A Note on Certain General Transformation Formulas for the Appell and the Horn Functions

Abstract

In a number of problems in applied mathematics, physics (theoretical and mathematical), statistics, and other fields the hypergeometric functions of one and several variables naturally appear. Hypergeometric functions in one and several variables have several known applications today. The Appell’s four functions and the Horn’s functions have shown to be particularly useful in providing solutions to a variety of problems in both pure and applied mathematics. The Hubbell rectangular source and its generalization, non-relativistic theory, and the hydrogen dipole matrix elements are only a few examples of the numerous scientific and chemical domains where Appell functions are used. The Appell series is also used in quantum field theory, especially in the evaluation of Feynman integrals. Additionally, since 1985, computational sciences such as artificial intelligence (AI) and information processing (IP) have used the well-known Horn functions as a key idea. In literature, there have been published a significant number of results of double series in particular of Appell and Horn functions in a series of interesting and useful research publications. We find three general transformation formulas between Appell functions F2 and F4 and two general transformation formulas between Appell function F2 and Horn function H4 in the present study, which are mostly inspired by their work and naturally exhibit symmetry. By using the generalizations of the Kummer second theorem in the integral representation of the Appell function F2, this is accomplished. As special cases of our main findings, both previously known and new results have been found.