On the Algebraic Independence of a Set of Generalized Constants (2508.12123v1)

Abstract: Neither the Euler-Mascheroni constant, \gamma=0.577215\ldots , nor the Euler-Gompertz constant, \delta=0.596347\ldots , is currently known to be irrational. However, it has been proved that these two numbers are disjunctively transcendental; that is, at least one of them must be transcendental. The two constants are related through a well-known equation of Hardy, which recently has been generalized to a pair of infinite sequences (\gamma^{\left(n\right)},\delta^{\left(n\right)}) based on moments of the Gumbel(0,1) probability distribution. In the present work, we demonstrate the algebraic independence of the set {\tfrac{\delta^{\left(n\right)}}{e}+\gamma^{\left(n\right)}}_{n\geq0}, and thus the transcendence of \tfrac{\delta^{\left(n\right)}}{e}+\gamma^{\left(n\right)} for all n\geq0. This further implies the disjunctive-transcendence of each pair (\gamma^{\left(n\right)},\tfrac{\delta^{\left(n\right)}}{e}).