Umbral insights into a hybrid family of hypergeometric and Mittag-Leffler functions

TL;DR

This paper introduces a new hybrid family of special functions combining hypergeometric and Mittag-Leffler functions using the umbral approach, providing new formulas, transforms, and insights into their properties.

Contribution

It employs the umbral method to define and analyze a novel hybrid family of functions, deriving their generating functions, differential relations, and integral transforms.

Findings

Derived generating functions and explicit representations.

Established Laplace and Sumudu transforms.

Analyzed zeros distribution and graphical patterns.

Abstract

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 formulae, and specific integral formulae. 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.