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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

${E}_{\beta ,\gamma }^{\delta }\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\delta \right)}_{n}}{{\Gamma }\left(\beta {n}_{+}\gamma \right)n!}{z}^{n},$

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 ${E}_{\beta ,\gamma }^{\delta }$ 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.

MSC:
 26A33 Fractional derivatives and integrals (real functions) 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$