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