Branched continued fraction representations of ratios of Horn's confluent function $\mathrm{H}_6$

Abstract

In this paper, we derive some branched continued fraction representations for the ratios of the Horn's confluent function $\mathrm{H}_6.$ The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. We establish the estimates of the rate of convergence for the branched continued fraction expansions in some region $\Omega$ (here, region is a domain (open connected set) together with all, part or none of its boundary). It is also proved that the corresponding branched continued fractions uniformly converge to holomorphic functions on every compact subset of some domain $\Theta,$ and that these functions are analytic continuations of the ratios of double confluent hypergeometric series in $\Theta.$ At the end, several numerical experiments are represented to indicate the power and efficiency of branched continued fractions as an approximation tool compared to double confluent hypergeometric series.

Keywords

• hypergeometric function
• branched continued fraction
• convergence

References

1. T. Antonova, R. Dmytryshyn and V. Kravtsiv: Branched continued fraction expansions of Horn’s hypergeometric function $H_3$ ratios, Mathematics, 9 (2) (2021), 148.
2. T. Antonova, R. Dmytryshyn and R. Kurka: Approximation for the ratios of the confluent hypergeometric function $\Phi^{(N)}_D$ by the branched continued fractions, Axioms, 11 (9) (2022), 426.
3. T. Antonova, R. Dmytryshyn and S. Sharyn: Generalized hypergeometric function ${}_3F_2$ ratios and branched continued fraction expansions, Axioms, 10 (4) (2021), 310.
4. T. M. Antonova, N. P. Hoyenko: Approximation of Lauricella’s functions $F_D$ ratio by Nörlund’s branched continued fraction in the complex domain, Mat. Metody Fiz. Mekh. Polya, 47 (2) (2004) 7–15. (In Ukrainian)
5. T. M. Antonova: On convergence of branched continued fraction expansions of Horn’s hypergeometric function $H_3$ ratios, Carpathian Math. Publ., 13 (3) (2021), 642–650.
6. P. Appell: Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles, C. R. Acad. Sci. Paris, 90 (1880), 296–298.
7. W. N. Bailey: Generalised Hypergeometric Series, Cambridge University Press, Cambridge (1935).
8. P. I. Bodnarchuk, V. Y. Skorobogatko: Branched Continued Fractions and Their Applications, Naukova Dumka, Kyiv (1974). (In Ukrainian)

9. D. I. Bodnar, I. B. Bilanyk, Estimation of the rates of pointwise and uniform convergence of branched continued fractions with inequivalent variables, J. Math. Sci., 265 (3) (2022), 423–437.
10. D. I. Bodnar, I. B. Bilanyk: On the convergence of branched continued fractions of a special form in angular domains, J. Math. Sci., 246 (2) (2020), 188–200.
11. D. I. Bodnar, I. B. Bilanyk: Parabolic convergence regions of branched continued fractions of the special form, Carpathian Math. Publ., 13 (3) (2021), 619–630.
12. D. I. Bodnar: Branched Continued Fractions, Naukova Dumka, Kyiv (1986). (In Russian)
13. D. I. Bodnar: Expansion of a ratio of hypergeometric functions of two variables in branching continued fractions, J. Math. Sci., 64 (32) (1993), 1155–1158.
14. D. I. Bodnar, N. P. Hoyenko Approximation of the ratio of Lauricella functions by a branched continued fraction, Mat. Studii, 20 (2) (2003), 210–214.
15. D. I. Bodnar, O. S. Manzii: Expansion of the ratio of Appel hypergeometric functions $F_3$ into a branching continued fraction and its limit behavior, J. Math. Sci., 107 (1) (2001), 3550–3554.
16. D. I. Bodnar: Multidimensional C-fractions, J. Math. Sci., 90 (5) (1998), 2352–2359.
17. Yu. A. Brychkov, N. V. Savischenko: On some formulas for the Horn functions $H_3(a,b;c;w,z),$ $H_6^{(c)}(a;c;w,z)$ and Humbert function $\Phi_3(b;c;w,z)$, Integral Transforms Spec. Funct., 32 (9) (2020), 661–676.
18. R. I. Dmytryshyn, I.-A. V. Lutsiv: Three- and four-term recurrence relations for Horn’s hypergeometric function $H_4$, Res. Math., 30 (1) (2022), 21–29.
19. R. I. Dmytryshyn: Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series, Proc. R. Soc. Edinb. Sect. A, 150 (5) (2020), 1853–1870.
20. R. I. Dmytryshyn: On the expansion of some functions in a two-dimensional g-fraction independent variables, J. Math. Sci., 181 (3) (2012), 320–327.
21. R. I. Dmytryshyn, S. V. Sharyn: Approximation of functions of several variables by multidimensional S fractions with independent variables, Carpathian Math. Publ., 13 (3) (2021), 592–607.
22. R. I. Dmytryshyn: The multidimensional generalization of g-fractions and their application, J. Comput. Appl. Math., 164–165 (2004), 265–284.
23. R. I. Dmytryshyn: Two-dimensional generalization of the Rutishauser qd-algorithm, J. Math. Sci., 208 (3) (2015), 301–309.
24. A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi: Higher Transcendental Functions, Vol. 1, McGraw-Hill Book Co., New York (1953).
25. H. Exton: Multiple Hypergeometric Functions and Applications, Halsted Press, Chichester (1976).
26. V. R. Hladun, N. P. Hoyenko, O. S. Manzij and 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 (3) (2022), 767–778.
27. J. Horn: Hypergeometrische Funktionen zweier Veränderlichen, Math. Ann., 105 (1931), 381–407.
28. N. Hoyenko, T. Antonova and S. Rakintsev: Approximation for ratios of Lauricella–Saran fuctions $\textit{F}_S$ with real parameters by a branched continued fractions, Math. Bul. Shevchenko Sci. Soc., 8 (2011), 28–42. (In Ukrainian)
29. N. Hoyenko, V. Hladun and O. Manzij: On the infinite remains of the Nórlund branched continued fraction for Appell hypergeometric functions, Carpathian Math. Publ., 6 (1) (2014), 11–25. (In Ukrainian)
30. J. A. Murphy, M. R. O’Donohoe: A two-variable generalization of the Stieltjes-type continued fraction, J. Comput. Appl. Math., 4 (3) (1978), 181–190.
31. M. Pétréolle, A. D. Sokal and B. X. Zhu: Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes-Rogers and Thron-Rogers polynomials, with coefficientwise Hankel-total positivity, arXiv, (2020), arXiv:1807.03271v2.
32. W. Siemaszko: Thile-type branched continued fractions for two-variable functions, J. Comput. Appl. Math., 6 (2) (1983), 121–125.