Expansion of a ratio of hypergeometric functions of two variables in branching continued fractions

We obtain two-dimensional analogs of the continued fractions of Gauss which are an expansion of the ratio of the Appel hypergeometric functions. It is proved that these fractions are those corresponding to the formal power series expansion of the given ratio. Convergence criteria are established for the branching continued fractions under consideration.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 40–44.