M‐Parameter Mittag–Leffler function, its various properties, and relation with fractional calculus operators

Abstract

A number of Mittag–Leffler functions are defined in the literature which have many applications across various areas of physical, biological, and applied sciences and are used in solving problems of fractional order differential, integral, and difference equations. This paper aims to define the m‐parameter Mittag–Leffler function, which can be reduced to various already known extensions of the Mittag–Leffler function. Important properties like recurrence relations, differentiation formulae, and integral representations are discussed for the newly defined m‐parameter Mittag–Leffler function. Moreover, the new m‐parameter Mittag–Leffler function is expressed in terms of some well‐known special functions such as generalized hypergeometric function, Mellin‐Barnes integral, Wright generalized hypergeometric function, and Fox H‐function. The paper also deals with various integral transforms like Euler‐Beta, Whittaker, Laplace, and Mellin transforms of the m‐parameter Mittag–Leffler function. Fractional differential and integral operators are considered to highlight certain intriguing properties of the m‐parameter Mittag–Leffler function. We also use the above function to define a generalization of Prabhakar integral and discuss its properties. Further, the relation of the newly defined m‐parameter Mittag–Leffler function with various other functions such as exponential, trigonometric, hypergeometric, and algebraic functions is obtained and represented graphically using MATHEMATICA 12.