Transcendence Results for $Γ^{(n)}(1)$ and Related Sequences of Generalized Constants

Abstract

Neither the Euler-Mascheroni constant, $γ=0.577215...$, nor the Euler-Gompertz constant, $δ=0.596347...$, is currently known to be irrational. However, it has been proved that at least one of them is transcendental. The two constants are related through a well-known equation of Hardy, equivalent to $γ+δ/e=\textrm{Ein}(1)$, which recently has been generalized to $γ^{(n)}+δ^{(n)}/e=η^{(n)},\:n\geq0$ for sequences of constants $γ^{(n)}$, $δ^{(n)}$, and $η^{(n)}$ (derived respectively from raw, conditional, and partial moments of the $\textrm{Gumbel}(0,1)$ probability distribution). Investigating $γ^{(n)}=(-1)^{n}Γ^{(n)}(1),\:n\geq1$ through $\textrm{Gumbel}(0,1)$ generating functions, we find that $γ^{(2n)}\in\mathbb{Q}[γ,γ^{(2)}$, $γ^{(3)},...,γ^{(2n-1)}]$ for $n\geq2$ and $γ^{(n)}$ is transcendental infinitely often. We then show, via a theorem of Shidlovskii, that the $η^{(n)}$ are algebraically independent, and therefore transcendental, for all $n\geq0$, implying that at least one element of each pair, $\left\{γ^{(n)},δ^{(n)}/e\right\}$ and $\left\{γ^{(n)},δ^{(n)}\right\}$, and at least two elements of the triple $\left\{γ^{(n)},δ^{(n)}/e,δ^{(n)}\right\}$ are transcendental for all $n\geq1$. Further analysis of the $γ^{(n)}$ and $η^{(n)}$ reveals that both the $δ^{(n)}/e$ and $δ^{(n)}$ are transcendental infinitely often with lower asymptotic densities of at least 1/2. Finally, we provide parallel results for the sequences $\widetildeδ^{(n)}$ and $\widetildeη^{(n)}$ satisfying the "non-alternating analogue" equation $γ^{(n)}+\widetildeδ^{(n)}/e=\widetildeη^{(n)}$.