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On the Śleszyńsky–Pringsheim Theorem for the Three-Dimensional Generalization of Continued Fractions

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We propose a three-dimensional generalization of continued fractions, establish a formula for the difference between approximants of the analyzed fraction, and deduce estimates for its tails. A majorant fraction is constructed and the absolute convergence of this generalized fraction is investigated.

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References

  1. D. I. Bodnar and Kh. I. Kuchmins’ka, “Development of the theory of branched continued fractions in 1996–2016,” Mat. Met. Fiz.-Mekh. Polya, 59, No. 2, 7–18 (2016); English translation: J. Math. Sci., 231, No. 4, 481–494 (2018); https://doi.org/10.1007/s10958-018-3828-7.

  2. W. B. Jones and W. J. Thron, Continued Fractions: Analytic Theory and Applications, Addison-Wesley, Reading, MA (1980); G.-C. Rota (editor), Encyclopedia of Mathematics and its Applications, Vol. 11.

  3. Kh. I. Kuchmins’ka, “Corresponding and associated branched continued fractions for a double power series,” Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 7, 613–617 (1978).

  4. Kh. Yo. Kuchmins’ka, Two-Dimensional Continued Fractions [in Ukrainian], Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv (2010).

  5. Kh. Yo. Kuchminska and S. M. Vozna, “Development of an N-multiple power series into an N-dimensional regular C-fraction,” Mat. Met. Fiz.-Mekh. Polya, 60, No. 3, 70–75 (2017); English translation: J. Math. Sci., 246, No. 2, 201–208 (2020); https://doi.org/10.1007/s10958-020-04730-3.

  6. Kh. Kuchminska, “Corresponding N -dimensional continued fractions for N -multiple power series,” in: J. Steuding and M. Pratsiovytyi (editors), Voronoi’s Impact on Modern Science: Proc. 6th Int. Conf. Analytic Number Theory Spat. Tessellations, Vol. 1, Drahomanov Kyiv National Pedagogical University Publ., Kyiv (2018), pp. 169–176.

  7. J. A. Murphy and M. R. O’Donohoe, “A two-variable generalization of the Stieltjes-type continued fractions,” J. Comput. Appl. Math., 4, No. 3, 181–190 (1978); https://doi.org/10.1016/0771-050X(78)90002-5.

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  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

    Kh. Yo. Kuchminska

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  1. Kh. Yo. Kuchminska
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Correspondence to Kh. Yo. Kuchminska.

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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 62, No. 4, pp. 60–71, October–December, 2019.

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Kuchminska, K.Y. On the Śleszyńsky–Pringsheim Theorem for the Three-Dimensional Generalization of Continued Fractions. J Math Sci 265, 408–422 (2022). https://doi.org/10.1007/s10958-022-06061-x

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  • DOI: https://doi.org/10.1007/s10958-022-06061-x

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