Fractional calculus of generalized k Mittag Leffler function and its applications to statistical distribution

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R E S E A R C H

Open Access

Fractional calculus of generalized

k

-Mittag-Lefﬂer function and its applications

to statistical distribution

Kottakkaran Sooppy Nisar

1

_{, Ashraf Fetoh Eata}

2,3

_{, Mujahed Al-Dhaifallah}

4

_{and Junesang Choi}

5*

*_{Correspondence:}

[email protected] 5_{Department of Mathematics,}

Dongguk University, Gyeongju, 38066, Republic of Korea Full list of author information is available at the end of the article

Abstract

We aim to investigate the MSM-fractional calculus operators, Caputo-type

MSM-fractional diﬀerential operator, and pathway fractional integral operator of the generalizedk-Mittag-Leﬄer function. We also investigate certain statistical

distribution associated with the generalizedk-Mittag-Leﬄer function. Certain particular cases of the derived results are considered and indicated to further reduce to some known results.

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

Keywords: gamma function; generalizedk-Mittag-Leﬄer functions; statistical distribution

1 Introduction and preliminaries Throughout this paper, letC,R,R+_,_Z–

, andNbe the sets of complex numbers, real num-bers, positive real numnum-bers, nonpositive integers, and positive integers, respectively, and letR+

:=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 this observation, they [] used the Gauss form of the gamma function (see [], Eq. (), p.) to introduce the so-calledk-gamma function

k(z) = lim n→∞

n!kn₍_nk₎zk–

(z)n,k

k∈R+;z∈C\kZ–_. (.)

Starting from this deﬁnition, they [] presented a number of properties for thek-gamma function. We recall some of them:

k(z+k) =zk(z) and k(k) = ; (.)

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the Euler integral form:

k(z) =

∞



tz–e–tkk _d_t _k∈R+_;(_z_{) > }_; _(.)

thek-Pochhammer symbol (λ)n,kdeﬁned (forλ,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 functionand thek-gamma functionk:

k(z) =k

z k–

z

k . (.)

In a number of subsequent works including [], the k-gamma function and k-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, and k-analogue of the Riemann-Liouvile fractional integral operator.

In , the Swedish mathematician Gosta Mittag-Leﬄer [] (see also []) introduced and investigated the so-called Mittag-Leﬄer function

Eα(z) :=

∞

n= zn

(αn+ )

z∈C;α∈R+_. (.)

Since then, the Mittag-Leffler functionEα(.) has been extended in a number of ways and, together with its extensions, applied in various research areas such as engineering and (in particular) statistics. The Mittag-Leffler functions and related distributions were given in []. Further, Mathai and Haubold [] established connections among generalized Mittag-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 and stochastic processes. Also, various pathways were investigated from the exponential and gamma densities to the Mittag-Leffler densities and then from the Mittag-Leffler densities to the Lèvi and Linnik densities []. Lin [] proved that the Mittag-Leffler distributions be-long to the class of distributions with completely monotone derivatives. The fundamental properties 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 density functions, are also given.

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ofEα:

Prabhakar [] introduced the functionEγ_α,βin the following form:

Eγ_α,β(z) =

Another generalization of the Mittag-Leﬄer functionEαwas given by Shukla and Prajap-ati []:

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

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

The Fox-Wright hypergeometric functionpq(z) is given by the series

pq(z) =pq

function for large values of the argument ofz∈Cwas studied in [], and under the con-dition

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The generalized hypergeometric functionpFqis deﬁned as follows []:

which obviously is a particular case of the Fox-Wright hypergeometric function pq(z)

(.) whenαi=  =βj(i= , , . . . ,p;j= , , . . . ,q).

Letλ,λ,ξ,ξ,γ ∈Cwith(γ) >  andx∈R+_{. Then the generalized fractional integral} operators involving the Appell functionsFare deﬁned as follows:

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

The corresponding fractional diﬀerential operators have their respective forms:

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

The following four results will be required (for the ﬁrst and second, see [, ]; for the third and fourth, see []).

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Lemma . Letλ,λ,ξ,ξ,γ,ρ∈Cbe such that(γ) > and

(ρ) >max(ξ),–λ–λ+γ,–λ–ξ+γ.

