On approximation of some Lauricella-Saran's hypergeometric functions $F_M$ and their ratios by branched continued fractions
Анотація
The paper considers the problem of approximating Lauricella-Saran's hypergeometric functions $F_M(a_1,a_2,b_1,b_2;a_1,c_2;z_1,z_2,z_3)$ by rational functions, which are approximants of branched continued fraction expansions - a special family functions. Under the conditions of positive definite values of the elements of the expansions, the domain of analytic continuation of these functions and their ratios is established. Here, the domain is an open connected set. It is also proven that under the above conditions, every branched continued fraction expansion converges to the function that is holomorphic in a given domain of analytic continuation at least as fast as a geometric series with a ratio less then unity.
Посилання
T. Antonova, C. Cesarano, R. Dmytryshyn, S. Sharyn, An approximation to Appell’s hypergeometric function $F_2$ by branched continued fraction, Dolomites Res. Notes Approx., 17 (2024), 22–31. http://dx.doi.org/10.14658/PUPJ-DRNA-2024-1-3
T. Antonova, R. Dmytryshyn, V. Goran, 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, 11 (2023), 4487. http://dx.doi.org/10.3390/math11214487
T. Antonova, R. Dmytryshyn, V. Kravtsiv, Branched continued fraction expansions of Horn’s hypergeometric function $H_3$ ratios, Mathematics, 9 (2021), 148. http://dx.doi.org/10.3390/math9020148
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/axioms12080738
T. Antonova, R. Dmytryshyn, R. Kurka, Approximation for the ratios of the confluent hypergeometric function $Phi_D^{(N)}$ by the branched continued fractions, Axioms, 11 (2022), 426. http://dx.doi.org/10.3390/axioms11090426
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
I.B. Bilanyk, D.I. Bodnar, O.G. Vozniak, Convergence criteria of branched continued fractions, Res. Math., 32 (2024), 53–69. http://doi.org/10.15421/242419
D.I. Bodnar, I.B. Bilanyk, Two-dimensional generalization of the Thron-Jones theorem on the parabolic domains of convergence of continued fractions, Ukr. Math. Zhurn., 74 (2022), 1155–1169. (in Ukrainian); Engl. transl.: Ukrainian Math. J., 74 (2023), 1317–1333. http://doi.org/10.1007/s11253-023-02138-1
D.I. Bodnar, O.S. Bodnar, I.B. Bilanyk, A truncation error bound for branched continued fractions of the special form on subsets of angular domains, Carpathian Math. Publ., 15 (2023), 437–448. http://doi.org/10.15330/cmp.15.2.437-448
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)
J. Choi, Recent advances in special functions and their applications, Symmetry, 15 (2023), 2159. http://doi.org/10.3390/sym15122159
R. Dmytryshyn, T. Antonova, M. Dmytryshyn, On the analytic extension of the Horn’s confluent function $mathrm{H}_6$ on domain in the space $mathbb{C}^2$, Constr. Math. Anal., 7 (2024), 11–26. http://dx.doi.org/10.33205/cma.1545452
R. Dmytryshyn, C. Cesarano, I.-A. Lutsiv, M. Dmytryshyn, Numerical stability of the branched continued fraction expansion of Horn’s hypergeometric function H4, Mat. Stud., 61 (2024), 51–60. http://dx.doi.org/10.30970/ms.61.1.51-60
R. Dmytryshyn, V. Goran, On the analytic extension of Lauricella–Saran’s hypergeometric function $F_K$ to symmetric domains, Symmetry, 16 (2024), 220. http://dx.doi.org/10.3390/sym16020220
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 H4, Ukr. Math. Zhurn., 76 (2024), 502–508. (in Ukrainian); Engl. transl.: Ukrainian 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
R. Dmytryshyn, On the analytic continuation of Appell’s hypergeometric function $F_2$ to some symmetric domains in the space $mathbb{C}^2$, Symmetry, 16 (2024), 1480. http://dx.doi.org/10.3390/sym16111480
R.I. Dmytryshyn, S.V. Sharyn, Approximation of functions of several variables by multidimensional S-fractions with independent variables, Carpathian Math. Publ., 13 (2021), 592–607. http://dx.doi.org/10.15330/cmp.13.3.592-607
R. Dmytryshyn, S. Sharyn, Representation of special functions by multidimensional A- and J-fractions with independent variables, Fractal Fract., 9 (2025), 89. http://dx.doi.org/10.3390/fractalfract9020089
P.-C. Hang, L. Hu, Full asymptotic expansions of the Humbert function Φ1, arXiv, (2025), arXiv:2504.09280. http://dx.doi.org/10.48550/arXiv.2504.09280
P.-C. 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
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 H6, 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), 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
G. Lauricella, Sulle funzioni ipergeometriche a pi`u variabili, Rend. Circ. Matem., 7 (1893), 111–158. http://dx.doi.org/10.1007/BF03012437
Y. Lutsiv, T. Antonova, R. Dmytryshyn, M. Dmytryshyn, On the branched continued fraction expansions of 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
I. Nyzhnyk, R. Dmytryshyn, T. Antonova, On branched continued fraction expansions of hypergeometric functions $F_M$ and their ratios, Modern Math. Methods, 3 (2025), 1–13.
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. Ryskan, T. Ergashev, On some formulas for the Lauricella function, Mathematics, 11 (2023), 4978. http://dx.doi.org/10.3390/math11244978
Sh. Saran, Transformations of certain hypergeometric functions of three variables, Acta Math., 93 (1955), 293–312. http://dx.doi.org/10.1007/BF02392525
X.-J. Yang, Theory and applications of special functions for scientists and engineers, Springer, Singapore, 2022.
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