On the Algebraic Independence of a Set of Generalized Constants

TL;DR

This paper proves the algebraic independence of a set of generalized constants related to the Euler-Mascheroni and Euler-Gompertz constants, establishing their transcendence and disjunctive transcendence for all sequence indices.

Contribution

It demonstrates the algebraic independence of specific generalized constants derived from moments of the Gumbel distribution, extending known transcendence results.

Findings

Proves algebraic independence of $\

$oxed{ ext{constants}}$

Establishes transcendence of $\

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 $\{\gamma^{\left(n\right)}+\delta^{\left(n\right)}/e\}_{n\geq0}$, 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 $n\geq1$.