Then

I_–λ,λ,ξ,ξ,γt–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 . Letλ,δ,γ,ρ∈Cwith(λ) > and(ρ) >max{,(δ–γ)}.Then

I_+λ,δ,γtρ–(x) = (ρ)(ρ+γ–δ)

(ρ–δ)(ρ+λ+γ)x

ρ–δ–_. _(.)

In particular,

I_+λ,γtρ–(x) = (ρ+γ)

(ρ+λ+γ)x

ρ– ₍_λ_{) > ,}₍_ρ_{) >}_max_{, –}₍_γ₎ _(.)

and

I_+λtρ–(x) = (ρ)

(ρ+λ)x

ρ– _min₍_λ_),₍_ρ₎_{> }_. _(.)

Lemma . Letλ,δ,γ,ρ∈Cwith(λ) > and(ρ) <  +min{(δ),(γ)}.Then

I_–λ,δ,γtρ–(x) = (δ–ρ+ )(γ–ρ+ )

( –ρ)(λ+δ+γ–ρ+ )x

ρ–δ–_. _(.)

In particular,

I_–λ,γtρ–(x) = (γ –ρ+ )

(λ+γ –ρ+ )x

ρ– ₍_λ_{) > ,}₍_ρ_{) <  +}₍_γ₎ _(.)

and

I_–λtρ–(x) = ( –ρ)

(λ–ρ+ )x

ρ– ₍_λ_{) > ,}₍_ρ_{) < }_. _(.)

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2 k-Mittag-Lefﬂer functions

Here 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 thek -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 thek-Mittag-Leﬄer function (.) slightly as follows:

Eη,τ_k_,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 fractional Fourier transform of orderα( <α≤). Daiya and Ram [] investigated the statistical density of thek-Mittag-Leffler function (.). Gupta and Parihar [] defined a further extension of thek-Mittag-Leffler functions,

Eη,δ,_k_,α,β,q_p(z) :=

∞

n=

(η)qn,k

k(αn+β)(δ)pn.k

zn

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, andk-Beta transform, and determined thek-Riemann-Liouville frac-tional integral and diﬀerentiation.

3 MSM fractional integral representations of (2.5)

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Theorem  Letλ,λ,ξ,ξ,γ,ρ,α,β,η,δ∈Cwith(γ) > and (ρ) >max,λ+λ+ξ–γ,λ–ξ.

Also,let k,p,q,x∈R+.Then

I_,+λ,λ,ξ,ξ,γtρ–E_kη,δ,_,α,β,q_p(t)(x) = (δ/k)

(η/k)

xρ–λ–λ+γ– kβk–

×

(η_k,q), (ρ, ), (ρ+γ–λ–λ–ξ, ), (β_k,α_k), (δ_k,p), (ρ+ξ, ), (ρ+γ –λ–λ, ),

(ρ+ξ–λ, ), (, ); (ρ+γ–λ–ξ, ); k

q–p–α

kx

. (.)

Proof LetLbe the left-hand side of (.). Then, using (.), we have

L=

I_,+λ,λ,ξ,ξ,γtρ–

∞

n=

(η)nq,k

k(αn+β)

tn

(δ)pn,k

(x).

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

L=

∞

n=

(η)nq,k

k(αn+β)(δ)pn,k

I_,+λ,λ,ξ,ξ,γtρ+n–(x).

Applying Lemma ., we obtain

L=

∞

n=

(η)nq,k

k(αn+β)(δ)pn,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)(

δ

k)

(η_k)kαnk+β–₍αn+β

k )(

δ

k+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 . Letλ,ξ,γ,ρ,α,β,η,δ∈Cwith(γ) > and

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Also,let k,p,q,x∈R+_._Then

I_,+λ,ξ,γtρ–E_kη,δ,_,α,β,q_p(t)(x)

=(δ/k)

(η/k) xρ–ξ–

kβk–

×

(η_k,q), (ρ+γ –ξ, ), (ρ, ), (, ); (β_k,α

k), (

δ

k,p), (ρ–ξ, ), (ρ+γ +λ, );

kq–p–αk_x

. (.)

Theorem  Letλ,λ,ξ,ξ,γ,ρ,α,β,η,δ∈Cbe such that(λ) > and

(ρ) >max(ξ),–λ–λ+γ,–λ–ξ+γ.

