Composition Formulae for the k-Fractional Calculus Operators Associated with k-Wright Function

Fractional calculus was introduced in 1695, but in the last two decades researchers have been able to use it properly on the account of availability of computational resources. In many areas of application of fractional calculus, the researchers found significant applications in science and engineering. In the literature, many applications of fractional calculus are available in astrophysics, biosignal processing, fluid dynamics, nonlinear control theory, stochastic dynamical system, and so on. Also, a number of researchers [1–10] have studied in-depth level of properties, applications, and various directions of extensions of Gauss hypergeometric function of fractional integration. Recently, in a series of research publications on generalized classical fractional calculus operators, research by Mubeen and Habibullah [11] has been published on the integral part of the Riemann-Liouville version and its applications; an alternative definition for the k-Riemann–Liouville fractional derivative was introduced by Dorrego [12]. (e leftand right-hand operators of Saigo k-fractional integration and differentiation associated with the k-Gauss hypergeometric function defined by Gupta and Parihar [13] (see also [14]) are as follows:


Introduction and Preliminaries
Fractional calculus was introduced in 1695, but in the last two decades researchers have been able to use it properly on the account of availability of computational resources. In many areas of application of fractional calculus, the researchers found significant applications in science and engineering. In the literature, many applications of fractional calculus are available in astrophysics, biosignal processing, fluid dynamics, nonlinear control theory, stochastic dynamical system, and so on. Also, a number of researchers [1][2][3][4][5][6][7][8][9][10] have studied in-depth level of properties, applications, and various directions of extensions of Gauss hypergeometric function of fractional integration.
Recently, in a series of research publications on generalized classical fractional calculus operators, research by Mubeen and Habibullah [11] has been published on the integral part of the Riemann-Liouville version and its applications; an alternative definition for the k-Riemann-Liouville fractional derivative was introduced by Dorrego [12]. e left-and right-hand operators of Saigo k-fractional integration and differentiation associated with the k-Gauss hypergeometric function defined by Gupta and Parihar [13] (see also [14]) are as follows: where 2 F 1,k ((τ, k), (δ, k); (c, k); x) is the k-Gauss hypergeometric function defined by [11] for x ∈ C, |x| < 1, R(c) > R(δ) > 0: e corresponding fractional differential operators have their respective forms as Remark 1. If we set k � 1 in equations (1), (2), (4), and (5), operators reduce to Saigo's fractional integral and derivative operators stated in [5], respectively. Now, we consider the following basic results for our study.

Saigo k-Fractional Integration in terms of k-Wright Function
In this section, we present the composition formulas of k-fractional integrals (1) and (2), involving the k-Wright function. with k-Wright function, then the following equation holds true: Proof. We indicate the R.H.S. of equation (22) by I 1 , and invoking equation (10), we obtain Now applying equation (6), we get Now, interpreting definition (10) on the aforementioned equation, we arrive at the desired result (22). □ Theorem 2. Let τ, δ, c, ∈ C; k ∈ R + , c, ε i , ς j ∈ R(ε i , ς j ≠ 0; i � 1, 2, . . . , p; j � 1, 2, . . . , q), and v > 0 such that R(τ) > 0 (11) is satisfied and I τ,δ,c − ,k be the right-sided integral operator of the generalized k-fractional integration associated with k-Wright function, then the following equation holds true: Proof. e finding is similar to that of eorem 1. So, we omit the details. □ Corollary 1. By assuming k � 1 in (22) and (25), the result becomes

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and

Saigo k-Fractional Differentiation in terms of k-Wright Function
In this section, we present the composition formulas of k-fractional derivatives (4) and (5), involving the k-Wright function. (11) is satisfied and D τ,δ,c 0+,k be the left-sided differential operator of the generalized k-fractional differentiation associated with k-Wright function, then the following equation holds true: Proof. For simplicity, let I 2 denote the left side of (28). Using definition (10), we obtain Now, applying equation (8), we obtain In accordance with (10), the required result is (28). is completes the proof of eorem 3. 2, . . . , p; j � 1, 2, . . . , q), and v > 0 such that R(τ) > 0, (11) is satisfied and D τ,δ,c − ,k be the right-sided differential operator of the generalized k-fractional differentiation associated with k-Wright function, then thefollowing equation holds true:

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Proof.

Special Cases and Concluding Remarks
Being very general, the results given in (22), (25), (28), and (31) can yield a wide number of special cases by assigning some appropriate values to the parameters involved. Now, as shown in the following, we are explaining a few corollaries.

Corollary 3.
If we put p � 1 and q � 2 in eorems 1 and 2, then we get the following interesting results on the right known as k-Mittag-Leffler function: and Corollary 4. If we put p � 1 and q � 3 in eorems 1 and 2, then we get the following interesting results on the right known as k-Bessel function of the first kind: and 6 Journal of Mathematics Corollary 5. If we put p � 3 and q � 3 in eorems 1 and 2, then we get the following results on the right known as k-hypergeomrtric function: Corollary 6. If we put p � 1 and q � 2 in eorems 3 and 4, then we get the following results on the right known as k-Mittag-Leffler function: Corollary 7. If we put p � 1 and q � 3 in eorems 3 and 4, then we get the following results on the right known as k-Bessel function of the first kind: Journal of Mathematics 7 Corollary 8. If we put p � 3 and q � 3 in eorems 3 and 4, then we get the following results on the right known as k-hypergeomrtric function: (45) e advantage of the generalized k-fractional calculus operators, which are also called by many authors as the general operator, is that they generalize Saigo's fractional calculus operators and classical Riemann-Liouville (R-L) operators. For k ⟶ 1, operators (1), (2), (4), and (5) reduce to Saigo's [5] fractional integral and differentiation operators. If we take δ � − τ, (1), (2), (4), and (5) reduce the operators to k-Riemann-Liouville as follows: Due to the most general character of the k-Wright function, numerous other interesting special cases from (22), (25), (28), and (31) can be given in the form of k-Struve function, k-Wright-type function, and many more, but due to lack of space, they are not represented here.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.