Generalized Integral Transform and Fractional Calculus Operators Involving a Generalized Mittag-Leffler (ML)-Type Function ^†

Abstract

In this paper, we consider a generalized Mittag-Leffler (ML)-type function and establish several integral formulas involving Jacobi and related transforms. We also establish some of the composition of generalized fractional derivative formulas associated with the generalized Mittag-Leffler (ML)-type function. Additionally, certain special cases of generalized fractional derivative formulas involving the Mittag-Leffler (ML)-type function are corollarily presented.

1. Introduction

In 1903, Gösta Mittag-Leffler [1] introduced the Mittag-Leffler (ML) function, which is represented by

$$\begin{array}{c}\hfill E_{\nu }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }(\nu k+1)},\Re \left(\nu \right)>0\end{array}$$

(1)

Wiman [2] generalized the Mittag-Leffler (ML) function (1) in 1905 and introduced the following definition:

$$\begin{array}{c}\hfill E_{\nu ,\kappa }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }(\nu k+\kappa )},\end{array}$$

(2)

where $\nu ,\kappa \in \mathbb{C}$ with $\Re \left(\nu \right)>0,\Re \left(\kappa \right)>0$.

Later, in 1971, Prabhakar [3] introduced the three-parameter Mittag-Leffler (ML) function, which can be defined as,

$$\begin{array}{c}\hfill E_{\nu ,\kappa }^{\gamma }\left(z\right)=\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k}{\mathsf{\Gamma }(\nu k+\kappa )}\frac{z^k}{k!},\end{array}$$

(3)

where $\kappa ,\nu ,\gamma \in \mathbb{C}$ with $\Re \left(\nu \right)>0,\Re \left(\kappa \right)>0,\Re \left(\gamma \right)>0$.

Wright [4] introduced an extension of the generalized hypergeometric function called the Fox–Wright function, which can be expressed in the following form:

$$\begin{array}{c}\hfill {}_f{\mathsf{\Psi }}_g\left(z\right)={\displaystyle {}_f{\mathsf{\Psi }}_g\left[\left.\begin{array}{c}{(m_a,M_a)}_{1,f}\\ {(n_b,N_b)}_{1,g}\end{array}\right|z\right]}=\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }(m_1+M_{1}k)\cdots \mathsf{\Gamma }(m_f+M_{f}k)}{\mathsf{\Gamma }(n_1+N_{1}k)\cdots \mathsf{\Gamma }(n_g+N_{g}k)}\frac{z^k}{k!},\end{array}$$

(4)

where $m_a,n_b\in \mathbb{C},a=1,\cdots ,f;b=1,\cdots ,g$, and the coefficients $M_1,\cdots ,M_f\in {\mathbb{R}}^{+}$ and $N_1,\cdots ,N_g\in {\mathbb{R}}^{+}$ satisfying the condition

$$\sum _{b=1}^g{N}_b-\sum _{a=1}^f{M}_a>-1.$$

(5)

In particular, when $M_a=N_b=1$, (4) reduces to

$${\displaystyle {}_f{\mathsf{\Psi }}_g\left[\left.\begin{array}{c}(m_1,1),\cdots ,(m_f,1)\\ (n_1,1),\cdots ,(n_g,1)\end{array}\right|z\right]}=\frac{{\prod }_{a=1}^f\mathsf{\Gamma }\left(m_a\right)}{{\prod }_{b=1}^g\mathsf{\Gamma }\left(n_b\right)}{\displaystyle {}_f{F}_g\left[\left.\begin{array}{c}{m}_1,\cdots ,m_f\\ n_1,\cdots ,n_g\end{array}\right|z\right]},$$

(6)

where ${}_f{F}_g(·)$ is the well-known generalized hypergeometric function [5].

