Some local properties of two-dimensional continued fractions

Abstract

By regarding a two-dimensional continued fraction as a function of its elements and applying recursion relations for its tails, we establish formulas for the first partial derivatives of the fraction, on the basis of which we construct linear approximations of limit-periodic two-dimensional continued fractions.

On the convergence of branched continued fractions

“…Analogs of Worpitzky's theorem, the Sleshins'kyi-Pringsheim theorem, parabolic theorems, and others [8,14,25] have been established for two-dimensional continued fractions. A survey of studies in the theory of convergence of two-dimensional continued fractions and branched continued fractions was given in [27].…”

“…If the fundamental inequalities hold for the two-dimensional continued fraction (12) with zi = z2 = 1, then its even and odd parts converge absolutely. Theorem 14 [25]. If for the two-dimensional continued fraction A multi-dimensional g-fraction is a branched continued fraction of the form i~,=l (1-gi{k-…”