THE ( k , s )-FRACTIONAL CALCULUS OF CLASS OF A FUNCTION

In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.


Introduction
Fractional calculus has gained appreciable fame and significance due to its numerous and boundless applications in Science; more particularly in engineering. For the recent development in the field of fractional calculus, one can refer to [5,13,17] and [6,7]. One more direction to such study was proposed by Atangana and Baleanu [1] by introducing derivatives which are based upon the generalized Mittag-Leffler function. Integral inequalities are considered to be of highly importance because they are useful in the research of different subjects of differential and integral equations (see [8]). We begin with the work of Diaz and Pariguan [3] which is defined as: (p) n,k = p (p + k), ..., (p + (n − 1)k) (n ∈ N, p ∈ C), 1, (n = 0, p ∈ C), (1.1) and where η ∈ C, k > 0, (z) > 0. They also define the following relations: and A new generalization of the k-fractional integral has been defined by Mubeen and Habibullah [11] as follows: (1.5) Clearly, it is observed that when k = 1 (1.5) coincide with the result of Riemann-Liouville fractional integration formula (see [5]). In fact, the following particular case: (1.6) In the same paper, they also define the following results: and Sarikaya et al. [16] have developed the (k, s)-fractional integral of order µ > 0 is defined by: . Also, they define, For more details about (k, s)-fractional integrals interesting readers can refer to [2,12,19]. Recently Nisar et al., [9,10] defined the following left and right sided (k, s)-fractional integral and differential operators of order µ as: and respectively. Substituting k = 1 and s = 0, then the above relation will coincide to the left and right sided (k, s)-fractional integral and derivatives see ( [5], [7]). The left and right sided (k, s)-fractional derivative operator s k D µ a+ defined in (1.12) is generalized by (k, s)-fractional derivative operator is denoted by s k D µ,ν a+ , where µ is the order such that 0 < ν < 1, we define as Obviously, when v = 0 then (1.15) approaches to the (k, s)-fractional derivatives operator s k D µ a+ (1.13) for n = 1. They [9] also defined the following lemma as: Lemma 1.1. For k > 0, with x > ρ, 0 < ν < 1 and (λ) > 0, then the following result for (k, s)-fractional derivatives operator s k D µ a+ in (1.15) hold true: (1.16) For the present investigation, we consider the following class of functions recently defined by Tunc et al. [20] σ,k where the coefficients σ(m)(m ∈ N 0 = N ∪ {0} is a bounded sequence of positive real numbers R + .

Generalized (k, s)-fractional integrals and differentials formulas
In this section, we present the generalized (k, s)-fractional integrals and differentials formulas involving a class of a function σ,k ρ,λ (x) as defined in (1.17). In this continuation of the study of generalized kfractional calculus, we define the following fractional integral operator.
By putting s = 0, then (2.1) can be written as see [20]. Similarly, when ω = 0 and k = 1 then (2.2) turns to: To prove the generalized (k, s)-fractional integral and differential formulas of a class of a function, we first prove the following derivative formula of class of functions.
Lemma 2.1. For k > 0, the following result holds true: Proof. Let S 1 be the L. H. S. of (2.4) then Changing the order of summation and differentiation, we have Using (1.4), we have . (2.7) Using (2.7) in (2.6), we get (2.8) Thus by using (2.8) in (2.5), we get the following required result.
The following theorem shows the (k, s)-fractional integral formulas of class of functions.
Proof. Let S 2 be L. H. S. side of (2.10) then Changing the order of summation and integration, we have Substituting τ s+1 = a s+1 + y(x s+1 − a s+1 ) in (2.11), then this implies dy, when τ → a, ⇒ y → 0 and τ → x, ⇒ y → 1, we have Next, we prove the (k, s)-fractional differential formula of class of functions.
This can be written as By applying (1.16), we get This completes the desired proof.
Next, we prove the result between (k, s)-fractional integral operator defined in (1.11) and the integral operaor defined in (2.1).
By changing the order of integration, we obtain s k I µ a+ s k ε ω,σ a+;ρ,λ f (x) Substituting τ s+1 − u s+1 = t s+1 , this implies τ s dτ = t s dt. Therefore (3.3) can be written as By the use of (1.11) and applying (2.10), we have s k I µ a+ s k ε ω,σ a+;ρ,λ f (x) Thus, we get To prove the second part, consider the right part of (3.2) and applying (2.1), we have s k ε ω,σ By changing the order of integration, we obtain s k ε ω,σ a+;ρ,λ s k I µ a+ f (x) Again making the use of (1.11) and applying (2.10), we have s k ε ω,σ Thus, we get This completes the desired proof.

Special Cases
If we consider k = 1 in our main results, then we obtain the following generalized fractional integral and differential formulas of (1.18).
Similarly, we can prove generalized fractional integral and differentials formulas for the remaining cases which are discussed in subsection 1.1.