Abstract:
This article presents a systematic investigation of a new class of Mittag-Leffler-type functions in three variables. These functions are a natural and significant extension of the classical Mittag-Leffler function, and are constructed to correspond analogously to the well-known Lauricella hypergeometric functions of three variables. Our study comprehensively explores the fundamental properties and analytical characteristics of these threevariable functions. A primary focus is the establishment of their precise interrelationships with other existing extensions and generalizations of the classical Mittag-Leffler function, thereby situating them within the broader landscape of special functions. Key analytical findings presented in this work include: The derivation of the exact three-dimensional regions of convergence for the series defining these functions. The formulation of elegant Euler-type integral representations, which provide a powerful tool for further analysis. A detailed exploration of their integral transforms, specifically the derivation of both one-dimensional and three-dimensional Laplace transforms.The examination of their intimate connections with fractional calculus, demonstrating their natural emergence as kernels and solutions in the context of the Riemann-Liouville fractional integral and differential operators. Furthermore, we delve into the associated differential equations, showing that these Mittag-Lefflertype functions serve as solutions to specific systems of partial differential equations. This work not only enriches the theory of special functions but also provides a robust mathematical framework for potential applications in fractional differential equations, anomalous diffusion, and other areas of mathematical physics.
Keywords:extended Mittag-Leffler type function, hypergeometric function, special (or higher transcendental) function, Lauricella function, integral representation, system of partial differential equation, one- and three-dimensional Laplace transform, Riemann-Liouville fractional integral, Riemann-Liouville fractional derivative, Appell and Kampé de Fériet functions, Srivastava-Daoust hypergeoemetric function.
Fulltext:
PDF file (615 kB)