On the domains of convergence of the branched continued fraction expansion of ratio $$$H_4(a,d+1;c,d;\mathbf{z})/H_4(a,d+2;c,d+1;\mathbf{z})$$$

R.I. Dmytryshyn; I.-A.V. Lutsiv; O.S. Bodnar

doi:10.15421/242311

On the domains of convergence of the branched continued fraction expansion of ratio $$$H_4(a,d+1;c,d;\mathbf{z})/H_4(a,d+2;c,d+1;\mathbf{z})$$$

R.I. Dmytryshyn (Vasyl Stefanyk Precarpathian National University), https://orcid.org/0000-0003-2845-0137
I.-A.V. Lutsiv (Vasyl Stefanyk Precarpathian National University), https://orcid.org/0000-0002-4100-2972
O.S. Bodnar (Ternopil Volodymyr Hnatiuk National Pedagogical University), https://orcid.org/0000-0003-4207-0624

Abstract

The paper considers the problem of establishing the convergence criteria of the branched continued fraction expansion of the ratio of Horn's hypergeometric functions $$$H_4$$$. To solve it, the technique of expanding the domain of convergence of the branched continued fraction from the known small domain of convergence to a wider domain of convergence is used. For the real and complex parameters of the Horn hypergeometric function $$$H_4$$$, a number of convergence criteria of the branched continued fraction expansion under certain conditions to its coefficients in various unbounded domains of the space have been established.

Keywords

Horn hypergeometric function; branched continued fraction; holomorphic function of several complex variables; convergence

MSC 2020

33C65; 32A17; 32A10; 40A99

Full Text:

PDF

References

Antonova T., Dmytryshyn R., Goran V. "On the analytic continuation of Lauricella-Saran hypergeometric function $$$F_K(a_1,a_2,b_1,b_2;a_1,b_2,c_3;\mathbf{z})$$$", Mathematics, 2023; 11(21): 4487. doi:10.3390/math11214487

Antonova T., Dmytryshyn R., Kravtsiv V. "Branched continued fraction expansions of Horn's hypergeometric function $$$H_3$$$ ratios", Mathematics, 2021; 9(2): p. 148. doi:10.3390/math9020148

Antonova T., Dmytryshyn R., Kril P., Sharyn S. "Representation of some ratios of Horn's hypergeometric functions $$$\mathrm{H}_7$$$ by continued fractions", Axioms, 2023; 12(8): 738. doi:10.3390/axioms12080738

Antonova T., Dmytryshyn R., Kurka R. "Approximation for the ratios of the confluent hypergeometric function $$$\Phi_D^{(N)}$$$ by the branched continued fractions", Axioms, 2023; 11(9): 426. doi:10.3390/axioms11090426

Antonova T., Dmytryshyn R., Lutsiv I.-A., Sharyn S. "On some branched continued fraction expansions for Horn's hypergeometric function $$$H_4(a,b;c,d;z_1,z_2)$$$ ratios", Axioms, 2023; 12(3): 299. doi:10.3390/axioms12030299

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): pp. 22-37. doi:10.33205/cma.1243021

Antonova T., Dmytryshyn R., Sharyn S. "Generalized hypergeometric function $$${}_3F_2$$$ ratios and branched continued fraction expansions", Axioms, 2021; 10(4): p. 310. doi:10.3390/axioms10040310

Antonova T.M., Dmytryshyn R.I. "Truncation error bounds for branched continued fraction $$$\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{a_{i(2)}}{1}{\atop+}\sum_{i_3=1}^{i_2}\frac{a_{i(3)}}{1}{\atop+}\ldots$$$, Ukrainian Math. J., 2020; 72(7): pp. 1018-1029. doi:10.1007/s11253-020-01841-7

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): pp. 3-14. doi:10.30970/ms.54.1.3-14

Antonova T.M. "On convergence of branched continued fraction expansions of Horn's hypergeometric function ratios", Carpathian Math. Publ., 2021; 13(3): pp. 642-650. doi:10.15330/cmp.13.3.642-650

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): pp. 501-518. doi:10.1007/s11253-022-02079-1

Antonova T.M., Sus' O.M. "Sufficient conditions for the equivalent convergence of sequences of different approximants for two-dimensional continued fractions", J. Math. Sci., 2018; 228(1): pp. 1-10. doi:10.1007/s10958-017-3601-3

