Skip to main content

Fractional calculus operators with Appell function kernels applied to Srivastava polynomials and extended Mittag-Leffler function

Abstract

This article aims to establish certain image formulas associated with the fractional calculus operators with Appell function in the kernel and Caputo-type fractional differential operators involving Srivastava polynomials and extended Mittag-Leffler function. The main outcomes are presented in terms of the extended Wright function. In addition, along with the noted outcomes, the implications are also highlighted.

Introduction and preliminaries

The Mittag-Leffler function is the most significant in the theory of special functions in literature; solutions are available to number of problems formulated in terms of fractional integral, differential, and difference equations. So it has become a subject of interest for many authors in the field of fractional calculus (FC) and its applications (see details [4, 9, 18, 22, 25, 27, 28, 31, 35, 4143]). Nowadays, there are several applications of the Mittag-Leffler function such as random walks, Levy flights, telegraph equation, kinetic equation, transport, and complex system. Some of the applications are in the field of applied sciences, like rheology, fluid flow, electric networks, diffusive transport, statistical distribution theory. Other significant applications are available in the area of fractals kinetics [3, 5, 6], medical sciences [11, 17, 21], fractal calculus and applications [1, 8, 24], and in the Haar wavelet and analytical approach [14, 19, 20, 36].

The Swedish mathematician Mittag-Leffler (M-L) [26] introduced the following function, known after his name as Mittag-Leffler function:

$$\begin{aligned} E_{\theta }(x)=\sum^{\infty }_{n=0} \frac{x^{n}}{\varGamma (\theta n+1)} \quad \bigl(x \in \mathbb{C}; \Re (\theta )>0\bigr). \end{aligned}$$
(1)

Throughout the paper, \(\mathbb{C}\), \(\mathbb{R}^{+}\), \(\mathbb{Z}_{0}^{-}\), and \(\mathbb{N}\) are the usual notations for the sets of complex numbers, positive real numbers, nonpositive integers, and positive integers, respectively. Further, \(\mathbb{N}_{0}:=\mathbb{N} \cup \{0\}\). Mathematician Wiman [44] generalized the M-L function (1) as follows:

$$\begin{aligned} E_{\theta ,\kappa }(t)=\sum^{\infty }_{n=0} \frac{t^{n}}{\varGamma (\theta n+\kappa )} \quad \bigl(t, \kappa \in \mathbb{C}; \Re (\theta )>0\bigr). \end{aligned}$$
(2)

Many authors have conferred and studied the above function in diverse areas of research (see, e.g., [10, 12, 13, 32, 33]).

Prabhakar [30] introduced the generalized Mittag-Leffler function as follows:

$$\begin{aligned} E_{\theta ,\kappa }^{\gamma }(x)=\sum _{n=0}^{\infty } \frac{(\gamma )_{n}}{\varGamma (\theta n+\kappa )}\frac{x^{n}}{n!} \quad \bigl(x, \kappa , \gamma \in \mathbb{C}; \Re (\theta )>0\bigr), \end{aligned}$$
(3)

where \((\gamma )_{\tau }\), \(\gamma , \tau \in \mathbb{C}\); the Pochhammer symbol is defined in terms of the well-known gamma function by

$$ (\gamma )_{\tau }:= \frac{\varGamma (\gamma +\tau )}{ \varGamma (\gamma )} = \textstyle\begin{cases} 1& (\tau =0; \gamma \in \mathbb{C}\setminus \{0\}), \\ \gamma (\gamma +1) \cdots (\gamma +n-1) & (\tau =n \in {\mathbb{N}}; \gamma \in \mathbb{C}). \end{cases} $$
(4)

We recall the definition of extended M-L function, which was introduced and investigated by Özarslan and Yilmaz [29] in the manner

$$ \begin{aligned} &E^{\gamma ;c}_{\theta ,\kappa }(x;p)= \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\gamma +n,c-\gamma )(c)_{n}}{\mathbf{B}(\gamma ,c-\gamma )\varGamma (\theta n+\kappa )} \frac{x^{n}}{n!} \\ &\bigl(x, \kappa \in \mathbb{C}; p \geq 0; \Re (c)>\Re (\gamma )>0, \Re (\theta )>0\bigr), \end{aligned} $$
(5)

where extended beta function \(\mathcal{B}_{p}(x,y)\) is defined as (see [7])

$$\begin{aligned} \mathcal{B}_{p} (x,y)= \int _{0}^{1} u^{x-1}(1-u)^{y-1} e^{- \frac{p}{u(1-u)}} \,du \quad \bigl(\min \bigl\{ \Re (p), \Re (x), \Re (y)\bigr\} >0 \bigr), \end{aligned}$$
(6)

where \(\mathcal{B}_{0} (j,k)=\mathbf{B}(j,k)\), the well-known beta function (see, e.g., [39, Sect. 1.1])

$$ \mathbf{B}(j, k) = \textstyle\begin{cases} \int _{0}^{1} u^{j -1} (1-u)^{k -1} \,du & (\min \{\Re (j), \Re (k)\}>0), \\ \frac{\varGamma (j) \varGamma (k)}{\varGamma (j+ k)} & (j, k \in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-} ). \end{cases} $$
(7)

Sharma and Devi [37] introduced and investigated the following extended Wright generalized hypergeometric function:

Ψs+1r+1[(ai,Ai)1,r,(γ,1);(bj,Bj)1,s,(c,1);x;p]=1Γ(cγ)k=0i=1rΓ(ai+kAi)j=1sΓ(bj+kBj)Bp(γ+k,cγ)xkk!((c)>(γ)>0,(p)>0;r,sN0;ai,bjC,Ai,BjR+,i=1,,r,j=1,,s),
(8)

where the empty product is perceived as unity, and it is presumed that the summation is convergent.

The Srivastava family of polynomials [38, p.1. Equation (1)] is defined as follows:

$$ S_{w}^{u} (x)=\sum _{s=0}^{[w/u]} \frac{(-w)_{u s} }{s!} A_{w,s} x^{s}, \quad w = 0,1,2,\ldots, u\in \mathbb{Z^{+}}, A_{w.s} (w,s)\ge 0, A_{w.s} (w,s) \in \mathbb{C}. $$
(9)

We recall the definitions of fractional integral operators associated with the Appell function \(F_{3}\) (see, e.g., [40, p. 53, Eq. (6)],) in the kernel [23, 34], which are defined for τ, \(\tau ^{\prime }\), ε, \(\varepsilon ^{\prime }\), \(\upsilon \in \mathbb{C}\), \(\Re (\upsilon )>0\), \(x\in \mathbb{R}^{+}\) as follows:

$$ \bigl( I_{0{+}}^{\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{ \prime },\upsilon }f \bigr) (x)= \frac{x^{-\tau }}{\varGamma (\upsilon )} \int _{0}^{x}(x-t)^{\upsilon -1}t^{-\tau ^{\prime }}F_{3} \biggl( \tau ,\tau ^{\prime }, \varepsilon ,\varepsilon ^{\prime }; \upsilon ;1-\frac{t}{x},1- \frac{x}{t} \biggr) f(t) \,\mathrm{d}t $$
(10)

and

$$ \bigl( I_{-}^{\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{ \prime },\upsilon }f \bigr) (x)= \frac{x^{-\tau ^{\prime }}}{\varGamma (\upsilon )}\int _{x}^{\infty }(t-x)^{\upsilon -1}t^{-\tau }F_{3} \biggl( \tau , \tau ^{\prime },\varepsilon ,\varepsilon ^{\prime }; \upsilon ;1- \frac{t}{x},1-\frac{x}{t} \biggr) f(t) \,\mathrm{d}t. $$
(11)

Fractional integral operators (10) and (11) were introduced by Marichev [23] and further extended and studied by Saigo and Maeda [34].

