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=Ein (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.
Michael R. Powers (Tue,) studied this question.