Truncation error bounds for branched continued fraction whose partial denominators are equal to unity

  • R. I. Dmytryshyn Vasyl Stefanyk Precarpathian National University
  • T. M. Antonova Lviv Polytechnic National University
Keywords: convergence, truncation error, branched continued fraction

Abstract

The paper deals with the problem of obtaining error bounds for branched continued fraction of the form $\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{a_{i(2)}}{1}{\atop+}\sum_{i_3=1}^{i_2}\frac{a_{i(3)}}{1}{\atop+}\ldots$. By means of fundamental inequalities method the truncation error bounds are obtained for the above mentioned branched continued fraction providing its elements belong to some rectangular sets of
a complex plane. Applications are considered for several classes of branched continued fraction expansions including the multidimensional \emph{S}-, \emph{A}-, \emph{J}-fractions with independent variables.

Author Biographies

R. I. Dmytryshyn, Vasyl Stefanyk Precarpathian National University

Vasyl Stefanyk Precarpathian National University

Department of Mathematical and Functional Analysis

Ivano-Frankivsk, Ukraine

T. M. Antonova, Lviv Polytechnic National University

Lviv Polytechnic National University

Department of Applied Mathematics

Lviv, Ukraine

References

T.M. Antonova, Speed of convergence of branched continued fractions of the special form, Volynskyi Mat. Visnyk, 6 (1999), 5–11. (in Ukrainian)

T.M. Antonova, D.I. Bodnar, Convergence domains for branched continued fractions of the special form, Approx. Theory and its Appl.: Proc. Inst. Math. NAS Ukr., 31 (2000), 19–32. (in Ukrainian)

T.M. Antonova, M.V. Dmytryshyn, S.M. Vozna, Some properties of approximants for branched continued fractions of the special form with positive and alternating-sign partial numerators, Carpathian Math. Publ., 10 (2018), №1, 3–13. doi: 10.15330/cmp.10.1.3-13

T.M. Antonova, R.I. Dmytryshyn, Truncation error bounds for branched continued fraction $sum_{i_1=1}^Nfrac{a_{i(1)}}{1}{atop+}sum_{i_2=1}^{i_1}frac{a_{i(2)}}{1}{atop+}sum_{i_3=1}^{i_2}frac{a_{i(3)}}{1}{atop+}ldots,$ Ukr. Mat. Zhurn., 72 (2020), №7, 877–885. (in Ukrainian) doi: 10.37863/umzh.v72i7.2342

T.M. Antonova, O.M. Sus’, On some sequences of the sets of uniform convergence for two-dimensional continued fractions, Mat. method. and fiz.-mech. polya, 58 (2015), №1, 47–56. (in Ukrainian); Engl. transl.: J. Math. Sci., 222 (2017), №1, 56–69. doi: 10.1007/s10958-017-3282-y

T.M. Antonova, S.M. Vozna, On one analogue of the method of fundamental inequalities for research of branched continued fractions of the special form, Visnyc Lviv Polytech. Ntl. Univ. Ser. Phys. and Math. Sci., 871 (2017), 5–12. (in Ukrainian)

O.E. Baran, Some domains convergence of branched continued fractions of the special form, Carpathian Math. Publ., 5 (2013), №1, 4–13. (in Ukrainian) doi: 10.15330/cmp.5.1.4-13

I.B. Bilanyk, A truncation error bound for some branched continued fractions of the special form, Mat. Stud., 52 (2019), №2, 115–123. doi: 10.30970/ms.52.2.115-123

I.B. Bilanyk, D.I. Bodnar, L.M. Byak, Representation of a quotient of solutions of a four-term linear recurrence relation in the form of a branched continued fraction, Carpathian Math. Publ., 11 (2019), №1, 33–41. doi: 10.15330/cmp.11.1.33-41

D.I. Bodnar, I.B. Bilanyk, On the convergence of branched continued fractions of a special form in angular domains, Mat. method. and fiz.-mech. polya, 60 (2017), №3, 60–69. (in Ukrainian); Engl. transl.: J. Math. Sci. 246

(2020), №2, 188–200. doi: 10.1007/s10958-020-04729-w

D.I. Bodnar, Branched continued fractions, Naukova Dumka, Kyiv, 1986. (in Russian)

D.I. Bodnar, R.I. Dmytryshyn, Multidimensional associated fractions with independent variables and multiple power series, Ukr. Math. Zhurn., 71 (2019), №3, 325–339. (in Ukrainian); Engl. transl.: Ukrainian Math. J., 71 (2019), №3, 370–386. doi: 10.1007/s11253-019-01652-5

M.M. Bubniak, Truncation-error bounds for the 1-periodic branched continued fraction of special form, Carpathian Math. Publ., 5 (2013), №2, 187–195. doi: 10.15330/cmp.5.2.187-195

R.I. Dmytryshyn, Associated branched continued fractions with two independent variables, Ukr. Math. Zhurn., 66 (2014), №9, 1175–1184. (in Ukrainian); Engl. transl.: Ukrainian Math. J., 66 (2015), №9, 1312–1323. doi: 10.1007/s11253-015-1011-6

R.I. Dmytryshyn, Convergence of some branched continued fractions with independent variables, Mat. Stud., 47 (2017), №2, 150–159. doi: 10.15330/ms.47.2.150-159

R.I. Dmytryshyn, Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series, Proc. Roy. Soc. Edinburgh Sect. A, (2019), 1–18. doi: 10.1017/prm.2019.2

R.I. Dmytryshyn, On some of convergence domains of multidimensional S-fractions with independent variables, Carpathian Math. Publ., 11 (2019), №1, 54–58. doi: 10.15330/cmp.11.1.54-58

R.I. Dmytryshyn, On the expansion of some functions in a two-dimensional g-fraction with independent variables, Mat. method. and fiz.-mech. polya, 53 (2010), №4, 56–69. (in Ukrainian); Engl. transl.: J. Math. Sci., 181 (2012), №3, 320–327. doi: 10.1007/s10958-012-0687-5

R.I. Dmytryshyn, The two-dimensional g-fraction with independent variables for double power series, J. Approx. Theory, 164 (2012), №12, 1520–1539. doi: 10.1016/j.jat.2012.09.002

R.I. Dmytryshyn, Two-dimensional generalization of the Rutishauser qd-algorithm, Mat. method. and fiz.-mech. polya, 56 (2013), №4, 6–11. (in Ukrainian); Engl. transl.: J. Math. Sci., 208 (2015), №3, 301–309. doi: 10.1007/s10958-015-2447-9

W.B. Jones, W.J. Thron, Continued fractions: Analytic theory and applications, Addison-Wesley Pub. Co., Reading, Mass., 1980.

Published
2020-10-05
How to Cite
1.
Dmytryshyn RI, Antonova TM. Truncation error bounds for branched continued fraction whose partial denominators are equal to unity. Mat. Stud. [Internet]. 2020Oct.5 [cited 2020Oct.27];54(1):3-14. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/37
Section
Articles