Fractional differential equations for the generalized Mittag-Leffler function

  • Praveen Agarwal
  • Qasem Al-Mdallal
  • Yeol Je Cho
  • Shilpi Jain
Open Access
Research
  • 217 Downloads
Part of the following topical collections:
  1. Advances in Fractional Differential Equations and Their Real World Applications

Abstract

In this paper, we establish some (presumably new) differential equation formulas for the extended Mittag-Leffler-type function by using the Saigo-Maeda fractional differential operators involving the Appell function \(F_{3}(\cdot)\) and results in terms of the Wright generalized hypergeometric-type function \({}_{m+1}\psi^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}} )}_{n+1}(z; p)\) recently established by Agarwal. Some interesting special cases are also pointed out.

Keywords

generalized Gamma function generalized beta functions generalized Mittag-Leffler function generalized Wright hypergeometric function fractional derivative operators 

MSC

26A33 33E12 33C05 33C15 33C20 33C65 33C90 

1 Introduction and preliminaries

Fractional calculus (derivative and integrals) is very old as the conventional calculus and has bern recently applied in various areas of engineering, science, finance, applied mathematics, and bio engineering (see, e.g., [1, 2]). Many differential equations involving various special functions have found significant importance and applications in various subfields of mathematical analysis. During the last few decades, a number of workers have studied, in depth, the properties, applications, and different extensions of various hypergeometric operators of fractional derivatives. A detailed account of such operators along with their properties and applications have been considered by several authors (see [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] and [23]).

A useful generalization of the hypergeometric fractional derivatives, including the Saigo operators [15, 16, 17], has been introduced by Marichev [13] as Mellin-type convolution operators with a special function \(F_{3}(\cdot )\) in the kernel (for more details, see Samko et al. [20, p. 194, Eq. (10.47) and Section 10.3]) and later extended and studied by Saigo and Maeda [18, p. 393, Eqs. (4.12) and (4.13)]. Note that the generalized fractional derivative operators (for Saigo-Maeda operators, see [18]) are defined as follows:
$$\begin{aligned} \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0+}f \bigr) (x)={}& \bigl(I^{-\tau', -\tau, -\upsilon', -\upsilon, -\sigma }_{0+}f \bigr) (x) \\ ={}& \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{-\tau', -\tau, -\upsilon'+k, -\upsilon, -\sigma+k}_{0+}f \bigr) (x), \end{aligned}$$
(1.1)
$$\begin{aligned} \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{-}f \bigr) (x)={}& \bigl(I^{-\tau', -\tau, -\upsilon', -\upsilon, -\sigma }_{-}f \bigr) (x) \\ ={}& \biggl(-\frac{d}{dx} \biggr)^{k} \bigl(I^{-\tau', -\tau, -\upsilon'+k, -\upsilon, -\sigma+k}_{0-}f \bigr) (x), \end{aligned}$$
(1.2)
where \(\Re(\sigma)>0\), \(k=[\Re(\sigma)]+1\), \(\tau, \tau', \upsilon, \upsilon', \sigma\in\mathbb{C}\), \(\mathbb{C}\) being the set of complex numbers, and \(x>0\).
Following Saigo et al. [18] and Saxena and Saigo [21], the left-hand sided and right-hand sided generalized differentiation for a power function are, respectively, given as follows (see, [24, p. 7, Eqs. (4.1) and (4.2)]):
$$\begin{aligned} \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0+}t^{\varrho -1} \bigr) (x) = \frac{\Gamma(\varrho)\Gamma(\varrho-\sigma +\tau+\tau'+\upsilon')\Gamma(\varrho+\tau-\upsilon)}{\Gamma(\varrho -\upsilon) \Gamma(\varrho-\sigma+\tau+\tau')\Gamma(\varrho-\sigma+\tau+\upsilon ')}x^{\tau+\tau'+\varrho-\sigma-1}, \end{aligned}$$
(1.3)
where \(\Re(\varrho)>\operatorname{max}\{0, \Re(-\tau+\upsilon), \Re(-\tau -\tau'-\upsilon'-\sigma)\}\), and
$$\begin{aligned} & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0-}t^{-\rho} \bigr) (x) \\ &\quad = \frac{\Gamma(\varrho+\sigma-\tau-\tau')\Gamma(\varrho-\tau'-\upsilon +\sigma)\Gamma(\varrho+\upsilon')}{ \Gamma(\varrho)\Gamma(\varrho-\tau-\tau'-\upsilon+\sigma)\Gamma(\varrho -\tau'+\upsilon')}x^{-\varrho+\tau+\tau'-\sigma}, \end{aligned}$$
(1.4)
where \(\Re(\varrho)>\operatorname{max}\{\Re(-\upsilon'), \Re(\tau+\tau '-\sigma), \Re(\tau'+\upsilon-\sigma)+\Re(\sigma)+1\}\).

