A NOVEL GENERALIZATION OF THE MITTAG-LEFFLER FUNCTION: PROPERTIES AND INTEGRAL TRANSFORMS
By
Naresh Bhati
Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur Rajasthan, India- 342005
Email: nareshbhati1978@gmail.com
(Received: May 23, 2025; In format: September 04, 2025; Revised: January 30, 2026; Accepted: June 10, 2026)
DOI: https://doi.org/10.58250/jnanabha.2026.56108
Abstract
In this paper, we introduce a new and more general form of the Mittag-Leffler function. This new generalization contains extra parameters that give the function greater flexibility and a richer analytical structure. After defining this generalized function, we study several of its basic and essential properties in detail. These include its convergence behaviour, different types of integral representations, and several important transforms such as the Laplace transform, Beta transform, Mellin transforms, and Whittaker transforms. We also show how this new generalized Mittag-Leffler function is related to many other known special functions in the literature. In addition, we examine how this function can be used to solve various fractional differential and integral equations. Overall, the results of this work extend many existing forms of the Mittag-Leffler function and bring them together into a unified framework. This provides a strong base for further theoretical research as well as practical applications in areas such as mathematical physics and engineering.
2020 Mathematical Sciences Classification: 33E12, 26A33, 35A22
Keywords and Phrases: Generalized Mittag-Leffler function, Generalized Hypergeometric Function, Foxs H-function.