Fractionalcalculusofgeneralized -Mittag … · 2017. 8. 23. · Nisaretal.AdvancesinDiﬀerenceEquations(2016)2016:304 DOI10.1186/s13662-016-1029-6 RESEARCH OpenAccess Fractionalcalculusofgeneralized

Nisar et al. Advances in Difference Equations (2016) 2016:304 DOI 10.1186/s13662-016-1029-6

R E S E A R C H Open Access

Fractional calculus of generalizedk-Mittag-Leffler function and its applicationsto statistical distributionKottakkaran Sooppy Nisar1, Ashraf Fetoh Eata2,3, Mujahed Al-Dhaifallah4 and Junesang Choi5*

*Correspondence:[email protected] of Mathematics,Dongguk University, Gyeongju,38066, Republic of KoreaFull list of author information isavailable at the end of the article

AbstractWe aim to investigate the MSM-fractional calculus operators, Caputo-typeMSM-fractional differential operator, and pathway fractional integral operator of thegeneralized k-Mittag-Leffler function. We also investigate certain statisticaldistribution associated with the generalized k-Mittag-Leffler function. Certainparticular cases of the derived results are considered and indicated to further reduceto some known results.

MSC: 33C10; 44A10; 44A20; 33E12

Keywords: gamma function; generalized k-Mittag-Leffler functions; statisticaldistribution

1 Introduction and preliminariesThroughout this paper, let C, R, R+, Z–

, and N be the sets of complex numbers, real num-bers, positive real numbers, nonpositive integers, and positive integers, respectively, andlet R+

:= R+ ∪ {}.

Díaz and Pariguan [] found that the expression

(x)n,k := x(x + k)(x + k) · · · (x + (n – )k)

(.)

has appeared repeatedly in a variety of contexts such as combinatorics of creation, anni-hilation operators, and perturbative computation of Feynman integrals. Motivated by thisobservation, they [] used the Gauss form of the gamma function (see [], Eq. (), p.) tointroduce the so-called k-gamma function

�k(z) = limn→∞

n!kn(nk)zk –

(z)n,k

(k ∈R

+; z ∈ C \ kZ–). (.)

Starting from this definition, they [] presented a number of properties for the k-gammafunction. We recall some of them:

�k(z + k) = z�k(z) and �k(k) = ; (.)

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the Euler integral form:

�k(z) =∫ ∞

tz–e– tk

k dt(k ∈R

+;�(z) > ); (.)

the k-Pochhammer symbol (λ)n,k defined (for λ,ν ∈C; k ∈R) by

(λ)ν,k :=�k(λ + νk)

�k(λ)(λ ∈C \ {})

=

⎧⎨

⎩ (ν = ),

λ(λ + k) · · · (λ + (n – )k) (ν = n ∈N);(.)

from (.), it is easy to find the following relationship between the gamma function � andthe k-gamma function �k :

�k(z) = kzk –�

(zk

). (.)

In a number of subsequent works including [], the k-gamma function andk-Pochhammer symbol have been used to extend and investigate such special func-tions and integral operators as (for example) the k-beta function, k-zeta function,k-hypergeometric function, k-Mittag-Letter functions, k-Wright function, andk-analogue of the Riemann-Liouvile fractional integral operator.

In , the Swedish mathematician Gosta Mittag-Leffler [] (see also []) introducedand investigated the so-called Mittag-Leffler function

Eα(z) :=∞∑

n=

zn

�(αn + )(z ∈C;α ∈R

+). (.)

Since then, the Mittag-Leffler function Eα (.) has been extended in a number of waysand, together with its extensions, applied in various research areas such as engineeringand (in particular) statistics. The Mittag-Leffler functions and related distributions weregiven in []. Further, Mathai and Haubold [] established connections among generalizedMittag-Leffler functions, pathway model, Tsallis statistics, superstatistics and power law,and the corresponding entropy measures. A statistical perspective of Mittag-Leffler func-tions and matrix-variate analogues was given by Mathai, who presented the involved re-sults in terms of statistical densities, which are useful in statistical distribution theory andstochastic processes. Also, various pathways were investigated from the exponential andgamma densities to the Mittag-Leffler densities and then from the Mittag-Leffler densitiesto the Lèvi and Linnik densities []. Lin [] proved that the Mittag-Leffler distributions be-long to the class of distributions with completely monotone derivatives. The fundamentalproperties of the Mittag-Leffler distributions and their extensions, including the tail be-havior of distribution and explicit expressions for moments of all orders and for the densityfunctions, are also given.

Here, for an easier reference, we give a brief history of some chosen extensions of theMittag-Leffler function Eα (.). Wiman [] presented the following generalization Eα,β


of Eα :

Eα,β (z) :=∞∑

n=

zn

�(αn + β)(α,β ∈C; min

{�(α),�(β)}

> ). (.)

Prabhakar [] introduced the function Eγ

α,β in the following form:

Eγ

α,β (z) =∞∑

n=

(γ )n

n!�(αn + β)zn (

α,β ,γ ∈ C; min{�(α),�(β),�(γ )

}>

). (.)