In certain areas of applied mathematics and mathematical physics, special functions and their generalizations are used for finding solutions of the initial or boundary value problems for partial differential equations and fractional differential equations. It is also important to mention here that the special and degenerated cases of hypergeometric functions; in particular, the Bessel, Mittage-Leffler, and Wright hypergeometric functions have an importance due to application point of view.

Recently, Parmar introduced the extended Mittag-Leffler type function in the following form [25, p. 1072, Eq. (16)]:
$$\begin{aligned} & E_{\xi,\zeta}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}; \lambda)}(z; p)=\sum _{n=0}^{\infty}\frac{\mathcal{B}^{(\{\kappa_{l}\}_{l\in\mathbb {N}_{0}})}(\lambda+n, 1-\lambda; p)}{B(\lambda, 1-\lambda)}\frac {z^{n}}{\Gamma(\xi n+\zeta)} \\ &\quad \bigl(z, \zeta, \lambda\in\mathbb{C}, \Re(\xi)>0, \Re(\zeta)>0, \Re( \lambda )>1; p\geq0 \bigr), \end{aligned}$$
(1.5)
where \(\mathcal{B}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}})}(\lambda+n, 1-\lambda; p)\) is the extended beta function defined by [26]
$$\begin{aligned} & \mathcal{B}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}})}(\alpha, \beta; p)= \int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1} \Theta \biggl(\{\kappa_{l}\} _{l\in\mathbb{N}_{0}}; -\frac{p}{t(1-t)} \biggr)\,dt \\ &\quad \bigl(\min \bigl\{ \Re(\alpha), \Re(\beta) \bigr\} >0; \Re(p)\geq0 \bigr), \end{aligned}$$
(1.6)
and \(\Theta(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}; z)\) is a function of an appropriately bounded sequence \(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}\) of arbitrary real or complex numbers defined as follows [26, p. 243, Eq. (2.1)]:
$$\begin{aligned} & \Theta \bigl(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}; z \bigr) \\ &\quad :=\textstyle\begin{cases} \sum_{l=0}^{\infty}\{\kappa_{l} \}_{l\in\mathbb {N}_{0}}\frac{z^{l}}{l!},& \vert z \vert < R; 0< R< \infty; \kappa_{0}:=1, \\ \mathcal{M}_{0} z^{\omega}\exp(z) [1+O ( \frac {1}{z} ) ],& \Re(z)\rightarrow\infty; \mathcal {M}_{0}>0; \omega\in\mathbb{C}. \end{cases}\displaystyle \end{aligned}$$
(1.7)

It can be easily seen that different selection of the sequence \(\{ \kappa_{l}\}_{l\in\mathbb{N}_{0}}\) would generate particular cases of (1.5) as explained in the following examples.

Example 1

If we set \(\kappa_{l}=\frac{(\rho)_{l}}{(\sigma)_{l}}\ (l \in \mathbb{N}_{0})\), then (1.5) results in to the extended generalized Mittag-Leffler function [25, p. 1072, Eq. (17)]
$$\begin{aligned} &E_{\xi,\zeta}^{(\rho, \sigma); \lambda}(z; p)=\sum _{n=0}^{\infty}\frac {\mathcal{B}^{(\rho, \sigma)}(\lambda+n, 1-\lambda; p)}{B(\lambda, 1-\lambda)}\frac{z^{n}}{\Gamma(\xi n+\zeta)} \\ &\quad \bigl(z, \zeta, \lambda\in\mathbb{C}, \Re(\rho)>0, \Re(\sigma)>0, \Re(\xi )>0, \Re(\zeta)>0, \Re(\lambda)>1; p\geq0 \bigr). \end{aligned}$$
(1.8)

Example 2

Setting \(\kappa_{l}=1\ (l \in\mathbb{N})\) in (1.5) (see [27] with \(c=1\)), we get the function
$$\begin{aligned} &E_{\xi,\zeta}^{\lambda}(z; p)=\sum _{n=0}^{\infty}\frac{B(\lambda+n, 1-\lambda; p)}{B(\lambda, 1-\lambda)}\frac{z^{n}}{\Gamma(\xi n+\zeta)} \\ &\quad \bigl(z, \zeta, \lambda\in\mathbb{C}, \Re(\xi)>0, \Re (\zeta)>0, \Re(\lambda)>1; p\geq0 \bigr). \end{aligned}$$
(1.9)

Example 3

Similarly, for \(p=0\), (1.9) immediately reduces to the Prabhakar-type [28] Mittage-Leffler function.

