On the numerical stability of the branched continued fraction expansion of the ratio $H_4(a,d+1;c,d;\mathbf{z})/H_4(a,d+2;c,d+1;\mathbf{z})$
Abstract
Continued fractions and their generalization, branched continued fractions, are the effective tools used to study special functions. In this aspect, an important problem of continued fractions and branched continued fractions is the study of their numerical stability. The backward recurrence algorithm is one of the main tools for computing approximants of both continued fraction and branched continued fractions. Like most recursive processes, it is prone to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors made in all the previous cycles. This paper considers numerical stability of branched continued fraction expansion of the one ratio of Horn's hypergeometric functions $H_4$ in the special case, namely, $H_4(a,d+1;c,d;\mathbf{z})/H_4(a,d+2;c,d+1;\mathbf{z}).$ For this purpose, the backward recurrence algorithm is investigated. It is proven that under certain conditions on the parameters $a,$ $c,$ and $d$ the some open bi-disc is the set of numerical stability for branched continued fraction expansion, and it is found the estimate of relative rounding error, produced by the backward recurrence algorithm in calculating an $n$th approximant of this expansion.
The results of this paper provide a toolkit for analyzing the numerical stability of algorithms that use branched continued fractions of the studied structure. Error estimates can be used to choose computation parameters, control accuracy, and ensure the reliability of results in applied problems that will use the aforementioned branched continued fractions.
References
T. Acar, O. Alagoz, A. Aral, D. Costarelli, M. Turgay, G. Vinti, Approximation by sampling Kantorovich series in weighted spaces of functions, Turkish J. Math., 46 (2022). http://dx.doi.org/2663–2676.10.55730/1300-0098.3293
T. Acar, O. Alagoz, A. Aral, D. Costarelli, M. Turgay, G. Vinti, Convergence of generalized sampling series in weighted spaces, Demonstr. Math., 55 (2022), 153–162. http://dx.doi.org/10.1515/dema-2022-0014
O. Alagoz, Weighted approximations by sampling type operators: recent and new results, Constr. Math. Anal., 7 (2024), 114–125. http://dx.doi.org/10.33205/cma.1528004
T. Antonova, R. Dmytryshyn, P. Kril, S. Sharyn, Representation of some ratios of Horn’s hypergeometric functions ${H}_7$ by continued fractions, Axioms, 12 (2023), 738. http://dx.doi.org/10.3390/math11214487
T. Antonova, R. Dmytryshyn, S. Sharyn, Generalized hypergeometric function $_3F_2$ ratios and branched continued fraction expansions, Axioms, 10 (2021), 310. http://dx.doi.org/10.3390/axioms10040310
T. Antonova, R. Dmytryshyn, I.-A. Lutsiv, S. Sharyn, On some branched continued fraction expansions for Horn’s hypergeometric function $H_4(a,b;c,d;z_1,z_2)$ ratios, Axioms, 12 (2023), 299. http://dx.doi.org/10.3390/axioms12030299
T. Antonova, R. Dmytryshyn, S. Sharyn, Branched continued fraction representations of ratios of Horn’s confluent function ${H}_6$, Constr. Math. Anal., 6 (2023), 22–37. http://dx.doi.org/10.33205/cma.1243021
W.W. Bell, Special functions for scientists and engineers, D. Van Nostrand Ltd, London, 1968.
