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Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. (English) Zbl 1213.26011
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. {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)].

26A33Fractional derivatives and integrals (real functions)
33C20Generalized hypergeometric series, ${}_pF_q$
33E12Mittag-Leffler functions and generalizations
47B38Operators on function spaces (general)
47G10Integral operators
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