In this work, we define a Mittag-Leffler (ML)-type confluent hypergeometric function introduced by Ghanim and Al-Janaby [6]:

$$\begin{array}{c}\hfill M_{\nu ,\kappa }^{\eta ,\gamma }\left(z\right)=\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }\left(\kappa \right)\mathsf{\Gamma }(\eta k+\gamma )}{\mathsf{\Gamma }\left(\gamma \right)\mathsf{\Gamma }(\nu k+\kappa )}\frac{z^k}{k!},\end{array}$$

(7)

where $\kappa ,\eta ,\nu ,\gamma \in \mathbb{C}$, such that $\Re \left(\nu \right)>0$. It is worth pointing out that series representation of Equation (7) returns a variety of connections with special functions, including a confluent hypergeometric function and generalized Mittag-Leffler (ML) functions (1)–(3).

2. Jacobi and Related Integral Transforms

The Jacobi integral transform [7] (p. 501) of a function $f\left(r\right)$ can be represented as follows:

$$\begin{array}{c}\hfill {\mathrm{J}}^{(\lambda ,\sigma )}\left[f\left(r\right);n\right]={\int }_{-1}^1{(1-r)}^{\lambda }{(1+r)}^{\sigma }{P}_n^{(\lambda ,\sigma )}\left(r\right)f\left(r\right)dr,\end{array}$$

(8)

where $min\left\{\Re \left(\lambda \right),\Re \left(\sigma \right)\right\}>-1;n\in {\mathbb{N}}_0,r\in \mathbb{C}$ and provided that the integral on the right-hand side in (8) exists. Here, $P_n^{(\lambda ,\sigma )}\left(r\right)$ is the classical orthogonal Jacobi polynomial [8] (Chapter 10) defined by

$$\begin{array}{cc}\hfill P_n^{(\lambda ,\sigma )}\left(r\right)& ={(-1)}^n{P}_n^{(\sigma ,\lambda )}(-r)\hfill \\ \hfill & =\left(\genfrac{}{}{0pt}{}{\lambda +n}{n}\right){\displaystyle {}_2{F}_1\left[\left.\begin{array}{c}-n,\lambda +\sigma +n+1\\ \lambda +1\end{array}\right|\frac{1-r}{2}\right]}.\hfill \end{array}$$

(9)

The Jacobi polynomials $P_n^{(\lambda ,\sigma )}\left(r\right)$ contain, as their special cases, other classical orthogonal polynomials, such as the Gegenbauer polynomials $C_n^c\left(r\right)$, the Legendre polynomials $P_n\left(r\right)$, and the Tchebycheff polynomials $T_n\left(r\right)$ and $U_n\left(r\right)$ of the first and second kind (see, [9]). The Legendre polynomials $P_n\left(r\right)$ and the Gegenbauer polynomials $C_n^c\left(r\right)$ have the following relationships:

$$\begin{array}{c}\hfill C_n^c\left(r\right)={\left(\genfrac{}{}{0pt}{}{c+n-\frac{1}{2}}{n}\right)}^{-1}\left(\genfrac{}{}{0pt}{}{2c+n-1}{n}\right){\mathrm{J}}^{\left(c-\frac{1}{2},c-\frac{1}{2}\right)}[f\left(r\right);n]\end{array}$$

(10)

and

$$\begin{array}{c}\hfill P_n\left(r\right)=C_n^{\frac{1}{2}}\left(r\right)=P_n^{(0,0)}\left(r\right),\end{array}$$

(11)

respectively. Thus, by applying the relationships in (10) and (11) and ignoring the constant binomial coefficients present in (10), the parameters $\lambda$ and $\sigma$ in (8) earlier can be suitably specialized to define the corresponding Gegenbauer transform ${\mathrm{G}}^{\left(c\right)}[f\left(r\right);n]$ and the Legendre transform $\mathrm{L}\left[f\right(r);n]$, as follows:

$$\begin{array}{cc}\hfill {\mathrm{G}}^{\left(c\right)}[f\left(r\right);n]& ={\left(\genfrac{}{}{0pt}{}{c+n-\frac{1}{2}}{n}\right)}^{-1}\left(\genfrac{}{}{0pt}{}{2c+n-1}{n}\right){\mathrm{J}}^{\left(c-\frac{1}{2},c-\frac{1}{2}\right)}[f\left(r\right);n]\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & ={\int }_{-1}^1{(1-r^2)}^{c-\frac{1}{2}}{C}_n^c\left(r\right)f\left(r\right)dr,\left(\Re \left(c\right)>-\frac{1}{2};n\in {\mathbb{N}}_0\right),\hfill \end{array}$$

(13)

and

$$\begin{array}{c}\hfill \mathrm{L}[f\left(r\right);n]={\mathrm{G}}^{\left(\frac{1}{2}\right)}[f\left(r\right);n]={\int }_{-1}^1{P}_n\left(r\right)f\left(r\right)dr,(n\in {\mathbb{N}}_0).\end{array}$$

(14)

Lemma 1. 

