\left[\begin{array}{cc}a,& b\\ c\end{array};x\right]\end{array}$$

(19)

and

$$\begin{array}{c}\hfill F_3(0,a^{\prime },b,b^{\prime };c;x;y)=F_3(a,a^{\prime },0,b^{\prime };c;x;y){=}_2{F}_1\left[\begin{array}{cc}{a}^{\prime },& b^{\prime }\\ c\end{array};y\right]\end{array}$$

(20)

In view of the obvious reduction formulas with first replacing the parameter a by $a+b$, and then setting $a^{\prime }=b^{\prime }=0,b=-d$ and $c=a$ in Equations (12), (13), (16) and (17) the generalized Marichev–Saigo–Maeda fractional integral and fractional derivative operators can be reduced to the Saigo fractional integral and derivative operators $I_{0,x}^{a,b,d},I_{x,\infty }^{a,b,d},D_{0,x}^{a,b,d}$, and $D_{x,\infty }^{a,b,d}$ involving the hypergeometric function ${}_2{F}_1$ in their kernel, which is defined by

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,b,d}f\right)\left(x\right)=\frac{x^{-a-b}}{\Gamma \left(a\right)}{\int }_0^x{(x-t)}_2^{a-1}{F}_1\left(a+b,-d;a;1-\frac{t}{x}\right)f\left(t\right)dt,\end{array}$$

(21)

and

$$\begin{array}{c}\hfill \left(I_{x,\infty }^{a,b,d}f\right)\left(x\right)=\frac{1}{\Gamma \left(a\right)}{\int }_x^{\infty }{(t-x)}_2^{a-1}{F}_1\left(a+b,-d;a;1-\frac{x}{t}\right)f\left(t\right)dt\end{array}$$

(22)

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,b,d}f\right)\left(x\right)=\left(I_{0,x}^{-a,-b,a+d}f\right)\left(x\right)\end{array}$$

(23)

and

$$\begin{array}{c}\hfill \left(D_{x,\infty }^{a,b,d}f\right)\left(x\right)=\left(I_{x,\infty }^{-a,-b,a+d}f\right)\left(x\right),\end{array}$$

(24)

where $a,b,d\in \mathbb{C}$ and $x>0$.

Several further consequences of the Marichev–Saigo–Maeda fractional integral and fractional derivative operators can easily be derived by first setting $b=-a$ and $b=0$ in Equations (21) and (22). We easily obtain the interesting consequences of the Marichev–Saigo–Maeda fractional integral and fractional derivative operators would involve the Erdélyi–Kober fractional integral operators ${\mathcal{E}}_{0,x}^{a,d}$ and ${\mathcal{K}}_{x,\infty }^{a,d}$, the Riemann–Liouville fractional integral operator ${\mathcal{R}}_{0,x}^a$ and the Weyl fractional integral operator ${\mathcal{W}}_{x,\infty }^a$, respectively:

$$\begin{array}{c}\hfill \left({\mathcal{R}}_{0,x}^{a}f\right)\left(x\right)=\frac{1}{\Gamma \left(a\right)}{\int }_0^x{(x-t)}^{a-1}f\left(t\right)dt=\left(I_{0,x}^{a,-a,d}f\right)\left(x\right),\end{array}$$

(25)

and

$$\begin{array}{c}\hfill \left({\mathcal{W}}_{x,\infty }^{a}f\right)\left(x\right)=\frac{1}{\Gamma \left(a\right)}{\int }_x^{\infty }{(t-x)}^{a-1}f\left(t\right)dt=\left(I_{x,\infty }^{a,-a,d}f\right)\left(x\right)\end{array}$$

(26)

and

$$\begin{array}{c}\hfill \left({\mathcal{E}}_{0,x}^{a,d}f\right)\left(x\right)=\frac{x^{-a-d}}{\Gamma \left(a\right)}{\int }_0^x{(x-t)}^{a-1}{t}^{d}f\left(t\right)dt=\left(I_{0,x}^{a,0,d}f\right)\left(x\right),\end{array}$$

(27)

and

$$\begin{array}{c}\hfill \left({\mathcal{K}}_{x,\infty }^{a,d}f\right)\left(x\right)=\frac{x^d}{\Gamma \left(a\right)}{\int }_x^{\infty }{(t-x)}^{a-1}{t}^{-a-d}f\left(t\right)dt=\left(I_{x,\infty }^{a,0,d}f\right)\left(x\right)\end{array}$$

(28)

The following lemmas proved in [4] are useful to prove our main results.

