Generalized Mittag-Leffler Function: Properties in Fractional Calculus and Integral Transforms

In this work, we introduce a new generalized Mittag-Leffler function defined via the extended Beta function and establish its integral and differential representations. Several fundamental properties are derived, including differentiation formulas and the Beta, Laplace, and Mellin transforms, together with relationships to the Wright function and generalized hypergeometric functions. Furthermore, the behavior of the associated Riemann-Liouville fractional integrals and derivatives of the proposed function is investigated. A number of interesting special cases are also presented to illustrate the generality and unifying nature of the main results. In the final section, we discuss potential applications of the newly defined Mittag-Leffler function, demonstrate its use in solving a fractional kinetic equation, and outline possible directions for future research.