The Jacobi transform of the power function $r^{\omega -1}$ is given by [10]

$$\begin{array}{cc}\hfill {\int }_{-1}^1{(1-r)}^{{\delta }_1-1}{(1+r)}^{{\delta }_2-1}{P}_n^{(\lambda ,\sigma )}\left(r\right)r^{\omega -1}dr& =2^{{\delta }_1+{\delta }_2-1}\left(\genfrac{}{}{0pt}{}{\lambda +n}{n}\right)B\left({\delta }_1,{\delta }_2\right)\hfill \\ & \hfill \times {\displaystyle F_{1:1,0}^{1:2,1}\left[\left.\begin{array}{c}{\delta }_1:-n,\lambda +\sigma +n+1;1-\omega \\ {\delta }_1+{\delta }_2:\lambda +1;-\end{array}\right|1,2\right]},\end{array}$$

(15)

where $min\left\{\Re \left({\delta }_1\right),\Re \left({\delta }_2\right)\right\}>0;\omega ,r\in \mathbb{C};n\in {\mathbb{N}}_0$ and $F_{l:m,n}^{p:q,r}(·)$ is the familiar Kampé de Fériet function [11].

Theorem 1. 

Under the assumptions defined in (7), the Jacobi transform of the Mittag-Leffler (ML)-type confluent hypergeometric function (7) can be expressed as

$$\begin{array}{cc}\hfill {\mathrm{J}}^{(\lambda ,\sigma )}\left[r^{\omega -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(ur\right);n\right]& =2^{\lambda +\sigma +1}\left(\genfrac{}{}{0pt}{}{\lambda +n}{n}\right)B\left(\lambda +1,\sigma +1\right)\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }\left(\kappa \right)\mathsf{\Gamma }(\eta k+\gamma )}{\mathsf{\Gamma }\left(\gamma \right)\mathsf{\Gamma }(\nu k+\kappa )}\hfill \\ & \times {\displaystyle F_{1:1,0}^{1:2,1}\left[\left.\begin{array}{c}\lambda +1:-n,\lambda +\sigma +n+1;1-\omega -k\\ \lambda +\sigma +2:\lambda +1;-\end{array}\right|1,2\right]}\frac{u^k}{k!},\hfill \end{array}$$

(16)

where $\omega \in \mathbb{C};n\in {\mathbb{N}}_0$ and $min\left\{\Re \left(\lambda \right),\Re \left(\sigma \right)\right\}>-1,\left|u\right|<1$.

Proof. 

To prove Theorem 1, we first apply Jacobi transform (8) in conjunction with (7). Upon reversing the order of summation and integration and making use of Lemma 1, this proves the Theorem 1. □

Corollary 1. 

If the hypothesis of the Theorem 1 is true, and substituting $\lambda =\sigma =c-\frac{1}{2}$, the following Gegenbauer transform formula is valid:

$$\begin{array}{cc}\hfill {\mathrm{G}}^{\left(c\right)}\left[r^{\omega -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(ur\right);n\right]& =2^{2c}\left(\genfrac{}{}{0pt}{}{2c+n-1}{n}\right)B\left(c+\frac{1}{2},c+\frac{1}{2}\right)\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }\left(\kappa \right)\mathsf{\Gamma }(\eta k+\gamma )}{\mathsf{\Gamma }\left(\gamma \right)\mathsf{\Gamma }(\nu k+\kappa )}\hfill \\ & \times {\displaystyle F_{1:1,0}^{1:2,1}\left[\left.\begin{array}{c}c+\frac{1}{2}:-n,2c+n;1-\omega -k\\ 2c+1:c+\frac{1}{2};-\end{array}\right|1,2\right]}\frac{u^k}{k!},\hfill \end{array}$$

(17)

where $\omega \in \mathbb{C};n\in {\mathbb{N}}_0,\Re \left(c\right)>-\frac{1}{2},\left|u\right|<1$.