Fractional differential operators corresponding to the above integrals (10) and (11) are defined as follows:

$$ \bigl( D_{0+}^{\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{ \prime },\upsilon }f \bigr) ( x ) = \biggl( \frac{\mathrm{d}}{\mathrm{d}x} \biggr) ^{ [ \Re ( \upsilon ) ] +1} \bigl( I_{0+}^{- \tau ^{\prime },-\tau ,-\varepsilon ^{\prime }+ [ \Re ( \upsilon ) ] +1,-\varepsilon ,-\upsilon + [ \Re ( \upsilon ) ] +1}f \bigr) ( x ) $$
(12)

and

$$ \bigl( D_{-}^{\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{ \prime },\upsilon }f \bigr) ( x ) = \biggl( - \frac{\mathrm{d}}{\mathrm{d}x} \biggr) ^{ [ \Re ( \upsilon ) ] +1} \bigl( I_{-}^{- \tau ^{\prime },-\tau ,-\varepsilon ^{\prime },-\varepsilon + [ \Re ( \upsilon ) ] +1,-\upsilon + [ \Re ( \upsilon ) ] +1}f \bigr) ( x ) . $$
(13)

The following lemmas [15, 34] shall be required in the sequel.

Lemma 1.1

Let \(\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{\prime },\upsilon , \varsigma \in \mathbb{C}\) be such that \(\Re {(\upsilon )}>0\) and

$$ \Re {(\tau )}>\max \bigl\{ 0,\Re {\bigl(\tau -\tau ^{\prime }-\varepsilon - \upsilon \bigr)},\Re {\bigl(\tau ^{\prime }-\varepsilon ^{\prime }\bigr)}\bigr\} . $$

Then there exists the relation

$$ \bigl( I_{0+}^{\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{ \prime },\upsilon } t^{\varsigma -1} \bigr) (x)= \frac{\varGamma ( \varsigma ) \varGamma ( \varsigma +\upsilon -\tau -\tau ^{\prime }-\varepsilon ) \varGamma ( \varsigma +\varepsilon {^{\prime }}-\tau {^{\prime }} ) }{\varGamma ( \varsigma +\varepsilon {^{\prime }} ) \varGamma ( \varsigma +\upsilon -\tau -\tau {^{\prime }} ) \varGamma ( \varsigma +\upsilon -\tau {^{\prime }}-\varepsilon ) }x^{\varsigma -\tau -\tau ^{\prime }+\upsilon -1}. $$
(14)

Lemma 1.2

Letτ, \(\tau ^{\prime }\), ε, \(\varepsilon ^{\prime }\), υ, \(\varsigma \in \mathbb{C}\)such that\(\Re (\upsilon )>0\)and

$$ \Re {(\varsigma )}>\max \bigl\{ \Re {(\varepsilon )}, \Re {\bigl(-\tau -\tau ^{ \prime }+\upsilon \bigr)}, \Re {\bigl(-\tau -\varepsilon ^{\prime }+ \upsilon \bigr)} \bigr\} . $$

Then

$$ \bigl( I_{-}^{\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{ \prime },\upsilon }t^{-\varsigma } \bigr) (x) = \frac{\varGamma ( -\varepsilon +\varsigma ) \varGamma ( \tau +\tau ^{\prime }-\upsilon +\varsigma ) \varGamma ( \tau +\varepsilon { ^{\prime }}-\upsilon +\varsigma ) }{\varGamma ( \varsigma ) \varGamma ( \tau -\varepsilon +\varsigma ) \varGamma ( \tau +\tau {^{\prime }+\varepsilon }^{{\prime }}-\upsilon +\varsigma ) }x^{-\tau -\tau ^{\prime }+ \upsilon -\varsigma }. $$

Lemma 1.3

Letτ, \(\tau ^{\prime }\), ε, \(\varepsilon ^{\prime }\), υ, \(\varsigma \in \mathbb{C}\)such that

$$ \Re ( \varsigma ) >\max \bigl\{ 0,\Re \bigl( -\tau + \varepsilon ^{\prime } \bigr) ,\Re \bigl( -\tau -\tau ^{\prime }- \varepsilon +\upsilon \bigr) \bigr\} . $$

Then

$$ \bigl( D_{0+}^{\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{ \prime },\upsilon }t^{\varsigma -1} \bigr) ( x ) = \frac{\varGamma ( \varsigma ) \varGamma ( -\varepsilon +\tau +\varsigma ) \varGamma ( \tau +\tau ^{\prime }+\varepsilon ^{{\prime }}-\upsilon +\varsigma ) }{\varGamma ( -\varepsilon +\varsigma ) \varGamma ( \tau +\tau ^{{\prime }}-\upsilon +\varsigma ) \varGamma ( \tau +{\varepsilon }^{{\prime }}-\upsilon +\varsigma ) }x^{\tau +\tau ^{\prime }- \upsilon +\varsigma -1}. $$
(15)

Lemma 1.4

Letτ, \(\tau ^{\prime }\), ε, \(\varepsilon ^{\prime }\), υ, \(\varsigma \in \mathbb{C}\)such that

$$ \Re ( \varsigma ) >\max \bigl\{ \Re \bigl( - \varepsilon ^{\prime } \bigr) ,\Re \bigl( \tau ^{\prime }+ \varepsilon -\upsilon \bigr) ,\Re \bigl( \tau + \tau ^{{\prime }}- \upsilon \bigr) + \bigl[ \Re ( \upsilon ) \bigr] +1 \bigr\} . $$

Then

$$ \bigl( D_{-}^{\tau ,\tau ^{\prime },\varepsilon ,\varepsilon ^{ \prime },\upsilon }t^{-\varsigma } \bigr) ( x ) = \frac{\varGamma ( \varepsilon ^{\prime }+\varsigma ) \varGamma ( -\tau -\tau ^{\prime }+\upsilon +\varsigma ) \varGamma ( -\tau ^{\prime }-\varepsilon +\upsilon +\varsigma ) }{\varGamma ( \varsigma ) \varGamma ( -\tau ^{\prime }+\varepsilon ^{\prime }+\varsigma ) \varGamma ( -\tau -\tau ^{\prime }-\varepsilon +\upsilon +\varsigma ) }x^{ \tau +\tau ^{\prime }-\upsilon -\varsigma }. $$
(16)

The left- and right-sided Saigo fractional integral formulas are defined for \(x>0\) and \(\tau , \varepsilon , \upsilon \in \mathbb{C}\), \(\Re (\tau )>0\), respectively, by (see [16])

$$ \bigl( I_{0+}^{\tau ,\varepsilon ,\upsilon }f \bigr) ( x ) = \frac{x^{-\tau -\varepsilon }}{\varGamma ( \tau ) } \int _{0}^{x} ( x-t ) ^{\tau -1}{}_{2}F_{1} \biggl( \tau +\varepsilon ,- \upsilon ;\tau ;1-\frac{t}{x} \biggr) f ( t ) \,dt $$
(17)

and

$$ \bigl( I_{- }^{\tau ,\varepsilon ,\upsilon }f \bigr) ( x ) =\frac{1}{\varGamma ( \tau ) } \int _{x}^{\infty } ( t-x ) ^{\tau -1}t^{-\tau -\varepsilon }{}_{2}F_{1} \biggl( \tau +\varepsilon ,-\upsilon ;\tau ;1-\frac{x}{t} \biggr) f ( t ) \,dt, $$
(18)

where \({}_{2}F_{1} ( \cdot ) \) is the Gauss hypergeometric series.