Another generalization of the Mittag-Leffler function Eα was given by Shukla and Prajap-ati []:

Eγ ,qα,β (z) =

∞∑

n=

(γ )qn

n!�(αn + β)zn

(α,β ,γ ∈ C; min

{�(α),�(β),�(γ )}

> ; q ∈ (, ) ∪N). (.)

We also recall the following two extensions of the Mittag-Leffler function (see [],Eqs. (.) and (.)):

Eη,δ,qα,β ,p(z) =

∞∑

n=

(η)qn

�(αn + β)(δ)pnzn

(p, q ∈R

+; min{�(α),�(β),�(η),�(δ)

}>

); (.)

Eμ,ρ,η,qα,β ,ν,σ ,δ,p(z) =

∞∑

n=

(μ)ρn(η)qn

�(αn + β)(ν)σn(δ)pnzn

(p, q ∈R

+; q ≤ �(α) + p;

min{�(α),�(β),�(η),�(δ),�(μ),�(ν),�(ρ),�(σ )

}>

). (.)

For more generalizations of the Mittag-Leffler functions, we refer the reader, for exam-ple, to [–, ].

The Fox-Wright hypergeometric function p�q(z) is given by the series

p�q(z) = p�q

[(ai,αi),p

(bj,βj),q

∣∣∣∣z

]

=∏q

j= �(βj)∏p

i= �(αi)

∞∑

k=

∏pi= �(ai + αik)

∏qj= �(bj + βjk)

zk

k!, (.)

where ai, bj ∈ C and αi,βj ∈ R (i = , , . . . , p; j = , , . . . , q). Asymptotic behavior of thisfunction for large values of the argument of z ∈C was studied in [], and under the con-dition

q∑

j=

βj –p∑

i=

αi > –, (.)

was found in [, ]. Properties of this generalized Wright function were investigatedin [] (see also [, ]).


The generalized hypergeometric function pFq is defined as follows []:

pFq

[(ap);(bq);

z

]

=∞∑

n=

∏pj=(aj)n

∏qj=(bj)n

zn

n!(p ≤ q, z ∈C; p = q + , |z| <

), (.)

which obviously is a particular case of the Fox-Wright hypergeometric function p�q(z)(.) when αi = = βj (i = , , . . . , p; j = , , . . . , q).

Let λ, λ′, ξ , ξ ′, γ ∈C with �(γ ) > and x ∈R+. Then the generalized fractional integral

operators involving the Appell functions F are defined as follows:

(Iλ,λ′ ,ξ ,ξ ′ ,γ

,+ f)(x) =

x–λ

�(γ )

∫ x

(x – t)γ –t–λ′

F

(λ,λ′, ξ , ξ ′;γ ; –

tx

, –xt

)f (t) dt (.)

and

(Iλ,λ′ ,ξ ,ξ ′ ,γ

– f)(x) =

x–λ′

�(γ )

∫ ∞

x(t – x)γ –t–λF

(λ,λ′, ξ , ξ ′;γ ; –

tx

, –xt

)f (t) dt. (.)

The generalized fractional integral operators of types (.) and (.) have been intro-duced by Marichev [] and later extended and studied by Saigo and Maeda []. Theseoperators are known as the Marichev-Saigo-Maeda operators (MSM-operators). Recently,Mondal and Nisar [] have investigated the Marichev-Saigo-Maeda fractional integraloperators involving generalized Bessel functions (see also []).

The corresponding fractional differential operators have their respective forms:

(Dλ,λ′ ,ξ ,ξ ′,γ

+ f)(x) =

(d

dx

)[�(γ )]+(I–λ′ ,–λ,–ξ ′+[�(γ )]+,–ξ ,–γ +[�(γ )]+

+ f)(x) (.)

and

(Dλ,λ′ ,ξ ,ξ ′,γ

– f)(x) =

(–

ddx

)[�(γ )]+(I–λ′ ,–λ,–ξ ′,–ξ+[�(γ )]+,–γ +[�(γ )]+

– f)(x). (.)

The fractional integral operators have many interesting applications in various fields in-cluding (for example) a certain class of complex analytic functions (see []). For somebasic results on fractional calculus, we refer to [–, ].

The following four results will be required (for the first and second, see [, ]; for thethird and fourth, see []).

Lemma . Let λ, λ′, ξ , ξ ′, γ , ρ ∈ C be such that �(γ ) > and

�(ρ) > max{

,�(λ + λ

′+ ξ – γ

),�(

λ′ – ξ ′)}.

Then

(Iλ,λ′ ,ξ ,ξ ′ ,γ

,+ tρ–)(x)

=�(ρ)�(ρ + γ – λ – λ′ – ξ )�(ρ + ξ ′ – λ′)

�(ρ + ξ ′)�(ρ + γ – λ – λ′)�(ρ + γ – λ′ – ξ )xρ–λ–λ′+γ –. (.)


Lemma . Let λ, λ′, ξ , ξ ′, γ , ρ ∈C be such that �(γ ) > and

�(ρ) > max{�(ξ ),�(

–λ – λ′ + γ),�(

–λ – ξ ′ + γ)}

.