D.I. Bodnar, O.S. Bodnar, M.V. Dmytryshyn, M.M. Popov, M.V. Martsinkiv, O.B. Salamakha, Research on the convergence of some types of functional branched continued fractions, Carpathian Math. Publ., 16 (2024), 448–460. http://doi.org/10.15330/cmp.16.2.448-460
D.I. Bodnar, Branched continued fractions, Naukova Dumka, Kyiv, 1986. (in Russian)
P.I. Bodnarchuk, V.Y. Skorobohatko, Branched continued fractions and their applications, Naukova Dumka, Kyiv, 1974. (in Ukrainian)
J. Bustamante, Approximation of bounded functions by positive linear operators, Modern Math. Methods, 2 (2024), 117–131. http://dx.doi.org/10.64700/mmm.30
S. Chaichenko, V. Savchuk, A. Shidlich, Approximation of functions by linear summation methods in the Orlicz-type spaces, J. Math. Sci., 249 (2020), 705–719. http://dx.doi.org/10.1007/s10958-020-04967-y
S.O. Chaichenko, A.L. Shidlich, Approximation of functions by linear methods in weighted Orlicz type spaces with variable exponent, Res. Math., 32 (2024), 70–87. http://doi.org/10.15421/242420
A. Cuyt, P. Van der Cruyssen, Rounding error analysis for forward continued fraction algorithms, Comput. Math. Appl., 11 (1985), 541–564. http://dx.doi.org/10.1016/0898-1221(85)90037-9
M. Dmytryshyn, V. Hladun, On the sets of stability to perturbations of some continued fraction with applications, Symmetry, 17(9) (2025), 1442. http://doi.org/10.3390/sym17091442
R. Dmytryshyn, T. Antonova, M. Dmytryshyn, On the analytic extension of the Horn’s confluent function ${H}_6$ on domain in the space $C^2$, Constr. Math. Anal., 7 (2024), 11–26. http://dx.doi.org/10.33205/cma.1545452
R. Dmytryshyn, C. Cesarano, M. Dmytryshyn, I.-A. Lutsiv, A priori bounds for truncation error of branched continued fraction expansions of Horn’s hypergeometric functions $H_4$ and their ratios, Res. Math., 33 (2025), 13–22. http://dx.doi.org/10.15421/242502
R. Dmytryshyn, C. Cesarano, I.-A. Lutsiv, On the analytical continuation of the ratio $H_4(alpha,delta+1;gamma,delta; z)/H_4(alpha,delta+2;gamma,delta+1;-z)$, Res. Math., 33 (2025), 65–76. https://dx.doi.org/10.15421/242515
R. Dmytryshyn, C. Cesarano, I.-A. Lutsiv, M. Dmytryshyn, Numerical stability of the branched continued fraction expansion of Horn’s hypergeometric function $H_4$, Mat. Stud., 61 (2024), №1, 51–60. http://dx.doi.org/10.30970/ms.61.1.51-60
R. Dmytryshyn, I.-A. Lutsiv, O. Bodnar, On the domains of convergence of the branched continued fraction expansion of ratio $H_4(a,d+1;c,d;mathbf{z})/H_4(a,d+2;c,d+1;mathbf{z})$, Res. Math., 31 (2023), 19–26. http://dx.doi.org/10.15421/242311
R. Dmytryshyn, I.-A. Lutsiv, M. Dmytryshyn, C. Cesarano, On some domains of convergence of branched continued fraction expansions of the ratios of Horn hypergeometric functions $H_4$, Ukr. Math. J., 76 (2024), 559–565. http://dx.doi.org/10.1007/s11253-024-02338-3
R. Dmytryshyn, I.-A. Lutsiv, M. Dmytryshyn, On the analytic extension of the Horn’s hypergeometric function $H_4$, Carpathian Math. Publ., 16 (2024), 32–39. http://dx.doi.org/10.15330/cmp.16.1.32-39
H. Exton, Multiple hypergeometric functions and applications, Halsted Press, Chichester, 1976.
C. Ferreira, J.L. Lopez, E. Perez Sinusia, Convergent and asymptotic expansions of the displacement elastodynamic integral in terms of known functions, J. Comput. Appl. Math., 460 (2025), 116395. http://dx.doi.org/10.1016/j.cam.2024.116395.
