Generalized Hypergeometric Function <sub>3</sub><i>F</i><sub>2</sub> Ratios and Branched Continued Fraction Expansions
The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction ex...
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2021-11-01
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| author | Tamara Antonova Roman Dmytryshyn Serhii Sharyn |
| author_facet | Tamara Antonova Roman Dmytryshyn Serhii Sharyn |
| author_sort | Tamara Antonova |
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| description | The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub><mo>.</mo></mrow></semantics></math></inline-formula> To illustrate this, we have given a few numerical experiments at the end. |
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| spelling | doaj.art-d5378c980df64b6a93ba4f06ec6e3f922023-11-23T03:49:53ZengMDPI AGAxioms2075-16802021-11-0110431010.3390/axioms10040310Generalized Hypergeometric Function <sub>3</sub><i>F</i><sub>2</sub> Ratios and Branched Continued Fraction ExpansionsTamara Antonova0Roman Dmytryshyn1Serhii Sharyn2Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Stepana Bandery Str., 79000 Lviv, UkraineFaculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, UkraineFaculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, UkraineThe paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub><mo>.</mo></mrow></semantics></math></inline-formula> To illustrate this, we have given a few numerical experiments at the end.https://www.mdpi.com/2075-1680/10/4/310generalized hypergeometric functionbranched continued fractionconvergencerational approximation |
| spellingShingle | Tamara Antonova Roman Dmytryshyn Serhii Sharyn Generalized Hypergeometric Function <sub>3</sub><i>F</i><sub>2</sub> Ratios and Branched Continued Fraction Expansions Axioms generalized hypergeometric function branched continued fraction convergence rational approximation |
| title | Generalized Hypergeometric Function <sub>3</sub><i>F</i><sub>2</sub> Ratios and Branched Continued Fraction Expansions |
| title_full | Generalized Hypergeometric Function <sub>3</sub><i>F</i><sub>2</sub> Ratios and Branched Continued Fraction Expansions |
| title_fullStr | Generalized Hypergeometric Function <sub>3</sub><i>F</i><sub>2</sub> Ratios and Branched Continued Fraction Expansions |
| title_full_unstemmed | Generalized Hypergeometric Function <sub>3</sub><i>F</i><sub>2</sub> Ratios and Branched Continued Fraction Expansions |
| title_short | Generalized Hypergeometric Function <sub>3</sub><i>F</i><sub>2</sub> Ratios and Branched Continued Fraction Expansions |
| title_sort | generalized hypergeometric function sub 3 sub i f i sub 2 sub ratios and branched continued fraction expansions |
| topic | generalized hypergeometric function branched continued fraction convergence rational approximation |
| url | https://www.mdpi.com/2075-1680/10/4/310 |
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