Then

(Iλ,λ′ ,ξ ,ξ ′ ,γ

– t–ρ)(x)

=�(–ξ + ρ)�(λ + λ′ – γ + ρ)�(λ + ξ ′ – γ + ρ)

�(ρ)�(λ – ξ + ρ)�(λ + λ′ + ξ ′ – γ + ρ)x–λ–λ′+γ –ρ . (.)

Lemma . Let λ, δ, γ , ρ ∈C with �(λ) > and �(ρ) > max{,�(δ – γ )}. Then

(Iλ,δ,γ

+ tρ–)(x) =�(ρ)�(ρ + γ – δ)

�(ρ – δ)�(ρ + λ + γ )xρ–δ–. (.)

In particular,

(Iλ,γ

+ tρ–)(x) =�(ρ + γ )

�(ρ + λ + γ )xρ– (�(λ) > ,�(ρ) > max

{, –�(γ )

})(.)

and

(Iλ

+tρ–)(x) =�(ρ)

�(ρ + λ)xρ– (

min{�(λ),�(ρ)

}>

). (.)

Lemma . Let λ, δ, γ , ρ ∈C with �(λ) > and �(ρ) < + min{�(δ),�(γ )}. Then

(Iλ,δ,γ

– tρ–)(x) =�(δ – ρ + )�(γ – ρ + )

�( – ρ)�(λ + δ + γ – ρ + )xρ–δ–. (.)

In particular,

(Iλ,γ

– tρ–)(x) =�(γ – ρ + )

�(λ + γ – ρ + )xρ– (�(λ) > ,�(ρ) < + �(γ )

)(.)

and

(Iλ

–tρ–)(x) =�( – ρ)

�(λ – ρ + )xρ– (�(λ) > ,�(ρ) <

). (.)

As mentioned before, k-extensions of the Mittag-Leffler functions have been given andinvestigated particularly in view of statistics. In this paper, we aim to investigate the MSM-fractional calculus operators, Caputo-type MSM-fractional differential operator, and thatpathway fractional integral operator of the generalized k-Mittag-Leffler function (.). Wealso investigate certain statistical distribution associated with the generalized k-Mittag-Leffler function (.), in which certain particular cases of the derived results are consideredand indicated to further reduce to some known results.


2 k-Mittag-Leffler functionsHere we introduce k-Mittag-Leffler functions and their extensions. The simplest k-extensions of the Mittag-Leffler functions (.) and (.) can be given by

Ek,α(z) :=∞∑

n=

zn

�k(αn + )(k ∈R

+;α ∈R+)

(.)

and

Ek,α,β(z) :=∞∑

n=

zn

�k(αn + β)(k ∈R

+; min{�(α),�(β)

}>

), (.)

respectively (see, e.g., [], Eq. ()). Dorrego and Cerutti [] introduced the k-Mittag-Leffler function

Eη

k,α,β(z) :=∞∑

n=o

(η)n,k

�k(αn + β)zn

n!(k ∈R

+;α,β ,η ∈C; min{�(α),�(β)

}>

)(.)

and investigated some properties associated with the definition itself and the Riemann-Liouville fractional calculus operators. Saxena et al. [] extended the k-Mittag-Lefflerfunction (.) slightly as follows:

Eη,τk,α,β(z) :=

∞∑

n=o

(η)nτ ,k

�k(αn + β)zn

n!(k ∈R

+;α,β ,η, τ ∈C; min{�(α),�(β)

}>

). (.)

They derived its Euler transform, Laplace transform, Whittaker transform, and fractionalFourier transform of order α ( < α ≤ ). Daiya and Ram [] investigated the statisticaldensity of the k-Mittag-Leffler function (.). Gupta and Parihar [] defined a furtherextension of the k-Mittag-Leffler functions,

Eη,δ,qk,α,β ,p(z) :=

∞∑

n=

(η)qn,k

�k(αn + β)(δ)pn.kzn

(k, p, q ∈R

+;α,β ,η, δ ∈C; min{�(α),�(β),�(η),�(δ)

}> ; q ≤ �(α) + p

), (.)

presented its properties, including differentiation, the fractional Fourier transform,Laplace transform, and k-Beta transform, and determined the k-Riemann-Liouville frac-tional integral and differentiation.

3 MSM fractional integral representations of (2.5)Here we present MSM fractional integral representations of the generalized k-Mittag-Leffler function (.) and consider some particular cases.


Theorem Let λ, λ′, ξ , ξ ′, γ , ρ , α, β , η, δ ∈C with �(γ ) > and

�(ρ) > max{

,�(λ + λ′ + ξ – γ

),�(

λ′ – ξ ′)}.