When \(\varepsilon =-\tau \), the operators in equations (17) and (18) coincide with the classical Riemann–Liouville fractional integrals of order \(\tau \in \mathbb{C}\) with \(x > 0\) as follows:

$$ \bigl( I_{0+}^{\tau }f \bigr) (x ) = \frac{1}{\varGamma ( \tau ) } \int _{0}^{x} \frac{1}{(x-t)^{1-\tau }}f(t)\,dt $$
(19)

and

$$ \bigl( I_{-}^{\tau }f \bigr) (x ) = \frac{1}{\varGamma ( \tau ) } \int _{x}^{\infty } \frac{1}{(t-x)^{1-\tau }}f(t)\,dt . $$
(20)

For \(\varepsilon =0\), the operators in equations (17) and (18) yield the so-called Erdélyi–Kober integrals of order \(\tau \in \mathbb{C}\) with \(x > 0\) as follows:

$$ \bigl( I_{\upsilon ,\tau }^{+}f \bigr) (x ) = \frac{x^{-\tau -\upsilon }}{\varGamma ( \tau ) } \int _{0}^{x} \frac{1}{(x-t)^{1-\tau }}t^{\upsilon }f(t) \,dt $$
(21)

and

$$ \bigl( K_{\upsilon ,\tau }^{-}f \bigr) (x ) = \frac{x^{\upsilon }}{\varGamma ( \tau ) } \int _{x}^{\infty } \frac{1}{(t-x)^{1-\tau }}t^{-\tau -\upsilon }f(t) \,dt . $$
(22)

We also need the following lemmas [16].

Lemma 1.5

Let\(\tau , \varepsilon , \upsilon \in \mathbb{C}\)be such that\(\Re (\tau )>0\), \(\Re (\varsigma )>\max [0, \Re (\varepsilon -\upsilon )]\). Then

$$ \bigl( I_{0+}^{\tau ,\varepsilon ,\upsilon } t^{\varsigma -1} \bigr) (x)= \frac{\varGamma ( \varsigma ) \varGamma ( \varsigma +\upsilon -\varepsilon )}{\varGamma ( \varsigma -\varepsilon ) \varGamma ( \varsigma +\tau +\upsilon )}x^{\varsigma -\varepsilon -1}. $$
(23)

In particular,

$$ \bigl( I_{0+}^{\tau } t^{\varsigma -1} \bigr) (x)= \frac{\varGamma ( \varsigma )}{\varGamma ( \varsigma +\tau )}x^{\varsigma + \tau -1}, \quad \Re (\tau )>0, \Re (\varsigma )>0 $$
(24)

and

$$ \bigl( I_{\upsilon ,\tau }^{+} t^{\varsigma -1} \bigr) (x)= \frac{\varGamma ( \varsigma +\upsilon )}{\varGamma ( \varsigma +\tau +\upsilon )}x^{\varsigma -1},\quad \Re (\tau )>0, \Re (\varsigma )>-\Re (\upsilon ). $$
(25)

Lemma 1.6

Let\(\tau , \varepsilon , \upsilon \in \mathbb{C}\)be such that\(\Re (\tau )>0\), \(\Re (\varsigma )<1+\min [\Re (\varepsilon ), \Re ( \upsilon )]\). Then

$$ \bigl( I_{-}^{\tau ,\varepsilon ,\upsilon } t^{\varsigma -1} \bigr) (x)= \frac{\varGamma ( \varepsilon -\varsigma +1 ) \varGamma ( \upsilon -\varsigma +1 )}{\varGamma ( 1-\varsigma ) \varGamma ( \tau +\varepsilon +\upsilon -\varsigma +1 )}x^{\varsigma -\varepsilon -1}. $$
(26)

In particular,

$$ \bigl( I_{-}^{\tau } t^{\varsigma -1} \bigr) (x)= \frac{\varGamma ( 1-\tau -\varsigma )}{\varGamma (1- \varsigma )}x^{\varsigma + \tau -1}, \quad 0< \Re (\tau )< 1-\Re ( \varsigma ) $$
(27)

and

$$ \bigl( K_{\upsilon ,\tau }^{- } t^{\varsigma -1} \bigr) (x)= \frac{\varGamma (\upsilon -\varsigma +1 )}{\varGamma (\tau +\upsilon -\varsigma +1 )}x^{\varsigma -1},\quad \Re (\varsigma )< 1+\Re ( \upsilon ). $$
(28)

Fractional calculus of extended Mittag-Leffler function

In this section, we present certain image formulas for product of Srivastava polynomials and generalized EMLF in view of the generalized fractional integral and differential calculus and consider some corollaries as particular cases.

Theorem 2.1

Let\(\tau ,\tau ',\varepsilon ,\varepsilon ', c,\theta ,\kappa , \upsilon , \chi , \varsigma \in \mathbb{C}\), \(x\in \mathbb{R}^{+}\)with\(\Re (\upsilon )>0\)and\(\Re (\varsigma )>\max \{ 0, \Re (\tau +\tau '+\varepsilon - \upsilon ),\Re (\tau -\varepsilon ') \} \)and\(p\geq 0\), then

(I0+τ,τ,ε,ε,υtς1Snm(t)Eθ,κχ,c(t;p))(x)=xςττ+υ1Γ(χ)s=0[n/m](n)m.ss!An,sxs×5ψ5[(c,1),(ς+s,1),(ς+s+υττε,1),(ς+s+ετ,1),(χ,1);(κ,θ),(ς+s+ε,1),(ς+s+υτε,1),(ς+s+υττ,1),(c,1);(x;p)].
(29)

Proof

Let \(I_{1}\) denote LHS of (29). Using (5) and (9) therein, we have

$$\begin{aligned} I_{1} =& \bigl(I_{0+}^{\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon }t^{\varsigma -1}S_{n}^{m} ( t )E^{\chi ,c}_{ \theta ,\kappa }(t;p) \bigr) (x) \\ =& \Biggl(I_{0+}^{\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon }t^{ \varsigma -1}\sum _{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} (t )^{s} \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )}\frac{t^{n}}{n!} \Biggr) (x). \end{aligned}$$

Under the valid conditions mentioned with this theorem, interchanging the integration and summation order allows us to write

$$\begin{aligned} I_{1} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{ \infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )n!} \bigl(I_{0+}^{\tau ,\tau ', \varepsilon ,\varepsilon ',\upsilon }t^{\varsigma +n+s-1} \bigr) (x). \end{aligned}$$

Applying Lemma (1.1), we get

$$\begin{aligned} I_{1} = &\sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa ){n!}} \\ &{}\times \frac{\varGamma (\varsigma +s+n)\varGamma (\varsigma +s+\upsilon -\tau -\tau '-\varepsilon +n)\varGamma (\varsigma +s+\varepsilon '-\tau '+n)}{\varGamma (\varsigma +s+\varepsilon '+n)\varGamma (\varsigma +s+\upsilon -\tau -\tau '+n)\varGamma (\varsigma +s+\upsilon -\tau '-\varepsilon +n)} \\ &{}\times x^{\varsigma +n+s+\upsilon -\tau -\tau '-1} \\ =&\sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\varGamma (\chi )\varGamma (c-\chi )} \frac{\varGamma (c+n)}{\varGamma (\theta n+\kappa ){n!}} \\ &{}\times \frac{\varGamma (\varsigma +s+n)\varGamma (\varsigma +s+\upsilon -\tau -\tau '-\varepsilon +n)\varGamma (\varsigma +s+\varepsilon '-\tau '+n)}{\varGamma (\varsigma +s+\varepsilon '+n)\varGamma (\varsigma +s+\upsilon -\tau -\tau '+n)\varGamma (\varsigma +s+\upsilon -\tau '-\varepsilon +n)} \\ &{}\times x^{\varsigma +s+n+\upsilon -\tau -\tau '-1} \\ =&\frac{x^{\varsigma +\upsilon -\tau -\tau '-1}}{\varGamma (\chi )} \sum_{s=0}^{[n/m]} \frac{ (-n )_{m,s} }{s!} A_{n,s} x^{s} \sum _{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\varGamma (c-\chi )} \frac{\varGamma (c+n)}{\varGamma (\theta n+\kappa )} \\ &{}\times \frac{\varGamma (\varsigma +s+n)\varGamma (\varsigma +s+\upsilon -\tau -\tau '-\varepsilon +n)\varGamma (\varsigma +s+\varepsilon '-\tau '+n)}{\varGamma (\varsigma +s+\varepsilon '+n)\varGamma (\varsigma +s+\upsilon -\tau -\tau '+n) \varGamma (\varsigma +s+\upsilon -\tau '-\varepsilon +n)} \frac{x^{n}}{n!}. \end{aligned}$$

