The problem of convergence of branched continued fractions

Authors

  • D.I. Bodnar West Ukrainian National University, Peremohy Square 3, 46009, Ternopil, Ukraine https://orcid.org/0000-0002-5811-7493
  • I.B. Bilanyk Ternopil Volodymyr Hnatiuk National Pedagogical University, 2 Maxyma Kryvonosa str., 46027, Ternopil, Ukraine https://orcid.org/0000-0002-1120-6317
  • O.S. Bodnar Ternopil Volodymyr Hnatiuk National Pedagogical University, 2 Maxyma Kryvonosa str., 46027, Ternopil, Ukraine https://orcid.org/0000-0003-4207-0624
  • O.B. Salamakha "SERPSTAT" LLC, 13 Black Sea Cossacks str., 65003, Odesa, Ukraine
  • S.V. Sharyn Vasyl Stefanyk Carpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0003-2547-1442
  • O.S. Kondur Vasyl Stefanyk Carpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0001-9342-1127
https://doi.org/10.15330/cmp.18.2.447-464

Keywords:

branched continued fraction, convergence, approximant, figured convergence
Published online: 2026-08-20

Abstract

Using the tails of the approximants of continued fractions and their multidimensional generalizations, branched continued fractions, canonical numerators and denominators have been defined, and the conditions for the degeneracy of approximants have been investigated. Certain types of figured approximants, specifically $\mathcal{A}$-, $\mathcal{B}$-, and $\mathcal{C}$-figured approximants and their applications, are considered. Various types of convergence for continued and branched continued fractions have been classified. By analogy with Leighton's example, a $\mathcal{C}$-figured-convergent two-dimensional branched continued fraction with independent variables has been constructed; in its structure, every fourth element in each constituent continued fraction is an arbitrary complex number different from $0$ and $-1/2$. Methods for investigating the convergence of branched continued fractions are analyzed.

Article metrics
How to Cite
(1)
Bodnar, D.; Bilanyk, I.; Bodnar, O.; Salamakha, O.; Sharyn, S.; Kondur, O. The Problem of Convergence of Branched Continued Fractions. Carpathian Math. Publ. 2026, 18, 447-464.

Most read articles by the same author(s)

1 2 > >>