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Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function. (English) Zbl 1144.26010

The authors consider the following generalization of the Mittag-Leffler function

E β,γ δ (z)= n=0 (δ) n Γ(βn + γ)n!z n ,

introduced in 1971 by T. R. Prabkahar. In its turn, this function is a special case of the generalized Wright function. The authors prove several formulas for the Riemann-Liouville fractional integrals and derivatives of the function E β,γ δ with a power weight. The main significance of these formulas is that the result is obtained also in terms of a generalized Mittag-Leffler function of such a type, with a power weight as well.

26A33Fractional derivatives and integrals (real functions)
33C20Generalized hypergeometric series, p F q