C. Ferreira, J.L. Lopez, E. Perez Sinusia, New series expansions for the ${H}$-function of communication theory, Integral Transforms Spec. Funct., 34 (2023), 879–890. http://dx.doi.org/10.1080/10652469.2023.2234556
C. Ferreira, J.L. Lopez, E. Perez Sinusia, Uniform convergent expansions of the error function in terms of elementary functions, Mediterr. J. Math., 20 (2023), 117. http://dx.doi.org/10.1007/s00009-023-02297-2
P.-Ch. Hang, M. Henkel, M.-J. Luo, Asymptotics of the Humbert functions $Psi_1$ and $Psi_2$, J. Approx. Theory, 314 (2026), 106233. http://dx.doi.org/10.1016/j.jat.2025.106233
P.-Ch. Hang, M.-J. Luo, Asymptotics of Saran’s hypergeometric function $F_K$, J. Math. Anal. Appl., 541 (2025), 128707. http://dx.doi.org/10.1016/j.jmaa.2024.128707
P.-Ch. Hang, M.-J. Luo, Asymptotics of the Humbert functions $Psi_1$ for two large arguments, SIGMA, 20 (2024), 074. http://dx.doi.org/10.3842/SIGMA.2024.074
O. Handera-Kalynovska, V. Kravtsiv, The Waring-Girard formulas for symmetric polynomials on spaces $l_p$, Carpathian Math. Publ., 16 (2024), 407–413. http://dx.doi.org/10.15330/cmp.16.2.407-413
V.R. Hladun, D.I. Bodnar, R.S. Rusyn, Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements, Carpathian Math. Publ., 16 (2024), 16–31. http://dx.doi.org/10.15330/cmp.16.1.16-31
V.R. Hladun, M.V. Dmytryshyn, V.V. Kravtsiv, R.S. Rusyn, Numerical stability of the branched continued fraction expansions of the ratios of Horn’s confluent hypergeometric functions ${H}_{6}$, Math. Model. Comput., 11 (2024), 1152–1166. http://doi.org/10.23939/mmc2024.04.1152
V.R. Hladun, N.P. Hoyenko, O.S. Manzij, L. Ventyk, On convergence of function $F_4(1,2;2,2;z_1,z_2)$ expansion into a branched continued fraction, Math. Model. Comput., 9 (2022), 767–778. http://dx.doi.org/10.23939/mmc2022.03.767
V. Hladun, V. Kravtsiv, M. Dmytryshyn, R. Rusyn, On numerical stability of continued fractions, Mat. Stud., 62 (2024), №2, 168–183. http://doi.org/10.30970/ms.62.2.168-183
V. Hladun, R. Rusyn, M. Dmytryshyn, On the analytic extension of three ratios of Horn’s confluent hypergeometric function ${H}_7$, Res. Math., 32 (2024), 60–70. http://dx.doi.org/10.15421/242405
V.R. Hladun, Some sets of relative stability under perturbations of branched continued fractions with complex elements and a variable number of branches, Mat. method. and fiz.-mech. polya, 57 (2014), 14–24. (in Ukrainian); Engl. transl.: J. Math. Sci., 215 (2016), 11–25. http://dx.doi.org/10.1007/s10958-016-2818-x
J. Horn, Hypergeometrische funktionen zweier ver¨anderlichen, Math. Ann., 105 (1931), 381–407. 20230135. https://doi.org/10.1515/dema-2023-013510.1007/BF01455825
N. Hoyenko, T. Antonova, S. Rakintsev, Approximation for ratios of Lauricella–Saran fuctions $F_S$ with real parameters by a branched continued fractions, Math. Bul. Shevchenko Sci. Soc., 8 (2011), 28–42. (in Ukrainian)
N.P. Hoyenko, V.R. Hladun, O.S. Manzij, On the infinite remains of the Norlund branched continued fraction for Appell hypergeometric functions, Carpathian Math. Publ., 6 (2014), 11–25. (in Ukrainian) http://dx.doi.org/10.15330/cmp.6.1.11-25
N.P. Hoyenko, O.S. Manzij, Expansion of Appel $F_1$ and Lauricella $F_D^{(N)}$ hypergeometric functions into branched continued fractions, Visnyk Lviv. Univ. Ser. Mech.-Math., 48 (1997), 17–26. (in Ukrainian)
R. Hryniv, V. Kravtsiv, T. Vasylyshyn, A. Zagorodnyuk, Symmetric and supersymmetric polynomials on $l_p$ and partition functions in quantum statistical physics, Phys. Scr., 100 (2025), 075208. http://dx.doi.org/10.1088/1402-4896/adde1e
W.B. Jones, W.J. Thron, Continued fractions: analytic theory and applications, Addison-Wesley Pub. Co., Reading, 1980.