Example 4

For \(\xi=\zeta=1\), (1.5), (1.8), and (1.9) can be expressed, respectively, in terms of the extended confluent hypergeometric functions as follows:
$$\begin{aligned} &E_{1, 1}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}; \lambda)}(z; p)= \Phi_{p}^{(\{ \kappa_{l}\}_{l\in\mathbb{N}_{0}})}(\lambda; 1; z), \end{aligned}$$
(1.10)
$$\begin{aligned} & E_{1, 1}^{(\rho, \sigma); \lambda}(z; p)= \Phi_{p}^{(\rho, \sigma)}(\lambda ; 1; z) \end{aligned}$$
(1.11)
and
$$\begin{aligned} E_{1, 1}^{\lambda}(z; p)= \Phi_{p}(\lambda; 1; z). \end{aligned}$$
(1.12)
Similarly, by (1.6) and (1.7) of the extended beta function \(\mathcal{B}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}})}(\lambda+n, 1-\lambda; p)\) and a function of an appropriately bounded sequence \(\{ \kappa_{l}\}_{l\in\mathbb{N}_{0}}\) of arbitrary real or complex numbers \(\Theta(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}; z)\), Agarwal [29] introduced and studied a further potentially useful extension of the Wright hypergeometric function as follows:
$$\begin{aligned} & _{m+1}\psi^{ (\{\kappa_{l}\}_{l\in\mathbb{N}_{0}} )}_{n+1}(z; p) \\ &\quad={} _{m+1}\psi^{ (\{\kappa_{l}\}_{l\in\mathbb{N}_{0}} )}_{n+1}\left [ \begin{matrix} (a_{i}, \alpha_{i})_{1,m}, & (\gamma, 1) \\ (b_{j}, \beta_{j})_{1,n}, & (c, 1) \end{matrix} \Big| (z; p) \right ] \\ &\quad =\frac{1}{\Gamma(c-\gamma)}\sum_{k=0}^{\infty} \frac{\prod_{i=1}^{m}\Gamma(a_{i}+k\alpha_{i})}{\prod_{j=1}^{n}\Gamma(b_{j}+k\beta_{j})} \frac{\mathcal{B}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}})}(\gamma+k, c-\gamma; p)z^{k}}{k!} \\ &\quad \bigl(z, \gamma\in\mathbb{C}, \Re(c)>\Re(\gamma)>0; p\geq0 \bigr). \end{aligned}$$
(1.13)

In this paper, we aim to establish certain (presumably) new fractional differential equation formulas involving the extended generalized Mittag-Leffler type function (1.9) and extended Wright Generalized hypergeometric function (1.13) by using the fractional differential operators (1.1) and (1.2), respectively. Some particular cases of our main findings are also pointed out.

2 Main results

In this section, we establish the fractional differential formulas involving the Saigo-Meada fractional derivative operators (1.1) and (1.2). These formulas are given in the following theorems.

Theorem 1

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon ', \sigma, \mu, \xi, \zeta, \lambda, t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(p)\geq0) \)be such that
$$ \Re(\mu)>\max \bigl\{ 0, \Re(-\tau+\upsilon), \Re \bigl(-\tau-\tau'- \upsilon'-\sigma \bigr) \bigr\} . $$
(2.1)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b] & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0+}t^{\mu-1}E_{\xi ,\zeta}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}; \lambda)}(t; p) \bigr) (x)\\ &\quad =\frac{x^{\mu+\tau+\tau'-\sigma-1}}{\Gamma(\lambda)} \\ & \qquad{}\times_{4}\psi^{ (\{\kappa_{l}\}_{l\in\mathbb{N}_{0}} )}_{4}\left [ \begin{matrix} (\mu,1), & (\mu-\upsilon+\tau, 1), & (\mu+\tau+\tau'+\upsilon'-\sigma, 1), & (\gamma, 1) \\ (\mu-\upsilon,1), & (\mu+\tau+\tau'-\sigma, 1), & (\mu-\sigma+\tau +\upsilon', 1), & (\zeta, \xi) \end{matrix} \Big|(x; p) \right ]. \end{aligned}} \end{aligned}$$
(2.2)

