A Family of Hybrid Functions Generated by the Composition of Bessel and Mittag–Leffler Functions

In this paper, we employ a symbolic technique to introduce a new family of Mittag–Leffler–Bessel functions (MLBFs), formed by compositionally combining the classical Bessel functions of the first kind with the three-parameter Mittag–Leffler function. We establish their fundamental analytic structure by presenting the generating function, series expansions, and symbolic operational rules. Recurrence relations and differential equations satisfied by the MLBFs are derived, followed by explicit series representations and several integral formulas. Using Saigo fractional operators, we investigate fractional calculus properties associated with this new class of hybrid functions. An application to a fractional kinetic equation illustrates the utility of MLBFs in fractional dynamical models. Furthermore, we obtain closed-form formulas for half-integer orders, providing simplified representations useful for computation.