标题： 关于一组广义常数的代数独立性 显示英文标题

标题： On the Algebraic Independence of a Set of Generalized Constants

摘要： 欧拉-马歇罗尼常数，\gamma=0.577215 \ldots ，以及欧拉-贡珀茨常数，\delta=0.596347 \ldots ，目前尚不知是否为无理数。然而，已经证明这两个数是选择性超越数；即至少其中一个必定是超越数。 这两个常数通过哈代的一个著名方程相关联，该方程最近已被推广到一对无限序列 (\gamma ^{\left (n\right )},\delta ^{\left (n\right )})，这基于Gumbel(0,1)概率分布的矩。 在本工作中，我们证明了集合 \{\tfrac{\delta^{\left(n\right)}}{e}+\gamma ^{\left (n\right )}\}_{n\geq 0}的代数独立性，因此对于所有 n\geq 0，\tfrac{\delta^{\left(n\right)}}{e}+\gamma ^{\left (n\right )}是超越的。 这进一步意味着每对 (\gamma ^{\left (n\right )},\tfrac{\delta^{\left(n\right)}}{e}) 的不相交超越性。 显示英文摘要

摘要： 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}).