Corollary 2. 

If the hypothesis of the Theorem 1 is true, and substituting $\lambda =\sigma =0$ or $c=\frac{1}{2}$, the following Legendre transform formula is valid:

$$\begin{array}{cc}\hfill \mathrm{L}\left[r^{\omega -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(ur\right);n\right]& =2\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }\left(\kappa \right)\mathsf{\Gamma }(\eta k+\gamma )}{\mathsf{\Gamma }\left(\gamma \right)\mathsf{\Gamma }(\nu k+\kappa )}\times {\displaystyle F_{1:1,0}^{1:2,1}\left[\left.\begin{array}{c}1:-n,n+1;1-\omega -k\\ 2:1;-\end{array}\right|1,2\right]}\frac{u^k}{k!},\hfill \end{array}$$

where $\omega \in \mathbb{C};n\in {\mathbb{N}}_0,\left|u\right|<1$.

3. Fractional Derivative Formulas

In this section, we develop a variety of fractional derivative formulas involving the Mittag-Leffler (ML)-type confluent hypergeometric function. In order to achieve this, we will review the given pairs of generalized left- and right-sided fractional derivative operator ${\mathcal{D}}_{0+}^{\delta ,\chi ,\rho }$ and ${\mathcal{D}}_{\infty -}^{\delta ,\chi ,\rho }$:

Definition 1. 

Let $\delta ,\chi ,\rho \in \mathbb{C}$. Then, the left-sided fractional integral operator ${\mathcal{I}}_{0+}^{\delta ,\chi ,\rho }$ and corresponding left-sided fractional derivative operator ${\mathcal{D}}_{0+}^{\delta ,\chi ,\rho }$ can be represented as [12],

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{0+}^{\delta ,\chi ,\rho }f\right)\left(x\right)=\frac{x^{-\delta -\chi }}{\mathsf{\Gamma }\left(\delta \right)}{\int }_0^x{(x-r)}^{\delta -1}{}_2{F}_1\left(\delta +\chi ,-\rho ;\delta ;1-\frac{r}{x}\right)f\left(r\right)dr,x>0,\end{array}$$

(18)

where $\Re \left(\delta \right)>0$ and

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_{0+}^{\delta ,\chi ,\rho }f\right)\left(x\right)& =\left({\mathcal{I}}_{0+}^{-\delta ,-\chi ,\delta +\rho }f\right)\left(x\right)={\left(\frac{d}{dx}\right)}^m\left\{\left({\mathcal{I}}_{0+}^{-\delta +\rho ,-\chi -\rho ,\delta +\rho -m}f\right)\left(x\right)\right\},\hfill \end{array}$$

(19)

where $m=[\Re (\delta \left)\right]+1;\Re \left(\delta \right)\ge 0$ and $\left[x\right]$ represents the greatest integer belonging to $x\in \mathbb{R}$.

Remark 1. 

On substituting $\chi =-\delta ,\chi =0$, operator (19) coincides with the familiar “Riemann–Liouville ($\mathcal{RL}$) fractional derivative operator ${}_{RL}{\mathcal{D}}_{0+}^{\delta }$” and the “left-sided Erdélyi-Kobar ($\mathcal{EK}$) fractional derivative operator ${}_{EK}{\mathcal{D}}_{0+}^{\delta ,\rho }$”, as given below (see [12]):

$$\left({\mathcal{D}}_{0+}^{\delta ,-\delta ,\rho }f\right)\left(x\right)=\left({}_{\mathcal{RL}}{\mathcal{D}}_{0+}^{\delta }f\right)\left(x\right)={\left(\frac{d}{dx}\right)}^m\left\{\frac{1}{\mathsf{\Gamma }(m-\delta )}{\int }_0^x{(x-r)}^{m-\delta -1}f\left(r\right)dr\right\},$$

(20)

and

$$\left({\mathcal{D}}_{0+}^{\delta ,0,\rho }f\right)\left(x\right)=\left({}_{\mathcal{EK}}{\mathcal{D}}_{0+}^{\delta ,\rho }f\right)\left(x\right)=x^{\rho }{\left(-\frac{d}{dx}\right)}^m\left\{\frac{1}{\mathsf{\Gamma }(m-\delta )}{\int }_0^x{(x-r)}^{m-\delta -1}{r}^{\delta +\rho }f\left(r\right)dr\right\},$$

(21)

where $\Re \left(\delta \right)\ge 0,m=[\Re (\delta \left)\right]+1;x>0$.

