FRACTIONAL CALCULUS OPERATORS OF THE GENERALIZED EXTENDED MITTAG-LEFFLER FUNCTION AND RELATED JACOBI TRANSFORMS

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) $.