Proof

Let \(\mathcal{I}\) be the left-hand side of (2.2). Using (1.5) and changing the order of integration and summation, which is verified under the conditions of the theorem, we have
$$\begin{aligned} \mathcal{I}={}&\sum_{n=0}^{\infty} \frac{\mathcal{B}^{(\{\kappa_{l}\}_{l\in \mathbb{N}_{0}})}(\lambda+n, 1-\lambda; p)}{B(\lambda, 1-\lambda)}\frac {1}{\Gamma(\xi n+\zeta)} \\ &{} \times \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0+}t^{\mu +n-1} \bigr) (x). \end{aligned}$$
(2.3)
Applying (1.1) and (1.3) with ρ replaced by \(\eta+n\) yields
$$\begin{aligned} \mathcal{I}={}&\sum_{n=0}^{\infty} \frac{\mathcal{B}^{(\{\kappa_{l}\}_{l\in \mathbb{N}_{0}})}(\lambda+n, 1-\lambda; p)}{B(\lambda, 1-\lambda)}\frac {1}{\Gamma(\xi n+\zeta)} \\ &{} \times\frac{\Gamma(\mu+n)\Gamma(\mu+n+\tau-\upsilon)\Gamma(\mu+n+\tau +\tau'+\upsilon'-\sigma)}{ \Gamma(\mu+n-\upsilon)\Gamma(\mu+n-\sigma+\tau+\tau')\Gamma(\mu +n-\sigma+\tau+\upsilon')}x^{\mu+n+\tau+\tau'-\sigma-1}, \end{aligned}$$
(2.4)
which, in view of (1.13), is equal to the right-hand side of (2.2). This completes the proof. □

Theorem 2

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon ', \sigma, \mu, \xi, \zeta, \lambda, t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(\lambda)>1, \Re(p)\geq0) \)be such that
$$\begin{aligned} \Re(\varrho)>\max \bigl\{ \Re \bigl(-\upsilon' \bigr), \Re \bigl( \tau+ \tau'-\sigma \bigr), \Re \bigl(\tau '+\upsilon- \sigma \bigr)+\Re(\sigma) \bigr\} . \end{aligned}$$
(2.5)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b] & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{-}t^{-\mu}E_{\xi ,\zeta}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}; \lambda)}(1/t; p) \bigr) (x)\\ &\quad=\frac{x^{-\mu+\tau+\tau'-\sigma}}{\Gamma(\lambda)} \\ &\qquad{} \times_{4}\psi^{ (\{\kappa_{l}\}_{l\in\mathbb{N}_{0}} )}_{4}\left [ \begin{matrix} (\mu+\upsilon',1), & (\mu-\upsilon-\tau'+\sigma, 1), & (\mu-\tau-\tau '+\sigma, 1), & (\gamma, 1) \\ (\mu,1), & (\mu-\tau-\tau'-\upsilon+\sigma, 1), & (\mu-\tau'+\upsilon, 1), & (\zeta, \xi) \end{matrix} \Big| \bigl(x^{-1}; p \bigr) \right ]. \end{aligned}} \end{aligned}$$
(2.6)

Proof

We establish the result by a similar argument as in the proof of Theorem 1 using (1.4) instead of (1.3). Therefore, we omit the details. □

3 Special cases

The results in Theorems 1 and 2 can be easily specialized to yield the corresponding formulas involving simpler functions like Mittag-Leffler-type functions and extended confluent hypergeometric functions given by (1.8)-(1.12) after appropriate selection of the sequence \(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}}\).

Setting \(\kappa_{l}=\frac{(\phi)_{l}}{(\varphi)_{l}}\ (l \in\mathbb{N}_{0})\), we obtain the following results from Theorems 1 and 2, respectively.