Making use of (8), we get the outcome needed. □

Corollary

Let\(\tau ,\varepsilon , c,\theta ,\kappa ,\upsilon ,\varsigma \in \mathbb{C}\), \(x\in \mathbb{R}^{+}\)with\(\Re (\upsilon )>0\)and\(\Re (\varsigma )>\max \{ 0,\Re (\varepsilon -\upsilon ) \} \)and\(p\geq 0\), then

(I0+τ,ε,υtς1Snm(t)Eθ,κχ,c(t;p))(x)=xςε1Γ(χ)s=0[n/m](n)m.ss!An,sxs×4ψ4[(c,1),(ς+s,1),(ς+s+υε,1),(χ,1);(c,1),(κ,θ),(ς+sε,1),(ς+s+υ+τ,1);(x;p)].
(30)

Corollary

Let\(\tau , c,\theta ,\kappa ,\upsilon ,\varsigma \in \mathbb{C}\), \(x\in \mathbb{R}^{+}\)with\(\Re (\upsilon )>0\)and\(\Re (\varsigma )>\Re (\upsilon )\)and\(p\geq 0\), then

(Iυ,τ+tς1Snm(t)Eθ,κχ,c(t;p))(x)=xς1Γ(χ)s=0[n/m](n)m.ss!An,sxs×3ψ3[(c,1),(ς+s+υ,1),(χ,1);(c,1),(κ,θ),(ς+s+υ+τ,1);(x;p)].
(31)

Theorem 2.2

Let\(\tau ,\tau ',\varepsilon ,\varepsilon ',\theta ,\kappa ,\upsilon , \chi , \varsigma \in \mathbb{C}\), \(x\in \mathbb{R}^{+}\)with\(\Re (\tau )>0\)and\(\Re (\varsigma )>\max \{ \Re (\varepsilon ), \Re (-\tau - \tau '+\upsilon ),\Re (-\tau -\varepsilon '+\upsilon ) \} \)and\(p\geq 0\), then

(Iτ,τ,ε,ε,υtςSnm(t)Eθ,κχ,c(1/t;p))(x)=xςττ+υΓ(χ)s=0[n/m](n)m.ss!An,sxs×5ψ5[(c,1),(ςsε,1),(τ+τυ+ςs,1),(τ+ευ+ςs,1),(χ,1);(c,1),(κ,θ),(ς,1),(τε+ςs,1),(τ+τευ+ςs,1),(υ,1);(1/x;p)].
(32)

Proof

Let \(I_{2}\) be LHS of (32), then using (5) and (9), we have

$$\begin{aligned} I_{2} =& \bigl(I_{-}^{\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon }t^{-\varsigma }S_{n}^{m} (t ) E^{\chi ,c}_{ \theta ,\kappa }(1/t;p) \bigr) (x) \\ =& \Biggl(I_{-}^{\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon }t^{- \varsigma }\sum _{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} (t)^{s}\sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )}\frac{t^{-n}}{n!} \Biggr) (x). \end{aligned}$$

Now, under the conditions of the validity of this theorem, switching the order of integration and summation, we get

$$\begin{aligned} I_{2} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{ \infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )n!} \bigl(I_{-}^{\tau ,\tau ', \varepsilon ,\varepsilon ',\upsilon }t^{-\varsigma +s-n} \bigr) (x). \end{aligned}$$

Applying Lemma (1.2), we get

$$\begin{aligned} I_{2} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa ){n!}} \\ &{}\times \frac{\varGamma (-\varepsilon +\varsigma -s+n)\varGamma (\tau +\tau '+\varsigma -s-\upsilon +n)\varGamma (\tau +\varsigma -s+\varepsilon '-\upsilon +n)}{\varGamma (\varsigma -s+n)\varGamma (\varsigma -s-\varepsilon +\tau +n)\varGamma (\varsigma -s+\tau +\tau '-\varepsilon '-\upsilon +n)} \\ &{}\times x^{-\varsigma +s-n+\upsilon -\tau -\tau '} \\ =&\sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\varGamma (\chi )\varGamma (c-\chi )} \frac{\varGamma (c+n)}{\varGamma (\theta n+\kappa ){n!}} \\ &{}\times \frac{\varGamma (-\varepsilon +\varsigma -s+n)\varGamma (\tau +\tau '+\varsigma -s-\upsilon +n)\varGamma (\tau +\varsigma -s+\varepsilon '-\upsilon +n)}{\varGamma (\varsigma -s+n)\varGamma (\varsigma -s-\varepsilon +\tau +n)\varGamma (\varsigma -s+\tau +\tau '-\varepsilon '-\upsilon +n)} \\ &{}\times x^{-\varsigma +s-n+\upsilon -\tau -\tau '} \\ =&\frac{x^{-\varsigma +\upsilon -\tau -\tau '}}{\varGamma (\chi )} \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} x^{s} \sum _{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\varGamma (c-\chi )} \frac{\varGamma (c+n)}{\varGamma (\theta n+\kappa )} \\ &{}\times \frac{\varGamma (-\varepsilon +\varsigma -s+n)\varGamma (\tau +\tau '+\varsigma -s-\upsilon +n)\varGamma (\tau +\varsigma -s+\varepsilon '-\upsilon +n)}{\varGamma (\varsigma -s+n)\varGamma (\varsigma -s-\varepsilon +\tau +n)\varGamma (\varsigma -s+\tau +\tau '-\varepsilon '-\upsilon +n)} \frac{x^{-n}}{n!}. \end{aligned}$$

Again, by employing (8), we obtain the desired result. □

Corollary

Let\(\tau ,\varepsilon , c,\theta ,\kappa , \chi , \upsilon ,\varsigma \in \mathbb{C}\), \(x\in \mathbb{R}^{+}\)with\(\Re (\upsilon )>0\)and\(\Re (\varsigma )>\max \{ \Re (\varepsilon ),\Re (-\upsilon ) \} \)and\(p\geq 0\), then

(Iτ,ε,υtςSnm(t)Eθ,κχ,c(1/t;p))(x)=xςεΓ(χ)s=0[n/m](n)m.ss!An,sxs×4ψ4[(c,1),(ε+ςs,1),(υ+ςs,1),(χ,1);(c,1),(κ,θ),(ςs,1),(τ+ε+υs+ς,1);(1/x;p)].
(33)

Corollary

Let\(\tau ,\varepsilon , c,\theta ,\kappa , \chi , \upsilon ,\varsigma \in \mathbb{C}\), \(x\in \mathbb{R}^{+}\)with\(\Re (\upsilon )>0\)and\(\Re (\varsigma )>\Re (-\upsilon )\)and\(p\geq 0\), then

(Kυ,τtςSnm(t)Eθ,κχ,c(1/t;p))(x)=xςΓ(χ)s=0[n/m](n)m.ss!An,sxs×3ψ3[(c,1),(υ+ςs,1),(χ,1);(c,1),(κ,θ),(τ+υs+ς,1);(1/x;p)].
(34)

Theorem 2.3

Let\(\tau ,\tau ',\varepsilon ,\varepsilon ', c,\theta ,\kappa , \chi , \upsilon ,\varsigma \in \mathbb{C}\)and\(\Re (\varsigma )>\max \{ 0,\Re (-\tau +\varepsilon ),\Re (- \tau -\tau '-\varepsilon '+\upsilon ) \} \)and\(p\geq 0\). Also let\(x\in \mathbb{R}^{+}\), then