Also,let k,p,q∈R+_._Then

I_–λ,λ,ξ,ξ,γtρ–E_kη,δ,_,α,β,q_p(t)(x) = (δ/k)

(η/k)

x–λ–λ+γ–ρ kβk–

×

(η_k,q), (–ξ+ρ, ), (λ+λ–γ +ρ, ), (β_k,α

k), (

δ

k,p), (ρ, ), (λ–ξ+ρ, ),

(λ+ξ–γ+ρ, ), (, ); (λ+λ+ξ–γ +ρ, ); k

q–p–α_k_x

. (.)

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

Corollary . Letλ,ξ,γ,ρ,α,β,η,δ∈Cbe such that(λ) > and

(ρ) >max(–ξ),(–γ).

I_–λ,ξ,γtρ–E_kη,δ,_,α,β,q_p(t)(x)

=(δ/k)

(η/k) xρ–ξ–

kβk _{– }

×

(η_k,q), (ξ–ρ+ , ), (γ–ρ+ , ), (, ); (β_k,α_k), (_kδ,p), ( –ρ, ), (λ+ξ+γ –ρ+ , ); k

q–p–α_k_x

. (.)

4 MSM-fractional differential operator of (2.5)

Here we derive the Marichev-Saigo-Maeda fractional diﬀerentiation of the generalized k-Mittag-Leﬄer function (.). The following lemmas will be required (see []).

Lemma . Letλ,λ,ξ,ξ,γ,ρ∈Cbe such that

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Then

Dλ,λ_+,ξ,ξ,γtρ–(x) = (ρ)(–ξ+λ+ρ)(λ+λ

₊_ξ_–_γ ₊_ρ₎

(–ξ+ρ)(λ+λ–γ+ρ)(λ+ξ–γ +ρ)x

λ+λ–γ+ρ–_. _(.)

Lemma . Letλ,λ,ξ,ξ,γ,ρ∈Cbe such that

(ρ) >max–ξ,λ+ξ–γ,λ+λ–γ+(γ)+ .

Then

Dλ,λ_– ,ξ,ξ,γt–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_. _(.)

Theorem  Letλ,λ,ξ,ξ,γ,ρ,α,β,η,δ∈Cbe such that

(ρ) >max,(–λ+ξ),–λ–λ–ξ+γ.

Also,let k,p,q∈R+.Then

Dλ,λ_+,ξ,ξ,γtρ–E_kη,δ,_,α,β,q_p(t)(x) = (δ/k)

(η/k)

xλ+λ–γ+ρ– kβk–

×

(η_k,q), (ρ, ), (–ξ+λ+ρ, ), (β_k,α

k), (

δ

k,p), (–ξ+ρ, ), (λ+λ–γ +ρ, ),

(λ+λ+ξ–γ+ρ, ), (, ); (λ+ξ–γ+ρ, ); k

q–p–α_k_x

. (.)

Proof LetL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

k(αn+β)(δ)pn,k

Dλ,λ_+,ξ,ξ,γtρ+n–(x).

Using (.), (.), and Lemma ., we get

L=

∞

n=

knq₍η

k+nq)(

δ

k)

(η_k)kαnk+β–(αn+β

k )(

δ

k+np)knp

× (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  Letλ,λ,ξ,ξ,γ,ρ,α,β,η,δ∈Cbe such that

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D_–λ,λ,ξ,ξ,γt–ρEη,δ,_k_,α,β,q_p(t)(x) =x λ+λ–γ–ρ

kβk–

(δ/k)

(η/k)

×

(η_k,q), (ξ+ρ, ), (–λ–λ+γ +ρ, ), (β_k,α_k), (_kδ,p), (ρ, ), (–λ+ξ+ρ, ),

(–λ–ξ+γ+ρ, ), (, ); (–λ–λ–ξ+γ +ρ, ); k

q–p–α_k_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

c

D_+λ,ξ,γf(x) =I_+–λ+[Re(λ)]+,–ξ–[Re(λ)]–,λ+γ–[Re(λ)]–f([Re(λ)]+)(x) and

c

D_–λ,ξ,γf(x) = (–)[Re(λ)]+I_––λ+[Re(λ)]+,–ξ–[Re(λ)]–,λ+γf([Re(λ)]+)(x), whereλ,ξ,γ ∈Cwith(λ) >  andx∈R+_.

