On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function H4(a,b;c,d;z1,z2) Ratios

The paper deals with the problem of representation of Horn’s hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn’s hypergeometric function H4 ratios are constructed. The method employed is a two-dimensional generalizati...

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Published in:	Axioms Vol. 12; no. 3; p. 299
Main Authors:	Antonova, Tamara, Dmytryshyn, Roman, Lutsiv, Ilona-Anna, Sharyn, Serhii
Format:	Journal Article
Language:	English
Published:	Basel MDPI AG 01.03.2023
Subjects:	Analytic functions branched continued fraction Computational mathematics convergence Domains holomorphic functions of several complex variables Horn function Horns Hypergeometric functions Mathematical analysis numerical approximation Partial differential equations Ratios
ISSN:	2075-1680, 2075-1680
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Abstract	The paper deals with the problem of representation of Horn’s hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn’s hypergeometric function H4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H,H⊂C2, and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn’s double hypergeometric series H4 is considered, and the expression of solutions of some systems of partial differential equations is indicated.
AbstractList	The paper deals with the problem of representation of Horn’s hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn’s hypergeometric function H4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H,H⊂C2, and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn’s double hypergeometric series H4 is considered, and the expression of solutions of some systems of partial differential equations is indicated. The paper deals with the problem of representation of Horn’s hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn’s hypergeometric function H4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H, H⊂ C 2, and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn’s double hypergeometric series H4 is considered, and the expression of solutions of some systems of partial differential equations is indicated.
Author	Lutsiv, Ilona-Anna Antonova, Tamara Dmytryshyn, Roman Sharyn, Serhii
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Cites_doi	10.15330/cmp.11.1.33-41 10.1007/BF02433970 10.1007/BF01455825 10.1023/A:1011977720316 10.15330/cmp.13.3.642-650 10.30970/ms.54.1.3-14 10.3390/axioms10040310 10.3390/math9020148 10.23939/mmc2022.03.767 10.1007/BF02433965 10.15330/cmp.13.3.608-618 10.1007/BF01571633 10.1007/s10958-022-06062-w 10.15330/cmp.11.1.54-58 10.1080/10652469.2021.1878356 10.1016/0377-0427(85)90019-6 10.1007/BF01472246 10.30970/ms.52.2.115-123 10.2991/978-94-91216-37-4_1 10.1007/s10958-020-04729-w 10.15330/cmp.10.1.58-64 10.1016/0771-050X(80)90005-4 10.15330/cmp.13.3.619-630 10.1007/s11253-020-01841-7 10.1007/BF01098839 10.1016/j.ejc.2020.103235 10.1016/0771-050X(78)90002-5 10.15421/242203 10.15330/cmp.6.1.11-25 10.1093/imanum/8.2.209 10.1016/0168-9274(83)90006-5 10.1007/BFb0072451 10.15330/cmp.12.2.353-359
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SubjectTerms	Analytic functions branched continued fraction Computational mathematics convergence Domains holomorphic functions of several complex variables Horn function Horns Hypergeometric functions Mathematical analysis numerical approximation Partial differential equations Ratios
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Title	On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function H4(a,b;c,d;z1,z2) Ratios
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