(D0+τ,τ,ε,ε,υtς1Snm(t)Eθ,κχ,c(t;p))(x)=xς+τ+τυ1Γ(χ)s=0[n/m](n)m.ss!An,sxs×5ψ5[(c,1),(ς+s,1),(ς+s+τε,1),(τ+τ+ευ+ς+s,1),(χ,1);(c,1),(κ,θ),(ς+sε,1),(τ+τυ+ς+s,1),(τ+ς+sυ+ε,1);(x;p)].
(35)

Proof

Let \(I_{3}\) be LHS of (35), then using (5), we have

$$\begin{aligned} I_{3} =& \bigl(D_{0+}^{\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon }t^{\varsigma -1}S_{n}^{m} (t )E^{\chi ,c}_{ \theta ,\kappa }(t;p) \bigr) (x) \\ =& \Biggl(D_{0+}^{\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon }t^{ \varsigma -1}\sum _{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} (t)^{s} \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )}\frac{t^{n}}{n!} \Biggr) (x). \end{aligned}$$

Employing the exchange in the orders of integration and summation, we obtain

$$\begin{aligned} I_{3} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{ \infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )n!} \bigl(D_{0+}^{\tau ,\tau ', \varepsilon ,\varepsilon ',\upsilon }t^{\varsigma +s+n-1} \bigr) (x). \end{aligned}$$

Applying Lemma (1.3), we get

$$\begin{aligned} I_{3} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa ){n!}} \\ &{}\times \frac{\varGamma (\varsigma +s+n)\varGamma (\varsigma +s+\tau -\varepsilon +n)\varGamma (\varsigma +s+\tau +\tau '+\varepsilon '-\upsilon +n)}{\varGamma (\varsigma +s-\varepsilon +n)\varGamma (\varsigma +s+\tau +\tau '-\upsilon +n)\varGamma (\varsigma +s+\tau +\varepsilon '-\upsilon +n)} \\ &{}\times x^{\varsigma +s+n-\upsilon +\tau +\tau '-1} \\ =&\frac{x^{\varsigma -\upsilon +\tau +\tau '-1}}{\varGamma (\chi )} \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} x^{s} \sum _{0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\varGamma (c-\chi )} \frac{\varGamma (c+n)}{\varGamma (\theta n+\kappa )} \\ &{}\times \frac{\varGamma (\varsigma +s+n)\varGamma (\varsigma +s+\tau -\varepsilon +n)\varGamma (\varsigma +s+\tau +\tau '+\varepsilon '-\upsilon +n)}{\varGamma (\varsigma +s-\varepsilon +n)\varGamma (\varsigma +s+\tau +\tau '-\upsilon +n)\varGamma (\varsigma +s+\tau +\varepsilon '-\upsilon +n)} \frac{x^{n}}{n!}. \end{aligned}$$

Finally, on exercising (8) therein, we easily figure out the desired result. □

Theorem 2.4

Let\(\tau ,\tau ',\varepsilon ,\varepsilon ', c,\theta ,\kappa , \chi , \upsilon ,\varsigma \in \mathbb{C}\)and

$$ \Re (\varsigma )>\max \bigl\{ \Re \bigl(-\varepsilon '\bigr),\Re \bigl(\tau '+ \varepsilon -\upsilon \bigr),\Re \bigl(\tau +\tau '-\upsilon \bigr)+\bigl[\Re (\upsilon )\bigr]+1 \bigr\} $$

and\(p\geq 0\). Also let\(x\in \mathbb{R}^{+}\), then

(Dτ,τ,ε,ε,υtςSnm(t)Eθ,κχ,c(1/t;p))(x)=xτ+τυςΓ(χ)s=0[n/m](n)m.ss!An,sxs×5ψ5[(c,1),(ε+ςs,1),(ςsττ+υ,1),(ςs+ετ,1),(χ,1);(c,1),(κ,θ),(ςs,1),(ςsτ+ε,1),(ςsττ+υε,1);(1/x;p)].
(36)

Proof

Let \(I_{4}\) be LHS of (36), then using (5), we have

$$\begin{aligned} I_{4} =& \bigl(D_{-}^{\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon }t^{-\varsigma }S_{n}^{m} (t )E^{\chi ,c}_{ \theta ,\kappa }(1/t;p) \bigr) (x) \\ =& \Biggl(D_{-}^{\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon }t^{- \varsigma } \sum _{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} (t)^{s} \sum_{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )}\frac{t^{-n}}{n!} \Biggr) (x). \end{aligned}$$

Interchanging the order of integration and summation, under the valid condition, we obtain

$$\begin{aligned} I_{4} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{ \infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )n!} \bigl(D_{-}^{\tau ,\tau ', \varepsilon ,\varepsilon ',\upsilon }t^{-\varsigma +s-n} \bigr) (x). \end{aligned}$$

Applying Lemma (1.4), we get

$$\begin{aligned} I_{4} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n= 0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa ){n!}} \\ &{}\times \frac{\varGamma (\varepsilon '+\varsigma -s+n)\varGamma (\varsigma -s-\tau -\tau '+\upsilon +n)\varGamma (\varsigma -s-\tau '+\varepsilon +\upsilon +n)}{\varGamma (\varsigma -s+n)\varGamma (\varsigma -s-\tau '+\varepsilon '+n)\varGamma (\varsigma -s-\tau -\tau '-\varepsilon '+\upsilon +n)} \\ &{}\times x^{\tau +\tau '-\varsigma -\upsilon +s-n} \\ =&\frac{x^{\tau +\tau '-\varsigma -\upsilon }}{\varGamma (\chi )}\sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} x^{s}\sum _{n=0}^{\infty } \frac{\mathcal{B}_{p} (\chi +n,c-\chi )}{\varGamma (c-\chi )} \frac{\varGamma (c+n)}{\varGamma (\theta n+\kappa )} \\ &{}\times \frac{\varGamma (\varepsilon '+\varsigma -s+n)\varGamma (\varsigma -s-\tau -\tau '+\upsilon +n)\varGamma (\varsigma -s-\tau '+\varepsilon +\upsilon +n)}{\varGamma (\varsigma -s+n)\varGamma (\varsigma -s-\tau '+\varepsilon '+n)\varGamma (\varsigma -s-\tau -\tau '-\varepsilon '+\upsilon +n)} \frac{x^{-n}}{n!}. \end{aligned}$$

Again, by using (8), we get the desired result. □

Caputo-type fractional differentiation of extended Mittag-Leffler function

The left- and right-sided Caputo-type fractional differential operators with Gauss hypergeometric function in the kernel are defined as follows:

$$\begin{aligned} \bigl({}^{c}D_{0+}^{\tau ,\varepsilon ,\upsilon }f \bigr) (x)=\bigl(I_{0+}^{-\tau + \varLambda ,-\varepsilon ,-\upsilon +\varLambda }f^{(\varLambda )}\bigr) (x) \end{aligned}$$
(37)

and

$$\begin{aligned} \bigl({}^{c}D_{-}^{\tau ,\varepsilon ,\upsilon }f \bigr) (x)=(-1)^{\varLambda }\bigl(I_{-}^{- \tau +\varLambda ,-\varepsilon +\varLambda ,\tau +\upsilon }f^{(\varLambda ) } \bigr) (x), \end{aligned}$$
(38)

where \(\tau ,\varepsilon , \upsilon ,\varsigma \in \mathbb{C}\), \(\varLambda =[\Re (\tau )]+1 \), and \(x\in \mathbb{R}^{+}\).