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

c

Dλ,λ+,ξ,ξ,γf

(x) =I+–λ,–λ,–ξ+[(γ)]+,–ξ,–γ+[(γ)]+f([(γ)]+)

(x)

and

c

Dλ,λ_– ,ξ,ξ,γf(x) = (–)[(γ)]+I–λ,–λ,–ξ,–ξ+[(γ)]+,–γ+[Re(γ)]+

– f([(λ)]+)

(x),

whereλ,λ,ξ,ξ,γ,ρ∈Cwith(γ) >  andx∈R+_.

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

Lemma . Letλ,λ,ξ,ξ,γ,ρ∈Cand m= [(γ)] + with

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Then

c

Dλ,λ_+,ξ,ξ,γtρ–(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 . Letλ,λ,ξ,ξ,γ,ρ∈Cand m= [(γ)] + with

(ρ) +m>max–ξ,λ+ξ–γ,λ+λ–γ+(γ)+ .

Then

c

Dλ,λ_– ,ξ,ξ,γt–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  Letλ,λ,ξ,ξ,γ,ρ,α,β,γ,η,δ∈Cand m= [(γ)] + with

(ρ) –m>max,(–λ+ξ),–λ–λ–ξ+γ.

c

D_+λ,λ,ξ,ξ,γtρ–E_kη,δ,_,α,β,q_p(t)(x) =x

λ+λ–γ+ρ+

kβk–

(δ/k)

(η/k)

×

(η_k,q), (ρ, ), (λ–ξ+ρ–m, ), (β_k,α_k), (_kδ,p), (–ξ–m+ρ, ), (λ+λ–γ +ρ, ),

(λ+λ+ξ–γ +ρ–m, ), (, ); (λ+ξ–γ+ρ–m, ); k

q–p–α

k_x

. (.)

Proof LetLbe the left-hand side of (.). Then using (.) and interchanging the order of summation and diﬀerentiation, which is veriﬁed under the conditions in this theorem, we have

L=

∞

n=

(η)nq,k

k(αn+β)(δ)pn,k

c

Dλ,λ+,δ,δ,γtρ+n–

.

Applying Lemma . together with (.) and (.), we get

L=

∞

n=

knq₍η

k+nq)(

δ

k)

(η_k)kαnk+β–(αn+β

k )(

δ

k+np)knp

× (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,

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Theorem  Letλ,λ,ξ,ξ,γ,ρ,α,β,γ,η,δ∈Cand m= [(γ)] + with (ρ) +m>max–ξ,λ+λ–γ+m.

c

D_–λ,λ,ξ,ξ,γt–ρEη,δ,_k_,α,β,q_p(t)(x) =x λ+λ–γ+ρ

kβk–

(δ/k)

(η/k)

×

(η_k,q), (ξ+ρ+m, ), (–λ–λ+γ+ρ, ), (β_k,α_k), (_kδ,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 , using Lemma . 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 as the left-hand-sided generalized Caputo fractional diﬀerentiationc_Dλ,ξ,γ

+ f, the left-hand-sided Caputo-type Erdelyi-Kober fractional diﬀerentiationc_D+

γ,λf, the right-hand-sided generalized Caputo fractional diﬀerentiationc_Dλ,ξ,γ

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

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 as follows (cf. []).

Letf ∈L(a,b),η∈Cwith(η),a∈R+, andσ<  be the pathway parameter. Then

P+(η,σ)f

(x) :=xη

[_a_(–x_σ₎]



 –a( –σ)t x

η

–σ

f(t) dt. (.)

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

f(x) =c|x|υ– –a( –σ)|x|ξ–λσ_,

provided thatx∈R,v,ξ∈R+,λ∈R+_,  –a( –σ)|x|ξ_{> . Here}_c_{is the normalizing} con-stant, andσis called the pathway parameter.

Forσ> , (.) can be written as follows:

P_+(η,σ)f(x) =xη

[_–_a_(–x_σ₎]



 +a(σ– )t x

η

–(σ–)

f(t) dt, (.)

and

f(x) =c|x|υ– +a(σ– )|x|ξ– λ

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provided thatx∈R,v,ξ∈R+_,_λ_∈_R+ .

Moreover, asσ→–, the operator (.) reduces to the Laplace integral transform, and whenσ =  anda= , replacingηbyη– , 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 applicafunc-tions in statistics. Recently, Nisar et al. [] studied the pathway frac-tional integral operator associated with the Struve function of the ﬁrst kind [] and the k-Mittag-Leﬄer function.

