A New Extension of Extended Caputo Fractional Derivative Operator

Abstract

Abstract. Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study.

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“…Some properties of these generalized beta are investigated, Kumar confluent, Appell's and Lauricella's hypergeometric functions, most of which are analogous with the classical and other related generalized beta functions. These generalized functions can be used to study theory of fractional integral and differential calculus (see for example, Pucheta, 2017;Rahman et al, 2018;Shadab et al, 2018;Nisar et al, 2019;Singhal and Mittal, 2020) and in the provision of the extended special function such as Mittag-Leffler, Bessel-Maitland and Wright (refer to, Khan et al, 2020a, b).…”

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