Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions.

*(English)*Zbl 1213.26011This is a survey of fractional calculus based on the fractional derivatives of the form

They coincide with the usual Riemann-Liouville derivatives ${D}_{a\pm}^{\alpha}f\left(x\right)$ up to finite-dimensional terms. Various known properties of such derivatives are presented together with their further developments and a number of applications.

Historical remark: Derivatives of form (1) and more general ones were first introduced and studied by *M. Dzherbashyan* and *A. Nersesyan* [Dokl. Akad. Nauk SSSR 121, 210–213 (1958; Zbl 0095.08504); Izv. Akad. Nauk Arm. SSR, Ser. Fiz.-Mat. Nauk 11, No.5, 85–106 (1958; Zbl 0086.05701)].

Reviewer: Stefan G. Samko (Faro)

##### MSC:

26A33 | Fractional derivatives and integrals (real functions) |

33C20 | Generalized hypergeometric series, ${}_{p}{F}_{q}$ |

33E12 | Mittag-Leffler functions and generalizations |

47B38 | Operators on function spaces (general) |

47G10 | Integral operators |