Prabhakar function and unified fractional kinetic equation in bicomplex space

Abstract: The Mittag-Leffler type functions arise naturally in the solution of fractional order integral and differential equations, especially in the investigations of the fractional generalization of the kinetic equation. This article introduces a bicomplex extension of the Prabhakar function, a generalization of the Mittag-Leffler function commonly used in fractional calculus. We explore the analyticity and determine the region of convergence for this new bicomplex Prabhakar function. Several fundamental properties are established, including its integral representations, recurrence formulas, and differential relations. Furthermore, we compute the bicomplex Laplace and Mellin transforms of the function, which are useful for solving differential and integral equations. Finally, we analyze a fractional kinetic equation where the bicomplex Prabhakar function appears both in the equation and in its solution, demonstrating its applicability in complex systems involving fractional dynamics.