Fractional calculus and generalized Mittag-Leffler type functions

Christian Lavault (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

In this paper, the generalized fractional integral operators of two generalized Mittag-Leffler type functions are investigated. The special cases of interest involve the generalized Fox–Wright function and the generalized M-series and K-function. In the next Section 2 we first recall some generalized fractional integral operators among the most widely used in fractional calculus. Section 3 is devoted to the definitions of M-series and K-function and their relations to special functions. In Sections 4 and 5, effective fractional calculus of the generalized M-series and the K-function is carried out. The last section briefly concludes and opens up new perspectives. The results established herein generalize recent properties of generalized Mittag-Leffler type functions using left-and right-sided generalized fractional differintegral operators. The note results also in important applications in physics and mathematical engineering.

Mots clés: Fox–Wright psi function; Generalized hypergeometric function; M -series and K-function; Mittag-Leffler type functions; Riemann–Liouville's; Saigo's and Saigo–Maeda's generalized fractional calculus operators

Domaines: Analyse fonctionnelle [math.FA]; Variables complexes [math.CV]; Analyse numérique [math.NA]; Physique mathématique [math-ph]

Christian Lavault. Fractional calculus and generalized Mittag-Leffler type functions. 2017. ⟨hal-01482060v2⟩