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

TL;DR

The paper investigates the uniqueness of the coefficient in the linear combination $\gamma+\alpha\delta$ of the Euler–Mascheroni constant $\gamma$ and the Euler–Gompertz constant $\delta$, focusing on the special value $\alpha^{*}=1/e$ for which $\gamma+\alpha\delta$ corresponds to $\textrm{Ein}(1)$. It leverages two canonical Borel-summable divergent series, $S_{\gamma}$ and $S_{\delta}$, and analyzes their Borel transforms, each with a single logarithmic singularity at $u=-1$, to show that $S_{\gamma}+\alpha S_{\delta}$ converges if and only if $\alpha=1/e$, due to precise cancellation of divergent terms (Stokes constants). The paper further extends the construction to generalized moments $\gamma^{(n)}$ and $\delta^{(n)}$, showing the same unique convergence property for $S^{(n)}(\alpha)=S_{\gamma}^{(n)}+\alpha S_{\delta}^{(n)}$, with $S_{\gamma}^{(n)}$ and $S_{\delta}^{(n)}$ formed via signed Stirling numbers and moment kernels, implying a consistent structural reason for the uniqueness across all $n$. Connections to probabilistic interpretations of the moments (via Gumbel distribution) and to transcendence results (e.g., $\textrm{Ein}(1)$ is transcendental) underscore the significance of the finding for understanding linear combinations of fundamental constants and the analytic behavior of associated $E$-functions.

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.