Corollary 1

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon ', \sigma, \mu, \xi, \zeta, \lambda, \phi, \varphi, t, p \in\mathbb {C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(p)\geq0) \)be such that
$$ \Re(\mu)>\max \bigl\{ 0, \Re(-\tau+\upsilon), \Re \bigl(-\tau-\tau'- \upsilon'-\sigma \bigr) \bigr\} . $$
(3.1)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b] & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0+}t^{\mu-1}E_{\xi ,\zeta}^{(\phi, \varphi); \lambda}(t; p) \bigr) (x) \\ &\quad =\frac{x^{\mu+\tau+\tau'-\sigma-1}\Gamma(\varphi)}{ \Gamma(\lambda) \Gamma(\phi)} \\ & \qquad{}\times_{5}\psi_{5}\left [ \begin{matrix} (\mu,1), & (\mu-\upsilon+\tau, 1), & (\mu+\tau+\tau'+\upsilon'-\sigma, 1), & (\gamma, 1), & (\phi, 1) \\ (\mu-\upsilon,1), & (\mu+\tau+\tau'-\sigma, 1), & (\mu-\sigma+\tau +\upsilon', 1), & (\zeta, \xi), & (\varphi, 1) \end{matrix} \Big|(x; p) \right ]. \end{aligned}} \end{aligned}$$
(3.2)

Corollary 2

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon ', \sigma, \mu, \xi, \zeta, \lambda, \phi, \varphi t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(\lambda)>1, \Re(p)\geq0 )\)be such that
$$\begin{aligned} \Re(\varrho)>\operatorname{max} \bigl\{ \Re \bigl(-\upsilon' \bigr), \Re \bigl(\tau+\tau'-\sigma \bigr), \Re \bigl(\tau'+ \upsilon-\sigma \bigr)+\Re(\sigma) \bigr\} . \end{aligned}$$
(3.3)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b] & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{-}t^{-\mu}E_{\xi ,\zeta}^{(\phi, \varphi); \lambda}(1/t; p) \bigr) (x)\\ &\quad =\frac{x^{-\mu+\tau +\tau'-\sigma}\Gamma(\varphi)}{ \Gamma(\lambda)\Gamma(\phi)} \\ &\qquad{} \times_{5}\psi_{5}\left [ \begin{matrix} (\mu+\upsilon',1), & (\mu-\upsilon-\tau'+\sigma, 1), & (\mu-\tau-\tau '+\sigma, 1), & (\gamma, 1), & (\phi, 1) \\ (\mu,1), & (\mu-\tau-\tau'-\upsilon+\sigma, 1), & (\mu-\tau'+\upsilon, 1), & (\zeta, \xi), & (\varphi, 1) \end{matrix} \Big| \bigl(x^{-1}; p \bigr) \right ].\end{aligned}} \end{aligned}$$
(3.4)

For \(\kappa_{l}=1\), Theorems 1 and 2 become as follows.

Corollary 3

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon ', \sigma, \mu, \xi, \zeta, \lambda, t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(p)\geq0) \)be such that
$$ \Re(\mu)>\max \bigl\{ 0, \Re(-\tau+\upsilon), \Re \bigl(-\tau-\tau'- \upsilon'-\sigma \bigr) \bigr\} . $$
(3.5)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b]& \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0+}t^{\mu-1}E_{\xi ,\zeta}^{\lambda}(t; p) \bigr) (x) \\ &\quad =\frac{x^{\mu+\tau+\tau'-\sigma-1}}{ \Gamma(\lambda)} \\ & \qquad{}\times_{4}\psi_{4}\left [ \begin{matrix} (\mu,1), & (\mu-\upsilon+\tau, 1), & (\mu+\tau+\tau'+\upsilon'-\sigma, 1), & (\gamma, 1) \\ (\mu-\upsilon,1), & (\mu+\tau+\tau'-\sigma, 1), & (\mu-\sigma+\tau +\upsilon', 1), & (\zeta, \xi) \end{matrix} \Big|(x; p) \right ]. \end{aligned}} \end{aligned}$$
(3.6)

Corollary 4

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon ', \sigma, \mu, \xi, \zeta, \lambda, t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(\lambda)>1, \Re(p)\geq0 )\)be such that
$$\begin{aligned} \Re(\varrho)>\operatorname{max} \bigl\{ \Re \bigl(-\upsilon' \bigr), \Re \bigl(\tau +\tau'-\sigma \bigr), \Re \bigl(\tau'+ \upsilon-\sigma \bigr)+\Re(\sigma) \bigr\} . \end{aligned}$$
(3.7)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b] & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{-}t^{-\mu}E_{\xi ,\zeta}^{\lambda}(1/t; p) \bigr) (x)\\ &\quad =\frac{x^{-\mu+\tau+\tau'-\sigma }\Gamma(\varphi)}{ \Gamma(\lambda)\Gamma(\phi)} \\ &\qquad{} \times_{4}\psi_{4}\left [ \begin{matrix} (\mu+\upsilon',1), & (\mu-\upsilon-\tau'+\sigma, 1), & (\mu-\tau-\tau '+\sigma, 1), & (\gamma, 1) \\ (\mu,1), & (\mu-\tau-\tau'-\upsilon+\sigma, 1), & (\mu-\tau'+\upsilon, 1), & (\zeta, \xi) \end{matrix} \Big| \bigl(x^{-1}; p \bigr) \right ]. \end{aligned}} \end{aligned}$$
(3.8)

