References

  1. Antonova T.M., Bodnar D.I. Convergence regions of branched continued fractions of the special form. In: Stepanets O.I. (Ed.) Approximation Theory and Its Applications, Proceedings of Institute of Mathematics of NAS of Ukraine, 31. Institute of Mathematics of NAS of Ukraine, Kyiv, 2000, 13–32. (in Ukrainian)
  2. Antonova T., Cesarano C., Dmytryshyn R., Sharyn S. An approximation to Appell’s hypergeometric function \(F_2\) by branched continued fraction. Dolomites Res. Notes Approx. 2024, 17 (1), 22–31. doi:10.14658/PUPJ-DRNA-2024-1-3
  3. Antonova T.M., Dmytryshyn R.I. Truncation error bounds for branched continued fraction whose partial denominators are equal to unity. Mat. Stud. 2020, 54 (1), 3–14. doi:10.30970/ms.54.1.3-14
  4. Antonova T., Dmytryshyn R., Sharyn S. Branched continued fraction representations of ratios of Horn’s confluent function \(\mathrm{H}_6\). Constr. Math. Anal. 2023, 6 (1), 22–37. doi:10.33205/cma.1243021
  5. Antonova T.M., Sus’ O.M, Vozna S.M. Convergence and estimation of the truncation error for the corresponding two-dimensional continued fractions. Ukrainian Math. J. 2022, 74 (4), 501–518. doi:10.1007/s11253-022-02079-1 (translation of Ukrain. Mat. Zh. 2022, 74 (4), 443–457. doi:10.37863/umzh.v74i4.7031 (in Ukrainian))
  6. Baran O.E. Approximation of functions of several variables by branched continued fractions with independent variables. PhD thesis (Math. Analysis), Pidstryhach IPPMM NASU, Lviv, 2014. (in Ukrainian)
  7. Baran O.E. Some circular regions of convergence for branched continued fractions of a special form. J. Math. Sci. 2015, 205 (4), 491–500. doi:10.1007/s10958-015-2262-3 (translation of Mat. Met. Fiz.-Mekh. Polya 2013, 56 (3), 7–14. (in Ukrainian))
  8. Baran O.E. Some domains of convergence of branched continued fractions of a special form. Carpathian Math. Publ. 2013, 5 (1), 4–13. doi:10.15330/cmp.5.1.4-13 (in Ukrainian)
  9. Bilanyk I.B. A truncation error bound for some branched continued fractions of the special form. Mat. Stud. 2019, 52 (2), 115–123. doi:10.30970/ms.52.2.115-123
  10. Bilanyk I.B., Bodnar D.I., Buyak L.M. Representation of a quotient of solutions of a four-term linear recurrence relation in the form of a branched continued fraction. Carpathian Math. Publ. 2019, 11 (1), 33–41. doi:10.15330/cmp.11.1.33-41
  11. Bilanyk I.B., Bodnar D.I., Vozniak O.G. Convergence criteria of branched continued fractions. Res. Math. 2024, 32 (2), 53–69. doi:10.15421/242419
  12. Bodnar D.I. Analytic theory of branched continued fractions: history, main results, unsolved problems. Mat. Met. Fiz.-Mekh. Polya 2007, 50 (3), 21–29. (in Ukrainian)
  13. Bodnar D.I. Branched Continued Fractions. Naukova Dumka, Kyiv, 1986. (in Russian)
  14. Bodnar D.I. Multidimensional generalizations of continued fractions. Mat. Met. Fiz.-Mekh. Polya 2003, 46 (3), 32–39. (in Ukrainian)
  15. Bodnar D.I., Bilanyk I.B. Estimation of the rates of pointwise and uniform convergence of branched continued fractions with inequivalent variables. J. Math. Sci. 2022, 265 (3), 423–437. doi:10.1007/s10958-022-06062-w (translation of Mat. Met. Fiz.-Mekh. Polya 2019, 62 (4), 72–82. (in Ukrainian))
