Abstract

Our aim is to obtain certain image formulas of the $p$--extended Mittag-Leffler function $\mathcal{E}_{\alpha,\beta,p}^{\gamma}(z) $ by using Saigo's hypergeometric fractional integral and differential operators. Corresponding assertions for the classical Riemann-Liouville (R-L) and Erd\'elyi-Kober (E-K) fractional integral and differential operators are established. All the results are represented in terms of the Hadamard product of the $p$--extended Mittag-Leffler function $\mathcal{E}_{\lambda,\mu,p}^{\gamma}(z) $ and Fox-Wright function $_{r}\Psi_{s}(z)$. We also established Jacobi and its particular assertions for the Gegenbauer and Legendre transforms of the $p$--extended Mittag-Leffler function $\mathcal{E}_{\alpha,\beta,p}^{\gamma}(z) $.

Keywords and Phrases

Fractional Calculus operators, Fox-Wright function, Generalized hypergeometric function, Extended Mittag-Leffler function, Gegenbauer and Legendre transforms.

A.M.S. subject classification

Primary 26A33, 33B20, 33C20; Secondary 26A09, 33B15, 33C05.

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