Similarly, putting \(p=0\) in Corollaries 3 and 4, we get the fractional differential formulas involving the Prabhakar-type [28] Mittag-Leffler function. We omit the details.

Following the same way, setting \(\xi=\zeta=1\), from Theorems 1 and 2 and Corollaries 1 and 2 we obtain the following interesting results.

Corollary 5

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon', \sigma, \mu, \lambda, \phi , \varphi, t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(p)\geq 0) \)be such that
$$ \Re(\mu)>\max \bigl\{ 0, \Re(-\tau+\upsilon), \Re \bigl(-\tau-\tau'- \upsilon'-\sigma \bigr) \bigr\} . $$
(3.9)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b] & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0+}t^{\mu-1} \Phi _{p}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}})}(\lambda; 1; t) \bigr) (x)\\ &\quad =\frac{x^{\mu+\tau+\tau'-\sigma-1}}{\Gamma(\lambda)} \\ & \qquad{}\times_{4}\psi^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}})}_{4}\left [ \begin{matrix} (\mu,1), & (\mu-\upsilon+\tau, 1), & (\mu+\tau+\tau'+\upsilon'-\sigma, 1), & (\gamma, 1) \\ (\mu-\upsilon,1), & (\mu+\tau+\tau'-\sigma, 1), & (\mu-\sigma+\tau +\upsilon', 1), & (1, 1) \end{matrix} \Big|(x; p) \right ]. \end{aligned}} \end{aligned}$$
(3.10)

Corollary 6

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon', \sigma, \mu, \lambda, t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(\lambda)>1, \Re (p)\geq0) \)be such that
$$\begin{aligned} \Re(\varrho)>\max \bigl\{ \Re \bigl(-\upsilon' \bigr), \Re \bigl( \tau+ \tau'-\sigma \bigr), \Re \bigl(\tau '+\upsilon- \sigma \bigr)+\Re(\sigma) \bigr\} . \end{aligned}$$
(3.11)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b]& \biggl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{-}t^{-\mu} \Phi _{p}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}})} \biggl(\lambda; 1; \frac{1}{t} \biggr) \biggr) (x)\\ &\quad =\frac{x^{-\mu+\tau+\tau'-\sigma}\Gamma(\varphi)}{ \Gamma(\lambda)\Gamma(\phi)} \\ &\qquad{} \times_{4}\psi_{4}\left [ \begin{matrix} (\mu+\upsilon',1), & (\mu-\upsilon-\tau'+\sigma, 1), & (\mu-\tau-\tau '+\sigma, 1), & (\gamma, 1) \\ (\mu,1), & (\mu-\tau-\tau'-\upsilon+\sigma, 1), & (\mu-\tau'+\upsilon, 1), & (1, 1) \end{matrix} \Big| \bigl(x^{-1}; p \bigr) \right ].\end{aligned}} \end{aligned}$$
(3.12)

Corollary 7

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon', \sigma, \mu, \lambda, \phi , \varphi, t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(p)\geq 0) \)be such that
$$ \Re(\mu)>\max \bigl\{ 0, \Re(-\tau+\upsilon), \Re \bigl(-\tau-\tau'- \upsilon'-\sigma \bigr) \bigr\} . $$
(3.13)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b] & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{0+}t^{\mu-1} \Phi _{p}^{(\phi, \varphi)}(\lambda; 1; t) \bigr) (x)\\ &\quad =\frac{x^{\mu+\tau+\tau '-\sigma-1}\Gamma(\varphi)}{ \Gamma(\lambda) \Gamma(\phi)} \\ & \qquad{}\times_{5}\psi_{5}\left [ \begin{matrix} (\mu,1), & (\mu-\upsilon+\tau, 1), & (\mu+\tau+\tau'+\upsilon'-\sigma, 1), & (\gamma, 1), & (\phi, 1) \\ (\mu-\upsilon,1), & (\mu+\tau+\tau'-\sigma, 1), & (\mu-\sigma+\tau +\upsilon', 1), & (1, 1), & (\varphi, 1) \end{matrix} \Big|(x; p) \right ]. \end{aligned}} \end{aligned}$$
(3.14)