Definition 2. 

Let $x>0,\delta ,\chi ,\rho \in \mathbb{C}$. Then, the right-sided fractional integral operator ${\mathcal{I}}_{\infty -}^{\delta ,\chi ,\rho }$ and corresponding right-sided fractional derivative operator ${\mathcal{D}}_{\infty -}^{\delta ,\chi ,\rho }$ can be defined as [12],

$$\begin{array}{c}\hfill \left({\mathcal{I}}_{\infty -}^{\delta ,\chi ,\rho }f\right)\left(x\right)=\frac{1}{\mathsf{\Gamma }\left(\delta \right)}{\int }_x^{\infty }{(r-x)}^{\delta -1}{r}^{-\delta -\chi }{}_2{F}_1\left(\delta +\chi ,-\rho ;\delta ;1-\frac{x}{r}\right)f\left(r\right)dr,x>0,\end{array}$$

(22)

where $\Re \left(\delta \right)>0$ and

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_{\infty -}^{\delta ,\chi ,\rho }f\right)\left(x\right)& =\left({\mathcal{I}}_{\infty -}^{-\delta ,-\chi ,\delta +\rho }f\right)\left(x\right)={\left(-\frac{d}{dx}\right)}^m\left\{\left({\mathcal{I}}_{0+}^{-\delta +\rho ,-\chi -\rho ,\delta +\rho -m}f\right)\left(x\right)\right\},\hfill \end{array}$$

(23)

where $m=[\Re (\delta \left)\right]+1;\Re \left(\delta \right)\ge 0.$

Remark 2. 

On substituting $\chi =-\delta ,\chi =0$, operator (23) coincides with the “Weyl fractional derivative operator ${}_W{\mathcal{D}}_{\infty -}^{\delta }$” and the “right-sided Erdélyi-Kobar ($\mathcal{EK}$) fractional derivative operator ${}_{EK}{\mathcal{D}}_{\infty -}^{\delta ,\rho }$”, as given below (see [12]):

$$\left({\mathcal{D}}_{\infty -}^{\delta ,-\delta ,\rho }f\right)\left(x\right)=\left({}_{\mathcal{W}}{\mathcal{D}}_{\infty -}^{\delta }f\right)\left(x\right)={\left(-\frac{d}{dx}\right)}^m\left\{\frac{1}{\mathsf{\Gamma }(m-\delta )}{\int }_x^{\infty }{(r-x)}^{m-\delta -1}f\left(r\right)dr\right\},$$

(24)

and

$$\left({\mathcal{D}}_{\infty -}^{\delta ,0,\rho }f\right)\left(x\right)=\left({}_{\mathcal{EK}}{\mathcal{D}}_{\infty -}^{\delta ,\rho }f\right)\left(x\right)=x^{\delta +\rho }{\left(\frac{d}{dx}\right)}^m\left\{\frac{1}{\mathsf{\Gamma }(m-\delta )}{\int }_x^{\infty }{(r-x)}^{m-\delta -1}{r}^{-\rho }f\left(r\right)dr\right\},$$

(25)

where $\Re \left(\delta \right)\ge 0,m=[\Re (\delta \left)\right]+1;x>0$.

Lemma 2. 