Lemma 1

([4]). Let $a,a^{\prime },b,b,c$ and $d\in \mathbb{C}$, $x>0$. Then, the Marichev–Saigo–Maeda fractional integral of the power $t^{d-1}$ is given as follows:

1. If $\Re \left(c\right)>0$ and $\Re \left(d\right)>max\{0,\Re (a+a^{\prime }+b-c),\Re (a^{\prime }-b^{\prime })\}$, then

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\right)\left(x\right)\\ \hfill =\frac{\Gamma \left(d\right)\Gamma (d+c-a-a^{\prime }-b)\Gamma (d+b^{\prime }-a^{\prime })}{\Gamma (d+b^{\prime })\Gamma (d+c-a-a^{\prime })\Gamma (d+c-a^{\prime }-b)}{x}^{d+c-a-a^{\prime }-1}\end{array}$$

(29)

2. If $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re (-b),\Re (a+a^{\prime }-c),\Re (a+b^{\prime }-c)\}$, then

$$\begin{array}{c}\hfill \left(I_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\right)\left(x\right)\\ \hfill =\frac{\Gamma (1-d-b)\Gamma (1-d-c+a+a^{\prime })\Gamma (1-d-c+a+b^{\prime })}{\Gamma (1-d)\Gamma (1-d-c+a+a^{\prime }+b^{\prime })\Gamma (1-d+a-b)}{x}^{d+c-a-a^{\prime }-1}\end{array}$$

(30)

Lemma 2

([4]). Let $a,a^{\prime },b,b^{\prime },c$ and $d\in \mathbb{C}$, $x>0$. Then, the Marichev–Saigo–Maeda fractional integral of the power $t^{d-1}$ is given as follows:

1. If $\Re \left(c\right)>0$ and $\Re \left(d\right)>max\{0,\Re (c-a-a^{\prime }-b^{\prime }),\Re (b-a)\}$, then

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\right)\left(x\right)\\ \hfill =\frac{\Gamma \left(d\right)\Gamma (d-c+a+a^{\prime }+b^{\prime })\Gamma (d-b+a)}{\Gamma (d-b)\Gamma (d-c+a+a^{\prime })\Gamma (d-c+a+b^{\prime })}{x}^{d-c+a+a^{\prime }-1}\end{array}$$

(31)

2. If $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re \left(b^{\prime }\right),\Re (c-a-a^{\prime }),\Re (c-a^{\prime }-b)\}$, then

$$\begin{array}{c}\hfill \left(D_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\right)\left(x\right)\\ \hfill =\frac{\Gamma (1-d+b^{\prime })\Gamma (1-d+c-a-a^{\prime })\Gamma (1-d+c-a^{\prime }-b)}{\Gamma (1-d)\Gamma (1-d+c-a-a^{\prime }-b)\Gamma (1-d-a^{\prime }+b^{\prime })}{x}^{d-c+a+a^{\prime }-1}\end{array}$$

(32)

3. Main Results

3.1. Marichev–Saigo–Maeda Fractional Integral of the Function $F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;z)$ and ${\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;z)$

In this section, we establish the Marichev–Saigo–Maeda fractional integral formulas involving the extended Gauss hypergeometric function and extended confluent hypergeometric function; the results are expressed as the Hadamard product of the generalized hypergeometric function and the extended Gauss hypergeometric function.

Theorem 1.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, $\left|t\right|<1$ be such that $\Re \left(c\right)>0$, and $\Re (d+n)>max\{0,\Re (a+a^{\prime }+b-c),\Re (a^{\prime }-b^{\prime })\}$; then, the following fractional integral formula holds true:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d+c-a-a^{\prime }-b)\Gamma (d+b^{\prime }-a^{\prime })}{\Gamma (d+b^{\prime })\Gamma (d+c-a-a^{\prime })\Gamma (d+c-a^{\prime }-b)}\\ \hfill \times F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d+c-a-a^{\prime }-b,d+b^{\prime }-a^{\prime };\\ d+b^{\prime },d+c-a-a^{\prime },d+c-a^{\prime }-b;\end{array}x\right]\end{array}$$

(33)

where ${}_3{F}_3\left(x\right)$ is a special case of the generalized hypergeometric function with $p=q=3$ [20].

Proof. 

