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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.