The left- and right-sided Caputo-type generalized fractional differential operators are given by

$$\begin{aligned} \bigl({}^{c}D_{0+}^{\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon }f \bigr) (x)=(-1)^{ \varLambda ' }\bigl(I_{0+}^{-\tau ,-\tau ',-\varepsilon ' +\varLambda ',- \varepsilon ,-\upsilon +\varLambda ',-\upsilon +\varLambda ' }f^{(\varLambda ')} \bigr) (x) \end{aligned}$$
(39)

and

$$\begin{aligned} \bigl({}^{c}D_{-}^{\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon }f \bigr) (x)=(-1)^{ \varLambda '}\bigl(I_{-}^{-\tau ,-\tau ',-\varepsilon ,-\varepsilon ' + \varLambda ',-\upsilon +\varLambda ',-\upsilon +\varLambda ' }f^{(\varLambda '} \bigr)) (x), \end{aligned}$$
(40)

where \(\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon ,\varsigma \in \mathbb{C}\), \(\varLambda '= [\Re (\upsilon )]+1 \), and \(x\in \mathbb{R}^{+}\).

The following lemmas are required to demonstrate the intended outcome.

Lemma 3.1

([15])

Let\(\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon ,\varsigma \in \mathbb{C}\)and\(\varLambda '=[\Re (\upsilon )]+1\)with\(\Re (\varsigma )-\varLambda ' >\max \{ 0, \Re (-\tau + \varepsilon ),\Re (-\tau -\tau '-\varepsilon '+\upsilon ) \} \)and\(p\geq 0\). Then

$$ \begin{aligned}[b] &\bigl({}^{c}D_{0+}^{\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon }t^{ \varsigma -1} \bigr) (x)\\ &\quad = \frac{\varGamma (\varsigma )\varGamma (\varsigma +\tau -\varepsilon -\varLambda ')\varGamma (\varsigma +\tau +\tau '+\varepsilon '-\upsilon -\varLambda ')}{\varGamma (\varsigma -\varepsilon -\varLambda ')\varGamma (\varsigma +\tau +\tau '-\upsilon )\varGamma (\varsigma +\tau +\varepsilon '-\upsilon -\varLambda ')}x^{ \varsigma -\upsilon +\tau +\tau '-1}. \end{aligned} $$

Lemma 3.2

[15] Let\(\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon ,\varsigma \in \mathbb{C}\)and\(\varLambda '=[\Re (\upsilon )]+1\)with\(\Re (\varsigma )+\varLambda '>\max \{ \Re (-\varepsilon '), \Re (\tau '+\varepsilon -\upsilon ),\Re (\tau +\tau '-\upsilon )+[ \Re (\upsilon )]+1 \} \). Then

$$\begin{aligned} \bigl({}^{c}D_{-}^{\tau ,\tau ',\varepsilon ,\varepsilon ',\upsilon }t^{- \varsigma } \bigr) (x)= \frac{\varGamma (\varsigma +\varepsilon '+\varLambda ')\varGamma (\varsigma -\tau -\tau '+\upsilon )\varGamma (\varsigma -\tau '-\varepsilon +\upsilon +\varLambda ')}{\varGamma (\varsigma )\varGamma (\varsigma -\tau '+\varepsilon '+\varLambda ')\varGamma (\varsigma -\tau -\tau '-\varepsilon +\upsilon +\varLambda ')}x^{ \tau +\tau '-\upsilon -\varsigma }. \end{aligned}$$

Theorem 3.3

Let\(\tau ,\tau ',\varepsilon ,\varepsilon ', c,\theta ,\kappa , \chi , \upsilon ,\varsigma \in \mathbb{C}\), \(x\in \mathbb{R}^{+}\), and\(\varLambda '=[\Re (\upsilon )+1]\)\(\Re (\varsigma )-\varLambda ' >\max \{ 0,\Re (-\tau + \varepsilon '),\Re (-\tau -\tau '-\varepsilon '+\upsilon ) \} \)and\(p\geq 0\), then

(cD0+τ,τ,ε,ε,υtς1Snm(t)Eθ,κχ,c(t;p))(x)=xς+τ+τυ+1Γ(χ)s=0[n/m](n)m.ss!An,sxs×5ψ5[(c,1),(ς,1),(ς+s+τεΛ,1),(τ+τ+ευΛ+ς+s,1),(χ,1);(c,1),(κ,θ),(ς+sεΛ,1),(τ+τυ+ς+s,1),(τ+ς+sυ+εΛ,1);(x;p)].
(41)

Proof

Let \(I_{5}\) be LHS of (41), then using (5), we have

$$\begin{aligned} I_{5} =& \bigl({}^{c}D_{0+}^{\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon }t^{\varsigma -1}S_{n}^{m} (t )E^{\chi ,c}_{ \theta ,\kappa }(t;p) \bigr) (x) \\ =& \Biggl({}^{c}D_{0+}^{\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon }t^{\varsigma -1}\sum _{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} (t)^{s} \sum_{n=0}^{ \infty } \frac{\mathbf{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi , c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )}\frac{t^{n}}{n!} \Biggr) (x). \end{aligned}$$

Interchange in the order of integration and summation under the verified condition in this theorem allows us to write

$$\begin{aligned} I_{5} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{ \infty } \frac{\mathbf{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )n!} \bigl({}^{c}D_{0+}^{\tau , \tau ',\varepsilon ,\varepsilon ',\upsilon }t^{\varsigma +s+n-1} \bigr) (x). \end{aligned}$$

Application of Lemma (3.1) leads to

$$\begin{aligned} I_{5} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{\infty } \frac{\mathbf{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa ){n!}} \\ &{}\times \frac{\varGamma (\varsigma +s+n)\varGamma (\varsigma +s+\tau -\varepsilon -\varLambda '+n)\varGamma (\varsigma +s+\tau +\tau '+\varepsilon '-\upsilon -\varLambda '+n)}{\varGamma (\varsigma +s-\varepsilon -\varLambda '+n)\varGamma (\varsigma +s+\tau +\tau '-\upsilon +n)\varGamma (\varsigma +s+\tau +\varepsilon '-\upsilon -\varLambda '+n)} \\ &{}\times x^{\varsigma +s+n-\upsilon +\tau +\tau '+1} \\ =&\frac{x^{\varsigma -\upsilon +\tau +\tau '+1}}{\varGamma (\chi )} \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} x^{s} \sum _{n=0}^{\infty } \frac{\mathbf{B}_{p} (\chi +n,c-\chi )}{\varGamma (c-\chi )} \frac{\varGamma (c+n)}{\varGamma (\theta n+\kappa )} \\ &{}\times \frac{\varGamma (\varsigma +s+n)\varGamma (\varsigma +s+\tau -\varepsilon -\varLambda '+n)\varGamma (\varsigma +s+\tau +\tau '+\varepsilon '-\upsilon -\varLambda '+n)}{\varGamma (\varsigma +s-\varepsilon -\varLambda '+n)\varGamma (\varsigma +s+\tau +\tau '-\upsilon +n)\varGamma (\varsigma +s+\tau +\varepsilon '-\upsilon -\varLambda '+n)} \frac{x^{n}}{n!}. \end{aligned}$$

In view of (8), we obtain the intended result. □

Theorem 3.4

Let\(\tau ,\tau ',\varepsilon ,\varepsilon ', c,\theta ,\kappa , \chi , \upsilon ,\varsigma \in \mathbb{C}\)and\(\varLambda '=[\Re (\upsilon )+1]\)with

$$ \Re (\varsigma )+\varLambda '>\max \bigl\{ \Re \bigl(-\varepsilon '\bigr), \Re \bigl(\tau +\tau '-\upsilon \bigr)+ \varLambda ' \bigr\} $$

and\(p\geq 0\). Also let\(x\in \mathbb{R}^{+}\), then

(cDτ,τ,ε,ε,υtςSnm(t)Eθ,κχ,c(1/t;p))(x)=xς+τ+τυΓ(χ)s=0[n/m](n)m.ss!An,sxs×5ψ5[(c,1),(ε+ς+s+m,1),(ττ+υ+ς+s,1),(ς+sτε+υ+m,1),(χ,1);(c,1),(κ,θ),(ς+s,1),(τ+ε+m+ς+s,1),(ττε+ς+s+υ+m,1);(1/x;p)].
(42)