From the definition of the Marichev–Saigo–Maeda fractional integral operator, we have:

$$\left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}f\left(t\right)\right)\left(x\right)=\frac{x^{-a}}{\Gamma \left(c\right)}{\int }_0^x{(x-t)}^{c-1}{t}^{-a^{\prime }}{F}_3\left(a,a^{\prime },b,b^{\prime };c;1-\frac{t}{x},1-\frac{x}{t}\right)f\left(t\right)dt,$$

(34)

Then, taking $f\left(t\right)=t^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)$ in the above equation, and using the definition of extended Gauss hypergeometric function (8), we obtain:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)=\left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\left(\sum _{n=0}^{\infty }\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}{\left(q_0\right)}_n\frac{z^n}{n!}\right)\right).\end{array}$$

(35)

Then, interchanging the order of integration and summation, we have:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)=\sum _{n=0}^{\infty }{\left(q_0\right)}_n\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}\frac{1}{n!}\left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d+n-1}\right)\left(x\right).\end{array}$$

(36)

Then, by the using Lemma (1), with d replaced by $d+n$, we have:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)=\sum _{n=0}^{\infty }{\left(q_0\right)}_n\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}\\ \hfill \times \frac{\Gamma (d+n)\Gamma (d+c-a-a^{\prime }-b+n)\Gamma (d+b^{\prime }-a^{\prime }+n)}{\Gamma (d+b^{\prime }+n)\Gamma (d+c-a-a^{\prime }+n)\Gamma (d+c-a^{\prime }-b+n)}\frac{x^{d+n+c-a-a^{\prime }-1}}{n!}.\end{array}$$

(37)

After simplification, the above equation reduces to

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)=x^{d+c-a-a^{\prime }-1}\sum _{n=0}^{\infty }{\left(q_0\right)}_n\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+k,q_2-p_1)}{B(q_1,q_2-q_1)}\\ \hfill \times \frac{\Gamma (d+n)\Gamma (d+c-a-a^{\prime }-b+n)\Gamma (d+b^{\prime }-a^{\prime }+n)}{\Gamma (d+b^{\prime }+n)\Gamma (d+c-a-a^{\prime }+n)\Gamma (d+c-a^{\prime }-b+n)}\frac{x^n}{n!}.\end{array}$$

(38)

After using the ${\left(a\right)}_n=\frac{\Gamma (a+n)}{\Gamma \left(a\right)}$, the above equation reduces to the following form:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)=x^{d+c-a-a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d+c-a-a^{\prime }-b)\Gamma (d+b^{\prime }-a^{\prime })}{\Gamma (d+b^{\prime })\Gamma (d+c-a-a^{\prime })\Gamma (d+c-a^{\prime }-b)}\\ \hfill \times \sum _{n=0}^{\infty }{\left(q_0\right)}_n\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}\frac{{\left(d\right)}_n{(d+c-a-a^{\prime }-b)}_n{(d+b^{\prime }-a^{\prime })}_n}{{(d+b^{\prime })}_n{(d+c-a-a^{\prime })}_n{(d+c-a^{\prime }-b)}_n}\frac{x^n}{n!}.\end{array}$$

(39)

then by interpreting the above equation with the help of the concept of the Hadamard product given by (10), we obtain our desired result.

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d+c-a-a^{\prime }-b)\Gamma (d+b^{\prime }-a^{\prime })}{\Gamma (d+b^{\prime })\Gamma (d+c-a-a^{\prime })\Gamma (d+c-a^{\prime }-b)}\\ \hfill \times F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d+c-a-a^{\prime }-b,d+b^{\prime }-a^{\prime };\\ d+b^{\prime },d+c-a-a^{\prime },d+c-a^{\prime }-b;\end{array}x\right]\end{array}$$

(40)

□

Theorem 2.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, $|\frac{1}{t}|<1$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re (-b),\Re (a+a^{\prime }-c),\Re (a+b^{\prime }-c)\}$; then, the following fractional integral formula holds true:

$$\begin{array}{c}\hfill \left(I_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;\frac{1}{t})\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma (1-d-b)\Gamma (1-d-c+a+a^{\prime })\Gamma (1-d-c+a+b^{\prime })}{\Gamma (1-d)\Gamma (1-d-c+a+a^{\prime }+b^{\prime })\Gamma (1-d+a-b)}\\ \hfill \times F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}1-d-b,1-d-c+a+a^{\prime },1-d-c+a+b^{\prime };\\ 1-d,1-d-c+a+a^{\prime }+b^{\prime },1-d+a-b;\end{array}x\right]\end{array}$$

(41)

where ${}_3{F}_3\left(x\right)$ is a special case of the generalized hypergeometric function with $p=q=3$ [20].

