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Truncation Error Bounds for the Branched Continued Fraction \( {\sum}_{i_{1=1}}^N\frac{a_{i(1)}}{1}+{\sum}_{i_{2=1}}^{i_1}\frac{a_{i(2)}}{1}+{\sum}_{i_{3=1}}^{i_2}\frac{a_{i(3)}}{1}+\cdots \)

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Ukrainian Mathematical Journal Aims and scope

We analyze the problem of estimation of the error of approximation of a branched continued fraction, which is a multidimensional generalization of a continued fraction. By the method of fundamental inequalities, we establish truncation error bounds for the branched continued fraction

$$ {\sum}_{i_{1=1}}^N\frac{a_{i(1)}}{1}+{\sum}_{i_{2=1}}^{i_1}\frac{a_{i(2)}}{1}+{\sum}_{i_{3=1}}^{i_2}\frac{a_{i(3)}}{1}+\cdots $$

whose elements belong to certain rectangular sets in the complex plane. The obtained results are applied to multidimensional S- and A-fractions with independent variables.

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Authors and Affiliations

  1. “L’vivs’ka Politekhnika” National University, Lviv, Ukraine

    T. M. Antonova

  2. V. Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine

    R. I. Dmytryshyn

Authors
  1. T. M. Antonova
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  2. R. I. Dmytryshyn
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Correspondence to R. I. Dmytryshyn.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 877–885, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.2342.

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Antonova, T.M., Dmytryshyn, R.I. Truncation Error Bounds for the Branched Continued Fraction \( {\sum}_{i_{1=1}}^N\frac{a_{i(1)}}{1}+{\sum}_{i_{2=1}}^{i_1}\frac{a_{i(2)}}{1}+{\sum}_{i_{3=1}}^{i_2}\frac{a_{i(3)}}{1}+\cdots \). Ukr Math J 72, 1018–1029 (2020). https://doi.org/10.1007/s11253-020-01841-7

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  • DOI: https://doi.org/10.1007/s11253-020-01841-7

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