An Analysis of Generalized Appell's and Lauricella's Hypergeometric Functions and Their Attributes

Building upon recent advancements by various researchers in the field of beta functions and hypergeometric functions with additional parameters, this paper introduces new generalizations of Appell functions F₁ and F₂, along with Lauricella's hypergeometric function F₃. These generalizations, inspired by the intriguing properties and potential applications uncovered in previous studies, provide a broader framework by incorporating additional parameters. We will establish specific properties for these generalized functions, including integral representations, transformation formulas, differential formulas, Mellin transforms and recurrence relations. Through this comprehensive analysis, we aim to provide a deeper understanding of the generalized Appell and Lauricella functions, highlighting their significance and potential applications in diverse mathematical and scientific contexts.