The proof of the Theorem (2) is the same as those of Theorem (1).

Theorem 3.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, be such that $\Re \left(c\right)>0$, and $\Re (d+n)>max\{0,\Re (a+a^{\prime }+b-c),\Re (a^{\prime }-b^{\prime })\}$; then, the following fractional integral formula holds true:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d+c-a-a^{\prime }-b)\Gamma (d+b^{\prime }-a^{\prime })}{\Gamma (d+b^{\prime })\Gamma (d+c-a-a^{\prime })\Gamma (d+c-a^{\prime }-b)}\\ \hfill \times {\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d+c-a-a^{\prime }-b,d+b^{\prime }-a^{\prime };\\ d+b^{\prime },d+c-a-a^{\prime },d+c-a^{\prime }-b;\end{array}x\right]\end{array}$$

(42)

where ${}_3{F}_3\left(x\right)$ is a special case of the generalized hypergeometric function with $p=q=3$ [20].

Proof. 

From the definition of the Marichev–Saigo–Maeda fractional integral operator, we have:

$$(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}f\left(t\right))\left(x\right)=\frac{x^{-a}}{\Gamma \left(c\right)}{\int }_0^x{(x-t)}^{c-1}{t}^{-a^{\prime }}{F}_3\left(a,a^{\prime },b,b^{\prime };c;1-\frac{t}{x},1-\frac{x}{t}\right)f\left(t\right)dt,$$

(43)

Then, taking $f\left(t\right)=t^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)$ in the above equation, and using the definition of the extended confluent hypergeometric function (9), we obtain:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)\right)\left(x\right)=\left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\left(\sum _{n=0}^{\infty }\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}\frac{z^n}{n!}\right)\right).\end{array}$$

(44)

Then, interchanging the order of integration and summation, we have:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)\right)\left(x\right)=\sum _{n=0}^{\infty }\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}\frac{1}{n!}\left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d+n-1}\right)\left(x\right).\end{array}$$

(45)

Then, by similar steps tot those in the proof of Theorem (1), we obtain our desired result.

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d+c-a-a^{\prime }-b)\Gamma (d+b^{\prime }-a^{\prime })}{\Gamma (d+b^{\prime })\Gamma (d+c-a-a^{\prime })\Gamma (d+c-a^{\prime }-b)}\\ \hfill \times {\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d+c-a-a^{\prime }-b,d+b^{\prime }-a^{\prime };\\ d+b^{\prime },d+c-a-a^{\prime },d+c-a^{\prime }-b;\end{array}x\right]\end{array}$$

(46)

□

Theorem 4.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re (-b),\Re (a+a^{\prime }-c),\Re (a+b^{\prime }-c)\}$; then, the following fractional integral formula holds true:

$$\begin{array}{c}\hfill \left(I_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;\frac{1}{t})\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma (1-d-b)\Gamma (1-d-c+a+a^{\prime })\Gamma (1-d-c+a+b^{\prime })}{\Gamma (1-d)\Gamma (1-d-c+a+a^{\prime }+b^{\prime })\Gamma (1-d+a-b)}\\ \hfill \times {\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}1-d-b,1-d-c+a+a^{\prime },1-d-c+a+b^{\prime };\\ 1-d,1-d-c+a+a^{\prime }+b^{\prime },1-d+a-b;\end{array}x\right]\end{array}$$

(47)

where ${}_3{F}_3\left(x\right)$ is a special case of the generalized hypergeometric function with $p=q=3$ and defined in [20].

The proof of Theorem (4) is the same as that of Theorem (3).

3.2. Marichev–Saigo–Maeda Fractional Derivative of the Function $F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;z)$ and ${\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;z)$

Here, we derive the Marichev–Saigo–Maeda fractional derivative formulas involving the extended Gauss hypergeometric function and extended confluent hypergeometric function; the results are expressed as the Hadamard product of the generalized hypergeometric function and the extended Gauss hypergeometric function.

Theorem 5.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, $\left|t\right|<1$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)>max\{0,\Re (c-a-a^{\prime }-b^{\prime }),\Re (b-a)\}$; then, the following fractional derivative formula holds true:

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d-c+a+a^{\prime }+b^{\prime })\Gamma (d-b+a)}{\Gamma (d-b)\Gamma (d-c+a+a^{\prime })\Gamma (d-c+a+b^{\prime })}\\ \hfill \times F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d-c+a+a^{\prime }+b^{\prime },d-b+a;\\ d-b,d-c+a+a^{\prime },d-c+a+b^{\prime };\end{array}x\right]\end{array}$$

(48)

where ${}_3{F}_3(.)$ is a special case of the generalized hypergeometric function [20].

