On the Uniqueness of Ein(1) among Linear Combinations of the Euler-Mascheroni and Euler-Gompertz Constants

Abstract

From a well-known equation of Hardy, one can derive a simple linear combination of the Euler-Mascheroni constant ($γ=0.577215\ldots$) and Euler-Gompertz constant ($δ=0.596347\ldots$): $γ+δ/e=\textrm{Ein}\left(1\right)$. Although neither $γ$ nor $δ$ is currently known to be irrational, this linear combination has been shown to be transcendental (by virtue of the fact that it appears as an algebraic point value of a particular E-function). Moreover, both pairs ($γ$,$δ$) and ($γ$,$δ/e$) are known to be disjunctively transcendental. In light of these observations, we investigate the impact of the coefficient $α$ in combinations of the form $γ+αδ$, and find that $α=1/e$ is the unique coefficient value such that canonical Borel-summable divergent series for $γ$ and $δ$ can be linearly combined to force conventional convergence of the resulting series. We further indicate how this uniqueness property extends to a sequence of generalized linear combinations, $γ^{\left(n\right)}+αδ^{\left(n\right)}$, with $γ^{\left(n\right)}$ and $δ^{\left(n\right)}$ given by (ordinary and conditional) moments of the Gumbel(0,1) probability distribution.