Corollary 8

Let\(x>0\), \(\tau, \tau', \upsilon, \upsilon', \sigma, \mu, \lambda, \phi , \varphi, t, p \in\mathbb{C}\)\((\Re(\xi)>0, \Re(\zeta)>0, \Re(\lambda )>1, \Re(p)\geq0) \)be such that
$$\begin{aligned} \Re(\varrho)>\max \bigl\{ \Re \bigl(-\upsilon' \bigr), \Re \bigl( \tau+ \tau'-\sigma \bigr), \Re \bigl(\tau '+\upsilon- \sigma \bigr)+\Re(\sigma) \bigr\} . \end{aligned}$$
(3.15)
Then the following formula holds:
$$\begin{aligned} {\begin{aligned}[b] & \bigl(D^{\tau, \tau', \upsilon, \upsilon', \sigma}_{-}t^{-\mu} \Phi _{p}^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}})}(\lambda; 1; t) \bigr) (x) \\ &\quad =\frac{x^{-\mu+\tau+\tau'-\sigma}\Gamma(\varphi)}{\Gamma(\lambda)\Gamma (\phi)} \\ &\qquad{} \times_{5}\psi_{5}\left [ \begin{matrix} (\mu+\upsilon',1), & (\mu-\upsilon-\tau'+\sigma, 1), & (\mu-\tau-\tau '+\sigma, 1), & (\gamma, 1), & (\phi, 1) \\ (\mu,1), & (\mu-\tau-\tau'-\upsilon+\sigma, 1), & (\mu-\tau'+\upsilon, 1), & (1, 1), & (\varphi, 1) \end{matrix} \Big| \bigl(x^{-1}; p \bigr) \right ]. \end{aligned}} \end{aligned}$$
(3.16)

4 Concluding remarks

In this study, we established some fractional differential formulas involving a family of Mittag-Leffler functions. Due to practical importance of the Mittag-Leffler functions, our results are of general character and hence encompass several cases of interest.

Notes

Acknowledgements

Authors would like to acknowledge and express their gratitude to the United Arab Emirates University, Al Ain, UAE for providing the financial support with Grant No. 31S240-UPAR (2) 2016.

