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

We aim to investigate the MSM-fractional calculus operators, Caputo-type MSM-fractional differential operator, and pathway fractional integral operator of the generalized k-Mittag-Leffler function. We also investigate certain statistical distribution associated with the generalized k-Mittag-Leffler function. Certain particular cases of the derived results are considered and indicated to further reduce to some known results.


Introduction and preliminaries
Throughout this paper, let C, R, R + , Z - , and N be the sets of complex numbers, real numbers, positive real numbers, nonpositive integers, and positive integers, respectively, and let 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, annihilation operators, and perturbative computation of Starting from this definition, they [] presented a number of properties for the k-gamma function. We recall some of them: k (z + k) = z k (z) and k (k) = ; (.) the Euler integral form: the k-Pochhammer symbol (λ) n,k defined (for λ, ν ∈ C; k ∈ R) by (λ) ν,k := k (λ + νk) k (λ) λ(λ + k) · · · (λ + (n -)k) (ν = n ∈ N); (.) from (.), it is easy to find the following relationship between the gamma function and the k-gamma function 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 functions 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.
(  .  ) Since then, the Mittag-Leffler function E α (.) 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.
Another generalization of the Mittag-Leffler function E α was given by Shukla and Prajapati []: 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 example, to [-, ].
The Fox-Wright hypergeometric function p q (z) is given by the series where a i , b j ∈ C and α i , β j ∈ R (i = , , . . . , p; j = , , . . . , q). Asymptotic behavior of this function for large values of the argument of z ∈ C was studied in [], and under the condition The generalized hypergeometric function p F q is defined as follows []: which obviously is a particular case of the Fox-Wright hypergeometric function p q (z) (.) when α i =  = β j (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: and The and The fractional integral operators have many interesting applications in various fields including (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 []).
In particular, In particular, As mentioned before, k-extensions of the Mittag-Leffler functions have been given and investigated particularly in view of statistics. In this paper, we aim to investigate the MSMfractional calculus operators, Caputo-type MSM-fractional differential operator, and that pathway fractional integral operator of the generalized k-Mittag-Leffler function (.). We also investigate certain statistical distribution associated with the generalized k-Mittag-Leffler function (.), in which certain particular cases of the derived results are considered and indicated to further reduce to some known results.

k-Mittag-Leffler functions
Here we introduce k-Mittag-Leffler functions and their extensions. The simplest kextensions of the Mittag-Leffler functions (.) and (.) can be given by and investigated some properties associated with the definition itself and the Riemann-Liouville fractional calculus operators. Saxena et al.
[] extended the k-Mittag-Leffler function (.) slightly as follows: They derived its Euler transform, Laplace transform, Whittaker transform, and fractional Fourier transform of order α ( < α ≤ ). Daiya presented its properties, including differentiation, the fractional Fourier transform, Laplace transform, and k-Beta transform, and determined the k-Riemann-Liouville fractional integral and differentiation.

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.
Also, let k, p, q, x ∈ R + . Then Proof Let L  be the left-hand side of (.). Then, using (.), we have Interchanging the summation and integration, which is verified under the conditions in this theorem, we get Applying Lemma ., we obtain Now, using relations (.) and (.), we get which, in view of (.), leads to the right-hand side of (.). This completes the proof.
Also, let k, p, q ∈ R + . Then
Lemma . Let λ, λ , ξ , ξ , γ , ρ ∈ C be such that Also, let k, p, q ∈ R + . Then 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 conditions in this theorem, we have Using (.), (.), and Lemma ., we get which, in view of (.), is equal to the right-hand side of (.).

Remark .
The results in Theorems  to  can be easily reduced to yield some corresponding formulas involving simpler factional calculus operators such as the Erdélyi-Kober fractional calculus operators.
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 []).

δ ∈ C and m = [ (γ )] +  with
Also, let k, p, q ∈ R + . Then Proof We establish the result by a similar argument as in the proof of Theorem , using Lemma . instead of Lemma .. We omit the details. ). Let f ∈ L(a, b), η ∈ C with (η), a ∈ R + , and σ <  be the pathway parameter. Then For a real scalar σ , the pathway model for scalar random variables is represented by the following probability density function (p.d.f.): Here c is the normalizing constant, and σ is called the pathway parameter.
For σ > , (.) can be written as follows: and provided that x ∈ R, v, ξ ∈ R + , λ ∈ R +  . Moreover, as σ → -, the operator (.) reduces to the Laplace integral transform, and when 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 particular cases, the interested reader may refer to the recent works [, , ].
It is observed that the pathway fractional integral operator Here we investigate the pathway integral operator of the generalized k-Mittag-Leffler function (.).
Using (.), we obtain 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 []).

Generalized k-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).