Umbral Insights into a Hybrid Family of Hypergeometric and Mittag-Leffler Functions

The umbral approach provides methods for comprehending and redefining special functions. This approach is employed efficiently in order to uncover intricacies and introduce new families of special functions. In this article, the umbral perspective is adopted to introduce a hybrid family of hypergeometric and Mittag-Leffler functions. The umbral-operational procedures are used to derive the generating functions, explicit representations, differential recurrence formulas, and specific integral formulas. Further, the Laplace and Sumudu transforms for the hypergeometric Mittag-Leffler functions are established. The graphical representation and pattern for distribution of zeros for suitable values of parameters are also presented.