Proof

Let \(I_{6}\) be LHS of (42), then using (5), we have

$$\begin{aligned} I_{6} =& \bigl({}^{c}D_{-}^{\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon }t^{-\varsigma }S_{n}^{m} (t ) E^{\chi ,c}_{ \theta ,\kappa }(1/t;p) \bigr) (x) \\ =& \Biggl({}^{c}D_{-}^{\tau ,\tau ',\varepsilon ,\varepsilon ', \upsilon }t^{-\varsigma }\sum _{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} (t)^{s}\sum_{n=0}^{ \infty } \frac{\mathbf{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )}\frac{t^{-n}}{n!} \Biggr) (x). \end{aligned}$$

Interchanging the order of integration and summation under the verified condition in this theorem, we obtain

$$\begin{aligned} I_{6} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{ \infty } \frac{\mathbf{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa )n!} \bigl({}^{c}D_{-}^{\tau , \tau ',\varepsilon ,\varepsilon ',\upsilon }t^{s-n-\varsigma } \bigr) (x). \end{aligned}$$

Applying Lemma (3.2), we get

$$\begin{aligned} I_{6} =& \sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} \sum_{n=0}^{\infty } \frac{\mathbf{B}_{p} (\chi +n,c-\chi )}{\mathbf{B}(\chi ,c-\chi )} \frac{(c)_{n}}{\varGamma (\theta n+\kappa ){n!}} \\ &{}\times \frac{\varGamma (\varsigma -s+\varepsilon '+\varLambda '+n)\varGamma (\varsigma -s-\tau -\tau '+\upsilon +n)\varGamma (\varsigma -s-\tau '-\varepsilon +\upsilon +\varLambda '+n)}{\varGamma (\varsigma -s+n)\varGamma (\varsigma -s-\tau '+\varepsilon '+\varLambda '+n)\varGamma (\varsigma -s-\tau -\tau '-\varepsilon +\upsilon +\varLambda '+n)} \\ &{}\times x^{-\varsigma +s-n-\upsilon +\tau +\tau '} \\ =&\frac{x^{-\varsigma -\upsilon +\tau +\tau '}}{\varGamma (\chi )}\sum_{s=0}^{[n/m]} \frac{ (-n )_{m.s} }{s!} A_{n,s} x^{s} \sum _{n=0}^{ \infty }\frac{\mathbf{B}_{p} (\chi +n,c-\chi )}{\varGamma (c-\gamma )} \frac{\varGamma (c+n)}{\varGamma (\theta n+\kappa )} \\ &{}\times \frac{\varGamma (\varsigma -s+\varepsilon '+\varLambda '+n)\varGamma (\varsigma -s-\tau -\tau '+\upsilon +n)\varGamma (\varsigma -s-\tau '-\varepsilon +\upsilon +\varLambda '+n)}{\varGamma (\varsigma -s+n)\varGamma (\varsigma -s-\tau '+\varepsilon '+\varLambda '+n)\varGamma (\varsigma -s-\tau -\tau '-\varepsilon +\upsilon +\varLambda '+n)}\\ &{}\times \frac{x^{-n}}{n!}. \end{aligned}$$

By using (8), we get the required result. □

Concluding remarks and discussion

In the current work, we have established the fractional calculus operators with Appell function kernels and Caputo-type fractional differential operators for the function involving the product of Srivastava’s polynomials and extended Mittag-Leffler function. Various special cases of the derived results in the paper can be evaluated by taking suitable values of parameters involved. Further, one can obtain a number of image formulas involving classical orthogonal polynomials as particular cases of our results, by giving special values to the coefficient \(A_{n,l} \), in the family of polynomials, which includes the polynomials viz. Laguerre, Hermite, Jacobi, Konhauser polynomials, and many others.

For example, if we put \(n=0\), then we see that the general class of polynomials \(S_{n}^{m} (x)\) reduces to unity, i.e., \(S_{0}^{m} (x) \to 1\), and we immediately obtain the result due to Araci et al. [2].

Also, if we set \(m=2\) and \(A_{n,l} =(-1)^{l} \), then the general class of polynomials stated in equation (9) reduces to

$$ S_{n}^{2} (x) \to x^{\frac{n}{2} } H_{n} \biggl(\frac{1}{2\sqrt{x} } \biggr), $$
(43)

where \(H_{n} (x)\) denotes the familiar Hermite polynomials defined as

$$ H_{n} (x)=\sum_{l=0}^{ \lfloor n/m \rfloor } (-1 )^{l} \frac{n!}{l!(n-2l)!} (2x )^{n-2l} . $$
(44)

Eventually, it is easy to discover a comprehensive representation of more generalized special functions that are widely used in applied sciences.

References

  1. Ahokposi, D.P., Atangana, A., Vermeulen, D.P.: Modelling groundwater fractal flow with fractional differentiation via Mittag-Leffler law. Eur. Phys. J. Plus 132, 165–175 (2017)

    Article  Google Scholar 

  2. Araci, S., Rahman, G., Ghaffar, A., Azeema, N.K.S.: Fractional calculus of extended Mittag-Leffler function and its applications to statistical distribution. Mathematics 7, 248 (2019). https://doi.org/10.3390/math7030248

    Article  Google Scholar 

  3. Atangana, A.: Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals 102, 396–406 (2017)

    MathSciNet  Article  Google Scholar 

  4. Baleanu, D., Kumar, D., Purohit, S.D.: Generalized fractional integrals of product of two H-functions and a general class of polynomials. Int. J. Comput. Math. 93, 1320–1329 (2016)

    MathSciNet  Article  Google Scholar 

  5. Brouers, F.: The fractal (BSf) kinetics equation and its approximations. J. Mod. Phys. 5, 1594–1601 (2014)

    Article  Google Scholar 

  6. Brouers, F., Sotolongo-Costa, O.: Generalized fractal kinetics in complex systems (application to biophysics and biotechnology). Phys. A, Stat. Mech. Appl. 368, 165–175 (2006)

    Article  Google Scholar 

  7. Chaudhry, M.A., Qadir, A., Rafique, M., Zubair, S.M.: Extension of Euler’s beta function. J. Comput. Appl. Math. 78, 19–32 (1997)

    MathSciNet  Article  Google Scholar 

  8. Chen, W., Liang, Y.: New methodologies in fractional and fractal derivatives modeling. Chaos Solitons Fractals 102, 72–77 (2017)

    MathSciNet  Article  Google Scholar 

  9. Chouhan, A., Khan, A.M., Saraswat, S.: A note on Marichev–Saigo–Maeda fractional integral operator. J. Fract. Calc. Appl. 5, 88–95 (2014)

    MathSciNet  Google Scholar 

  10. Dorrego, G.A., Cerutti, R.A.: The k-Mittag-Leffler function. Int. J. Contemp. Math. Sci. 7, 705–716 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Ghanbari, B., Kumar, S., Kumar, R.: A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative. Chaos Solitons Fractals 133, 109619 (2020). https://doi.org/10.1016/j.chaos.2020.109619

    MathSciNet  Article  Google Scholar 

  12. Gorenflo, R., Kilbas, A.A., Rogosin, S.V.: On the generalized Mittag-Leffler type functions. Integral Transforms Spec. Funct. 7, 215–224 (1998)

    MathSciNet  Article  Google Scholar 

  13. Gorenflo, R., Mainardi, F., Srivastava, H.M.: Special functions in fractional relaxation oscillation and fractional diffusion-wave phenomena. In: Proceedings of the Eighth International Colloquium on Differential Equations, pp. 195–202. VSP Publishers, London (1998)