Proof. 

Applying the Marichev–Saigo–Maeda fractional derivative operator defined in (16) to function $f\left(t\right)=t^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)$, and using the definition of extended Gauss hypergeometric function (8), we obtain:

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)=\left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\left(\sum _{n=0}^{\infty }\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}{\left(q_0\right)}_n\frac{z^n}{n!}\right)\right).\end{array}$$

(49)

Then, interchanging the order of integration and summation because the Marichev–Saigo–Maeda fractional differential operator exists, involving the extended hypergeometric function with conditions $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, $\left|t\right|<1$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)>max\{0,\Re (c-a-a^{\prime }-b^{\prime }),\Re (b-a)\}$, so the convergence is uniform, we have:

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)=\sum _{n=0}^{\infty }{\left(q_0\right)}_n\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}\frac{1}{n!}\left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d+n-1}\right)\left(x\right).\end{array}$$

(50)

Then, by using Lemma (2), and following the same procedure as in the proof of Theorem (1), we obtain our desired result.

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d-c+a+a^{\prime }+b^{\prime })\Gamma (d-b+a)}{\Gamma (d-b)\Gamma (d-c+a+a^{\prime })\Gamma (d-c+a+b^{\prime })}\\ \hfill \times F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d-c+a+a^{\prime }+b^{\prime },d-b+a;\\ d-b,d-c+a+a^{\prime },d-c+a+b^{\prime };\end{array}x\right]\end{array}$$

(51)

□

Theorem 6.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, $|\frac{1}{t}|<1$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re \left(b^{\prime }\right),\Re (c-a-a^{\prime }),\Re (c-a^{\prime }-b)\}$, then the following fractional derivative formula holds true:

$$\begin{array}{c}\hfill \left(D_{0,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{F}_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;\frac{1}{t})\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma (1-d+b^{\prime })\Gamma (1-d+c-a-a^{\prime })\Gamma (1-d-a^{\prime }-b+c)}{\Gamma (1-d)\Gamma (1-d-a-a^{\prime }-b+c)\Gamma (1-d-a^{\prime }+b^{\prime })}\\ \hfill \times F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}1-d+b^{\prime },1-d+c-a-a^{\prime },1-d-a^{\prime }-b+c;\\ 1-d,1-d-a-a^{\prime }-b+c,1-d-a^{\prime }+b^{\prime };\end{array}x\right]\end{array}$$

(52)

where ${}_3{F}_3(.)$ is a special case of the generalized hypergeometric function [20].

The proof of Theorem (6) is the same as that of Theorem (5).

Theorem 7.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)>max\{0,\Re (c-a-a^{\prime }-b^{\prime }),\Re (b-a)\}$, then the following fractional derivative formula holds true:

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d-c+a+a^{\prime }+b^{\prime })\Gamma (d-b+a)}{\Gamma (d-b)\Gamma (d-c+a+a^{\prime })\Gamma (d-c+a+b^{\prime })}\\ \hfill \times {\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d-c+a+a^{\prime }+b^{\prime },d-b+a;\\ d-b,d-c+a+a^{\prime },d-c+a+b^{\prime };\end{array}x\right]\end{array}$$

(53)

where ${}_3{F}_3(.)$ is a special case of ${}_p{F}_q(.)$, when $p=q=3$ [20].

Proof. 

Applying the Marichev–Saigo–Maeda fractional derivative operator defined in (16) to function $f\left(t\right)=t^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)$, and using the definition of the extended confluent hypergeometric function (9), we obtain:

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)\right)\left(x\right)=\left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\left(\sum _{n=0}^{\infty }\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}\frac{z^n}{n!}\right)\right).\end{array}$$

(54)

Then, interchanging the order of integration and summation, which is valid under the conditions of Theorem (7), we have:

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)\right)\left(x\right)=\sum _{n=0}^{\infty }\frac{B_{(s_1,s_2)}^{\left(s\right)}(q_1+n,q_2-p_1)}{B(q_1,q_2-q_1)}\frac{1}{n!}\left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d+n-1}\right)\left(x\right).\end{array}$$

(55)

Then, by using Lemma (2), and following the same procedure as in the proof of Theorem (1), we obtain our desired result.

