Composition Formulae for the 
 k
 -Fractional Calculus Operator with the 
 S
 -Function

In this study, the S-function is applied to Saigo’s 
 
 k
 
 -fractional order integral and derivative operators involving the 
 
 k
 
 -hypergeometric function in the kernel; outcomes are described in terms of the 
 
 k
 
 -Wright function, which is used to represent image formulas of integral transformations such as the beta transform. Several special cases, such as the fractional calculus operator and the 
 
 S
 
 -function, are also listed.


Introduction and Preliminaries
Fractional calculus was first introduced in 1695, but only in the last two decades have researchers been able to use it efficiently due to the availability of computing tools. Significant uses of fractional calculus have been discovered by scholars in engineering and science. In literature, many applications of fractional calculus are available in astrophysics, biosignal processing, fluid dynamics, nonlinear control theory, and stochastic dynamical system. Furthermore, research studies in the field of applied science [1,2], and on the application of fractional calculus in real-world problems [3,4], have recently been published. A number of researchers [5][6][7][8][9][10][11][12][13][14][15] have also investigated the structure, implementations, and various directions of extensions of the fractional integration and differentiation in detail. A detailed description of such fractional calculus operators, as well as their characterization and application, can be found in research monographs [16,17].

Saigo k-Fractional Integration in Terms of k-Wright Function
In this section, the results are displayed based on the k-fractional integrals associated with the S-function. (17) is satisfied and I ϑ,ς,c 0+,k is the left-sided integral operator of the generalized k-fractional integration associated with S-function, then (18) holds true: Proof. We indicate the R.H.S. of equation (18) by I 1 ; invoking equation (10), we have Now, applying equation (6) and (11), we obtain Using (12) and some important simplifications on the above equation, we obtain Interpreting the definition of Wright hypergeometric function (16) on the above equation, we arrive at the desired result (18). (17) is satisfied and I ϑ,ς,c − ,k is the right-sided integral operator of the generalized k-fractional integration associated with S-function, then (22) holds true: Proof. e proof is parallel to that of eorem 1. erefore, we omit the details. □ e results given in (18) and (22), being very general, can yield a large number of special cases by assigning some suitable values to the involved parameters. Now, we demonstrate some corollaries as follows. (18) leads to the subsequent result of S-function: (18), we obtain the subsequent result in term of S-function as Corollary 3. If we set ε � 1, c ′ � 1, and k � 1, in equation (18), we obtain the following formula: Journal of Mathematics Corollary 4. Letting p � q � 0 in equation (22), then Corollary 5. Setting ε � 1, k � 1, then equation (22) becomes Corollary 6. If we put ε � 1, c ′ � 1, and k � 1 in equation (22), then equation becomes

Saigo k-Fractional Differentiation in Terms of k-Wright Function
In this section, the results are displayed based on the k-fractional derivatives associated with the S-function. (17) is satisfied and D ϑ,ς,c 0+,k is the left-sided differential operator of the generalized k-fractional integration associated with S-function, then (29) holds true: Proof. For the sake of convenience, let the left-hand side of (29) be denoted by I 2 . Using definition (10), we arrive at 6 Journal of Mathematics Now, applying equation (8) and (11), we obtain Using (12) and simplifications on the above equation, we obtain In accordance with (16), we obtain the required result (29). is completed the proof of eorem 3. 1, 2, . . . , p), b j (j � 1, 2, . . . , q), R(ϑ ′ ) > kR(ε), and p < q + 1. If condition (17) is satisfied and D ϑ,ς,c − ,k is the rightsided differential operator of the generalized k-fractional integration associated with S-function, then (33) holds true:

□
Proof. e proof is parallel to that of eorem 3. erefore, we omit the details. □ e results given in (29) and (33) are reduced as special cases by assigning some suitable values to the involved parameters. Now, we demonstrate some corollaries as follows.

Image Formulas Associated with Integral Transforms
In this section, we establish some theorems involving the results obtained in previous sections pertaining with the integral transform. Here, we defined k-beta function as follows. e k-beta function [32] is defined as ey have the following important identities: Now, we define k-beta function in the form (42) and R(ε + c − ς) > 0; then, the leading fractional order integral holds true: Proof. Let I 3 be the left-hand side of (43), and using (42), we have which, using (10) and changing the order of integration and summation, is valid under the conditions of eorem 1 and yields From Lemma 1 and substituting (41) in (45), we obtain Journal of Mathematics Using the definition of (16) in the right-hand side of (46), we arrive at result (43).
, with R(ς) ≠ R(c); then, the following fractional integral holds true: Proof. e proof is similar of eorem 5. erefore, we omit the details.
, and R(ε + c + ς) > 0; then, the following fractional derivative holds true: Proof. Let I 4 be the left-hand side of (48), and using the definition of Beta transform, we have which, using (10) and changing the order of integration and summation, is reasonable under the conditions of eorem 3 and yields From Lemma 3 and substituting equation (41) in (50), we obtain Γ k b q k + nk Γ k (ε + c + vn)Γ k (ε + ς + n − nk + vn)Γ k (g + h + vn)n! cx v+1/k− 1 n . (51) Using the definition of (16) in the above equation, we obtain the required result (48). is completed the proof of eorem 7.

Conclusion
e strength of generalized k-fractional calculus operators, also known as general operators by many scholars, is that they generalize classical Riemann-Liouville (R-L) operators and Saigo's fractional calculus operators. For k ⟶ 1, operators (1) to (5) reduce to Saigo's [9] fractional integral and differentiation operators. If we set δ � − ϑ, operators (1) to (5) reduce to k-Riemann-Liouville operators as follows: On the account of the most general character of the S-function, numerous other interesting special cases of results (18), (22), (29), 2and (33) can be obtained, but for lack of space, they are not represented here.

Data Availability
No data were used to support this study.

Conflicts of Interest
ere are no conflicts of interest regarding the publication of this article.