Some properties of approximants for branched continued fractions of the special form with positive and alternating-sign partial numerators

  • T.M. Antonova Lviv Polytechnic National University, 12 Bandera str., 79013, Lviv, Ukraine
  • M.V. Dmytryshyn Ternopil National Economic University, 11 Lvivska str., 46020, Ternopil, Ukraine
  • S.M. Vozna Lviv Polytechnic National University, 12 Bandera str., 79013, Lviv, Ukraine https://orcid.org/0000-0002-5354-2124
Keywords: branched continued fractions of the special form, ordinary approximants, figured approximants, convergence
Published online: 2018-07-03

Abstract


The paper deals with research of convergence for one of the generalizations of continued fractions -- branched continued fractions of the special form with two branches. Such branched continued fractions, similarly as the two-dimensional continued fractions and the branched continued fractions with two independent variables are connected with the problem of  the correspondence between a formal double power series and a sequence of the rational approximants of a function of two variables.

Unlike continued fractions, approximants of which are constructed unambiguously, there are many ways to construct approximants of branched continued fractions of the general and the special form. The paper examines the ordinary approximants and one of the structures of figured approximants of the studied branched continued fractions, which is connected with the problem of correspondence.

We consider some properties of approximants of such fractions, whose partial numerators are positive and alternating-sign  and partial denominators are equal to one. Some necessary and sufficient conditions for figured convergence are established. It is proved that under these conditions from the convergence of the sequence of figured approximants it follows the convergence of the sequence of ordinary approximants  to the same limit.

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How to Cite
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Antonova T., Dmytryshyn M., Vozna S. Some Properties of Approximants for Branched Continued Fractions of the Special Form With Positive and Alternating-Sign Partial Numerators. Carpathian Math. Publ. 2018, 10 (1), 3-13.