SOME PROPERTIES OF APPROXIMANTS FOR BRANCHED CONTINUED FRACTIONS OF THE SPECIAL FORM WITH POSITIVE AND ALTERNATING-SIGN PARTIAL NUMERATORS
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 1
Abstract
The paper deals with research of convergence for one of the generalizations of continued frac- tions — branched continued fractions of the special form with two branches. Such branched con- tinued 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 spe- cial 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 suf- ficient 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.
Authors and Affiliations
T. Antonova, M. Dmytryshyn, S. Vozna
Keywords
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- DOI 10.15330/cmp.10.1.3-13
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