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Zbl 1213.26011
Tomovski, Živorad; Hilfer, Rudolf; Srivastava, H.M.
Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. (English)
[J] Integral Transforms Spec. Funct. 21, No. 11-12, 797-814 (2010). ISSN 1065-2469; ISSN 1476-8291/e

This is a survey of fractional calculus based on the fractional derivatives of the form $$D_{a\pm}^{\alpha,\beta}f(x)=\pm I_{a\pm}^{\beta(1-\alpha)}\frac{d}{dx} I_{a\pm}^{(1-\beta)(1-\alpha)}f(x).\eqno(1)$$ They coincide with the usual Riemann-Liouville derivatives $D_{a\pm}^{\alpha}f(x)$ up to finite-dimensional terms. Various known properties of such derivatives are presented together with their further developments and a number of applications. \par {Historical remark}: Derivatives of form (1) and more general ones were first introduced and studied by {\it M. Dzherbashyan} and {\it 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)].
[Stefan G. Samko (Faro)]

MSC 2000:: *26A33 Fractional derivatives and integrals (real functions)
33C20 Generalized hypergeometric series
33E12 Mittag-Leffler functions and generalizations
47B38 Operators on function spaces
47G10 Integral operators

Keywords: Riemann-Liouville fractional derivative operator; generalized Mittag-Leffler function; Hardy-type inequalities; Laplace transform method; Volterra differintegral equations; fractional differential equations; fractional kinetic equations; Lebesgue integrable functions; Fox-Wright hypergeometric functions

Citations: Zbl 0095.08504; Zbl 0086.05701

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