Convergence criteria of branched continued fractions

The convergence criteria of branched continued fractions with N branches of branching and branched continued fractions of the special form are analyzed. The classical theorems of convergence of continued fractions that have become the subject of multidimensional generalizations are formulated. The convergence conditions of branched continued fractions of the general form with positive elements are reviewed. The problem the solution of which caused changes in the structure of such branched continued fractions is formulated. A multidimensional generalization of the convergence criterion of branched continued fractions of the special form is stated. A multidimensional generalization of Worpitzky's and van Vleck's convergence theorems, the Śleszyński-Pringsheim theorem for the considered types of branched continued fractions are considered. The obtained multidimensional analogs of the theorems are analyzed, and other conditions of convergence, in particular, of branched continued fractions with real elements, multidimensional Leighton's and Wall's theorems, and others are given.