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d-c+a+a^{\prime }+b^{\prime })\Gamma (d-b+a)}{\Gamma (d-b)\Gamma (d-c+a+a^{\prime })\Gamma (d-c+a+b^{\prime })}\\ \hfill \times {\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d-c+a+a^{\prime }+b^{\prime },d-b+a;\\ d-b,d-c+a+a^{\prime },d-c+a+b^{\prime };\end{array}x\right]\end{array}$$

(56)

□

Theorem 8.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re \left(b^{\prime }\right),\Re (c-a-a^{\prime }),\Re (c-a^{\prime }-b)\}$ then the following fractional derivative formula holds true:

$$\begin{array}{c}\hfill \left(D_{0,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;\frac{1}{t})\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma (1-d+b^{\prime })\Gamma (1-d+c-a-a^{\prime })\Gamma (1-d-a^{\prime }-b+c)}{\Gamma (1-d)\Gamma (1-d-a-a^{\prime }-b+c)\Gamma (1-d-a^{\prime }+b^{\prime })}\\ \hfill \times {\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}1-d+b^{\prime },1-d+c-a-a^{\prime },1-d-a^{\prime }-b+c;\\ 1-d,1-d-a-a^{\prime }-b+c,1-d-a^{\prime }+b^{\prime };\end{array}x\right]\end{array}$$

(57)

where ${}_3{F}_3(.)$ is a special case of ${}_p{F}_q(.)$, when $p=q=3$ and defined in [20].

The proof of Theorem (8) is the same as that of Theorem (7).

3.3. Examples of the Marichev–Saigo–Maeda Fractional Integral of the Function $F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;z)$ and ${\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;z)$

Due to the high degree of symmetry, we can assume that $r_1=r_2=1$ and $r=0$, then a new extension of the Gauss hypergeometric function (8) and a new extension of the confluent hypergeometric function (9) can be reduced to the Gauss hypergeometric function (4) and confluent hypergeometric function (6). Then, from the formulae established in (33), (41), (42) and (47), we have the following results.

Corollary 1.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\left|t\right|<1$ be such that $\Re \left(c\right)>0$, and $\Re (d+n)>max\{0,\Re (a+a^{\prime }+b-c),\Re (a^{\prime }-b^{\prime })\}$, then the following fractional integral formula holds true:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{}_2{F}_1(q_0,q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d+c-a-a^{\prime }-b)\Gamma (d+b^{\prime }-a^{\prime })}{\Gamma (d+b^{\prime })\Gamma (d+c-a-a^{\prime })\Gamma (d+c-a^{\prime }-b)}\\ \hfill \times {}_2{F}_1(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d+c-a-a^{\prime }-b,d+b^{\prime }-a^{\prime };\\ d+b^{\prime },d+c-a-a^{\prime },d+c-a^{\prime }-b;\end{array}x\right]\end{array}$$

(58)

where ${}_3{F}_3(a_1,a_2,a_3,b_1,b_2,b_3;x)$ is a special case of the generalized hypergeometric function ${}_p{F}_q(a_1,\dots \dots ,$$a_p,b_1,\dots \dots ,b_q;x)$, when $p=q=3$ and defined in [20].

Corollary 2.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $|\frac{1}{t}|<1$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re (-b),\Re (a+a^{\prime }-c),\Re (a+b^{\prime }-c)\}$, then the following fractional integral formula holds true:

$$\begin{array}{c}\hfill \left(I_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{}_2{F}_1(q_0,q_1,q_2;\frac{1}{t})\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma (1-d-b)\Gamma (1-d-c+a+a^{\prime })\Gamma (1-d-c+a+b^{\prime })}{\Gamma (1-d)\Gamma (1-d-c+a+a^{\prime }+b^{\prime })\Gamma (1-d+a-b)}\\ \hfill \times {}_2{F}_1(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}1-d-b,1-d-c+a+a^{\prime },1-d-c+a+b^{\prime };\\ 1-d,1-d-c+a+a^{\prime }+b^{\prime },1-d+a-b;\end{array}x\right]\end{array}$$

(59)

where ${}_3{F}_3(a_1,a_2,a_3,b_1,b_2,b_3;x)$ is a special case of the generalized hypergeometric function ${}_p{F}_q(a_1,\dots \dots ,$$a_p,b_1,\dots \dots ,b_q;x)$, when $p=q=3$ and defined in [20].

