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 ($\gamma =0.577215\dots$) and Euler-Gompertz constant ($\delta =0.596347\dots$): $\gamma +\delta /e=\text{Ein}\left(1\right)$. Although neither $\gamma$ nor $\delta$ 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 ($\gamma$,$\delta$) and ($\gamma$,$\delta /e$) are known to be disjunctively transcendental. In light of these observations, we investigate the impact of the coefficient $\alpha$ in combinations of the form $\gamma +\alpha \delta$, and find that $\alpha =1/e$ is the unique coefficient value such that canonical Borel-summable divergent series for $\gamma$ and $\delta$ 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, ${\gamma }^{\left(n\right)}+\alpha {\delta }^{\left(n\right)}$, with ${\gamma }^{\left(n\right)}$ and ${\delta }^{\left(n\right)}$ given by (ordinary and conditional) moments of the Gumbel(0,1) probability distribution.