On the Algebraic Independence of a Set of Generalized Constants

Abstract

Neither the Euler-Mascheroni constant, $\gamma=0.577215\ldots$, nor the Euler-Gompertz constant, $\delta=0.596347\ldots$, 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 $\{\gamma^{\left(n\right)}+\delta^{\left(n\right)}/e\}_{n\geq0}$, and thus the transcendence of $\gamma^{\left(n\right)}+\delta^{\left(n\right)}/e$ for all $n\geq0$. 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 $n\geq1$.