A basic study of a fractional integral operator with extended Mittag-Le ﬄ er kernel

: In this present paper, the basic properties of an extended Mittag-Le ﬄ er function are studied. We present some fractional integral and di ﬀ erential formulas of an extended Mittag-Le ﬄ er function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Le ﬄ er function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.


Introduction
In the last few years, many mathematical models were developed for various real-world problems by utilizing the field of fractional calculus with boundary conditions.Therefore, this field has received more attention in a variety of fields in diverse domains see [13,14,18,32,33].In the development of fractional calculus, many researchers focused on developing certain new fractional integral operators and their applications in diverse fields.In practical applications, certain various types of fractional operators such as Riemann-Liouville, Caputo, Riesz [11,35] and Hilfer [12] fractional operators are introduced.Freshly, the researchers have studied certain new fractional integral and derivative operators and their possible applications in various disciplines of sciences.Very recently Khalil et al. [9] have introduced the notion of fractional conformable derivative (FCD) operators with some shortcomings.Abdeljawad [1] investigated the properties of the fractional conformable derivative operators.In [8], Jarad et al. introduced the fractional conformable integral and derivative operators.Anderson and Unless [4] developed the idea of conformable derivative by employing local proportional derivatives.Abdeljawad and Baleanu [2] investigated certain monotonicity results for fractional difference operators with discrete exponential kernels.Abdeljawad and Baleanu [3] have established fractional derivative operators with exponential kernel and their discrete versions.In [5], Atangana and Baleanu defined a new fractional derivative operator with the non-local and non-singular kernel.Caputo and Fabrizio [6] defined fractional derivative without a singular kernel.Certain properties of fractional derivative without a singular kernel can be found in the work of Losada and Nieto [10].In [7], Jarad et al. defined generalized fractional derivatives generated by a class of local proportional derivatives.
On the other side, the researchers studied the fractional operators and have investigated and studied fractional integral inequalities to examine the existence and uniqueness of such type of fractional boundary value problems [46][47][48][49][50][51][52].
The special functions such as Mittag-Leffler function and its extensions appear as solution of fractional order differential and integral equations.Various interesting applications of such functions can be found in the study of the telegraph equations, kinetic equation, Levy flights, random walks, complex system, non-equilibrium statistical mechanics, super diffuse transport and quantum mechanics.The interested readers may refer to the work [15,[26][27][28] and the references cited therein.

Preliminaries
In this section, we present some basic and well known results.In [53], the Gosta Mittag-Leffler function E ρ (z) is defined by The Wiman Mittag-Leffler function E ρ,ϕ (z) is defined in [54] by 2) The researchers extensively studied E ρ (z) and E ρ,ϕ (z) [17, 20-22, 24, 25, 34].The numerical investigation of such functions in the complex plane can be found in the work of [23,36].
Prabhakar [31] has proposed the following generalization of Mittag-Leffler function by where (δ) n denotes the well-known Pochhammer symbol.
Rahman et al. [29] recently defined an extension of Mittag-Leffler function by where (δ; p, v) n is an extended Pochhammer symbol recently defined by Srivastava et al. [44] by (2.5) An extension of gamma function Γ v (δ; p) is defined by where K v+ 1 2 ( δ t ) is the modified Bessel function of the first kind (see, e.g., Chaudhry and Zubair [16]).If we consider v = 0 in (2.5), then we get the following extension of Pochhammer symbol defined by Srivastava et al. [45] as where Γ(δ; p) is defined by Similarly, if we consider p = 0 in (2.6), then we get the following classical Pochhammer symbol defined by where Γ(δ) is defined by The extensions and applications of these Mittag-Leffler functions were extensively studied by the researchers ( [19,28,[37][38][39][40][41]43]).
The Lebesgue measurable space for real (complex) valued functions is defined by The Riemann-Liouville type (left and right sided) fractional integral operators are respectively defined in [11,35] by and The Riemann-Liouville fractional derivatives (left and right sided) for h(x) ∈ L(x 1 , x 2 ), κ ∈ C, (κ) > 0 and n = [ (κ)] + 1 are defined in [11,35] by respectively.The generalized differential operator D κ,v x 1 + of order 0 < κ < 1 and type 0 < v < 1 with respect to x can be found in ( [25,35]) as; Clearly, if we put v = 0 then (2.10) reduces to the operator D κ x 1 + defined in (2.9).Moreover, we consider the following well known results.

Fractional integral and differential formulas of an extended Mittag-Leffler function
In this section, we present some fractional integration and differentiation formulas of an extended Mittag-Leffler function (2.4).We consider the following well-known result recently presented by Rahman et al. [29].
To prove (3.4), we have By applying (2.12), we get which completes the required proof.
If we consider v = 0, then we get the following result of Parmar et al. [30].

Extension of Prabhakar type fractional integral and its properties
In this section, we estimate a new extension of Prabhakar type fractional integral and its properties. and respectively.Substituting v = 0, then (4.1) and (4.2) reduce to the following operators and When we substitute p = 0, then (4.3) and (4.4) reduce to the following Prabhakar operators and If we consider ω = 0, then the integral operators in (4.5) and (4.6) reduce to the well-known Riemann-Liouville fractional integral operators.
Next, we present the following results.

Conclusions
In this article, we have investigated fractional integral and differential formulas for the extended Mittag-Leffler function.Also, we defined a further extension of Prabhakar fractional operator and investigated its various properties.The special cases of our result can be found in the literature cited therein.One can easily investigate many of its applications and develop various fractional integral inequalities by utilizing the newly defined Prabhakar fractional operator.