Research of absolute and figured absolute convergence of the branched continued fractions of the special form

Authors

  • Тамара Миколаївна Антонова Lviv Polytechnic National University Str. St. Bandery, Lviv, Ukraine, 79000, Ukraine
  • Світлана Миколаївна Возна Lviv Polytechnic National University Str. St. Bandery, Lviv, Ukraine, 79000, Ukraine

DOI:

https://doi.org/10.15587/1729-4061.2015.54116

Keywords:

continued fractions, branched continued fractions, approximation, absolute convergence, figured absolute convergence, majorant fraction

Abstract

The paper deals with the issue of convergence of one of the generalizations of continued fractions - branched continued fraction (BCF) of the special form with two branches, which has been proposed by the Polish mathematician W. Siemaszko in solving the problem of compliance between the formal double power series and a sequence of rational approximations of the function of two variables.

Unlike continued fractions, approximations of which are constructed unambiguously, there are many ways to construct approximations of BCF of the general and special form. The paper examines the conventional approximations of one of the structures of figured approximations of the studied BCF, which is associated with the problem of compliance. The formulas of difference between two approximations (two conventional, two figured, figured and conventional) were given.

Using the majorant method and known results of the theory of convergence of continued fractions, the theorem on some sufficient conditions for absolute and absolute figured convergence of BCF of the special form with two branches towards the same border was proved. It was shown that the values of the studied BCF and its approximations belong to a circle, the radius of which depends on the values of BCF elements on the first floor and the constants appearing in the formulation of the theorem.

By choosing different values for these constants, it is possible to obtain various signs of absolute and figured absolute convergence of BCF of the special form.

Author Biographies

Тамара Миколаївна Антонова, Lviv Polytechnic National University Str. St. Bandery, Lviv, Ukraine, 79000

PhD sciences, associate professor

The department of Applied mathematica

Світлана Миколаївна Возна, Lviv Polytechnic National University Str. St. Bandery, Lviv, Ukraine, 79000

PhD sciences

The department of Applied mathematica

References

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Published

2015-12-22

How to Cite

Антонова, Т. М., & Возна, С. М. (2015). Research of absolute and figured absolute convergence of the branched continued fractions of the special form. Eastern-European Journal of Enterprise Technologies, 6(4(78), 19–26. https://doi.org/10.15587/1729-4061.2015.54116

Issue

Vol. 6 No. 4(78) (2015): Mathematics and Cybernetics - applied aspects

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Mathematics and Cybernetics - applied aspects

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Copyright (c) 2015 Тамара Миколаївна Антонова, Світлана Миколаївна Возна

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According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.