  16. Bodnar D.I., Bilanyk I.B. On the convergence of branched continued fractions of a special form in angular domains. J. Math. Sci. 2020, 246 (2), 188–200. doi:10.1007/s10958-020-04729-w (translation of Mat. Met. Fiz.-Mekh. Polya 2017, 60 (3), 60–69. (in Ukrainian))
  17. Bodnar D.I., Bilanyk I.B. Parabolic convergence regions of branched continued fractions of the special form. Carpathian Math. Publ. 2021, 13 (3), 619–630. doi:10.15330/cmp.13.3.619-630
  18. Bodnar D.I., Bilanyk I.B. Two-dimensional generalization of the Thron-Jones theorem on the parabolic domains of convergence of continued fractions. Ukranian Math. J. 2023, 74 (9), 1317–1333. doi:10.1007/s11253-023-02138-1 (translation of Ukrain. Mat. Zh. 2022, 74 (9), 1155–1169. doi:10.37863/umzh.v74i9.7096 (in Ukrainian))
  19. Bodnar D.I., Bodnar O.S., Bilanyk I.B. A truncation error bound for branched continued fractions of the special form on subsets of angular domains. Carpathian Math. Publ. 2023, 15 (2), 437–448. doi:10.15330/cmp.15.2.437-448
  20. Bodnar D.I., Bodnar O.S., Dmytryshyn M.V., Popov M.M., Martsinkiv M.V., Salamakha O.B. Research on the convergence of some types of functional branched continued fractions. Carpathian Math. Publ. 2024, 16 (2), 448–460. doi:10.15330/cmp.16.2.448-460
  21. Bodnar D.I., Dmytryshyn R.I. Multidimensional associated fractions with independent variables and multiple power series. Ukranian Math. J. 2019, 71 (3), 370–386. doi:10.1007/s11253-019-01652-5 (translation of Ukrain. Mat. Zh. 2019, 71 (3), 325–339. (in Ukrainian))
  22. Bodnar D.I., Oleksiv I.Ya. Sufficient criterion for the convergence of branching continued fractions with positive terms. Mat. Met. Fiz.-Mekh. Polya 1975, 2, 160–161. (in Russian)
  23. Bodnar D.I., Oleksiv I.Ya. On the convergence of branching continued fractions with nonnegative terms. Ukranian Math. J. 1976, 28 (3), 290–293. doi:10.1007/BF01089177 (translation of Ukrain. Mat. Zh. 1976, 28 (3), 373–377. (in Ukrainian))
  24. Bodnar D.I., Voznyak O.H., Mykhal’chuk R.I. A criterion of convergence of a branched continued fraction with positive elements. J. Math. Sci. 2017, 222 (1), 70–80. doi:10.1007/s10958-017-3283-x (translation of Mat. Met. Fiz.-Mekh. Polya 2015, 58 (1), 57–64. (in Ukrainian))
  25. Bodnar D.I., Zators’kyi R.A. Generalization of continued fractions. I. J. Math. Sci. 2012, 183 (1), 54–64. doi:10.1007/s10958-012-0797-0 (translation of Mat. Met. Fiz.-Mekh. Polya 2011, 54 (1), 57–64. (in Ukrainian))
  26. Bodnar D.I., Zators’kyi R.A. Generalization of continued fractions. II. J. Math. Sci. 2012, 184 (1), 46–55. doi:10.1007/s10958-012-0851-y (translation of Mat. Met. Fiz.-Mekh. Polya 2011, 54 (2), 43–50. (in Ukrainian))
  27. Bodnar O.S., Dmytryshyn R.I. On the convergence of multidimensional S-fractions with independent variables. Carpathian Math. Publ. 2018, 10 (1), 58–64. doi:10.15330/cmp.10.1.58-64
  28. Bodnar O.S., Dmytryshyn R.I., Sharyn S.V. On the convergence of multidimensional S-fractions with independent variables. Carpathian Math. Publ. 2020, 12 (2), 353–359. doi:10.15330/cmp.12.2.353-359
  29. Bodnarchuk P.I., Skorobogatko V.Ya. Branched Continued Fractions and Their Applications. Naukova Dumka, Kyiv, 1974. (in Ukrainian)
  30. Cuyt A., Petersen V.B., Verdonk B., Waadeland H., Jones W.B. Handbook of Continued Fractions for Special Functions. Springer, Dordrecht, 2008.