Also, let k, p, q, x ∈R+. Then

(Iλ,λ′ ,ξ ,ξ ′ ,γ

,+ tρ–Eη,δ,qk,α,β ,p(t)

)(x) =

�(δ/k)�(η/k)

xρ–λ–λ′+γ –

kβk –

× �

[( η

k , q), (ρ, ), (ρ + γ – λ – λ′ – ξ , ),( β

k , αk ), ( δ

k , p), (ρ + ξ ′, ), (ρ + γ – λ – λ′, ),

(ρ + ξ ′ – λ′, ), (, );(ρ + γ – λ′ – ξ , );

kq–p– αk x

]. (.)

Proof Let L be the left-hand side of (.). Then, using (.), we have

L =

(

Iλ,λ′ ,ξ ,ξ ′ ,γ,+ tρ–

∞∑

n=

(η)nq,k

�k(αn + β)tn

(δ)pn,k

)

(x).

Interchanging the summation and integration, which is verified under the conditions inthis theorem, we get

L =∞∑

n=

(η)nq,k

�k(αn + β)(δ)pn,k

(Iλ,λ′ ,ξ ,ξ ′,γ

,+ tρ+n–)(x).

Applying Lemma ., we obtain

L =∞∑

n=

(η)nq,k


× �(ρ + n)�(ρ + γ – λ – λ′ – ξ + n)�(ρ + ξ ′ – λ′ + n)�(ρ + ξ ′ + n)�(ρ + γ – λ – λ′ + n)�(ρ + γ – λ′ – ξ + n)

× xρ+n–λ–λ′+γ –.

Now, using relations (.) and (.), we get

L = xρ–λ–λ′+γ –

×∞∑

n=

knq�( η

k + nq)�( δk )

�( η

k )kαn+β

k –�( αn+β

k )�( δk + np)knp

× �(ρ + n)�(ρ + γ – λ – λ′ – ξ + n)�(ρ + ξ ′ – λ′ + n)�(n + )�(ρ + ξ ′ + n)�(ρ + γ – λ – λ′ + n)�(ρ + γ – λ′ – ξ + n)

xn

n!,

which, in view of (.), leads to the right-hand side of (.). This completes the proof. �

Corollary . Let λ, ξ , γ , ρ , α, β , η, δ ∈C with �(γ ) > and

�(ρ) > max{

,�(ξ – γ )}

.


Also, let k, p, q, x ∈R+. Then

(Iλ,ξ ,γ

,+ tρ–Eη,δ,qk,α,β ,p(t)

)(x)

=�(δ/k)�(η/k)

xρ–ξ–

kβk –

× �

[( η

k , q), (ρ + γ – ξ , ), (ρ, ), (, );( β

k , αk ), ( δ

k , p), (ρ – ξ , ), (ρ + γ + λ, );kq–p– α

k x]

. (.)

Theorem Let λ, λ′, ξ , ξ ′, γ , ρ , α, β , η, δ ∈C be such that �(λ) > and

�(ρ) > max{�(ξ ),�(

–λ – λ′ + γ),�(

–λ – ξ ′ + γ)}

.

Also, let k, p, q ∈R+. Then

(Iλ,λ′ ,ξ ,ξ ′ ,γ

– tρ–Eη,δ,qk,α,β ,p(t)

)(x) =

�(δ/k)�(η/k)

x–λ–λ′+γ –ρ

kβk –

× �

[( η

k , q), (–ξ + ρ, ), (λ + λ′ – γ + ρ, ),( β

k , αk ), ( δ

k , p), (ρ, ), (λ – ξ + ρ, ),

(λ + ξ ′ – γ + ρ, ), (, );(λ + λ′ + ξ ′ – γ + ρ, );

kq–p– αk x

]. (.)

Proof We can establish (.) by a similar argument as in the proof of (.), using Lemma .instead of Lemma .. We omit the details. �

Corollary . Let λ, ξ , γ , ρ , α, β , η, δ ∈C be such that �(λ) > and

�(ρ) > max{�(–ξ ),�(–γ )

}.


(Iλ,ξ ,γ

– tρ–Eη,δ,qk,α,β ,p(t)

)(x)

=�(δ/k)�(η/k)

xρ–ξ–

kβk –

× �

[( η

k , q), (ξ – ρ + , ), (γ – ρ + , ), (, );( β

k , αk ), ( δ

k , p), ( – ρ, ), (λ + ξ + γ – ρ + , );kq–p– α

k x]

. (.)

4 MSM-fractional differential operator of (2.5)Here we derive the Marichev-Saigo-Maeda fractional differentiation of the generalizedk-Mittag-Leffler function (.). The following lemmas will be required (see []).

Lemma . Let λ, λ′, ξ , ξ ′, γ , ρ ∈C be such that

�(ρ) > max{

,�(–λ + ξ ),�(–λ – λ′ – ξ ′ + γ

)}.


Then

(Dλ,λ′ ,ξ ,ξ ′,γ

+ tρ–)(x) =�(ρ)�(–ξ + λ + ρ)�(λ + λ′ + ξ ′ – γ + ρ)

�(–ξ + ρ)�(λ + λ′ – γ + ρ)�(λ + ξ ′ – γ + ρ)xλ+λ′–γ +ρ–. (.)