Authors’ contributions

All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

References

  1. 1.
    Al-Mdallal, QM, Hajji, MA: A convergent algorithm for solving higher-order nonlinear fractional boundary value problems. Fract. Calc. Appl. Anal. 18(6), 1423-1440 (2015) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Al-Mdallal, QM, Omer, AS: Fractional-order Legendre-collocation method for solving fractional initial value problems. Appl. Math. Comput. 321, 74-84 (2018) MathSciNetGoogle Scholar
  3. 3.
    Agarwal, P: Some inequalities involving Hadamard-type k-fractional integral operators. Math. Methods Appl. Sci. 40(11), 3882-3891 (2017) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Agarwal, P: Certain properties of the generalized Gauss hypergeometric functions. Appl. Math. Inf. Sci. 8(5), 2315-2320 (2014) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Agarwal, P: Fractional integration of the product of two multivariables H-function and a general class of polynomials. In: Advances in Applied Mathematics and Approximation Theory, Springer Proceedings in Mathematics & Statistics, vol. 41, pp. 359-374 (2013) CrossRefGoogle Scholar
  6. 6.
    Agarwal, P: Further results on fractional calculus of Saigo operators. Appl. Appl. Math. 7(2), 585-594 (2012) MathSciNetMATHGoogle Scholar
  7. 7.
    Agarwal, P: Generalized fractional integration of the -function. Le Matematiche LXVII, 107-118 (2012) MathSciNetMATHGoogle Scholar
  8. 8.
    Agarwal, P, Jain, S: Further results on fractional calculus of Srivastava polynomials. Bull. Math. Anal. Appl. 3(2), 167-174 (2011) MathSciNetMATHGoogle Scholar
  9. 9.
    Agarwal, P, Jain, S, Mansour, T: Further extended Caputo fractional derivative operator and its applications. Russ. J. Math. Phys. 24(4), 415-425 (2017) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kim, DS, Kim, T, Mansour, T, Seo, J-J: Degenerate Mittag-Leffler polynomials. Appl. Math. Comput. 274, 258-266 (2016) MathSciNetGoogle Scholar
  11. 11.
    Kim, DS, Kim, T, Rim, S-H: Some identities involving Gegenbauer polynomials. Adv. Differ. Equ. 2012, 219 (2012) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kim, T, Kim, DS, Dolgy, DV: Some identities on Bernoulli and Hermite polynomials associated with Jacobi polynomials. Discrete Dyn. Nat. Soc. 2012, Article ID 584643 (2012) MathSciNetMATHGoogle Scholar
  13. 13.
    Marichev, OI: Volterra equation of Mellin convolution type with a Horn function in the kernel. Izv. AN BSSR Ser. Fiz.-Mat. Nauk 1, 128-129 (1974) (In Russian) Google Scholar
  14. 14.
    Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  15. 15.
    Saigo, M: A remark on integral operators involving the Gauss hypergeometric functions. Math. Rep. Kyushu Univ. 11, 135-143 (1978) MathSciNetMATHGoogle Scholar
  16. 16.
    Saigo, M: A certain boundary value problem for the Euler-Darboux equation I. Math. Jpn. 24(4), 377-385 (1979) MathSciNetMATHGoogle Scholar
  17. 17.
    Saigo, M: A certain boundary value problem for the Euler-Darboux equation II. Math. Jpn. 25(2), 211-220 (1980) MathSciNetMATHGoogle Scholar
  18. 18.
    Saigo, M, Maeda, N: More generalization of fractional calculus. In: Rusev, P, Dimovski, I, Kiryakova, V (eds.) Transform Methods and Special Functions, Varna, 1996. Proc. 2nd Intern. Workshop, pp. 386-400. IMI-BAS, Sofia (1998) Google Scholar
  19. 19.
    Saigo, M, Kilbas, AA: Generalized fractional calculus of the H-function. Fukuoka Univ. Sci. Rep. 29, 31-45 (1999) MathSciNetMATHGoogle Scholar
  20. 20.
    Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach Sci. Publ., New York (1993) MATHGoogle Scholar
  21. 21.
    Saxena, RK, Saigo, M: Generalized fractional calculus of the H-function associated with the Appell function. J. Fract. Calc. 19, 89-104 (2001) MathSciNetMATHGoogle Scholar
  22. 22.
    Srivastava, HM, Tomovski, Z: Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211, 198-210 (2009) MathSciNetMATHGoogle Scholar
  23. 23.
    Ruzhansky, M, Cho, YJ, Agarwal, P, Area, I: Advances in Real and Complex Analysis with Applications. Birkhäuser, Basel (2017) CrossRefMATHGoogle Scholar
  24. 24.
    Kataria, KK, Vellaisamy, P: The generalized k-Wright function and Marichev-Saigo-Maeda fractional operators. arXiv:1408.4762v1 [math.CA] (17 Aug 2014)
  25. 25.
    Parmar, RK: A class of extended Mittag-Leffler functions and their properties related to integral transform and fractional calculus. Matematiche 3, 1069-1082 (2015) MathSciNetMATHGoogle Scholar
  26. 26.
    Srivastava, HM, Parmar, RK, Chopra, P: A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions. Axioms 1, 238-258 (2012) CrossRefMATHGoogle Scholar
  27. 27.
    Özarslan, MA, Yilmaz, B: The extended Mittag-Leffler function and its properties. J. Inequal. Appl. 2014, 85 (2014) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Prabhakar, TR: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7-15 (1971) MathSciNetMATHGoogle Scholar
  29. 29.
    Agarwal, P: Certain properties of extended Wright generalized hypergeometric type function. Submitted Google Scholar

Copyright information

© The Author(s) 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Praveen Agarwal
    • 1
    • 2
  • Qasem Al-Mdallal
    • 3
  • Yeol Je Cho
    • 4
    • 5
  • Shilpi Jain
    • 6
  1. 1.Department of MathematicsAnand International College of EngineeringJaipurIndia
  2. 2.International Center for Basic and Applied SciencesJaipurIndia
  3. 3.Department of Mathematical SciencesUAE UniversityAl AinUnited Arab Emirates
  4. 4.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China
  5. 5.Department of Mathematics EducationGyeongsang National UniversityJinjuKorea
  6. 6.Department of MathematicsPoornima College of EngineeringJaipurIndia

Personalised recommendations