On the dual valued generalized hypergeometric function and its special cases

TL;DR

This paper investigates dual-valued hypergeometric functions, analyzing their properties, convergence, differential equations, and special cases, with a focus on dual numbers as parameters and variables.

Contribution

It introduces the calculus of dual-valued functions and extends classical hypergeometric functions to the dual number context, exploring their fundamental properties and special cases.

Findings

Derived properties of dual hypergeometric functions

Analyzed convergence regions and differential equations

Discussed dual confluent and Gauss hypergeometric functions

Abstract

This paper explores the calculus of dual-valued functions and investigates the gamma function, beta function and generalized hypergeometric functions by incorporating dual numbers as parameters and variables. We examine its fundamental properties, including regions of convergence, differential equations, and integral representations. Furthermore, we provide an in-depth discussion on the various properties of the dual confluent and Gauss hypergeometric functions.