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Pré-Publication, Document De Travail Année : 2024

Stirling-Ramanujan constants are exponential periods

Résumé

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 appear 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 $\mathbb Q(t,e^{-t})$
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Dates et versions

hal-04437773 , version 1 (05-02-2024)
hal-04437773 , version 2 (20-02-2024)

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Domaine public

Identifiants

  • HAL Id : hal-04437773 , version 2

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Vicente Muñoz, Ricardo Pérez-Marco. Stirling-Ramanujan constants are exponential periods. 2024. ⟨hal-04437773v2⟩
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