    Google Scholar 

  14. Jleli, M., Kumar, S., Kumar, R., Samet, B.: Analytical approach for time fractional wave equations in the sense of Yang–Abdel–Aty–Cattani via the homotopy perturbation transform method. Alex. Eng. J. (2019). https://doi.org/10.1016/j.aej.2019.12.022

    Article  Google Scholar 

  15. Kataria, K.K., Vellaisamy, P.: The generalized k-Wright function and Marichev–Saigo–Maeda fractional operators. J. Anal. 23, 75–87 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Kilbas, A.A., Sebastian, N.: Generalized fractional integration of Bessel function of the first kind. Integral Transforms Spec. Funct. 19, 869–883 (2008)

    MathSciNet  Article  Google Scholar 

  17. Kosmidis, K., Macheras, P.: On the dilemma of fractal or fractional kinetics in drug release studies: a comparison between Weibull and Mittag-Leffler functions. Int. J. Pharm. 43, 269–273 (2018)

    Article  Google Scholar 

  18. Kumar, D., Purohit, S.D., Choi, J.: Generalized fractional integrals involving product of multivariable H-function and a general class of polynomials. J. Nonlinear Sci. Appl. 9, 8–21 (2016)

    MathSciNet  Article  Google Scholar 

  19. Kumar, S., Kumar, A., Abbas, S., Qurashi, M.A., Baleanu, D.: A modified analytical approach with existence and uniqueness for fractional Cauchy reaction–diffusion equations. Adv. Differ. Equ. (2020). https://doi.org/10.1186/s13662-019-2488-3

    MathSciNet  Article  Google Scholar 

  20. Kumar, S., Kumar, R., Agarwal, R.P., Samet, B.: A study on fractional Lotka Volterra population model by using Haar wavelet and Adams Bashforth–Moulton methods. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6297

    Article  Google Scholar 

  21. Kumar, S., Kumar, R., Singh, J., Nisar, K.S., Kumar, D.: An efficient numerical scheme for fractional model of HIV-1 infection of CD4+ T-cells with the effect of antiviral drug therapy. Alex. Eng. J. (2019). https://doi.org/10.1016/j.aej.2019.12.046

    Article  Google Scholar 

  22. Kumar, S., Nisar, K.S., Kumar, R., Cattani, C., Samet, B.: A new Rabotnov fractional-exponential function based fractional derivative for diffusion equation under external force. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6208

    Article  Google Scholar 

  23. Marichev, O.I.: Volterra equation of Mellin convolution type with a horn function in the kernel. Vescì Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk 1, 128–129 (1974)

    Google Scholar 

  24. Meilanov, R.P., Yanpolov, M.S.: Features of the phase trajectory of a fractal oscillator. Tech. Phys. Lett. 28, 30–32 (2002)

    Article  Google Scholar 

  25. Misra, V.N., Suthar, D.L., Purohit, S.D.: Marichev–Saigo–Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function. Cogent Math. 4, 1320830 (2017). https://doi.org/10.1080/23311835.2017.1320830

    MathSciNet  Article  MATH  Google Scholar 

  26. Mittag-Leffler, G.M.: Sur la nouvelle fonction \(E_{\alpha } ( x )\). C. R. Acad. Sci. Paris 137, 554–558 (1903)

    MATH  Google Scholar 

  27. Mondal, S.R., Nisar, K.S.: Marichev–Saigo–Maeda fractional integration operators involving generalized Bessel functions. Math. Probl. Eng. 2014, 274093 (2014)

    MathSciNet  Article  Google Scholar 

  28. Nisar, K.S., Eata, A.F., Dhaifallah, M.D., Choi, J.: Fractional calculus of generalized k-Mittag-Leffler function and its applications to statistical distribution. Adv. Differ. Equ. (2016). https://doi.org/10.1186/s13662-016-1029-6

    MathSciNet  Article  MATH  Google Scholar 

  29. Özarslan, M.A., Yilmaz, B.: The extended Mittag-Leffler function and its properties. J. Inequal. Appl. 2014, 85 (2014)

    MathSciNet  Article  Google Scholar 

  30. Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)

    MathSciNet  MATH  Google Scholar 

  31. Purohit, S.D., Suthar, D.L., Kalla, S.L.: Marichev–Saigo–Maeda fractional integration operators of the Bessel function. Matematiche 67, 21–32 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Rahman, G., Agarwal, P., Mubeen, S., Arshad, M.: Fractional integral operators involving extended Mittag-Leffler function as its kernel. Bol. Soc. Mat. Mex. 24, 381–392 (2017)

    MathSciNet  Article  Google Scholar 

  33. Rahman, G., Baleanu, D., Al-Qurashi, M., Purohit, S.D., Mubeen, S., Arshad, M.: The extended Mittag-Leffler function via fractional calculus. J. Nonlinear Sci. Appl. 10, 4244–4253 (2017)

    MathSciNet  Article  Google Scholar 

  34. Saigo, M., Maeda, N.: More generalization of fractional calculus. In: Transform Methods & Special Functions, vol. 96, pp. 386–400. Bulgarian Academy of Sciences, Sofia (1998)

    Google Scholar 

  35. Saxena, R.K., Ram, J., Suthar, D.L.: Generalized fractional calculus of the generalized Mittag-Leffler functions. J. Indian Acad. Math. 31(1), 165–172 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Sharma, B., Kumar, S., Cattani, C., Baleanu, D.: Nonlinear dynamics of Cattaneo–Christov heat flux model for third-grade power-law fluid. J. Comput. Nonlinear Dyn. (2019). https://doi.org/10.1115/1.4045406

    Article  Google Scholar 

  37. Sharma, S.C., Devi, M.: Certain properties of extended Wright generalized hypergeometric function. Ann. Pure Appl. Math. 9, 45–51 (2015)

    Google Scholar 

  38. Srivastava, H.M.: On an extension of the Mittag-Leffler function. Yokohama Math. J. 16, 77–88 (1968)

    MathSciNet  MATH  Google Scholar 

  39. Srivastava, H.M., Choi, J.: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)

    MATH  Google Scholar 

  40. Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted, Chichester (1984)

    MATH  Google Scholar 

  41. Suthar, D.L., Amsalu, H.: Generalized fractional integral operators involving Mittag-Leffler function. Appl. Appl. Math. 12(2), 1002–1016 (2017)

    MathSciNet  MATH  Google Scholar 

  42. Suthar, D.L., Habenom, H., Tadesse, H.: Generalized fractional calculus formulas for a product of Mittag-Leffler function and multivariable polynomials. Int. J. Appl. Comput. Math. 4(1), 1–12 (2018)

    MathSciNet  Article  Google Scholar 

  43. Suthar, D.L., Purohit, S.D.: Unified fractional integral formulae for the generalized Mittag-Leffler functions. J. Sci. Arts 27(2), 117–124 (2014)

    MathSciNet  Google Scholar 

  44. Wiman, A.: Über den fundamentalsatz in der theorie der funktionen \(E_{\alpha }(x)\). Acta Math. 29, 191–201 (1905)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Not applicable.

Availability of data and materials

Not applicable.

Funding

Not applicable.

Author information

Authors and Affiliations

Authors

Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Kottakkaran Sooppy Nisar.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nisar, K.S., Suthar, D.L., Agarwal, R. et al. Fractional calculus operators with Appell function kernels applied to Srivastava polynomials and extended Mittag-Leffler function. Adv Differ Equ 2020, 148 (2020). https://doi.org/10.1186/s13662-020-02610-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-020-02610-3

MSC

  • 26A33
  • 33B15
  • 33C05
  • 33C99
  • 44A10

Keywords

  • Extended Mittag-Leffler function
  • Srivastava polynomial
  • Wright-type hypergeometric functions
  • Extended Wright-type hypergeometric functions