On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function <i>H</i><sub>4</sub>(<i>a</i>,<i>b</i>;<i>c</i>,<i>d</i>;<i>z</i><sub>1</sub>,<i>z</i><sub>2</sub>) 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&qu...
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MDPI AG
2023-03-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/3/299 |
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| author | Tamara Antonova Roman Dmytryshyn Ilona-Anna Lutsiv Serhii Sharyn |
| author_facet | Tamara Antonova Roman Dmytryshyn Ilona-Anna Lutsiv Serhii Sharyn |
| author_sort | Tamara Antonova |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>4</mn></msub></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">C</mi><mn>2</mn></msup><mo>,</mo></mrow></semantics></math></inline-formula> and that this function is an analytic continuation of this ratio in the domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>.</mo></mrow></semantics></math></inline-formula> The application to the approximation of functions of two variables associated with Horn’s double hypergeometric series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>4</mn></msub></semantics></math></inline-formula> is considered, and the expression of solutions of some systems of partial differential equations is indicated. |
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| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T06:56:34Z |
| publishDate | 2023-03-01 |
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| series | Axioms |
| spelling | doaj.art-820a29c3aaaf4ccb829a0e95f0318d212023-11-17T09:35:33ZengMDPI AGAxioms2075-16802023-03-0112329910.3390/axioms12030299On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function <i>H</i><sub>4</sub>(<i>a</i>,<i>b</i>;<i>c</i>,<i>d</i>;<i>z</i><sub>1</sub>,<i>z</i><sub>2</sub>) RatiosTamara Antonova0Roman Dmytryshyn1Ilona-Anna Lutsiv2Serhii Sharyn3Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Stepan Bandera Str., 79013 Lviv, UkraineFaculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, UkraineFaculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, UkraineFaculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, UkraineThe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>4</mn></msub></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">C</mi><mn>2</mn></msup><mo>,</mo></mrow></semantics></math></inline-formula> and that this function is an analytic continuation of this ratio in the domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>.</mo></mrow></semantics></math></inline-formula> The application to the approximation of functions of two variables associated with Horn’s double hypergeometric series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>4</mn></msub></semantics></math></inline-formula> is considered, and the expression of solutions of some systems of partial differential equations is indicated.https://www.mdpi.com/2075-1680/12/3/299Horn functionbranched continued fractionholomorphic functions of several complex variablesnumerical approximationconvergence |
| spellingShingle | Tamara Antonova Roman Dmytryshyn Ilona-Anna Lutsiv Serhii Sharyn On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function <i>H</i><sub>4</sub>(<i>a</i>,<i>b</i>;<i>c</i>,<i>d</i>;<i>z</i><sub>1</sub>,<i>z</i><sub>2</sub>) Ratios Axioms Horn function branched continued fraction holomorphic functions of several complex variables numerical approximation convergence |
| title | On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function <i>H</i><sub>4</sub>(<i>a</i>,<i>b</i>;<i>c</i>,<i>d</i>;<i>z</i><sub>1</sub>,<i>z</i><sub>2</sub>) Ratios |
| title_full | On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function <i>H</i><sub>4</sub>(<i>a</i>,<i>b</i>;<i>c</i>,<i>d</i>;<i>z</i><sub>1</sub>,<i>z</i><sub>2</sub>) Ratios |
| title_fullStr | On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function <i>H</i><sub>4</sub>(<i>a</i>,<i>b</i>;<i>c</i>,<i>d</i>;<i>z</i><sub>1</sub>,<i>z</i><sub>2</sub>) Ratios |
| title_full_unstemmed | On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function <i>H</i><sub>4</sub>(<i>a</i>,<i>b</i>;<i>c</i>,<i>d</i>;<i>z</i><sub>1</sub>,<i>z</i><sub>2</sub>) Ratios |
| title_short | On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function <i>H</i><sub>4</sub>(<i>a</i>,<i>b</i>;<i>c</i>,<i>d</i>;<i>z</i><sub>1</sub>,<i>z</i><sub>2</sub>) Ratios |
| title_sort | on some branched continued fraction expansions for horn s hypergeometric function i h i sub 4 sub i a i i b i i c i i d i i z i sub 1 sub i z i sub 2 sub ratios |
| topic | Horn function branched continued fraction holomorphic functions of several complex variables numerical approximation convergence |
| url | https://www.mdpi.com/2075-1680/12/3/299 |
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