Corollary 3.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$ be such that $\Re \left(c\right)>0$, and $\Re (d+n)>max\{0,\Re (a+a^{\prime }+b-c),\Re (a^{\prime }-b^{\prime })\}$, then the following fractional integral formula holds true:

$$\begin{array}{c}\hfill \left(I_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\Phi (q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d+c-a-a^{\prime }-b)\Gamma (d+b^{\prime }-a^{\prime })}{\Gamma (d+b^{\prime })\Gamma (d+c-a-a^{\prime })\Gamma (d+c-a^{\prime }-b)}\\ \hfill \times \Phi (q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d+c-a-a^{\prime }-b,d+b^{\prime }-a^{\prime };\\ d+b^{\prime },d+c-a-a^{\prime },d+c-a^{\prime }-b;\end{array}x\right]\end{array}$$

(60)

where ${}_3{F}_3(a_1,a_2,a_3,b_1,b_2,b_3;x)$ is a special case of the generalized hypergeometric function ${}_p{F}_q(a_1,\dots \dots ,$$a_p,b_1,\dots \dots ,b_q;x)$, when $p=q=3$ and defined in [20].

Corollary 4.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re (-b),\Re (a+a^{\prime }-c),\Re (a+b^{\prime }-c)\}$, then the following fractional integral formula holds true:

$$\begin{array}{c}\hfill \left(I_{x,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\Phi (q_1,q_2;\frac{1}{t})\right)\left(x\right)\\ \hfill =x^{d+c-a-a^{\prime }-1}\frac{\Gamma (1-d-b)\Gamma (1-d-c+a+a^{\prime })\Gamma (1-d-c+a+b^{\prime })}{\Gamma (1-d)\Gamma (1-d-c+a+a^{\prime }+b^{\prime })\Gamma (1-d+a-b)}\\ \hfill \times \Phi (q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}1-d-b,1-d-c+a+a^{\prime },1-d-c+a+b^{\prime };\\ 1-d,1-d-c+a+a^{\prime }+b^{\prime },1-d+a-b;\end{array}x\right]\end{array}$$

(61)

where ${}_3{F}_3(a_1,a_2,a_3,b_1,b_2,b_3;x)$ is a special case of the generalized hypergeometric function ${}_p{F}_q(a_1,\dots \dots ,$$a_p,b_1,\dots \dots ,b_q;x)$, when $p=q=3$ and defined in [20].

3.4. Examples of the Marichev–Saigo–Maeda Fractional Derivative of the Function $F_{(s_1,s_2)}^{\left(s\right)}(q_0,q_1,q_2;z)$ and ${\Phi }_{(s_1,s_2)}^{\left(s\right)}(q_1,q_2;z)$

If we put $r_1=r_2=1$ and $r=0$, then a new extension of the Gauss hypergeometric function (8) and a new extension of the confluent hypergeometric function (9) can be reduced to the Gauss hypergeometric function (4) and the confluent hypergeometric function (6). Then, from the formulae established in (48), (52), (53) and (57), we have the following results.

Corollary 5.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\left|t\right|<1$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)>max\{0,\Re (c-a-a^{\prime }-b^{\prime }),\Re (b-a)\}$, then the following fractional derivative formula holds true:

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{}_2{F}_1(q_0,q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d-c+a+a^{\prime }+b^{\prime })\Gamma (d-b+a)}{\Gamma (d-b)\Gamma (d-c+a+a^{\prime })\Gamma (d-c+a+b^{\prime })}\\ \hfill \times {}_2{F}_1(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d-c+a+a^{\prime }+b^{\prime },d-b+a;\\ d-b,d-c+a+a^{\prime },d-c+a+b^{\prime };\end{array}x\right]\end{array}$$

(62)

where ${}_3{F}_3(a_1,a_2,a_3,b_1,b_2,b_3;x)$ is a special case of the generalized hypergeometric function ${}_p{F}_q(a_1,\dots \dots ,$$a_p,b_1,\dots \dots ,b_q;x)$, when $p=q=3$ and defined in [20].

