Stirling-Ramanujan constants are exponential periods
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910620534702080 |
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| author | Muñoz, Vicente Perez-Marco, Ricardo |
| author_facet | Muñoz, Vicente Perez-Marco, Ricardo |
| contents | Ramanujan studied a general class of Stirling constants that are the resummation of some natural divergent series. These constants include the classical Euler-Mascheroni, Stirling and Glaisher-Kinkelin constants. We find natural integral representations for all these constants that appear as exponential periods in the field $\mathbb Q (t,e^{-t})$ which reveals their natural transalgebraic nature. We conjecture that all these constants are transcendental numbers. Euler-Mascheroni's and Stirling's integral formula are classical, but the integral formula for Glaisher-Kinkelin appears to be new, as well as the integral formulas for the higher Stirling-Ramanujan constants. The method presented generalizes naturally to prove that many other constants are exponential periods over the field $\mathbb Q(t,e^{-t})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stirling-Ramanujan constants are exponential periods Muñoz, Vicente Perez-Marco, Ricardo Number Theory Mathematical Physics Classical Analysis and ODEs Complex Variables 40A05, 40G99, 11J81, 33B15, 11B68 Ramanujan studied a general class of Stirling constants that are the resummation of some natural divergent series. These constants include the classical Euler-Mascheroni, Stirling and Glaisher-Kinkelin constants. We find natural integral representations for all these constants that appear as exponential periods in the field $\mathbb Q (t,e^{-t})$ which reveals their natural transalgebraic nature. We conjecture that all these constants are transcendental numbers. Euler-Mascheroni's and Stirling's integral formula are classical, but the integral formula for Glaisher-Kinkelin appears to be new, as well as the integral formulas for the higher Stirling-Ramanujan constants. The method presented generalizes naturally to prove that many other constants are exponential periods over the field $\mathbb Q(t,e^{-t})$. |
| title | Stirling-Ramanujan constants are exponential periods |
| topic | Number Theory Mathematical Physics Classical Analysis and ODEs Complex Variables 40A05, 40G99, 11J81, 33B15, 11B68 |
| url | https://arxiv.org/abs/2402.02660 |