On the analytic extension of the Horn's confluent function $\mathrm{H}_6$ on domain in the space $\mathbb{C}^2$

Year 2024, , 11 - 26, 16.12.2024

Roman Dmytryshyn , Tamara Antonova , Marta Dmytryshyn

https://doi.org/10.33205/cma.1545452

Abstract

The paper considers the problem of representation and extension of Horn's confluent functions by a special family of functions - branched continued fractions. In a new region, an estimate of the rate of convergence for branched continued fraction expansions of the ratios of Horn's confluent functions $\mathrm{H}_6$ with real parameters is established. Here, region is a domain (open connected set) together with all, part or none of its boundary. Also, a new domain of the analytical continuation of the above-mentioned ratios is established, using their branched continued fraction expansions whose elements are polynomials in the space $\mathbb{C}^2$. These expansions can be used to approximate the solutions of certain differential equations and analytic functions, which are represented by the Horn's confluent functions $\mathrm{H}_6.$

Keywords

Horn, branched continued fraction, holomorphic functions of several complex variables, analytic continuation, convergence

References

• T. Antonova, C. Cesarano, R. Dmytryshyn and S. Sharyn: An approximation to Appell’s hypergeometric function F2 by branched continued fraction, Dolomites Res. Notes Approx., 17 (1) (2024), 22–31.
• T. Antonova, R. Dmytryshyn and V. Goran: On the analytic continuation of Lauricella-Saran hypergeometric function FK(a1, a2, b1, b2; a1, b2, c3; z), Mathematics, 11 (21) (2023), Article ID: 4487.
• T. Antonova, R. Dmytryshyn and V. Kravtsiv: Branched continued fraction expansions of Horn’s hypergeometric function H3 ratios, Mathematics, 9 (2) (2021), Article ID: 148.
• T. Antonova, R. Dmytryshyn and R. Kurka: Approximation for the ratios of the confluent hypergeometric function Φ(N)D by the branched continued fractions, Axioms, 11 (9) (2022), Article ID: 426.
• T. Antonova, R. Dmytryshyn and S. Sharyn: Branched continued fraction representations of ratios of Horn’s confluent function H6, Constr. Math. Anal., 6 (1) (2023), 22–37.
• T. M. Antonova: On convergence of branched continued fraction expansions of Horn’s hypergeometric function H3 ratios, Carpathian Math. Publ., 13 (3) (2021), 642–650.
• T. Antonova: On structure of branched continued fractions, Carpathian Math. Publ., 16 (2) (2024), 391–400.
• Z. e. a. Bentalha: Representation of the Coulomb matrix elements by means of Appell hypergeometric function F2, Math. Phys. Anal. Geom., 21 (2018), Article ID: 10.
• D. I. Bodnar, O. S. Bodnar and I. B. Bilanyk: A truncation error bound for branched continued fractions of the special form on subsets of angular domains, Carpathian Math. Publ., 15 (2) (2023), 437–448.
• 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. J., 74 (9) (2023), 1317–1333.
• D. I. Bodnar: Branched Continued Fractions, Naukova Dumka, Kyiv (1986). (In Russian)
• 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.
• D. I. Bodnar, O. S. Manzii: Expansion of the ratio of Appel hypergeometric functions F3 into a branching continued fraction and its limit behavior, J. Math. Sci., 107 (1) (2001), 3550–3554.
• D. I. Bodnar: Multidimensional C-fractions, J. Math. Sci., 90 (5) (1998), 2352–2359.
• R. Dmytryshyn, C. Cesarano, I.-A. Lutsiv and M. Dmytryshyn: Numerical stability of the branched continued fraction expansion of Horn’s hypergeometric function H4, Mat. Stud., 61 (1) (2024), 51–60.
• R. Dmytryshyn, V. Goran: On the analytic extension of Lauricella–Saran’s hypergeometric function FK to symmetric domains, Symmetry, 16 (2) (2024), Article ID: 220.
• R. Dmytryshyn, I.-A. Lutsiv and O. Bodnar: On the domains of convergence of the branched continued fraction expansion of ratio H4(a, d + 1; c, d; z)/H4(a, d + 2; c, d + 1; z), Res. Math., 31 (2) (2023), 19–26.
• R. Dmytryshyn, I.-A. Lutsiv and M. Dmytryshyn: On the analytic extension of the Horn’s hypergeometric function H4, Carpathian Math. Publ. 16 (1) (2024), 32–39.
• R. I. Dmytryshyn: The multidimensional generalization of g-fractions and their application, J. Comput. Appl. Math., 164–165 (2004), 265–284.
• H. Exton: Multiple Hypergeometric Functions and Applications, Halsted Press, Chichester (1976).
• C. F. Gauss: Disquisitiones generales circa seriem infinitam 1 + αβ/1·γ x + α(α+1)β(β+1)/1·2·γ(γ+1) xx + α(α+1)(α+2)β(β+1)(β+2)/1·2·3·γ(γ+1)(γ+3) x3+etc, In Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, Classis Mathematicae (1812), H. Dieterich: Gottingae, Germany, 2 (1813), 3–46.
• V. R. Hladun, D. I. Bodnar and R. S. Rusyn: Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements, Carpathian Math. Publ., 16 (1) (2024), 16–31.
• V. R. Hladun, N. P. Hoyenko, O. S. Manzij and L. Ventyk: On convergence of function F4(1, 2; 2, 2; z1, z2) expansion into a branched continued fraction, Math. Model. Comput., 9 (3) (2022), 767–778.
• V. Hladun, R. Rusyn and M. Dmytryshyn: On the analytic extension of three ratios of Horn’s confluent hypergeometric function H7, Res. Math., 32 (1) (2024), 60–70.
• J. Horn: Hypergeometrische Funktionen zweier Veränderlichen, Math. Ann., 105 (1931), 381–407.
• 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)
• B. Kol, R. Shir: The propagator seagull: general evaluation of a two loop diagram, J. High Energy Phys., 2019 (2019), Article ID: 83.
• I. Kovalyov: Rational generalized Stieltjes functions, Constr. Math. Anal., 5 (3) (2022), 154–167.
• N. Ortner, P, Wagner: On the singular values of the incomplete Beta function, Constr. Math. Anal., 5 (2) (2022), 93–104.
• 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.
• J. B. Seaborn: Hypergeometric Functions and Their Applications, Springer, New York (1991).
• B. V. Shabat: Introduce to complex analysis. Part II. Functions of several variables, American Mathematical Society, Providence (1992).
• V. S. Vladimirov: Methods of the theory of functions of many complex variables, The MIT Press, Cambridge (1966).
• H. S. Wall: Analytic Theory of Continued Fractions, D. Van Nostrand Co., New York (1948).