Corollary 6.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $|\frac{1}{t}|<1$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re \left(b^{\prime }\right),\Re (c-a-a^{\prime }),\Re (c-a^{\prime }-b)\}$, then the following fractional derivative formula holds true:

$$\begin{array}{c}\hfill \left(D_{0,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}{}_2{F}_1(q_0,q_1,q_2;\frac{1}{t})\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma (1-d+b^{\prime })\Gamma (1-d+c-a-a^{\prime })\Gamma (1-d-a^{\prime }-b+c)}{\Gamma (1-d)\Gamma (1-d-a-a^{\prime }-b+c)\Gamma (1-d-a^{\prime }+b^{\prime })}\\ \hfill \times {}_2{F}_1(q_0,q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}1-d+b^{\prime },1-d+c-a-a^{\prime },1-d-a^{\prime }-b+c;\\ 1-d,1-d-a-a^{\prime }-b+c,1-d-a^{\prime }+b^{\prime };\end{array}x\right]\end{array}$$

(63)

where ${}_3{F}_3(a_1,a_2,a_3,b_1,b_2,b_3;x)$ is a special case of the generalized hypergeometric function ${}_p{F}_q(a_1,\dots \dots ,$$a_p,b_1,\dots \dots ,b_q;x)$, when $p=q=3$ and defined in [20].

Corollary 7.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$ be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)>max\{0,\Re (c-a-a^{\prime }-b^{\prime }),\Re (b-a)\}$, then the following fractional derivative formula holds true:

$$\begin{array}{c}\hfill \left(D_{0,x}^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\Phi (q_1,q_2;t)\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma \left(d\right)\Gamma (d-c+a+a^{\prime }+b^{\prime })\Gamma (d-b+a)}{\Gamma (d-b)\Gamma (d-c+a+a^{\prime })\Gamma (d-c+a+b^{\prime })}\\ \hfill \times \Phi (q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}d,d-c+a+a^{\prime }+b^{\prime },d-b+a;\\ d-b,d-c+a+a^{\prime },d-c+a+b^{\prime };\end{array}x\right]\end{array}$$

(64)

where ${}_3{F}_3(a_1,a_2,a_3,b_1,b_2,b_3;x)$ is a special case of the generalized hypergeometric function ${}_p{F}_q(a_1,\dots \dots ,$$a_p,b_1,\dots \dots ,b_q;x)$, when $p=q=3$ and defined in [20].

Corollary 8.

Let $x>0,a,a^{\prime },b,b^{\prime },c,d\in \mathbb{C}$ and $\Re \left(q_2\right)>\Re \left(q_1\right)>0$, $\Re \left(s_1\right)>0$, $\Re \left(s_2\right)>0$, $s\ge 0$, be such that $\Re \left(c\right)>0$ and $\Re \left(d\right)<1+min\{\Re \left(b^{\prime }\right),\Re (c-a-a^{\prime }),\Re (c-a^{\prime }-b)\}$, then the following fractional derivative formula holds true:

$$\begin{array}{c}\hfill \left(D_{0,\infty }^{a,a^{\prime },b,b^{\prime },c}{t}^{d-1}\Phi (q_1,q_2;\frac{1}{t})\right)\left(x\right)\\ \hfill =x^{d-c+a+a^{\prime }-1}\frac{\Gamma (1-d+b^{\prime })\Gamma (1-d+c-a-a^{\prime })\Gamma (1-d-a^{\prime }-b+c)}{\Gamma (1-d)\Gamma (1-d-a-a^{\prime }-b+c)\Gamma (1-d-a^{\prime }+b^{\prime })}\\ \hfill \times \Phi (q_1,q_2;x)\ast {}_3{F}_3\left[\begin{array}{c}1-d+b^{\prime },1-d+c-a-a^{\prime },1-d-a^{\prime }-b+c;\\ 1-d,1-d-a-a^{\prime }-b+c,1-d-a^{\prime }+b^{\prime };\end{array}x\right]\end{array}$$

(65)

where ${}_3{F}_3(a_1,a_2,a_3,b_1,b_2,b_3;x)$ is a special case of the generalized hypergeometric function ${}_p{F}_q(a_1,\dots \dots ,$$a_p,b_1,\dots \dots ,b_q;x)$, when $p=q=3$ and defined in [20].

4. Concluding Remarks

Motivated by the demonstrated usages and the potential for application of the many operators in fractional calculus, and also the considerably large spectrum of special functions in mathematics, physics, and engineering, we have introduced here new results for the Marichev–Saigo–Maeda fractional integral and Marichev–Saigo–Maeda fractional derivative involving extended hypergeometric functions. Our results are expressed as the Hadamard product of the extended hypergeometric functions and generalized hypergeometric functions. Some special cases of our main results are also derived easily due to the high degree of symmetry of the involved functions. The results presented in this paper are very useful in the theory of approximation, which is used to find several inequalities of the Marichev–Saigo–Maeda fractional operators involving extended hypergeometric functions. We conclude this investigation by emphasizing the possibility of further research involving q-extensions of hypergeometric functions.