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

Abstract

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