Derivatives of Mittag-Leffler functions: theory, computation and applications

Properties of Mittag-Leffler (ML) functions, and their fundamental role in fractional calculus, are well known. In contrast, derivatives of these functions, along with their properties, applications, and computational aspects, have been less thoroughly explored. This paper presents an in-depth analysis of the derivatives of ML functions. Starting from their relationships with different instances of the same functions and fractional derivatives of some elementary functions, we describe their asymptotic behavior for large arguments. This is of importance to analyze the behavior of solutions of multi-term fractional-order differential equations, for which the representation in terms of derivatives of ML functions is shown. In particular, we emphasize the role of a special instance of ML derivatives introduced by Podlubny in his 1999 book. After an efficient procedure for their accurate numerical computation is described, some applications of the derivatives of ML functions in circuits and systems are discussed.