Further study on the conformable fractional Gauss hypergeometric function

TL;DR

This paper provides an extensive mathematical analysis of the conformable fractional Gauss hypergeometric function, including solutions, generating functions, differential forms, and integral representations, advancing the theoretical understanding of this special function.

Contribution

It introduces new solutions, generating functions, and integral representations for the conformable fractional Gauss hypergeometric function, expanding the theoretical framework of fractional hypergeometric functions.

Findings

Solved the conformable fractional Gauss hypergeometric equation at key singular points

Established various generating functions for CFGHF

Developed differential forms, operators, and integral representations

Abstract

This paper presents a somewhat exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric equation (CFGHE) about the fractional regular singular points $x=1$ and $x=\infty$. Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and the contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF.