2025/09/30 by Michael R. Powers, Powers, Michael R.
Mathematics · #11J81 #11J85 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2510.00315
openalex publication_date 2025/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
From a well-known equation of Hardy, one can derive a simple linear combination of the Euler-Mascheroni constant (γ=0.577215…) and Euler-Gompertz constant (δ=0.596347…): γ+δ/e=\textrmEin(1). 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(n)+αδ(n), with γ(n) and δ(n) given by (ordinary and conditional) moments of the Gumbel(0,1) probability distribution.