Stirling-Ramanujan constants are exponential periods

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