D. Kaliuzhnyi-Verbovetskyi, V. Pivovarchik, Recovering the shape of a quantum caterpillar tree by two spectra, Mech. Math. Methods, 5 (2023), №1, 14–24. http://dx.doi.org/10.31650/2618-0650-2023-5-1-14-24
H. Karsli, Approximation process of a positive linear operator of hypergeometric type, Demonstr. Math., 57 (2024), 20230135. 20230135. https://doi.org/10.1515/dema-2023-013510.1515/dema-2023-0135
H. Karsli, Historical Backround of wavelets and orthonormal systems: Recent results on positive linear operators reconstructed via wavelets: Recent results on positive linear operators reconstructed via wavelets, Modern Math. Methods, 2 (2024), 132–154. http://dx.doi.org/10.64700/mmm.44
V. Kravtsiv, Block-supersymmetric polynomials on spaces of absolutely convergent series, Symmetry, 16 (2024), 179. http://dx.doi.org/10.3390/sym16020179
V. Kravtsiv, A. Zagorodnyuk, Descriptions of spectra of algebras of bounded-type block-symmetric analytic functions, Symmetry, 17 (2025), 1974. http://dx.doi.org/10.3390/sym17111974
V. Kravtsiv, A.V. Zagorodnyuk, Spectra of algebras of block-symmetric analytic functions of bounded type, Mat. Stud., 58 (2022), №1, 69–81. http://dx.doi.org/10.30970/ms.58.1.69-81
Kh.Yo. Kuchminska, Two-dimensional continued fractions, Pidstryhach Institute for Appl. Probl. in Mech. and Math., NAS of Ukraine, Lviv, 2010. (in Ukrainian)
D. Leviatan, O.V. Motorna, I.O. Shevchuk, More on fast decreasing trigonometric polynomials, Res. Math., 32 (2024), 101–114. https://doi.org/10.15421/242422
L. Lorentzen, H. Waadeland, Continued fractions, V.1, Convergence Theory, Atlantis Press, Amsterdam, 2008.
Y. Lutsiv, T. Antonova, R. Dmytryshyn, M. Dmytryshyn, On the branched continued fraction expansionsof the complete group of ratios of the generalized hypergeometric function $_4F_3$, Res. Math., 32 (2024), 115–132. http://dx.doi.org/10.15421/242423
O. Manziy, V. Hladun, L. Ventyk, The algorithms of constructing the continued fractions for any rations of the hypergeometric Gaussian functions, Math. Model. Comput., 4 (2017), 48–58. http://dx.doi.org/10.23939/mmc2017.01.048
A. Pinti, O. Tulai, Y. Chaikovskyi, M. Stetsko, M. Dmytryshyn, L. Alekseyenko, International Communication and Innovation in Investment in Flagship, Mortgage and Social Projects to Guarantee Security and Minimize Risks of Public Finances, In: R.K. Hamdan (eds) Integrating Big Data and IoT for Enhanced Decision-Making Systems in Business, Studies in Big Data, 177, Springer, Cham, 2026. http://doi.org/10.1007/978-3-031-97609-4_35
K.V. Pozharska, A.S. Romanyuk, S.Ya. Yanchenko, Estimates of the characteristics of nonlinear approximation of classes of periodic functions of many variables, Carpathian Math. Publ., 17 (2025), 447–460. http://dx.doi.org/10.15330/cmp.17.2.447-460.
J. Prestin, V. Savchuk, A. Shidlich, Approximation on hexagonal domains by Taylor-Abel-Poisson means, J. Math. Anal. Appl., 529 (2024), 12753674. http://dx.doi.org/10.1016/j.jmaa.2023.127536
J.B. Seaborn, Hypergeometric functions and their applications, Springer, New York, 1991.
A.S. Romanyuk, S.Y. Yanchenko, Approximation of the classes of periodic functions of one and many variables from the Nikol’skii–Besov and Sobolev spaces, Ukr. Math. J., 74 (2022), 967–980. http://dx.doi.org/10.1007/s11253-022-02110-5
A. Zagorodnyuk, N. Baziv, Y. Chopyuk, T. Vasylyshyn, I. Burtnyak, V. Kravtsiv, Symmetric and supersymmetric polynomials and their applications in the blockchain technology and neural networks, Proceedings of 2023 IEEE World Conference on Applied Intelligence and Computing (AIC), Sonbhadra, India, (2023), 508–513. http://dx.doi.org/10.1109/AIC57670.2023.10263936
Copyright (c) 2025 M. V. Dmytryshyn, C. Cesarano, O. Kondur, I.-A. Lutsiv
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.