Here we investigate the pathway integral operator of the generalizedk-Mittag-Leﬄer function (.).

Proof LetL be the left-hand side of (.). Using (.) and interchanging the order of the integration and summation, which is veriﬁed under the conditions in this theorem, we get

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

L=xγ

which, with the aid of (.), is seen to reach the right-hand side of (.).

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Corollary . Letρ,γ,β,η∈Cwithmin{(ρ),(β),(η)}> and(_–σγ ) > –.Also,let w,σ∈Rwithσ< ,p∈R+_._Then

P(η,σ_+ )tβ–E_kγ,_,ρ,,wtρ(x)

=xη+β( + η –σ) [a( –σ)]βE

γ ρ,β++_–η_σ

wxρ (a( –σ))ρ

. (.)

We give Theorem  by considering the caseσ>  and using equation (.), without its proof, since the proof is similar to that in Theorem .

Theorem  Letρ,γ,β,η∈Cwithmin{(ρ),(β),(η)}> and( –_ση_–) > .Also,let

k,w,σ∈Rwithσ> ,p,q∈R+,andδ∈C\Z–_.Then

P(η,σ_+ )tβk–Eη,δ,q

k,ρ,β,p

wtρk(x) =xγ+

β

kk(–

γ

–σ) ( – γ σ–)

[–a( –σ)]βk

×Eη,δ,_k_,ρ,β+q _k_(– γ σ–),p

w

x –a( –σ)

ρ

k

. (.)

The particular case of Theorem  whenδ=k=q=  reduces to the following known result (see []).

Corollary . Letρ,γ,β,η∈Cwithmin{(ρ),(β),(η)}> and( –_ση_–) > .Also,

let w,σ∈Rwithσ> ,p∈R+.Then

P_+(η,σ)tβ–E_,ρ,βγ, wtρ(x)

=x

η+β_{( –} η σ–) [–a(σ– )]β E

γ

ρ,β+(–_ση_–)

w

x –a(σ– )

ρ

. (.)

7 Generalizedk-Mittag-Lefﬂer function and statistical distribution

In this section, we investigate the density function for (.) stated in Theorem . We also consider some particular cases of Theorem , which are connected with some possible known results (if any).

Theorem  Let k,p,q,μ,x∈R+ _with _{ <}_μ_≤_ _{and q}_≤_μ₊_p_. _Also_,_let _γ_,_δ_∈_C_with

min{(γ),(δ)}> .Let

Fx(x) =  –Ekγ,δ,,μ,qk,p

–xμ.

Then the density function f(x)ofFx(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,k

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Proof Using (.), we have

Diﬀerentiating each side of (.) with respect toxgives the density function

f(x) =

which, upon replacingnbyn+ , yields

f(x) =xμ–

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

f(x) =xμ–

which is just the desired result (.).

Next, employing the same process as in the proof of (.), we ﬁnd from (.) that

f(x) =xμ–

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

f(x) =(γ)q,k

which, in view of (.), can be expressed as the desired result (.).

Here we consider some particular known cases of Theorem .

Takingδ=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) =  –Eμ,γ,q

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f(x) = (γ)qxμ–Eμ,μ,γ+q,,q

–xμ.

Settingq=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) =  –Eμ,,γ,δp

–xμ.

f(x) = γ (δ)p

xμ–E_μ,μ,γ+,δ+_p p–xμ. (.)

Here

Eη,δ_α,β,_p(z) :=Eη,δ,_α,β,_p(z). (.)

Settingq=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) =  –Eμ,,,δp

–xμ.

f(x) =  (δ)p

xμ–E_μ,μ,,δ+p_p–xμ. (.)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have contributed equally to this manuscript. They read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Al dawaser,}

Riyadh region 11991, Saudi Arabia. 2_{Department of Applied Statistics & Insurance, Mansoura University, Mansoura, Egypt.} 3_{College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Al dawaser, Riyadh region 11991, Saudi Arabia.} 4_{Electrical Engineering Department, College of Engineering-Wadi Aldawaser, Prince Sattam bin Abdulaziz University,}

Wadi Al dawaser, Riyadh region 11991, Saudi Arabia.5_{Department of Mathematics, Dongguk University, Gyeongju,} 38066, Republic of Korea.

Acknowledgements

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

Received: 20 July 2016 Accepted: 17 November 2016

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