Abstract
In this paper, we aim to determine some results of the generalized Bessel–Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel–Maitland functions and the Mittag-Leffler functions are considered. We develop Saigo and Riemann–Liouville fractional integral operators by using the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. We establish a new operator Zν,η,ρ,γ,w,a+μ,ξ,m,σϕ and its inverse operator Dν,η,ρ,γ,w,a+μ,ξ,m,σϕ, involving the generalized Bessel–Maitland function as its kernel, and also discuss its convergence and boundedness. Moreover, the Riemann–Liouville operator and the integral transform (Laplace) of the new operator have been developed.
Highlights
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
An integral operator which involves generalized Bessel–Maitland function (19) as its kernel is defined for μ, ], η, w, c, ρ ∈ C, R(μ) > 0, R(]) ≥ − 1, R(η) > 0, R(w) > 0, R(c) > 0, R(ρ) > 0, ξ, m, σ ≥ 0, and m, ξ > R(μ) + σ as follows: s
Summary
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]. Discrete Dynamics in Nature and Society e Saigo fractional integral operators are defined [21]. Samko et al [22] defined the Riemann–Liouville fractional operators for R(a) > 0 and n [Re(a)] + 1 as. An integral operator which involves generalized Bessel–Maitland function (19) as its kernel is defined for μ, ], η, w, c, ρ ∈ C, R(μ) > 0, R(]) ≥ − 1, R(η) > 0, R(w) > 0, R(c) > 0, R(ρ) > 0, ξ, m, σ ≥ 0, and m, ξ > R(μ) + σ as follows:. If we put w 0 and replace ] by ] − 1, 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. Where the inverse operator (Dημ,],w,a+ φ)(s) is described and discussed by Polito and Tomovski in [26]
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