Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

Abstract

We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann-Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.

arXiv comment14 pages; accepted for publication in Communications in Nonlinear Science and Numerical Simulation

References in corpus (8)

• On some new properties of fractional derivatives with Mittag-Leffler kernel
• The Prabhakar or three parameter Mittag--Leffler function: theory and application
• Hilfer-Prabhakar Derivatives and Some Applications
• Models of dielectric relaxation based on completely monotone functions
• Prabhakar-like fractional viscoelasticity
• Some Properties of Prabhakar-type Fractional Calculus Operators
• A comment on some new definitions of fractional derivative
• Fractional calculus and generalized Mittag-Leffler type functions

Cited by in corpus (7)

• On fractional calculus with general analytic kernels
• On some analytic properties of tempered fractional calculus
• General fractional calculus and Prabhakar's theory
• A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators
• A complex analysis approach to Atangana-Baleanu fractional calculus
• Mittag-Leffler functions in superstatistics
• Infinite series representation of fractional calculus: theory and applications