An Extension of Caputo’s k-Fractional Derivative Operator and its Applications

Abstract

This research introduces a novel extension of the Caputo fractional derivative operator, characterized by a new parameter k >0. We establish several properties of the Caputo k-fractional derivative operator and present a series of results related to its application. Additionally, we extend the concept of k-hypergeometric functions and derive their integral representations utilizing the k-fractional derivative operator. Our work further includes the development of linear and bilinear generating relations for the extended k-hypergeometric functions, as well as the Mellin transform of selected extended k-fractional derivatives.

Keywords: Caputo fractional derivative operator; k-fractional derivative operator; k-hypergeometric functions; linear generating relations; mellin transform; extended k-fractional derivatives.

2010 Mathematics Subject Classification. 26A30; 26A45; 26A48