Generalized Mittag-Leffler-type function of arbitrary order and its properties related to integral transforms and fractional calculus

This paper introduces a novel generalization of the Mittag-Leffler function, delving into its fundamental characteristics. The analysis encompasses a thorough exploration of its properties, including the derivation of recurrence relations, differential formulas, and various integral representations such as the Euler, Laplace, Mellin, Whittaker, and Mellin–Barnes transforms. Furthermore, the study establishes connections to other significant special functions, expressing the new generalization in terms of the Fox-Wright function, the generalized hypergeometric function, and the H-function. The paper also defines associated fractional integral and differential operators, highlighting the function’s relevance to fractional calculus. Several noteworthy special cases are derived from the main results, demonstrating the breadth and adaptability of this new function. This research provides a comprehensive framework for understanding the properties of this generalized Mittag-Leffler function and suggests its potential for applications in diverse areas, particularly within the realm of fractional analysis and its related fields.