Convergence of some branched continued fractions with independent variables

R. I. Dmytryshyn

doi:10.15330/ms.47.2.150-159

In this paper, we investigate a convergence of associated multidimensional fractions and multidimensional \emph{J}-fractions with independent variables that are closely related to each other; the coefficients of its partial numerators are positive constants or are non-zero complex \linebreak constants from parabolic regions. We have established the uniform convergence of the sequences of odd and even approximants of the above mentioned fractions to holomorphic functions on compact subsets of certain domains of $\mathbb{C}^N.$ And also, we have proved that a condition of convergence for the considered branched continued fractions in certain subsets of $\mathbb{C}^N$ is the divergence of the series composed of its coefficients. Moreover, we have established that the convergence is uniform to a holomorphic function on all compact subsets of domains of $\mathbb{C}^N,$ which are interior of the above mentioned subsets.

1. Antonova T.M., Bodnar D.I. Convergence domains for branched continued fractions of the special form// Approx. Theor. and its Appl.: Pr. Inst. Math. NAS Ukr. – 2000. – V.31. – P. 19–32. (in Ukrainian)

2. Antonova T.M. Multidimensional generalization of the theorem on parabolic domains of convergence of continued fractions// Mat. Met. Fiz.-Mekh. Polya. – 1999. – V.42, №4. – P. 7–12. (in Ukrainian)

3. Baran O.E. Approximation of functions of multiple variables branched continued fractions with inequivalent variables. – Ph.D. dissertation. Mathematical Analysis. Inst. for App. Problem. of Mech. and Math., Ntl. Acad. of Sci. of the Ukr., Lviv, 2014, 166 p.

4. Bodnar D.I., Branched Continued Fractions, Kiev: Naukova Dumka, 1986. – 176 p. (in Russian)

5. Dmytryshyn R.I. Associated branched continued fractions with two independent variables// Ukr. Math. Zh. – 2014. – V.66, №9. – P. 1175–1184, (in Ukrainian); Engl. Transl.: Ukr. Math. J. – 2015. – V.66, №9. – P. 1312–1323.

6. Dmytryshyn R.I. On some convergence domains for multidimensional J-fraction with independent variables// Mat. Visn. Nauk. Tov. Im. Shevchenka. – 2011. – V.8. – P. 69–77. (in Ukrainian)

7. Dmytryshyn R.I. On the convergence multidimensional g-fraction with independent variables// Mat. Met. Fiz.-Mekh. Polya. – 2005. – V.48, №4. – P. 87–92. (in Ukrainian)

8. Dmytryshyn R.I. The two-dimensional g-fraction with independent variables for double power series// J. Approx. Theory. – 2012. – V.164, №12. – P. 1520–1539.

9. Dmytryshyn R.I. Two-dimensional generalization of the Rutishauser qd-algorithm// J. Math. Sci. – 2015. – V.208, №3. – P. 301–309.

10. Wall H.S. Analytic Theory of Continued Fractions. – New York: Van Nostrand, 1948. – xiii+433 p.

	Convergence of some branched continued fractions with independent variables
Author	R. I. Dmytryshyn dmytryshynr@hotmail.com Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Abstract	In this paper, we investigate a convergence of associated multidimensional fractions and multidimensional \emph{J}-fractions with independent variables that are closely related to each other; the coefficients of its partial numerators are positive constants or are non-zero complex \linebreak constants from parabolic regions. We have established the uniform convergence of the sequences of odd and even approximants of the above mentioned fractions to holomorphic functions on compact subsets of certain domains of $\mathbb{C}^N.$ And also, we have proved that a condition of convergence for the considered branched continued fractions in certain subsets of $\mathbb{C}^N$ is the divergence of the series composed of its coefficients. Moreover, we have established that the convergence is uniform to a holomorphic function on all compact subsets of domains of $\mathbb{C}^N,$ which are interior of the above mentioned subsets.
Keywords	convergence, uniform convergence, branched continued fraction, associated multidimensional fraction, multidimensional J-fraction
DOI	doi:10.15330/ms.47.2.150-159
Reference	1. Antonova T.M., Bodnar D.I. Convergence domains for branched continued fractions of the special form// Approx. Theor. and its Appl.: Pr. Inst. Math. NAS Ukr. – 2000. – V.31. – P. 19–32. (in Ukrainian) 2. Antonova T.M. Multidimensional generalization of the theorem on parabolic domains of convergence of continued fractions// Mat. Met. Fiz.-Mekh. Polya. – 1999. – V.42, №4. – P. 7–12. (in Ukrainian) 3. Baran O.E. Approximation of functions of multiple variables branched continued fractions with inequivalent variables. – Ph.D. dissertation. Mathematical Analysis. Inst. for App. Problem. of Mech. and Math., Ntl. Acad. of Sci. of the Ukr., Lviv, 2014, 166 p. 4. Bodnar D.I., Branched Continued Fractions, Kiev: Naukova Dumka, 1986. – 176 p. (in Russian) 5. Dmytryshyn R.I. Associated branched continued fractions with two independent variables// Ukr. Math. Zh. – 2014. – V.66, №9. – P. 1175–1184, (in Ukrainian); Engl. Transl.: Ukr. Math. J. – 2015. – V.66, №9. – P. 1312–1323. 6. Dmytryshyn R.I. On some convergence domains for multidimensional J-fraction with independent variables// Mat. Visn. Nauk. Tov. Im. Shevchenka. – 2011. – V.8. – P. 69–77. (in Ukrainian) 7. Dmytryshyn R.I. On the convergence multidimensional g-fraction with independent variables// Mat. Met. Fiz.-Mekh. Polya. – 2005. – V.48, №4. – P. 87–92. (in Ukrainian) 8. Dmytryshyn R.I. The two-dimensional g-fraction with independent variables for double power series// J. Approx. Theory. – 2012. – V.164, №12. – P. 1520–1539. 9. Dmytryshyn R.I. Two-dimensional generalization of the Rutishauser qd-algorithm// J. Math. Sci. – 2015. – V.208, №3. – P. 301–309. 10. Wall H.S. Analytic Theory of Continued Fractions. – New York: Van Nostrand, 1948. – xiii+433 p.
Pages	150-159
Volume	47
Issue	2
Year	2017
Journal	Matematychni Studii
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