Asian Journal of Mathematics and Computer Research

This paper establishes two image formulae for generalized fractional differentiation involving an extended Mittag-Leffler type function. The work uses Marichev-Saigo-Maeda fractional differential operators, whose kernels involve Appell’s function, and expresses the derived identities in terms of a generalized Wright hypergeometric function. The preliminary section recalls the extended Mittag-Leffler type function, the associated extended beta function, the Wright hypergeometric function and relevant special cases, thereby preparing the notation and parameter conditions used in the main results. Two principal theorems are then formulated: the first concerns the left-sided generalized fractional differential operator, whereas the second concerns the corresponding right-sided operator. In each case, the proof applies the definition of the extended Mittag-Leffler type function, the relevant Marichev-Saigo-Maeda operator and known power-function formulae, followed by the rearrangement of terms into the generalized Wright hypergeometric form. Several corollaries are obtained by specifying parameter values and by reducing the extended Mittag-Leffler type function to known forms. These reductions show that previously reported fractional-calculus identities can be recovered as special cases. The results therefore provide a unified presentation of related fractional differential formulae within the stated analytic setting. The study is theoretical and contributes a structured set of fractional differential identities for extended special functions only.