Lemma . Let λ, λ′, ξ , ξ ′, γ , ρ ∈C be such that

�(ρ) > max{�(

–ξ ′),�(λ′ + ξ – γ

),�(

λ + λ′ – γ)

+[�(γ )

]+

}.

Then

(Dλ,λ′ ,ξ ,ξ ′,γ

– t–ρ)(x) =

�(ξ ′ + ρ)�(–λ – λ′ + γ + ρ)�(–λ′ – ξ + γ + ρ)�(ρ)�(–λ′ + ξ ′ + ρ)�(–λ – λ′ – ξ + γ + ρ)

xλ+λ′–γ –ρ . (.)

Theorem Let λ, λ′, ξ , ξ ′, γ , ρ , α, β , η, δ ∈C be such that

�(ρ) > max{

,�(–λ + ξ ),�(–λ – λ′ – ξ ′ + γ

)}.


(Dλ,λ′ ,ξ ,ξ ′,γ

+ tρ–Eη,δ,qk,α,β ,p(t)

)(x) =

�(δ/k)�(η/k)

xλ+λ′–γ +ρ–

kβk –

× �

[( η

k , q), (ρ, ), (–ξ + λ + ρ, ),( β

k , αk ), ( δ

k , p), (–ξ + ρ, ), (λ + λ′ – γ + ρ, ),

(λ + λ′ + ξ ′ – γ + ρ, ), (, );(λ + ξ ′ – γ + ρ, );

kq–p– αk x

]. (.)

Proof Let L be the left-hand side of (.). Taking the MSM differential operator on (.)and interchanging the differentiation and summation, which is verified under the condi-tions in this theorem, we have

L =∞∑

n=

(η)nq,k


(Dλ,λ′ ,ξ ,ξ ′,γ

+ tρ+n–)(x).

Using (.), (.), and Lemma ., we get

L =∞∑

n=

knq�( η

k + nq)�( δk )

�( η

k )kαn+β

k –�( αn+β


× �(ρ + n)�(–ξ + λ – ρ + n)�(λ + λ′ + ξ ′ – γ + ρ + n)�(–ξ – ρ + n)�(λ + λ′ – γ + ρ + n)�(λ + ξ ′ – γ + ρ + n)

× xλ+λ′–γ +ρ+n–,

which, in view of (.), is equal to the right-hand side of (.). �

Theorem Let λ, λ′, ξ , ξ ′, γ , ρ , α, β , η, δ ∈C be such that

�(ρ) > max{�(

–ξ ′),�(λ′ + ξ – γ

),�(

λ + λ′ – γ)

+[�(γ )

]+

}.



(Dλ,λ′ ,ξ ,ξ ′,γ

– t–ρEη,δ,qk,α,β ,p(t)

)(x) =

xλ+λ′–γ –ρ

kβk –

�(δ/k)�(η/k)

× �

[( η

k , q), (ξ ′ + ρ, ), (–λ – λ′ + γ + ρ, ),( β

k , αk ), ( δ

k , p), (ρ, ), (–λ′ + ξ ′ + ρ, ),

(–λ – ξ + γ + ρ, ), (, );(–λ – λ′ – ξ + γ + ρ, );

kq–p– αk x

]. (.)

Proof The proof runs parallel to that of Theorem , using Lemma . instead of Lem-ma .. We omit the details. �

Remark . The results in Theorems to can be easily reduced to yield some cor-responding formulas involving simpler factional calculus operators such as the Erdélyi-Kober fractional calculus operators.

5 Caputo-type MSM fractional differentiation of (2.5)Rao et al. [] introduced the Caputo-type fractional derivatives that have the Gauss hy-pergeometric function in the kernel. The left- and right-hand sided Caputo fractional dif-ferential operators associated with the Gauss hypergeometric function are defined, re-spectively, by

(cDλ,ξ ,γ+ f

)(x) =

(I–λ+[Re(λ)]+,–ξ–[Re(λ)]–,λ+γ –[Re(λ)]–

+ f ([Re(λ)]+))(x)

and

(cDλ,ξ ,γ– f

)(x) = (–)[Re(λ)]+(I–λ+[Re(λ)]+,–ξ–[Re(λ)]–,λ+γ

– f ([Re(λ)]+))(x),

where λ, ξ , γ ∈C with �(λ) > and x ∈R+.

The left- and right-hand sided Caputo-type MSM fractional differential operators asso-ciated with the Appell function F are defined, respectively, by

(cDλ,λ′ ,ξ ,ξ ′ ,γ+ f

)(x) =

(I–λ′ ,–λ,–ξ ′+[�(γ )]+,–ξ ,–γ +[�(γ )]+

+ f ([�(γ )]+))(x)

and

(cDλ,λ′ ,ξ ,ξ ′ ,γ– f

)(x) = (–)[�(γ )]+(I–λ′ ,–λ,–ξ ′,–ξ+[�(γ )]+,–γ +[Re(γ )]+

– f ([�(λ)]+))(x),

where λ, λ′, ξ , ξ ′, γ , ρ ∈ C with �(γ ) > and x ∈R+.

In this section, we investigate the Caputo-type MSM fractional differential operator ofthe generalized k-Mittag-Leffler function (.). The following lemmas will be required(see []).

