Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $π$ and Values of the Gamma Function
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916123884126208 |
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| author | Hakimoglu, Cetin |
| author_facet | Hakimoglu, Cetin |
| contents | The beta integral is applied to accelerate the hypergeometric function $2 F 1\left\{1, B; C ; w\right\}$ to derive new infinite series for constants such as $π$ and values of the gamma function. A compendium of new infinite series is given. Ramanujan-like formulas for pi are also derived based on elementary inverse trigonometric functions, including a formula with rational values that adds 2.5 digits per terms, which makes the series much more compact than similar formulas in the existing literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08693 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $π$ and Values of the Gamma Function Hakimoglu, Cetin Classical Analysis and ODEs 40A25 (Primary) 11B65, 11Y60, 65B10 (Secondary) The beta integral is applied to accelerate the hypergeometric function $2 F 1\left\{1, B; C ; w\right\}$ to derive new infinite series for constants such as $π$ and values of the gamma function. A compendium of new infinite series is given. Ramanujan-like formulas for pi are also derived based on elementary inverse trigonometric functions, including a formula with rational values that adds 2.5 digits per terms, which makes the series much more compact than similar formulas in the existing literature. |
| title | Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $π$ and Values of the Gamma Function |
| topic | Classical Analysis and ODEs 40A25 (Primary) 11B65, 11Y60, 65B10 (Secondary) |
| url | https://arxiv.org/abs/2402.08693 |