Explicit bounds for Bell numbers and their ratios
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913480676737024 |
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| author | Grunwald, Jerzy Serafin, Grzegorz |
| author_facet | Grunwald, Jerzy Serafin, Grzegorz |
| contents | In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main results correspond to two asymptotic forms expressed by means of the Lambert $W$ function. As an application, some straightforward elementary bounds are derived. Additionally, an absolute convergence rate of the ratio of the consecutive Bell numbers is derived. The main challenge was to obtain satisfactory constants, as the Bell numbers grow rapidly, while the convergence rates are rather slow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14182 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit bounds for Bell numbers and their ratios Grunwald, Jerzy Serafin, Grzegorz Number Theory Classical Analysis and ODEs Combinatorics 11B73, 05A16, 26D07 In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main results correspond to two asymptotic forms expressed by means of the Lambert $W$ function. As an application, some straightforward elementary bounds are derived. Additionally, an absolute convergence rate of the ratio of the consecutive Bell numbers is derived. The main challenge was to obtain satisfactory constants, as the Bell numbers grow rapidly, while the convergence rates are rather slow. |
| title | Explicit bounds for Bell numbers and their ratios |
| topic | Number Theory Classical Analysis and ODEs Combinatorics 11B73, 05A16, 26D07 |
| url | https://arxiv.org/abs/2408.14182 |