Extended Conformable K?Hypergeometric Function and Its Application

Abstract

The extended conformable k-hypergeometric function finds various applications in physics due to its ability to describe complex mathematical relationships arising in different physical scenarios. Here are a few instances of its uses in physics, including nuclear physics, fluid dynamics, quantum mechanics, and astronomy. The main objectives of this paper are to introduce the extended conformable k-hypergeometric and confluent hypergeometric functions by utilizing the new definition of the (?, k)-beta function and studying its important properties, like integral representation, summation formula, derivative formula, transform formula, and generating function. Also, introduce the extension of the Riemann-Liouville fractional derivative and establish some results related to the newly defined fractional operator, such as the Mellin transform and relations to extended (?, k)-hypergeometric functions.