Abstract
The aim of this paper is to introduce a presumably and remarkably altered integral operator involving the extended generalized Bessel-Maitland function. Particular properties are considered for the extended generalized Bessel-Maitland function connected with fractional integral and differential operators. The integral operator connected with operators of the fractional calculus is also observed. We point out important links to known findings from some individual cases with our key outcomes.
1. Introduction and Preliminaries
The Bessel-Maitland function is a generalization of Bessel function introduced by Ed. Maitland Wright [1] through a series representation as follows:
In fact, Watson’s book [2] finds the application of the Bessel-Maitland function in the diverse field of engineering, chemical and biological sciences, and mathematical physics.
Further, Pathak [3] defined generalization of the Bessel-Maitland function in the form as follows:where ; , and is known as generalized Pochhammer symbol which is defined as
Since the implementation of the Bessel-Maitland function in 1983, a number of extensions and generalizations have been introduced and examined with different applications (see information in [4–9]).
By motivation of these investigations and applications of the Bessel-Maitland function, Suthar et al. [7] defined the generalized Bessel-Maitland function (2) in the following manner:which is known as extended generalized Bessel-Maitland function; here, is the extended beta function (see [10]).
For , (5) reduces to beta function (see, e.g., [11], Section 1.1).
Remark 1. (i)The particular case of equation (4), when , reduces to (2) and when , reduces to (1).(ii)When and in (4) reduce to the extended Mittag–Leffler function defined by Ozarslan and Yilmaz ([12], equation (4)).
Definition 1. The space of Lebesgue measurable of real or complex valued function for our study of the significance of fractional calculus is defined as follows:
Definition 2. The Riemann–Liouville (R-L) fractional integral operators and are defined respectively as (see, e.g., [13]) follows:where , and .
Definition 3. For and , the Riemann–Liouville fractional differential operators are defined by (see, e.g., [13])Also, of order and class with reference to which is the generalized form of (9) (see [13–15]) is defined as follows:On setting in (10), it reduces specified in (9) to the fractional differential operator.
We found the following baseline findings for our study.
Lemma 1 (Mathai and Haubold [16]). If , then
Lemma 2 (Srivastava and Manocha [17]). If a function is analytic and has a power series representation in the disc , then
Lemma 3 (Srivastava and Tomovski [18]). Let and . Then, the subsequent result holds true for as follows:
We also provided the subsequent established facts and rules in this article.
Fubini’s theorem (Dirichlet formula) (Samko et al. [15])
We define the following integral operator in terms of extended generalized Bessel function for and for our further analysis of fractional calculus, then the integral operatorwhere .
If we put to the operator, then (15) reduces
If , then (16) reduces the integral operator to the R-L fractional integral operator described in (7).
2. Integral Operators with Extended Generalized Bessel-Maitland Function in the Kernel
In this part, we consider the composition of the fractional integral and derivative of Riemann–Liouville and the fractional derivative of Hilfer with the extended generalized Bessel-Maitland function defined by (4).
Theorem 1. Suppose , , and , then the following result holds true:
Proof. Using (4), we see thatUsing the identity,and after simplifying, we haveFinally, it can be expressed by using (4) again, and we obtain
Corollary 1. Suppose , , and , then the following result holds true:
Theorem 2. If , then
Proof. (i)Using (4) and (7), we have By use of (11), we have This occupies in the (23) statement.(ii)On using (9), we have and using (23), this takes the following form: Applying (17), we get This completes the desired proof (24).(iii)By using (4), we obtain By applying (13), we getwhich brings in the necessary proof.
Corollary 2. If , then Theorem 2 reduces respectively to
3. Some Properties of the Operator
In this section, we derive several continuity properties of the generalized fractional integral operator.
Theorem 3. If , and , then
Proof. From (4) and (15), we obtainThis completes the desired proof.
Corollary 3. If , and , then
Theorem 4. If , and , thenwhere
Proof. From (4), (6), and (15), we haveBy exchanging the integration order and using Dirichlet formula (14), we haveSetting , we obtainThis can also be written asThis completes the desired proof.
Corollary 4. If , and , thenwhere
Theorem 5. If , and , for any function , then the result holds true:
Proof. From (7) and (15), we haveBy interchanging the order of integration and using (14), we haveSetting , we obtainBy applying (7) and (23), we getthus, using (15), we getTo demonstrate the second part, we start from the right side of (45), and using (7) and (15), we haveBy interchanging the order of integration and using (14), we obtainSetting , we haveAgain, by using (7) and applying (23), we getFinally, using (15), we obtainThus, (50) and (55) complete the desired proof of (45).
Corollary 5. If , and , for any function , then Theorem 5 takes the form:
4. Concluding Remark and Discussion
The newly defined integral operators involving the extended generalized Bessel-Maitland function is investigated here. Various special cases of the paper’s related results may be analyzed by taking appropriate values of the relevant parameters. For example, as given in remarks (i) and (ii), we obtain the undeniable result due to Gauhar et al. [19, 20]. For a number of other special cases, we refer to [21] and leave the findings to interested readers.
Data Availability
No data used to support the study.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the present investigation. All authors read and approved the final manuscript.