A New Extension Of Gauss Hypergeometric Function

Abstract

In this paper, we introduce a novel extension of the Gauss hypergeometric function, aiming to broaden its applicability and theoretical underpinnings. Our main objective is to define the extended Gauss hypergeometric function, denoted as , where , and  are parameters representing the order and degree of the function, respectively. We define it and its fundamental properties which are convergence criteria, analytic continuation, and integral representations for its further study. Besides this, we also address the ways and procedures of how to use the extended Gauss hypergeometric function in different mathematical problems as well as its flexibility and applicability in solving various mathematical problems. Through the examinations of its series representations, integral transforms, and differential equations, we reveal its structural characteristics and behavior under different parameters, uncovering its asymptotic behaviors and special cases. Furthermore, we made a complete study of the mathematical features of the extended Gauss hypergeometric function such as its symmetry properties, transformation formulas and connection to other special functions. We prove the validity of the theoretical results and illustrate the effectiveness of the extended Gauss hypergeometric function in practical applications by means of numerical experiments and computational simulations. The paper closes by indicating the development of mathematical knowledge in which a new extension of the Gauss hypergeometric function is introduced, and its properties and applications are explored. The theoretical framework and analytical methods created by this paper lay the foundation for future research and innovation in the area of special functions and mathematical analysis.