An Extended k-type Hypergeometric Functions

TL;DR

This paper introduces a new extended k-type hypergeometric function and explores its properties, including integral representations, differential formulas, and fractional calculus, expanding the mathematical framework of hypergeometric functions.

Contribution

It presents a novel extended k-type hypergeometric function and investigates its fundamental properties, which have not been previously documented.

Findings

Derived integral representations of the extended k-type hypergeometric function.

Established differential formulas for the new function.

Developed fractional integral and derivative formulas for the extended function.

Abstract

Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing (presumably new) extended $k-$type hypergeometric function $_{2}f_{1}^{k}[a, b; c; \omega; z]$ and study various properties including integral representations, differential formulas and fractional integral and derivative formula.