Estimates of Classes of Generalized Special Functions and Their Application in the Fractional ðk, sÞ-Calculus Theory

In this article, we aim to develop new ðk, sÞ-fractional integral and differential operators containing S-functions as kernels in a form of generalized k-Mittag-Leffer functions. We also set up various properties of such operators. Furthermore, we consider a variety of implications of the major outcomes that will be very useful in the implementation of scientific, engineering, and technical problems.


Introduction and Preliminaries
More focus has been given in recent years to the development of fractional calculus applications. The fractional calculus is very important in the development of integration and differentiation with the fractional calculus powers of real numbers or complex numbers (for example, integral and differential operators). The properties and application of the fractional calculus operator are described by [1,2]. For more modern fractional calculus developments, the reader can refer to [3][4][5][6][7]. Some new results for ψ-Hilfer fractional pantograph-type differential equation depending on ψ-Riemann-Liouville integral are studied by Foukrach et al. [8].
In the frame of fractional derivatives, Alqahtani et al. [9] studied nonlinear F-contractions on b-metric spaces and differential equations with Mittag-Leffler kernel. Many scholars have computed numerous fractional integral inequalities containing the various fractional integration and differentiation operators over the past few years (see [10,11]). The k symbols are well known from a number of sources related to the measurement of finite differences (see [12,13]). In the literature, k-fractional integral operators have recently been considered by different scholars.
For this function, we begin the literature with the following properties. The Pochhammer k-symbols and k-gamma function were introduced by Diaz and Pariguan (see [14]) ζ ð Þ n,k = 1, n = 0, ζ ∈ ℂ ð Þ , In the same paper, they spell out the relations The k-fractional integral is develop by [15] as When we choose k = 1, then I ς k ð f ðxÞÞ shows the result of the Riemann-Liouville (R-L) fractional integration formula.
We have Formulas for k-fractional integral are developed by [15] as The R-L ðk, sÞ-fractional integral of order ς > 0 was elucidated by [16].
where x ∈ ½a, b, k > 0 and s ∈ R \ f−1g. In the same paper, they defined the following result: In recent years, the applications of the fractional calculus are given by the researchers (see [17,18]). By using generalized k-fractional integrals, Gruss-type integral inequalities for generalized R-L k-fractional integrals, and ðk, sÞ-R-L fractional integral inequalities for continuous random variables, analytical properties of ðk, sÞ-Riemann-Liouville fractional integral several researchers have also provided such results including Hermite-Hadamard-type inequalities by using the definition of ðk, sÞ-fractional integrals [19][20][21][22][23].

ðk, sÞ-Fractional Integrals and Differentials of S-Function
In this section, we develop ðk, sÞ-fractional integration and differentiation operators containing S-function as its kernel. Also, we study ðk, sÞ-fractional calculus; we define integral operators in terms of ðk, sÞ as follows.
where x > ρ. Substituting s = 0, then (20) reduces to the operator In particular, the integral operator in (21) decreases to the well-known R-L fractional integral operator defined as τ = 0 and k = 1.
The integral operator is described as s k I ς a+ and s k I ς a− , and also, ðk, sÞ-fractional order left side and right side fractional differential operators are described as ðk, sÞ-D ς ρ+,k and D ς ρ−,k . Also, we used left-and right-sided R-L ðk, sÞ-fractional integral operators s k I ς a+ and s k I ς a− . Similarly, the left-and rightsided R-L ðk, sÞ-fractional differential operators are s k D ς a+,k and s k D ς a− , respectively. By using the Lebesgue measurable integral of a real or complex valued function, we can describe both R-L ðk, sÞ-fractional integral operators. The Lebesgue measurable integral of a valued function that is denoted and defined as real or complex form Definition 2. For ψðxÞ ∈ Lðρ, νÞ, ς ∈ ℂ, RðςÞ > 0, k > 0, then we define the R-L left-sided ðk, sÞ-fractional integral operator of order ς as The R-L right-sided ðk, sÞ-fractional integral operator of order ς is defined as Definition 3. For k > 0 ; s ∈ Rf−1g ; ς ∈ ℂ, RðςÞ > 0 and n = ½RðςÞ + 1, then the R-L left-and right-sided ðk, sÞ-fractional differential operators are defined as respectively. Substituting k = 1 and s = 0, then the R-L left-and right-sided ðk, sÞ-fractional integrals and derivatives will reduce to the well-known R-L left-side and rightside fractional integrals and derivatives; see [35,36].
Proof. We obtain from equation (8) s k I Journal of Function Spaces x s d dx which by applying the relation given by (2) which is the desired proof.
and using (38), this take the following form: Applying (32), we have This completes the desired proof. Now, to prove (40), we have Using (28), we get which completes the desired proof.