On the Algebraic Independence of a Set of Generalized Constants

Abstract

Neither the Euler-Mascheroni constant, $\gamma =0.577215\dots$, nor the Euler-Gompertz constant, $\delta =0.596347\dots$, is currently known to be irrational. However, it has been proved that these two numbers are disjunctively transcendental; that is, at least one of them must be transcendental. The two constants are related through a well-known equation of Hardy, which recently has been generalized to a pair of infinite sequences $({\gamma }^{\left(n\right)},{\delta }^{\left(n\right)})$ based on moments of the Gumbel(0,1) probability distribution. In the present work, we demonstrate the algebraic independence of the set $\{{\gamma }^{\left(n\right)}+{\delta }^{\left(n\right)}/e{\}}_{n\ge 0}$, and thus the transcendence of ${\gamma }^{\left(n\right)}+{\delta }^{\left(n\right)}/e$ for all $n\ge 0$. This further implies the disjunctive transcendence of both pairs $({\gamma }^{\left(n\right)},{\delta }^{\left(n\right)}/e)$ and $({\gamma }^{\left(n\right)},{\delta }^{\left(n\right)})$ for all $n\ge 1$.