On Several Results Associated with the Apéry-like Series
In 1979, Apéry proved the irrationality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantic...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/5/1022 |
| _version_ | 1797598260591853568 |
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| author | Prathima Jayarama Dongkyu Lim Arjun K. Rathie |
| author_facet | Prathima Jayarama Dongkyu Lim Arjun K. Rathie |
| author_sort | Prathima Jayarama |
| collection | DOAJ |
| description | In 1979, Apéry proved the irrationality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> and other infinite series involving central binomial coefficients. The purpose of this work is to present the first 20 results related to the Apéry-like series in the form of 4 lemmas, each containing 5 results. The Sherman’s results are applied to attain this. Thereafter, these 20 results are further used to establish up to 104 results pertaining to the Apéry-like series in the form of 4 theorems, with 26 results each. These findings are finally been described in terms of the generalized hypergeometric functions. Symmetry occurs naturally in the generalized hypergeometric functions. |
| first_indexed | 2024-03-11T03:16:55Z |
| format | Article |
| id | doaj.art-d9abb3e805734c83960bcf148bfc6def |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T03:16:55Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-d9abb3e805734c83960bcf148bfc6def2023-11-18T03:29:50ZengMDPI AGSymmetry2073-89942023-05-01155102210.3390/sym15051022On Several Results Associated with the Apéry-like SeriesPrathima Jayarama0Dongkyu Lim1Arjun K. Rathie2Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576104, Karnataka, IndiaDepartment of Mathematics Education, Andong National University, Andong 36729, Republic of KoreaDepartment of Mathematics, Vedant College of Engineering & Technology, Rajasthan Technical University, Bundi 323021, Rajasthan, IndiaIn 1979, Apéry proved the irrationality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> and other infinite series involving central binomial coefficients. The purpose of this work is to present the first 20 results related to the Apéry-like series in the form of 4 lemmas, each containing 5 results. The Sherman’s results are applied to attain this. Thereafter, these 20 results are further used to establish up to 104 results pertaining to the Apéry-like series in the form of 4 theorems, with 26 results each. These findings are finally been described in terms of the generalized hypergeometric functions. Symmetry occurs naturally in the generalized hypergeometric functions.https://www.mdpi.com/2073-8994/15/5/1022central binomail coefficientsreciprocalsApéry-like seriescombinatoricspochhammer symbolgamma function |
| spellingShingle | Prathima Jayarama Dongkyu Lim Arjun K. Rathie On Several Results Associated with the Apéry-like Series Symmetry central binomail coefficients reciprocals Apéry-like series combinatorics pochhammer symbol gamma function |
| title | On Several Results Associated with the Apéry-like Series |
| title_full | On Several Results Associated with the Apéry-like Series |
| title_fullStr | On Several Results Associated with the Apéry-like Series |
| title_full_unstemmed | On Several Results Associated with the Apéry-like Series |
| title_short | On Several Results Associated with the Apéry-like Series |
| title_sort | on several results associated with the apery like series |
| topic | central binomail coefficients reciprocals Apéry-like series combinatorics pochhammer symbol gamma function |
| url | https://www.mdpi.com/2073-8994/15/5/1022 |
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