Literature Cited
D. I. Bodnar,Branched Continued Fractions [in Russian], Naukova Dumka, Kiev (1986).
D. I. Bodnar, “Questions of the analytic theory of branched continued fractions” (doctoral dissertation), L'vov (1989).
D. I. Bodnar, “Some applications of branched continued fractions in computational mathematics,” in:Computational Mathematics in Modern Scientific-Technological Progress. Computational Methods in Algebra. Applied Mathematics, Data Processing Systems and Automatic Control [in Russian], proceedings of the conference of 26–28 September 1974, Kiev (1974), pp. 94–103.
D. I. Bodnar, “Corresponding branched continued fractions with linear partial numerators for a double power series,”Ukr. Mat. Zh.,43, No. 4, 474–482 (1991).
D. I. Bodnar, H. Waadeland, Kh. I. Kuchmins'ka, and O. M. Sus', “On the stability of branched continued fractions,”Mat. Met. Fiz.-Mekh. Polya, No. 37, 3–7 (1994).
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).
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 (1976).
D. I. Bodnar and V. Ya. Skorobogat'ko, “An algorithm for expanding a real number in a regular branched continued fraction and an application of it in electrical engineering,” in:Proceedings of the Republic Conference on “Number Theory and its Applications”, Tashkent, 26–28 September 1990, p. 17.
P. I. Bodnarchuk, “A study of the theory of fractional-rational approximations and solution of rigid systems of ordinary differential equations,” (doctoral dissertation), L'vov (1988).
P. I. Bodnarchuk and Kh. I. Kuchminskaya, “An interpolation and functional formula in the form of branched continued fractions for functions of several variables,”Mat. Met. Fiz.-Mekh. Polya, No. 2, 31–36 (1975).
P. I. Bodnarchuk and V. Ya. Skorobogat'ko,Branched Continued Fractions and their Applications [in Ukrainian], Naukova Dumka, Kiev (1974).
G. F. Voronoi,Collected Works [in Russian], Vol. 1, Ukrainian Academy of Sciences (1952).
N. S. Dronyuk, “Expansion cf certain functions in branched continued fractions,” in:Proceedings of the Second Conference of Young Ukrainian Mathematicians [in Ukrainian], Naukova Dumka, Kiev (1966), pp. 185–189.
Z. I. Krupka and V. I. Shmoilov, “On the parallel computation of algorithms represented by branched continued fractions,”Multiproc. Comp. Struc., No. 2, 78–80 (1980).
Kh. I. Kuchmins'ka, “Corresponding and associated branched continued fractions for a double power series,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 7, 614–618 (1978).
Kh. I. Kuchminskaya, “Interpolation and approximation of functions by continued and branched continued fraction,” (kandidat dissertation), L'vov (1976).
Kh. I. Kuchminskaya, “On the approximation of functions by continued and branched continued fractions,”Mat. Met. Fiz.-Mekh. Polya, No. 12, 3–10 (1980).
Kh. I. Kuchminskaya and D. I. Bodnar, “Computational stability of expansions of functions of several variables in branched continued fractions,”Homog. Num. Comp. Int. Struc., No. 8, 145–151 (1977).
Continued Fractions, their Generalizations and Applications [in Ukrainian], Proceedings of the International School in Verkhne Sinevidne, 18–25 September 1994, Pidstrigach Institute of Applied Problems of Mechanics and Mathematics, Urkainian Acacdemy of Sciences, L'viv (1994).
E. M. Maksymiv and M. V. Kutniv, “A nonlinear explicit method of numerical integration of differential equations,”Vestnik L'vov. Politekh. Inst., No. 141, 57–58 (1980).
V. F. Marko, “The connection of the generalized Jacobi algorithm with branched continued fractions,”Visnik L'viv. Politekh. Inst. Mat. Mekh., No. 87, 14–17 (1974).
Yu. V. Mel'nichuk, “OnP-adic continued fractions,” in:Continued Fractions and their Applications [in Russian], Ukrainian Academy of Sciences Mathematical Institute (1976), pp. 29–31.
