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On the convergence of branched continued fractions

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Abstract

We give a survey of research on the theory of convergence of branched continued fractions.

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Literature Cited

  1. T. M. Antonova, “Sufficient conditions for convergence and stability of integral continued fractions,” Kand. Diss., L'viv (1996).

  2. T. M. Antonova, “A multidimensional analog of the theorem on the uniform simple parabolic region of convergence of continued fractions,”Volyn. Mat. Visn., No. 2, 6–8 (1996).

    Google Scholar 

  3. T. M. Antonova and D. I. Bodnar, “Necessary conditions for convergence of branched continued fractions,” in:Proceedings of the Seventh Kravchuk International Conference [in Ukrainian], Kiev (1998), p. 23.

  4. O. S. Baran, “An analog of the Worpitzky convergence criterion for branched continued fractions of special form,”Mat. Met. Fiz.-Mekh. Polya,39, No. 2, 35–38 (1996).

    MATH  Google Scholar 

  5. D. I. Bodnar, “An analog of the Worpitzky convergence criterion for branched continued fractions,”Mat. Sb., Naukova Dumka, Kiev (1976), pp. 40–43.

    Google Scholar 

  6. D. I. Bodnar,Branched Continued Fractions [in Russian], Naukova Dumka, Kiev (1986).

    Google Scholar 

  7. D. I. Bodnar, “Convergence criteria of Pringsheim type for branched continued fractions,”Ukr. Mat. Zh.,41, No. 11, 1559–1563 (1989).

    MathSciNet  Google Scholar 

  8. D. I. Bodnar and Kh. I. Kuchminskaya, “Absolute convergence of the even and odd parts of a two-dimensional corresponding continued fraction,”Mat. Met. Fiz.-Mekh. Polya, No. 18, 30–34 (1983).

    MathSciNet  Google Scholar 

  9. D. I. Bodnar and I. Ya. Oleksiv, “On the convergence of branched continued fractions with nonnegative terms,”Ukr. Mat. Zh.,28, No. 3, 373–377.

  10. D. I. Bodnar, “Convergence criteria for branched continued fractions with nonnegative components,”Mat. Met. Fiz.-Mekh. Polya,40, No. 2, 7–13 (1997).

    MATH  Google Scholar 

  11. D. I. Bodnar, “On the Koch criterion for branched continued fractions,”Mat. Met. Fiz.-Mekh. Polya, No. 36, 10–13 (1992).

    Google Scholar 

  12. D. I. Bodnar and R. I. Dmytryshyn, “On multidimensional generalized g-fractions,”Dop. Nats. Akad. Nauk Ukr., No. 12, 11–17 (1997).

    Google Scholar 

  13. D. I. Bodnar and N. P. Goyenko, “On the convergence of the even part of the continued-fraction expansion of a ratio of Lauricella hypergeometric functionF D(N),”Mat. Met. Fiz.-Mekh. Polya,40, No. 4, 7–9 (1997).

    Google Scholar 

  14. D. I. Bodnar and Kh. I. Kuchmins'ka, “A parabolic region of convergence for two-dimensional continued fractions,”Mat. Stud., No. 4, 29–36 (1995).

    MathSciNet  Google Scholar 

  15. P. I. Bodnarchuk and V. Ya. Skorobogat'ko,Branched Continued Fractions and their Applications [in Ukrainian], Naukova Dumka, Kiev (1974).

    Google Scholar 

  16. E. A. Boltarovich, “An analog of the Leyton-Wall convergence criterion for branched continued fractions,” in:Methods of Studying Differential and Integral Operators [in Russian], Naukova Dumka, Kiev (1989), pp. 32–36.

    Google Scholar 

  17. R. I. Dmytryshyn, “A priori estimates for the approximation errors of a multidimensionalg-fraction,”Mat. Met. Fiz.-Mekh. Polya,40, No. 4, 10–12 (1997).

    MATH  Google Scholar 

  18. Kh. I. Kuchmins'ka, “Corresponding and adjoint branched continued fractions for a double power series,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 7, 614–618 (1978).

    Google Scholar 

  19. Kh. I. Kuchminskaya and D. I. Bodnar, “Computational stability of the branched continued-fraction expansions of functions of several variables,”Odnorod. Tsifr. Vychisl. Integr. Strukt., No. 8, 145–151 (1977).

    Google Scholar 

  20. O. S. Manzyi, “Expansion of a ratio of Appel hypergeometric functions in a branched continued fraction,” in:Proceedings of the Seventh International Kravchuk Conference [in Ukrainian], Kiev (1998), p. 317.

  21. N. A. Nedashkovskii, “Sufficient criteria for convergence of branched continued fractions,”Mat. Met. Fiz.-Mekh. Polya, No. 19, 29–33 (1984).

    MATH  MathSciNet  Google Scholar 

  22. V. Ya. Skorobogat'ko, “Criteria for convergence of branched continued fractions,”Dop. Akad. Nauk Ukr. RSR, Sr. A, No. 1, 27–29 (1972).

    MathSciNet  Google Scholar 

  23. V. Ya. Skorobogat'ko,Theory of Branched Continued Fractions and its Applications in Computational Mathematics [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  24. V. Ya. Skorobogat'ko, N. S. Dronyuk, O. I. Bobyk, and B. I. Ptashnyk, “Branched continued fractions,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 2, 131–133 (1967).

    Google Scholar 

  25. O. M. Sus', “Some problems in the analytic theory of two-dimensional continued fractions,” Kand. Diss., L'viv (1996).

  26. D. Bodnar, “Sur la convergence des fractions continues branchées avec des termes positifs,”Det Kongelige Norske Videnskabers Selskab, Skrifter, No. 1, 1–21 (1994).

    MATH  Google Scholar 

  27. D. Bodnar, Kh. Kuchmins'ka, and O. Sus', “A survey of the analytic theory of branched continued fractions,”Comm. Anal. Th. Cont. Frac.,2, 4–23 (1993).

    MathSciNet  Google Scholar 

  28. A. Cuyt and B. Verdonk, “A review of branched continued fraction theory of the construction of multivariate rational approximations,”Appl. Numer. Math.,4, 263–271 (1988).

    MathSciNet  Google Scholar 

  29. A. Cuyt and B. Verdonk, “Multivariate rational interpolation,”Computing,34, 41–61 (1985).

    Article  MathSciNet  Google Scholar 

  30. J. Murphy and M. O'Donohoe, “A two-variable generalization of the Stieltjes-type continued fractions,”J. Comp. Appl. Math., No. 4, 181–190 (1978).

    Article  MathSciNet  Google Scholar 

  31. W. Siemaszko, “Branched continued fractions for double power series,”J. Comp. Appl. Math.,6, No. 2, 121–125 (1980).

    MATH  MathSciNet  Google Scholar 

  32. W. Siemaszko, “Thiele-type branched continued fractions for two-variable functions,”J. Comp. Appl. Math.,9, 137–153 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  33. H. A. Waadeland, “Worpitzky boundary theorem forN-branched continued fractions,”Comm. Anal. Th. Cont. Frac.,2, 24–29 (1993).

    MathSciNet  Google Scholar 

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  1. D. I. Bodnar
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Translated fromMatematychni Metody ta Fizyko-Mechanichni Polya, Vol. 41, No. 1, 1998, pp. 117–126.

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Bodnar, D.I. On the convergence of branched continued fractions. J Math Sci 97, 3862–3871 (1999). https://doi.org/10.1007/BF02364926

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