Some fractional operators with the generalized Bessel-Maitland function (Q2194587)

Summary: 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 \(\mathcal{Z}_{\nu, \eta, \rho, \gamma, w, a^+}^{\mu, \xi, m, \sigma}\phi\) and its inverse operator \(D_{\nu, \eta, \rho, \gamma, w, a^+}^{\mu, \xi, m, \sigma}\phi \), 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.