R. I. Mikhal'chuk and M. S. Syavavko, “A continuous analog of continued fractions,”Ukr. Mat. Zh.,4, No. 1, 559–564 (1982).
N. A. Nedashkovskii, “On convergence and computational stability of branched continued fractions of certain types,”Mat. Met. Fiz.-Mekh. Polya, No. 20, 27–31 (1984).
N. A. Nedashkovskii, “A direct method of solving linear algebraic equations by branched continued fractions,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 8, 24–28 (1980).
M. O. Nedashkovs'kii, “Solution of systems of linear algebraic equations with polynomial elements and models of parallel computations,” (doctoral dissertation), Ternopil' (1995).
O. V. Ogirko, “A nonlinear method of approximate computation of definite integrals,”Mat. Met. Fiz.-Mekh. Polya, No. 11, 28–31 (1980).
T. N. Odnovolova (Antonova), “Some estimates of the error in computing integral continued fractions,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 19–22 (1984).
F. O. Pasichnyak, “Expansion of algebraic irrationalities in a branched continued fraction,” in:Continued Fractions and their Applications [in Russian], Ukrainian Academy of Sciences Mathematical Institute, Kiev (1976), pp. 85–86.
Ya. N. Pelekh, Z. I. Krupka, and M. T. Solodyak, “Application of continued fractions to the solution of the equations that describe the electromagnetic field in ferromagnetic bodies,” in:Methods of Studying Differential and Integral Equations [in Russian], Naukova Dumka, Kiev (1989), pp. 165–171.
I. P. Pustomel'nikov, “Some applications of branched continued fractions in the theory of differential equations,” (kandidat dissertation), L'vov (1970).
V. Ya. Skorobogat'ko, “Convergence criteria for branched continued fractions,”Dop. Akad. Nauk Ukr. RSR. Ser. A, No. 1, 27–29 (1972).
V. Ya. Skorobogat'ko,Theory of Branched Continued Fractions and its Applications in Computational Mathematics [in Russian], Nauka, Moscow (1983).
V. Ya. Skorobogat'ko, N. S. Dronyuk, O. I. Bobik, and B. I. Ptashnik, “Branched continued fractions,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 2, 131–133 (1967).
V. Ya. Skorobogat'ko, N. S. Dronyuk, O. I. Bobik, and B. I. Ptashnik, “Branched continued fractions and their applications,” in:Proceedings of the Second Conference of Young Ukrainian Mathematicians [in Ukrainian], Naukova Dumka, Kiev (1966), pp. 561–565.
R. V. Slonevskii, “Elements of the theory of branched continued fractions and its application to the solution of differential equations and Markov processes,” (kandidat dissertation), L'vov (1972).
O. M. Sus', “Some local properties of two-dimensional continued fractions,”Mat. Met. Fiz.-Mekh. Polya, No. 38, 29–33 (1995).
M. S. Syavavko,Integral Continued Fractions, [in Ukrainian], Naukova Dumka, Kiev (1994).
M. S. Syavavko, “Theory and application of integral continued fractions with respect to measure,” (doctoral dissertation), L'vov (1990).
M. S. Syavavko and Yu. R. Batyuk, “Some convergence criteria for continued fractions for functionals,”Visn. L'viv. Politekh. Inst., No. 119, 144–146 (1977).
V. P. Terskikh,The Method of Continued Fractions in Application to the Study of Vibrations of Mechanical Systems [in Russian], Vol. 2, Sudpromgiz, Leningrad (1955).
N. V. Tkach, “Systems of exact equations for the mass opertor of quasi-particles interacting with phonons,”Teor. Mat. Fiz. 61, No. 3, 400–407 (1984).
A. N. Khovanskii,Application of Continued Fractions and their Generalizations to Questions of Approximation Theory, [in Russian], Gostekhizdat, Moscow (1956).
P. L. Chebyshev,Complete Works [in Russian] Vol. 2, USSR Academy of Sciences, Moscow-Leningrad (1947).
V. Ya. Skorobogat'ko, ed.,Continued Fractions and their Applications [in Russian], Ukrainian Academy of Sciences Institute of Mathematics, Kiev (1976).