Let $\delta ,\chi ,\rho ,\kappa \in \mathbb{C}$ with $\Re \left(\delta \right)\ge 0,x>0$. Then, we have the following fractional derivative formulas [12]:

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_{0+}^{\delta ,\chi ,\rho }{r}^{\kappa -1}\right)\left(x\right)& =\frac{\mathsf{\Gamma }\left(\kappa \right)\mathsf{\Gamma }(\kappa +\delta +\chi +\rho )}{\mathsf{\Gamma }(\kappa +\chi )\mathsf{\Gamma }(\kappa +\rho )}{x}^{\kappa +\chi -1},\hfill \end{array}$$

(26)

where $\Re \left(\kappa \right)>-min\left\{0,\Re (\delta +\chi +\rho )\right\}$ and

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_{\infty -}^{\delta ,\chi ,\rho }{r}^{\kappa -1}\right)\left(x\right)& =\frac{\mathsf{\Gamma }(1-\chi -\kappa )\mathsf{\Gamma }(1-\kappa +\delta +\rho )}{\mathsf{\Gamma }(1-\kappa )\mathsf{\Gamma }(1-\kappa +\rho -\chi )}{x}^{\kappa +\chi -1},\hfill \end{array}$$

(27)

where $\Re \left(\kappa \right)<1+min\left\{-\Re (\chi +\rho ),\Re (\delta +\rho )\right\}.$

Theorem 2. 

Let $\delta ,\nu ,\chi ,\eta \rho ,\gamma ,\kappa \in \mathbb{C}$, such that $\Re \left(\delta \right)\ge 0,\Re \left(\nu \right)>0$ and $x>0$. Then, the following left-sided fractional derivative formula is valid:

$$\begin{array}{cc}\hfill & \left[{\mathcal{D}}_{0+}^{\delta ,\chi ,\rho }{r}^{\kappa -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(u{r}^{\nu }\right)\right]\left(x\right)=x^{\kappa +\chi -1}\frac{\mathsf{\Gamma }\left(\kappa \right)}{\mathsf{\Gamma }\left(\gamma \right)}{\displaystyle {}_2{\mathsf{\Psi }}_2\left[\left.\begin{array}{c}(\gamma ,\eta ),(\kappa +\delta +\chi +\rho ,\nu )\\ (\kappa +\chi ,\nu ),(\kappa +\rho ,\nu )\end{array}\right|u{x}^{\nu }\right]},\hfill \end{array}$$

(28)

where $\Re \left(\kappa \right)>-min\left\{0,\Re (\delta +\chi +\rho )\right\}$

Proof. 

Using (7), we have

$$\begin{array}{cc}\hfill \left[{\mathcal{D}}_{0+}^{\delta ,\chi ,\rho }{r}^{\kappa -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(u{r}^{\nu }\right)\right]\left(x\right)& =\sum _{n=0}^{\infty }\frac{\mathsf{\Gamma }\left(\kappa \right)\mathsf{\Gamma }(\eta n+\gamma )}{\mathsf{\Gamma }\left(\gamma \right)\mathsf{\Gamma }(\nu n+\kappa )}\frac{u^n}{n!}\times \left({\mathcal{D}}_{0+}^{\delta ,\chi ,\rho }{r}^{\nu n+\kappa -1}\right)\left(x\right).\hfill \end{array}$$

(29)

Using (26), this proves Theorem 2. □

Corollary 3. 

If the assumptions stated in Theorem 2 are true, and substituting $\chi =-\delta$ and $\chi =0$, the following fractional derivative formulas are valid:

$$\begin{array}{cc}\hfill \left[{}_{\mathcal{RL}}{\mathcal{D}}_{0+}^{\delta }{r}^{\kappa -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(u{r}^{\nu }\right)\right]\left(x\right)& =x^{\kappa -\delta -1}\frac{\mathsf{\Gamma }\left(\kappa \right)}{\mathsf{\Gamma }(\kappa -\delta )}{M}_{\nu ,\kappa -\delta }^{\eta ,\gamma }\left(u{x}^{\nu }\right).\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \left[{}_{\mathcal{EK}}{\mathcal{D}}_{0+}^{\delta ,\rho }{r}^{\kappa -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(u{r}^{\nu }\right)\right]\left(x\right)& =x^{\kappa -1}\frac{\mathsf{\Gamma }\left(\kappa \right)}{\mathsf{\Gamma }\left(\gamma \right)}{\displaystyle {}_2{\mathsf{\Psi }}_2\left[\left.\begin{array}{c}(\gamma ,\eta ),(\kappa +\delta +\rho ,\nu )\\ (\kappa ,\nu ),(\kappa +\rho ,\nu )\end{array}\right|u{x}^{\nu }\right]}.\hfill \end{array}$$

(31)

Theorem 3. 