Lemma . Let λ, λ′, ξ , ξ ′, γ , ρ ∈ C and m = [�(γ )] + with

�(ρ) – m > max{

,�(–λ + ξ ),�(–λ – λ′ – ξ ′ + γ

)}.


Then

(cDλ,λ′ ,ξ ,ξ ′ ,γ+ tρ–)(x) =

�(ρ)�(λ – ξ + ρ – m)�(λ + λ′ + ξ ′ – γ + ρ – m)�(–ξ + ρ – m)�(λ + λ′ – γ + ρ)�(λ + ξ ′ – γ + ρ – m)

× xλ+λ′–γ +ρ+. (.)

Lemma . Let λ, λ′, ξ , ξ ′, γ , ρ ∈ C and m = [�(γ )] + with

�(ρ) + m > max{�(

–ξ ′),�(λ′ + ξ – γ

),�(

λ + λ′ – γ)

+[�(γ )

]+

}.

Then

(cDλ,λ′ ,ξ ,ξ ′ ,γ– t–ρ

)(x) =

�(ξ ′ + ρ + m)�(–λ – λ′ + γ + ρ)�(–λ′ – ξ + γ + ρ + m)�(ρ)�(–λ′ + ξ ′ + ρ + m)�(–λ – λ′ – ξ + γ + ρ + m)

× xλ+λ′–γ –ρ . (.)

Theorem Let λ, λ′, ξ , ξ ′, γ , ρ , α, β , γ , η, δ ∈C and m = [�(γ )] + with

�(ρ) – m > max{

,�(–λ + ξ ),�(–λ – λ′ – ξ ′ + γ

)}.


(cDλ,λ′ ,ξ ,ξ ′ ,γ+ tρ–Eη,δ,q

k,α,β ,p(t))(x) =

xλ+λ′–γ +ρ+

kβk –

�(δ/k)�(η/k)

× �

[( η

k , q), (ρ, ), (λ – ξ + ρ – m, ),( β

k , αk ), ( δ

k , p), (–ξ – m + ρ, ), (λ + λ′ – γ + ρ, ),

(λ + λ′ + ξ ′ – γ + ρ – m, ), (, );(λ + ξ ′ – γ + ρ – m, );

kq–p– αk x

]. (.)

Proof Let L be the left-hand side of (.). Then using (.) and interchanging the orderof summation and differentiation, which is verified under the conditions in this theorem,we have

L =∞∑

n=

(η)nq,k


(cDλ,λ′ ,δ,δ′ ,γ+ tρ+n–).

Applying Lemma . together with (.) and (.), we get

L =∞∑

n=

knq�( η

k + nq)�( δk )

�( η

k )kαn+β

k –�( αn+β


× �(ρ + n)�(λ – ξ + ρ + n – m)�(λ + λ′ + ξ ′ – γ + ρ + n – m)�(–ξ + ρ + n – m)�(λ + λ′ – γ + ρ + n)�(λ + ξ ′ – γ + ρ + n – m)

× xλ+λ′–γ +ρ++n,

which, in view of (.), leads to the right-hand side of (.). �


Theorem Let λ, λ′, ξ , ξ ′, γ , ρ , α, β , γ , η, δ ∈C and m = [�(γ )] + with

�(ρ) + m > max{�(

–ξ ′),�(λ + λ′ – γ

)+ m

}.


(cDλ,λ′ ,ξ ,ξ ′ ,γ– t–ρEη,δ,q

k,α,β ,p(t))(x) =

xλ+λ′–γ +ρ

kβk –

�(δ/k)�(η/k)

× �

[( η

k , q), (ξ ′ + ρ + m, ), (–λ – λ′ + γ + ρ, ),( β

k , αk ), ( δ

k , p), (ρ, ), (–λ′ + ξ ′ + ρ + m, ),

(–λ′ – ξ + γ + ρ + m, ), (, );(–λ – λ′ – ξ + γ + ρ + m, );

kq–p– αk

x

]. (.)

Proof We establish the result by a similar argument as in the proof of Theorem , usingLemma . instead of Lemma .. We omit the details. �

Remark . The results in Theorems and can be easily specialized to yield the corre-sponding formulas involving simpler Caputo-type fractional derivatives with (.) such asthe left-hand-sided generalized Caputo fractional differentiation cDλ,ξ ,γ

+ f , the left-hand-sided Caputo-type Erdelyi-Kober fractional differentiation cD+

γ ,λf , the right-hand-sidedgeneralized Caputo fractional differentiation cDλ,ξ ,γ

– f , and the right-hand-sided Caputo-type Erdélyi-Kober fractional differentiation cD–

γ ,λf (see, e.g., []).

6 Pathway integral representation of (2.5)Recently, Nair [] introduced the pathway fractional integral operator by using the path-way idea of Mathai [], developed further by Mathai and Haubold [] and defined asfollows (cf. []).

