Three- and four-term recurrence relations for Horn's hypergeometric function $H_4$

Three- and four-term recurrence relations for hypergeometric functions of the second order (such as hypergeometric functions of Appell, Horn, etc.) are the starting point for constructing branched continued fraction expansions of the ratios of these functions. These relations are essential for obtaining the simplest structure of branched continued fractions (elements of which are simple polynomials) for approximating the solutions of the systems of partial differential equations, as well as some analytical functions of two variables. In this study, three- and four-term recurrence relations for Horn's hypergeometric function $H_4$ are derived. These relations can be used to construct branched continued fraction expansions for the ratios of this function and they are a generalization of the classical three-term recurrent relations for Gaussian hypergeometric function underlying Gauss' continued fraction.