Arithmetic Properties of Several Generalized-Constant Sequences, with Implications for ${Γ^{\left(n\right)}\left(1\right)}$
Authors
Michael R. Powers
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 by a well-known equation of Hardy, equivalent to $γ+ δ/e = \mathrm{Ein}(1)$, which recently has been generalized to $γ^{(n)} + δ^{(n)}/e = η^{(n)}$; $n \ge 0$ for sequences of constants $γ^{(n)}$, $δ^{(n)}$, and $η^{(n)}$ (given respectively by raw, conditional, and partial moments of the Gumbel(0,1) probability distribution). Investigating the $γ^{(n)}$ through recurrence relations, we find that at least one of the pair {$γ,γ^{(2)}$} and at least two of each of the sets {$γ,γ^{(2)},γ^{(3)},γ^{(4)}$}, {$γ,γ^{(3)},γ^{(4)},\ldots,γ^{(6)}$}, and {$γ,γ^{(4)},γ^{(5)},\ldots,γ^{(8)}$} are transcendental, implying analogous results for the sequence $Γ^{(n)}(1)=\left(-1\right)^{n}γ^{\left(n\right)}$. We then show, via a theorem of Shidlovskii, that the $η^{(n)}$ are algebraically independent, and therefore transcendental, for all $n \ge 0$, implying that at least one of each pair, {$γ^{(n)},δ^{(n)}/e$} and {$γ^{(n)},δ^{(n)}$}, and at least two of the triple {$γ^{(n)},δ^{(n)}/e,δ^{(n)}$}, are transcendental for all $n \ge 1$. Further analysis of the $γ^{(n)}$ and $η^{(n)}$ reveals that the values $δ^{(n)}/e$ are transcendental infinitely often, with the density of the set of transcendental terms having asymptotic lower bound $1/2-o(1)$. Finally, we provide parallel results for the sequences $\tildeδ^{(n)}$ and $\tildeη^{(n)}$ satisfying the "non-alternating analogue" equation $γ^{(n)} + \tildeδ^{(n)}/e = \tildeη^{(n)}$.