Let f ∈ L(a, b), η ∈C with �(η), a ∈R+, and σ < be the pathway parameter. Then

(P(η,σ )

+ f)(x) := xη

∫ [ xa(–σ ) ]

[ –

a( – σ )tx

] η–σ

f (t) dt. (.)

For a real scalar σ , the pathway model for scalar random variables is represented by thefollowing probability density function (p.d.f.):

f (x) = c|x|υ–[ – a( – σ )|x|ξ ] λ–σ ,

provided that x ∈ R, v, ξ ∈ R+, λ ∈ R

+, – a( – σ )|x|ξ > . Here c is the normalizing con-

stant, and σ is called the pathway parameter.

For σ > , (.) can be written as follows:

(P(η,σ )

+ f)(x) = xη

∫ [ x–a(–σ ) ]

[ +

a(σ – )tx

] η–(σ–)

f (t) dt, (.)

and

f (x) = c|x|υ–[ + a(σ – )|x|ξ ]– λ(σ–) , (.)


provided that x ∈ R, v, ξ ∈R+, λ ∈R

+.

Moreover, as σ → –, the operator (.) reduces to the Laplace integral transform, andwhen σ = and a = , replacing η by η – , the operator (.) reduces to the Riemann-Liouville fractional integral operator. For more details on the pathway model and its par-ticular cases, the interested reader may refer to the recent works [, , ].

It is observed that the pathway fractional integral operator (.) can lead to other inter-esting examples of fractional calculus operators regarding some probability density func-tions and applications in statistics. Recently, Nisar et al. [] studied the pathway frac-tional integral operator associated with the Struve function of the first kind [] and thek-Mittag-Leffler function.

Here we investigate the pathway integral operator of the generalized k-Mittag-Lefflerfunction (.).

Theorem Let ρ , γ , β , η ∈ C with min{�(ρ),�(β),�(η)} > and �( γ

–σ) > –. Also, let

k, w,σ ∈R with σ < , p, q ∈R+, and δ ∈C \Z–

. Then

P(γ ,σ )+

[t

βk –Eη,δ,q

k,ρ,β ,p(wt

ρk)]

(x) = xγ + βk k(+ γ

–σ ) �( + γ

–σ)

[a( – σ )]βk

× Eη,δ,qk,ρ,β+k(+ γ

–σ )k,p

[w

(x

a( – σ )

) ρk]

. (.)

Proof Let L be the left-hand side of (.). Using (.) and interchanging the order ofthe integration and summation, which is verified under the conditions in this theorem, weget

L = xγ

∞∑

n=

(η)nq,k


∫ xa(–σ )

[ –

a( – σ )tx

] γ–σ

tβk + ρ

k – dt.

Evaluating the inner integral using the beta function (see, e.g., [], p.), we get

L = xγ

∞∑

n=

(η)nq,k


(x

a( – σ )

) ρk n+ β

k �( + γ

–σ)�( ρ

k n + β

k )�( ρ

k n + β

k + + γ

–σ)

=xγ + β

k �( + γ

–σ)

[a( – σ )]βk

∞∑

n=

(η)nq,k[w( xa(–σ ) )

ρk ]n

�( ρ

k n + β

k + + γ

–σ)(δ)pn,k

k– ρk n�( ρ

k n + β

k )�( ρ

k n + β

k ).

Using (.), we obtain

L =xγ + β

k �( + γ

–σ)k+ γ

–σ

[a( – σ )]βk

∞∑

n=

(η)nq,k[w( xa(–σ ) )

ρk ]n

�k(ρn + β + ( + γ

–σ)k)(δ)pn,k

,

which, with the aid of (.), is seen to reach the right-hand side of (.). �

Setting δ = q = and k = in Theorem , we obtain the following known result (see []).


Corollary . Let ρ , γ , β , η ∈C with min{�(ρ),�(β),�(η)} > and �( γ

–σ) > –. Also, let

w,σ ∈R with σ < , p ∈R+. Then

P(η,σ )+

[tβ–Eγ ,

k,ρ,,(wtρ

)](x)

= xη+β�( + η

–σ)

[a( – σ )]βEγ

ρ,β++ η–σ

[wxρ

(a( – σ ))ρ

]. (.)

We give Theorem by considering the case σ > and using equation (.), without itsproof, since the proof is similar to that in Theorem .

Theorem Let ρ , γ , β , η ∈C with min{�(ρ),�(β),�(η)} > and �( – η

σ– ) > . Also, letk, w,σ ∈R with σ > , p, q ∈R

+, and δ ∈C \Z– . Then

P(η,σ )+

[t

βk –Eη,δ,q

k,ρ,β ,p(wt

ρk)]

(x) = xγ + βk k(– γ

–σ ) �( – γ

σ– )

[–a( – σ )]βk

× Eη,δ,qk,ρ,β+k(– γ

σ– ),p

[w

(x

–a( – σ )

) ρk]

. (.)

The particular case of Theorem when δ = k = q = reduces to the following knownresult (see []).

