An algorithm for the expansion of a given formal double power series in the associated branched continued fraction with two independent variables is constructed and the conditions for the existence of this expansion are established.
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References
D. I. Bodnar, Branched Continued Fractions [in Russian], Naukova Dumka, Kiev (1986).
L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland, Amsterdam–London–New-York–Tokyo (1992).
W. B. Jones and W. J. Thron, “Continued fractions: Analytic theory and applications,” Encycl. Math. Its Appl., 11, Addison-Wesley, London, Amsterdam, Don Mills, Ontario, Sydney, Tokyo (1980).
N. P. Hoyenko, “Correspondence principle and convergence of sequences of analytic functions of several variables,” Mat. Visn. Nauk. Tovar. Shevchenka, 4, 42–48 (2007).
Kh. Yo. Kuchmins’ka, Two-Dimensional Continued Fractions [in Ukrainian], Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, L’viv (2010).
O. E. Baran and R. I. Dmytryshyn, “Some types of branched continued fraction corresponding to multiple power series,” Approx. Theor. Its Appl., Proc. Inst. Math., Nat. Acad. Sci. Ukr., 31, 82–92 (2000).
D. I. Bodnar, “Corresponding branched continued fractions with linear partial numerators for double power series,” Ukr. Mat. Zh., 43, No. 4, 474–482 (1991).
A. Cuyt and B. Verdonk, “A review of branched continued fraction theory for the construction of multivariate rational approximations,” Appl. Numer. Math., 4, 263–271 (1988).
R. I. Dmytryshyn, “On the expansion of some functions in a two-dimensional g -fraction with independent variables,” J. Math. Sci., 181, No. 3, 320–327 (2012).
R. I. Dmytryshyn, “The multidimensional generalization of g -fractions and their application,” J. Comput. Appl. Math., 164-165, 265–284 (2004).
R. I. Dmytryshyn, “The two-dimensional g -fraction with independent variables for double power series,” J. Approx. Theory, 164, No. 12, 1520–1539 (2012).
Kh. Yo. Kuchmins’ka, “Corresponding and associated branched continued fractions for double power series,” Dop. AN URSR. Ser. A, No. 7, 614–617 (1978).
J. F. 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).
W. Siemaszko, “Branched continued fractions for double power series,” J. Comput. Appl. Math., 6, No. 2, 121–125 (1980).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 9, pp. 1175–1184, September, 2014.
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Dmytryshyn, R.I. Associated Branched Continued Fractions with Two Independent Variables. Ukr Math J 66, 1312–1323 (2015). https://doi.org/10.1007/s11253-015-1011-6
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DOI: https://doi.org/10.1007/s11253-015-1011-6