  31. Cuyt A., Verdonk B. A review of branched continued fraction theory for the construction of multivariate rational approximants. Appl. Numer. Math. 1988, 4 (2-4), 263–271. doi:10.1016/0168-9274(83)90006-5
  32. Dmytryshyn R., Antonova T., Dmytryshyn M. On the analytic extension of the Horn’s confluent function \(\mathrm{H}_6\) on domain in the space \(\mathbb{C}^2\). Constr. Math. Anal. 2024, 7, 11–26. doi:10.33205/cma.1545452
  33. Dmytryshyn R.I. Associated branched continued fractions with two independent variables. Ukranian Math. J. 2015, 66 (9), 1312–1323. doi:10.1007/s11253-015-1011-6 (translation of Ukrain. Mat. Zh. 2014, 66 (9), 1175–1184. (in Ukrainian))
  34. Dmytryshyn R., Cesarano C., Dmytryshyn M., Lutsiv I.-A. A priori bounds for truncation error of branched continued fraction expansions of Horn’s hypergeometric functions \(H_4\) and their ratios. Res. Math. 2025, 33 (1), 13–22. doi:10.15421/242502
  35. Dmytryshyn R., Goran V. On truncation error bounds of branched continued fraction expansions of some ratios of Lauricella-Saran’s hypergeometric functions \(F_K\). Mat. Stud. 2026, 65 (1), 48–57. doi:10.30970/ms.65.1.48-57
  36. Dmytryshyn R.I. Convergence of multidimensional A- and J-fractions with independent variables. Comput. Methods Funct. Theory 2022, 22 (2), 229–242. doi:10.1007/s40315-021-00377-6
  37. Dmytryshyn R.I. Convergence of some branched continued fractions with independent variables. Mat. Stud. 2017, 47 (2), 150–159. doi:10.15330/ms.47.2.150-159
  38. Dmytryshyn R.I. Some classes of functional branched continued fractions with independent variables and multiple power series. Doctoral diss. (Math. Analysis). Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, 2018. (in Ukrainian)
  39. Dmytryshyn R., Lutsiv I.A., Dmytryshyn M., Cesarano C. On some domains of convergence of branched continued fraction expansions of the ratios of Horn hypergeometric functions \(H_4\). Ukranian Math. J. 2024, 76 (4), 559–565. doi:10.1007/s11253-024-02338-3 (translation of Ukrain. Mat. Zh. 2024, 76 (4), 502–508. (in Ukrainian))
  40. Dmytryshyn R.I., Sharyn S.V. Approximation of functions of several variables by multidimensional S-fractions with independent variables. Carpathian Math. Publ. 2021, 13 (3), 592–607. doi:10.15330/cmp.13.3.592-607
  41. O’Donohoe M. Applications of continued fractions in one and more variables. PhD Thesis, Brunel University, London, 1974.
  42. Hladun V.R., Bodnar D.I., Rusyn R.S. Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements. Carpathian Math. Publ. 2024, 16 (1), 16–31. doi:10.15330/cmp.16.1.16-31
  43. Hladun V., Rusyn R., Dmytryshyn M. On the analytic extension of three ratios of Horn’s confluent hypergeometric function \(\mathrm{H}_7\). Res. Math. 2024, 32 (1), 60–70. doi:10.15421/242405
  44. Jacobsen L. General convergence of continued fractions. Trans. Amer. Math. Soc. 1986, 294 (2), 477–485. doi:10.2307/2000194
  45. Jones W.B., Thron W.J. Continued Fractions: Analytic Theory and Applications. Addison-Wesley Pub. Co., Reading, 1980.
  46. Kuchminska H.Y. Two-dimensional continuous fractions. Pidstryhach IAPMM NASU, Lviv, 2010. (in Ukrainian)
  47. Leighton W., Wall H.S. On the transformation and convergence of continued fractions. Amer. J. Math. 1936, 58 (2), 267–281. doi:10.2307/2371036
  48. Lorentzen L., Waadeland H. Continued Fractions with Applications. Elsevier Sci. Publ. B.V., Amsterdam, 1992.
  49. Lorentzen L., Waadeland H. Continued Fractions, Vol. 1: Convergence Theory. Atlantis Press/World Scientific, Amsterdam, 2008.
  50. Manziy O., Hladun V., Ventyk L. The algorithms of constructing the continued fractions for any ratios of the hypergeometric Gaussian functions. Math. Model. Comput. 2017, 4 (1), 48–58. doi:10.23939/mmc2017.01.048
  51. Perron O. The Theory of Continued Fractions, Vol. II: Analytic-Function-Theoretic Continued Fractions. Teubner, Stuttgart, 1957. (in German)
  52. Pringsheim A. On the Convergence of Infinite Continued Fractions. Bayer. Akad. Wiss., Math.-Naturwiss. Kl., Sitzungsber 1898, 28, 295–324. (in German)
  53. Skorobogatko V.Ya. Convergence criterion for a branching continued fraction. Dopov. Akad. Nauk Ukr. RSR Ser. A 1972, 1, 27–29. (in Ukrainian)
  54. Skorobogatko V.Ya. The theory of branched continued fractions and its application in computational mathematics. Nauka, Moscow, 1983. (in Russian)
  55. Skorobogatko V.Ya., Dronyuk N.S., Bobyk O.I., Ptashnyk B.I. Branched continued fractions. Dopov. Akad. Nauk Ukr. RSR Ser. A 1967, 2, 131–133. (in Ukrainian)
  56. Wall H.S. Analytic Theory of Continued Fractions. D. Van Nostrand Co., New York, 1948.