Abstract
In this article, we aim to develop new -fractional integral and differential operators containing -functions as kernels in a form of generalized -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.
1. 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 [3–7]. Some new results for -Hilfer fractional pantograph-type differential equation depending on -Riemann–Liouville integral are studied by Foukrach et al. [8]. In the frame of fractional derivatives, Alqahtani et al. [9] studied nonlinear -contractions on -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 symbols are well known from a number of sources related to the measurement of finite differences (see [12, 13]). In the literature, -fractional integral operators have recently been considered by different scholars. For this function, we begin the literature with the following properties. The Pochhammer -symbols and -gamma function were introduced by Diaz and Pariguan (see [14])
In the same paper, they spell out the relations
The -fractional integral is develop by [15] as
When we choose , then shows the result of the Riemann-Liouville (R-L) fractional integration formula.
We have
Formulas for -fractional integral are developed by [15] as
The R-L -fractional integral of order was elucidated by [16]. where and . 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 -fractional integrals, Gruss-type integral inequalities for generalized R-L -fractional integrals, and -R-L fractional integral inequalities for continuous random variables, analytical properties of -Riemann-Liouville fractional integral several researchers have also provided such results including Hermite-Hadamard-type inequalities by using the definition of -fractional integrals [19–23].
The -function is given by [24] , and .
The Pochhammer symbol defined in terms of gamma function as (also see [7])
For some more result on -function, the reader can refer to [25–29].
1.1. Special Cases
(i)Choosing in equation (9), it becomes generalized -Mittag-Leffler function, defined by Saxena et al. [30].
where . (ii)For , equation (9) yields
where . (iii)When , equation (9) yieldswhere . (iv)For , equation (14) yields -function defined by Sharma [31]where . (v)When (equation (16)), it reduces to generalized -series defined by [32] (see also [33])where . (vi)When (equation (18)), it reduces to generalized Mittag-Leffler function, defined by [34]where .
2. -Fractional Integrals and Differentials of -Function
In this section, we develop -fractional integration and differentiation operators containing -function as its kernel. Also, we study -fractional calculus; we define integral operators in terms of as follows.
Definition 1. If and , then where . Substituting , 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 and . The integral operator is described as and , and also, -fractional order left side and right side fractional differential operators are described as - and . Also, we used left- and right-sided R-L -fractional integral operators and . Similarly, the left- and right-sided R-L -fractional differential operators are and , respectively. By using the Lebesgue measurable integral of a real or complex valued function, we can describe both R-L -fractional integral operators. The Lebesgue measurable integral of a valued function that is denoted and defined as real or complex form
Definition 2. For , then we define the R-L left-sided -fractional integral operator of order as The R-L right-sided -fractional integral operator of order is defined as
Definition 3. For and , then the R-L left- and right-sided -fractional differential operators are defined as respectively. Substituting and , then the R-L left- and right-sided -fractional integrals and derivatives will reduce to the well-known R-L left-side and right-side fractional integrals and derivatives; see [35, 36].
Definition 4. The generalized -fractional derivative operator is denoted by where is the order such that and is defined as Obviously, when , then (27) reduces to the R-L -fractional derivative operators (24).
Lemma 5. For , the following result for -fractional derivative operator defined in (27) holds true: with , and .
Proof. We obtain from equation (8) which by applying the relation given by (2) yields which is the desired proof.
Theorem 6. For , the following result always holds true: where .
Proof. The proof is obvious by applying where .
Corollary 7. Choosing in equation (32), it becomes generalized -Mittag-Leffler function and we get a result earlier given by Nisar et al. [37].
Corollary 8. Choosing in equation (32) and using equation (12), we get
Corollary 9. Choosing in equation (32) and using (14) yield
Corollary 10. Choosing in equation (34) then -function converted in to -function and using (16) yield
Corollary 11. Choosing in equation (35) and using equation (16), it becomes generalized -series defined by Sharma and Jain [32]
Corollary 12. Choosing in equation (36), it becomes generalized Mittag-Leffler function [34]
Theorem 13. Suppose and , then
Proof. By using equation (7), we get Substituting , this implies ; we get This completes the proof of (38).
Now, to prove (39), 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.
Corollary 14. Choosing in equations (38), (39), and (40), it reduces to generalized -Mittag-Leffler function and we get a result earlier given by Nisar et al. [37].
Corollary 15. Choosing in equations (38), (39), and (40) and using equation (12), we get
Corollary 16. Choosing in equations (38), (39), and (40) and using (14) yield
Corollary 17. Choosing in equations (49), (50), and (51) then -function reduces to -function, and using (16), it yields
Corollary 18. Choosing in equations (52), (53), and (54) then S-function becomes -series defined by [32], and using (16), it yields
Corollary 19. Choosing in equations (55), (56), and (57) then -function turns into generalized Mittag-Leffler function defined by Mittag-Leffler [34], and using (16), it yields If we substitute in (38), (39), and (40), then we have some special results of the -function.
Corollary 20. Suppose and , , then If we substitute in (59), (60), and (61), then we have some special results of the -function.
Corollary 21. Suppose and , then
3. Some Properties of the Operator
Theorem 22. For and and , we have
Proof. From (20), Therefore, we have which completes the desired proof.
Theorem 23. Suppose let and and , then exit for any .
Proof. Assume that and such that for all . It is obvious that where As is measurable on , we can write So, we get Again, integral the above term, we have The function is then such that can be integrated into by Toneli’s theorem. Thus, by Fubini’s theorem is an integrable function on , as a function of . Thus, exists.
Theorem 24. For and and , the following result holds: for any .
Proof. From (20) and (24), we observe By changing the order of integration, we obtain Using (38), we have Thus, we get To prove the second part of the theorem, consider the RHS of (70); then, by applying (20), we get By interchanging the order of integration, we have Again, by making use of (24) and using (38), we get Thus, (74) and (77) complete the proof of (70).
4. Concluding Remarks
In this article, we develop some new results of -fractional integral and differential operators involving -function under the extension of left and right R-L -fractional integral and differential operators. Also, we find some special cases of functions like -Mittag-Leffler function, -function, and -series. If we set , then we obtained results of the generalized -Mittag-Leffler function given by [37], and if we set , the results obtained are reduced to the well-known results given by [38]. If we select and , then the results obtained are reduced to the Mittag-Leffler function’s well-known results.
Data Availability
No data were used to support this study.
Conflicts of Interest
There is no conflict of interest regarding the publication of this article.