D. Bodnar, Kh. Kuchmins'ka, and O. Sus', “A survey of analytic theory of branched continued fractions”Comm. Anal. Theory Cont. Frac.,2, 4–23 (1993).
V. Brun, “Mehrdimensionale Algorithmen, welche die Eulersche Kettenbrüchentwicklung der Zahlen verallgemeinern,” in:Sammelband der zu Ehren des 250 Geburtstages Leonard Eulers Festschrift, Akademie-Verlag, Berlin (1959), pp. 87–100.
A. Cuyt and B. Verdonk, “A review of branched continued fraction theory for the construction of multivariate rational approximations,”Appl. Num. Math. 4, 263–271 (1988).
A. Cuyt and B. Verdonk, “Multivariate rational interpolation,”Computing,34, 41–61 (1985).
L. Euler, “De invertine quotiunque mediarum proportionalium citra radicum extractionem,”Novi Comm. Acad. Petropol.,14, No. 1, 188–224 (1771).
M. Jujiwara, “A problem of diophantine approximations in the old Japanese mathematics,”Proc. Acad. Tokyo,15, 101–104 (1939).
C. G. J. Jacobi, “Allgemeine Theorie der kettenbrüchähnlichen Algorithmen, in welchen jede Zahl ausdrei vorhergehenden gebildet wird,”J. reine angew. Math.,69, 29–64 (1868).
W. B. Jones and W. J. Thron,Continued Fractions: Analytic Theory and Applications, Addison-Wesley, Reading, MA (1980).
Kh. Kuchmins'ka, “Convergence criteria of two-dimensional continued fractions”, in:Nonlinear Numerical Methods and Rational Approximation II (Annie Cuyt, ed.) Kluwer Academic Publishers, Dordrecht (1994).
Kh. Kuchminskaya “On approximation of functions by two-dimensional continued fractions”, in:Rational Approximation and its Applications in Mathematics and Physics, (Lect. Notes Math.),1237, 205–216 (1987).
Kh. Kuchminskaya and W. Siemaszko, “Rational approximation and interpolation of functions by branched continued fractions”, in:Rational Approximation and its Applications in Mathematics and Physics, (Lect. Notes Math. 1237, 24–40 (1987).
L. Lorentzen and H. Waadeland,Continued Fractions with Applications, North-Holland, Amsterdam, (1992)
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).
T. V. Pasichnik, “An operator continued fraction for solution of the equation with second kind nonlinearity”,Comm. Anal. Theory Cont. Frac.,4, 50–58 (1995).
O. Perron, “Grundlagen für eine Theorie des Jacobischen Kettenbrüch-algorithmus”,Math. Ann.,64, 1–76 (1907).
M. Pindor and G. Turchetti, “Pade approximants and variational series for operator series”,Nuovo Cimento,A 71, 171–186 (1982).
I. Pratje, “Iteration der Joukowski Abbildung und ihre Strecken-Komplexe”Mitt. Math. Sem. Giessen, No. 48, 1–54 (1954).
W. Siemaszko, “Branched continued fractions for double power series”,J. Comp. Appl. Math. 6, No. 2. 121–125 (1980).
W. Siemaszko, “Thiele-type branched continued fractions for two-variable functions”,J. Comp. Appl. Math.,9, 137–153 (1983).
S. Swain, “Continued fraction solution to systems of linear equations”,J. Phys. A. Math. Gen.,9, No. 11, 1811–1821 (1976).
G. Szekeres, “Multidimensional continued fraction”,Ann. Univ. Sci. Budapest. Sec. Math.,13, 113–140 (1970).
H. S. Wall,Analytic Theory of Continued Fractions, Van Nostrand, New York (1948).
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In memory of Vitalii Yakovich Skorobogat'ko (18 July 1927–7 April 1996). J Math Sci 90, 2323–2333 (1998). https://doi.org/10.1007/BF02433961
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DOI: https://doi.org/10.1007/BF02433961