Conjunctions of Three "Euler Constants" in Poisson-Related Expressions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913920384499712 |
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| author | Powers, Michael R. |
| author_facet | Powers, Michael R. |
| contents | Three mathematical constants bear the name of the venerable Leonhard Euler: Euler's number, $e=2.718281\ldots$; the Euler-Mascheroni constant, $γ=0.577216\ldots$; and the Euler-Gompertz constant, $δ=0.596347\ldots$. In the present work, we consider two joint appearances of these constants, one in a well-known equation of Hardy (interpretable in connection with inverse second moments of the Poisson probability distribution), and the other from a sequence of probabilities generated by recursively conditional Exponential (i.e., Poisson-event waiting-time) distributions. In both cases, we explore generalizations of the initial observations to offer more comprehensive results, including extensions of Hardy's equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02054 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conjunctions of Three "Euler Constants" in Poisson-Related Expressions Powers, Michael R. Number Theory Probability 11B83, 60E05 Three mathematical constants bear the name of the venerable Leonhard Euler: Euler's number, $e=2.718281\ldots$; the Euler-Mascheroni constant, $γ=0.577216\ldots$; and the Euler-Gompertz constant, $δ=0.596347\ldots$. In the present work, we consider two joint appearances of these constants, one in a well-known equation of Hardy (interpretable in connection with inverse second moments of the Poisson probability distribution), and the other from a sequence of probabilities generated by recursively conditional Exponential (i.e., Poisson-event waiting-time) distributions. In both cases, we explore generalizations of the initial observations to offer more comprehensive results, including extensions of Hardy's equation. |
| title | Conjunctions of Three "Euler Constants" in Poisson-Related Expressions |
| topic | Number Theory Probability 11B83, 60E05 |
| url | https://arxiv.org/abs/2503.02054 |