Corollary . Let ρ , γ , β , η ∈ C with min{�(ρ),�(β),�(η)} > and �( – η

σ– ) > . Also,let w,σ ∈R with σ > , p ∈R

+. Then

P(η,σ )+

[tβ–Eγ ,

,ρ,β(wtρ

)](x)

=xη+β�( – η

σ– )[–a(σ – )]β

Eγ

ρ,β+(– ησ– )

[w

(x

–a(σ – )

)ρ]. (.)

7 Generalized k-Mittag-Leffler function and statistical distributionIn this section, we investigate the density function for (.) stated in Theorem . We alsoconsider some particular cases of Theorem , which are connected with some possibleknown results (if any).

Theorem Let k, p, q,μ, x ∈ R+ with < μ ≤ and q ≤ μ + p. Also, let γ , δ ∈ C withmin{�(γ ),�(δ)} > . Let

Fx(x) = – Eγ ,δ,qk,μ,k,p

(–xμ

).

Then the density function f (x) of Fx(x) is given as follows:

f (x) = μxμ–(γ )q,k

∞∑

n=

(n + )(γ + qk)nq,k

�k(μn + μ + k)(–xμ)n

(δ)(n+)p,k(.)

=(γ )q,k

(δ)p,kxμ–Eγ +qk,δ+pk,q

k,μ,μ,p(–xμ

). (.)


Proof Using (.), we have

Fx(x) =∞∑

n=

(–)n+(γ )nq,k

�k(μn + k)xμn

(δ)pn,k. (.)

Differentiating each side of (.) with respect to x gives the density function

f (x) =∞∑

n=

(–)n+(γ )nq,k

�k(μn + k)μnxμn–

(δ)pn,k,

which, upon replacing n by n + , yields

f (x) = xμ–∞∑

n=

(–)n(γ )nq+q,k

�k(μn + μ + k)(μn + μ)xμn

(δ)p(n+),k. (.)

Applying the relation

(γ )n+j,k = (γ )j,k(γ + jk)n,k (.)

to the factor in the numerator of (.) and using (.) on the denominator, we get

f (x) = xμ–∞∑

n=

(–)n(γ )q,k(γ + qk)nq,k

(μn + μ)�k(μn + μ)(μn + μ)xμn

(δ)p(n+),k

= (γ )q,kxμ–∞∑

n=

(γ + qk)nq,k

�k(μn + μ)(–xμ)n

(δ)p(n+),k

= xμ–(γ )q,k

∞∑

n=

(γ + qk)nq,k�k (μn+μ+k)

(μn+μ)

(–xμ)n

(δ)p(n+),k, (.)

which is just the desired result (.).Next, employing the same process as in the proof of (.), we find from (.) that

f (x) = xμ–∞∑

n=

(–)n(γ )nq+q,k

�k(μn + μ + k)(μn + μ)xμn

(δ)p(n+),k.

Applying (.) to both numerator and denominator, we get

f (x) =(γ )q,k

(δ)p,kxμ–

∞∑

n=

(γ + qk)nq,k

�k(μn + μ)(–xμ)n

(δ + pk)np,k, (.)

which, in view of (.), can be expressed as the desired result (.). �

Here we consider some particular known cases of Theorem .Taking δ = p = k = in Theorem , we get the following result (cf. [], Eq. ()).

Corollary . Let q,μ, x ∈R+ with < μ ≤ and q ≤ μ + . Also, let �(γ ) > . Let

Fx(x) = – Eγ ,qμ,

(–xμ

).



f (x) = (γ )qxμ–Eγ +q,,qμ,μ,

(–xμ

).

Setting q = k = γ = in Theorem , we obtain the following result (cf. [], Eq. ()).

Corollary . Let p,μ, x ∈ R+ with < μ ≤ and ≤ μ + p. Also, let γ , δ ∈ C with

min{�(γ ),�(δ)} > . Let

Fx(x) = – Eγ ,δμ,,p

(–xμ

).


f (x) =γ

(δ)pxμ–Eγ +,δ+p

μ,μ,p(–xμ

). (.)

Here

Eη,δα,β ,p(z) := Eη,δ,

α,β ,p(z). (.)

Setting q = k = γ = in Theorem , we get the following result (see also []).

Corollary . Let p,μ, x ∈R+ with < μ ≤ and ≤ μ + p. Also, let �(δ) > . Let

Fx(x) = – E,δμ,,p

(–xμ

).


f (x) =

(δ)pxμ–E,δ+p

μ,μ,p(–xμ

). (.)

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe authors have contributed equally to this manuscript. They read and approved the final manuscript.

Author details1Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Al dawaser,Riyadh region 11991, Saudi Arabia. 2Department of Applied Statistics & Insurance, Mansoura University, Mansoura, Egypt.3College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Al dawaser, Riyadh region 11991, Saudi Arabia.4Electrical Engineering Department, College of Engineering-Wadi Aldawaser, Prince Sattam bin Abdulaziz University,Wadi Al dawaser, Riyadh region 11991, Saudi Arabia. 5Department of Mathematics, Dongguk University, Gyeongju,38066, Republic of Korea.

AcknowledgementsThis project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under theresearch project No. 2016/01/6637.

Received: 20 July 2016 Accepted: 17 November 2016

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