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arXiv · 2402.02660

Stirling-Ramanujan constants are exponential periods

Vicente Muñoz·Ricardo Perez-Marco
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Abstract

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 $\mathbb Q(t,e^{-t})$.

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BibTeXRIS

Vicente Muñoz, Ricardo Perez-Marco. 2024-02-05. Stirling-Ramanujan constants are exponential periods. https://doi.org/10.4310/cntp.240904005850

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