Prabhakar function and unified fractional kinetic equation in bicomplex space

Source 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 of the bicomplex Prabhakar function and establish its region of convergence under the stated conditions. 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, the unified fractional kinetic equation is solved in terms of the bicomplex Prabhakar function.