On The Extended Incomplete Pochhammer Symbols and Hypergeometric Functions

TL;DR

This paper introduces extended incomplete Pochhammer symbols and uses them to define new hypergeometric functions, exploring their properties such as integral representations, derivatives, generating functions, and fractional calculus relationships.

Contribution

It presents novel extended incomplete Pochhammer symbols and hypergeometric functions, along with their fundamental properties and special cases.

Findings

Derived integral representations of the new functions

Established derivative and generating function formulas

Explored fractional integral and derivative relationships

Abstract

In this paper, we first introduce certain forms of extended incomplete Pochhammer symbols which are then used to define families of extended incomplete generalized hypergeometric functions. For these functions, we investigate various properties including the integral representations, derivative formula, certain generating function and fractional integrals (and derivatives) relationships. Some special cases of the main results are also deduced.