On a new version of bicomplex Mittag-Leffler functions and their applications in fractional kinetic equations

The Mittag-Leffler function, often referred to as the ”queen function” of fractional calculus, plays a fundamental role in solving fractional differential equations across various disciplines, including physics and engineering. In this paper, we introduce a novel extension of the classical Mittag-Leffler function to the bicomplex domain, defining the bicomplex $k$-Mittag-Leffler function. We establish its fundamental properties, derive key representations and explore its analytical structure within bicomplex analysis. Furthermore, we investigate its interaction with the bicomplex $k$-Riemann–Liouville fractional integration and differentiation operators, demonstrating its effectiveness in solving bicomplex fractional differential equations. In particular, we provide explicit solutions to bicomplex fractional kinetic equations (FKE), showcasing the applicability of our proposed function. This generalization not only deepens the understanding of fractional calculus in bicomplex spaces but also lays the groundwork for future advancements in both theoretical research and applied mathematics. Our findings contribute to the growing body of work on special functions in bicomplex analysis and open new perspectives for applications in complex systems and mathematical physics.