Some Fractional Operators with the Generalized Bessel–Maitland Function

Department of Mathematics, University of Sargodha, Sargodha, Pakistan Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Pakistan Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia


Introduction
During the last few years, many types of research studies developed the class of generalized fractional integrals containing a variety of special functions [1][2][3][4][5]. Watson [6] discussed applications of the Bessel function with some fields of applied sciences, biology, chemistry, physical sciences, and engineering. e generalization and extensions of the Bessel-Maitland function [7][8][9][10][11][12][13][14] dealt with special cases that gave useful results in different areas of mathematics. e recent work in the field of fractional calculus theory, differential equations of the Mittag-Leffler function, Sturm-Liouville problems in theoretical sense, Gronwall's inequality, and exponential kernels of the differential operator [15][16][17][18] have found many applications in various subfields of mathematical analysis. e series representation of the Bessel-Maitland function [19] is defined as (− s) n n!Γ(αn + β + 1) e Saigo fractional integral operators are defined [21] for s > 0, a, c, d ∈ C, and R(a) > 0: Samko et al. [22] defined the Riemann-Liouville fractional operators for R(a) > 0 and n � [Re(a)] + 1 as e Gauss hypergeometric function defined by Saigo [21] for all a, c, d ∈ C, a ≠ 0, and |s| < 1 is where (b) n , (− d) n , and (a) n are Pochhammer's symbols.
Definition 3. An integral operator which involves generalized Bessel-Maitland function (19) as its kernel is defined for Remark 2. If we put w � 0 and replace ] by ] − 1, then it will become a left-sided Riemann-Liouville fractional integral operator. e new fractional operator (22) can be discussed to improve the results of some inequalities such as Polya-Szego inequality, Chebyshev inequality, and Hadamard inequality in the field of analysis.

Convergence and Boundedness of the New Fractional Integral Operator
In this section, we discuss the convergence and boundedness of the fractional integral operator involving the generalized Bessel-Maitland function as its kernel in the form of a theorem.
and m, ξ > R(μ) + σ; then, the following relation holds: where Proof. Let K n denote the nth term of (36); then, Hence, |K n+1 /K n | ⟶ 0 as n ⟶ ∞, and ξ, σ < ρ + Re(μ) which means that the right-hand side of (36) is convergent and finite under the given condition. e condition of boundedness of the integral operator (Z According to equations (19) and (22), we have

The Generalized Bessel-Maitland Function with Some Fractional Integral Operators
In this section, we derive some results of Saigo fractional integral operators with the generalized Bessel-Maitland function, and these results are established in terms of the Fox-Wright function. Also, we develop the composition of Riemann-Liouville operators with the generalized Bessel-Maitland function.

Riemann-Liouville Fractional Operators and Laplace Transform of the New Operator
In this section, we discuss the Riemann-Liouville fractional integral and differential operators with the fractional integral operator. Also, we developed a result which deals with the Laplace transform of the new fractional integral operator.

Inverse Operator with Some Special Functions
In this section, we discuss some applications of the inverse fractional operator. We derive some results of the inverse fractional operator with the Mittag-Leffler function and Bessel-Maitland function, and results can be seen in the form of Wright functions.

Conclusion
In this paper, we discussed some relations of generalized Bessel-Maitland functions and the Mittag-Leffler functions and also developed Saigo and Riemann-Liouville fractional integral operators with the generalized Bessel-Maitland function, and results can be seen in the form of Fox-Wright functions. Also, we established a new operator Z μ,ξ,m,σ ],η,ρ,c,w,a + ϕ and also discussed its convergence and boundedness. Moreover, we discussed the Riemann-Liouville fractional operator and the integral transform (Laplace) of the new operator and also developed some important applications of the left inverse operator.

Data Availability
No data were used to support this study since they are more of simulation.

Conflicts of Interest
e authors declare that they have no conflicts of interest.