Let $\delta ,\chi ,\rho ,\kappa ,\nu ,\eta ,\gamma \in \mathbb{C}$, such that $\Re \left(\delta \right)\ge 0,\Re \left(\nu \right)>0$ and $x>0,t>0$. Then, the following right-sided fractional derivative formula is valid:

$$\begin{array}{cc}\hfill & \left[{\mathcal{D}}_{\infty -}^{\delta ,\chi ,\rho }{r}^{\kappa -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(\frac{u}{r^{\nu }}\right)\right]\left(x\right)=x^{\kappa +\chi -1}\frac{\mathsf{\Gamma }\left(\kappa \right)}{\mathsf{\Gamma }\left(\gamma \right)}{\displaystyle {}_3{\mathsf{\Psi }}_3\left[\left.\begin{array}{c}(\gamma ,\eta ),(1-\chi -\kappa ,\nu ),(1-\kappa +\delta +\rho ,\nu )\\ (\kappa ,\nu ),(1-\kappa ,\nu ),(1-\kappa +\rho -\chi ,\nu )\end{array}\right|\frac{u}{x^{\nu }}\right]},\hfill \end{array}$$

(32)

where $\Re \left(\kappa \right)<1+min\left\{-\Re (\chi +\rho ),\Re (\delta +\rho )\right\}$

Proof. 

Using (7), we have

$$\begin{array}{cc}\hfill \left[{\mathcal{D}}_{\infty -}^{\delta ,\chi ,\rho }{r}^{\kappa -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(\frac{u}{r^{\nu }}\right)\right]\left(x\right)& =\sum _{n=0}^{\infty }\frac{\mathsf{\Gamma }\left(\kappa \right)\mathsf{\Gamma }(\eta n+\gamma )}{\mathsf{\Gamma }\left(\gamma \right)\mathsf{\Gamma }(\nu n+\kappa )}\frac{u^n}{n!}\times \left({\mathcal{D}}_{\infty -}^{\delta ,\chi ,\rho }{r}^{\kappa -\nu n-1}\right)\left(x\right).\hfill \end{array}$$

(33)

Using (27), this proves Theorem 3. □

Corollary 4. 

If the assumptions stated in Theorem 3 are true, and substituting $\chi =-\delta$ and $\chi =0$, the following fractional derivative formulas are valid:

$$\begin{array}{cc}\hfill \left[{}_{\mathcal{W}}{\mathcal{D}}_{\infty -}^{\delta }{r}^{\kappa -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(\frac{u}{r^{\nu }}\right)\right]\left(x\right)& =x^{\kappa -\delta -1}\frac{\mathsf{\Gamma }\left(\kappa \right)}{\mathsf{\Gamma }\left(\gamma \right)}{\displaystyle {}_2{\mathsf{\Psi }}_2\left[\left.\begin{array}{c}(\gamma ,\eta ),(1+\delta -\kappa ,\nu )\\ (\kappa ,\nu ),(1-\kappa ,\nu )\end{array}\right|\frac{u}{x^{\nu }}\right]}.\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & \left[{}_{\mathcal{EK}}{\mathcal{D}}_{\infty -}^{\delta ,\rho }{r}^{\kappa -1}{M}_{\nu ,\kappa }^{\eta ,\gamma }\left(\frac{u}{r^{\nu }}\right)\right]\left(x\right)=x^{\kappa -1}\frac{\mathsf{\Gamma }\left(\kappa \right)}{\mathsf{\Gamma }\left(\gamma \right)}{\displaystyle {}_2{\mathsf{\Psi }}_2\left[\left.\begin{array}{c}(\gamma ,\eta ),(1-\kappa +\delta +\rho ,\nu )\\ (\kappa ,\nu ),(1-\kappa +\rho ,\nu )\end{array}\right|\frac{u}{x^{\nu }}\right]}.\hfill \end{array}$$

(35)