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): pp. 423-437. doi:10.1007/s10958-022-06062-w

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): pp. 188-200. doi:10.1007/s10958-020-04729-w

Bodnar D.I., Bilanyk I.B. "Parabolic convergence regions of branched continued fractions of the special form", Carpathian Math. Publ., 2021; 13(3): pp. 619-630. doi:10.15330/cmp.13.3.619-630

Bodnar D.I., Bilanyk I.B. "Two-dimensional generalization of the Thron-Jones theorem on the parabolic domains of convergence of continued fractions", Ukrainian Math. J., 2023; 74(9): pp. 1317-1333. doi:10.1007/s11253-023-02138-1

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): pp. 437-448. doi:10.15330/cmp.15.2.437-448

Bodnar D.I. Branched Continued Fractions, Naukova Dumka, Kyiv, 1986. (in Russian)

Bodnar D.I., Manzii O.S. "Expansion of the ratio of Appel hypergeometric functions $$$F_3$$$ into a branching continued fraction and its limit behavior", J. Math. Sci., 2001; 107(1): pp. 3550-3554. doi:10.1023/A:1011977720316

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): pp. 353-359. doi:10.15330/cmp.12.2.353-359

Bodnarchuk P.I., Skorobogatko V.Y. Branched continued fractions and their applications, Naukova Dumka, 1974.

Dmytryshyn R.I. "Associated branched continued fractions with two independent variables", Ukrainian Math. J., 2015; 66(9): pp. 1312-1323. doi:10.1007/s11253-015-1011-6

Dmytryshyn R.I., Lutsiv I.-A.V. "Three- and four-term recurrence relations for Horn's hypergeometric function $$$H_4$$$", Res. Math., 2022; 30(1): pp. 21-29. doi:10.15421/242203

Dmytryshyn R.I. "Convergence of multidimensional A- and J-fractions with independent variables", Comput. Methods Funct. Theory, 2022; 22(2): pp. 229-242. doi:10.1007/s40315-021-00377-6

Dmytryshyn R.I. "Convergence of some branched continued fractions with independent variables", Mat. Stud., 2017; 47(2): pp. 150-159. doi:10.15330/ms.47.2.150-159

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): pp. 592-607. doi:10.15330/cmp.13.3.592-607

Dmytryshyn R.I. "The multidimensional generalization of g-fractions and their application", J. Comp. Appl. Math., 2004; 164-165: pp. 265-284. doi:10.1016/S0377-0427(03)00642-3

Dmytryshyn R.I. "Two-dimensional generalization of the Rutishauser qd-algorithm", J. Math. Sci., 2015; 208(3): pp. 301-309. doi:10.1007/s10958-015-2447-9

Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G. Higher transcendental functions. Vol. 1, McGraw-Hill Book Co., 1953.

Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G. Higher transcendental functions. Vol. 2, McGraw-Hill Book Co., 1953.

Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G. Higher transcendental functions. Vol. 3, McGraw-Hill Book Co., 1953.

Exton H. Multiple Hypergeometric Functions and Applications, Halsted Press, 1976.

Hladun V.R., Hoyenko N.P., Manzij O.S., Ventyk L. "On convergence of function $$$F_4(1, 2; 2, 2; z_1, z_2)$$$ expansion into a branched continued fraction", Math. Model. Comput., 2022; 9(3): pp. 767-778. doi:10.23939/mmc2022.03.767

Horn J. "Hypergeometrische funktionen zweier veränderlichen", Math. Ann., 1931; 105(1): pp. 381-407. (in German) doi:10.1007/BF01455825

Hoyenko N.P., Manzij O.S. "Expansion of Appel $$$F_1$$$ and Lauricella $$$F_D^{(N)}$$$ hypergeometric functions into branched continued fractions", Visnyk Lviv. Univ. Ser. Mech.-Math., 1997; 48: pp. 17-26.

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

Manzii O.S. "Investigation of expansion of the ratio of Appel hypergeometric functions $$$F_3$$$ into a branching continued fraction", Approximation Theory and Its Applications: Proceedings of IM NAS of Ukraine, 2000; 31: pp. 344-353.