A simple algorithm for expanding a formal triple power series as a three-dimensional associated continued fractions

Kh.Yo. Kuchminska; R.M. Deyneka

doi:10.15421/242517

A simple algorithm for expanding a formal triple power series as a three-dimensional associated continued fractions

Kh.Yo. Kuchminska (Pidstryhach Institute for Applied Problems in Mechanics and Mathematics of NAS Ukraine), https://orcid.org/0000-0001-5350-8180
R.M. Deyneka (Lviv Polytechnic National University), https://orcid.org/0000-0002-2062-2312

Abstract

We propose an algorithm for expanding a formal triple power series into a three-dimensional corresponding to this series associated continued fraction, which generalized one of the simplest algorithms for expanding a formal power series into a corresponding continued fraction, namely the Viskovatov algorithm.

Keywords

triple power series; Viskovatov's algorithm; three-dimensional associated continued fraction

MSC 2020

Pri 41A20, 65D15, Sec 11A55, 11J70, 30B70

Full Text:

PDF

References

Bodnar D.I., Kuchminska Kh.Yo. "Development of the theory of branched continued fractions in 1996–2016", J. Math. Sci., 2018; 231(4): pp. 481-494. doi:10.1007/s10958-018-3828-7

Bodnar D.I., Dmytryshyn R.I. "Multidimensional associated fractions with independent variables and multiple power series", Ukrainian Math. J., 2019; 71(3): pp. 370-386. doi:10.1007/s11253-019-01652-5

Cuyt A. "A review of multivariate Padé approximation theory", J. Comput. Appl. Math., 1985; 12-13: pp. 221-232. doi:10.1016/0377-0427(85)90019-6

Jones W.B., Thron W.J. Continued fractions: analytic theory and applications, Addison-Wesley Pub. Co., 1980.

Deyneka R., Tykhan M., Markina O. "Non-destructive testing of ferromagnetic materials using hand inductive sensor", Arch. Mater. Sci. Eng., 2019; 1(98): pp. 32-41. doi:10.5604/01.3001.0013.3392

Komatsu T. "Branched continued fractions associated with Hosoya index of the tree graph", MATCH Commun. Math. Comput. Chem., 2020; 84(2): pp. 399-428.

Kuchminska Kh.Yo. "Corresponding and associated branched continued fractions for double power series", Dopov. AN Ursr, ser.A, 1978; 7: pp. 614-618. (in Ukrainian)

Kuchminska Kh. "Corresponding $$$N$$$-dimensional continued fractions for $$$N$$$-multiple power series", in: Steuding J., Pratsiovytyi M (eds.) Voronoї’s Impact on Modern Science. Proc. 6th Int. Conf. Analytic Number Theory, Spatial Tessellations, 2 Volumes, Nat. Pedagog. Drahomanov Univ. Publ. 2018; 1: pp. 169-176.

Kuchminska Kh.Yo. Two-dimensional continued fractions, Ya.S. Pidstryhach Inst. Appl. Problems Mech. Math. NAS Ukraine, Lviv, 2010; (in Ukrainian)

Kuchminska Kh.Yo., Deyneka R.M. "Three-dimensional generalization of corresponding regular $$$C$$$-fraction", Math. Meth. Phys.-Mech. Fields, 2021; 64(3): pp. 5-15. doi:10.1007/s10958-024-06958-9

Kuchminska Kh.Yo., Vozna S.M. "Development of an $$$N$$$-multiple power series into an $$$N$$$-dimensional regular $$$C$$$-fraction", J. Math. Sci., 2020; 246(2): pp. 201-208. doi:10.1007/s10958-020-04730-3

Murphy J.A., O'Donohoe M.R. "A two-variable generalization of the Stieltjes-type continued fraction", J. Comput. Appl. Math., 1978; 4(3): pp. 181-190. doi:10.1016/0771-050x(78)90002-5

Siemaszko W. "Branched continued fractions for double power series", J. Comput. Appl. Math., 1980; 6(2): pp. 121-125. doi:10.1016/0771-050x(80)90005-4

Skorobogatko V.Ya. Theory of branching continued fractions and its application in computational mathematics, Nauka, 1983.