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： 既不是欧拉-马歇罗尼常数，$\gamma=0.577215\ldots$，也不是欧拉-戈姆珀茨常数，$\delta=0.596347\ldots$，目前已被证明是无理数。然而，已经证明这两个数是不相交超越的；也就是说，其中至少有一个必须是超越数。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基于Gumbel(0,1)概率分布的矩的一对无限序列$(\gamma^{\left(n\right)},\delta^{\left(n\right)})$。在本工作中，我们证明了集合$\{\gamma^{\left(n\right)}+\delta^{\left(n\right)}/e\}_{n\geq0}$的代数独立性，因此对于所有$n\geq0$，$\gamma^{\left(n\right)}+\delta^{\left(n\right)}/e$的超越性。 这进一步意味着对于所有$n\geq1$，对$(\gamma^{\left(n\right)},\delta^{\left(n\right)}/e)$和$(\gamma^{\left(n\right)},\delta^{\left(n\right)})$的析取超越性。 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 $\{\gamma^{\left(n\right)}+\delta^{\left(n\right)}/e\}_{n\geq0}$, and thus the transcendence of $\gamma^{\left(n\right)}+\delta^{\left(n\right)}/e$ for all $n\geq0$. This further implies the disjunctive transcendence of both pairs $(\gamma^{\left(n\right)},\delta^{\left(n\right)}/e)$ and $(\gamma^{\left(n\right)},\delta^{\left(n\right)})$ for all $n\geq1$.