Research on the convergence of some types of functional branched continued fractions

An analysis of research on the problem of convergence of various types of functional branched continued fractions has been carried out. Branched continued fractions with $N$ branching branches and branched continued fractions with independent variables are considered. The definition and, in our opinion, characteristic criteria of convergence of multidimensional generalizations of C-, S-, g-, J-fractions are given, both for branched continued fractions of the general form with $N$ branching branches and branched continued fractions with independent variables. Such multidimensional generalizations of continued fractions arise, in particular, in the development of various classes of hypergeometric functions of several variables, in particular, the functions of Appel, Lauricella, Horn, etc.