Arithmetic Properties of Several Generalized-Constant Sequences, with Implications for ${Γ^{\left(n\right)}\left(1\right)}$

Michael R. Powers

Paper

Arithmetic Properties of Several Generalized-Constant Sequences, with Implications for ${Γ^{\left(n\right)}\left(1\right)}$

Abstract

Neither the Euler-Mascheroni constant,

, nor the Euler-Gompertz constant,

, 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

, which recently has been generalized to

;

for sequences of constants

, and

(given respectively by raw, conditional, and partial moments of the Gumbel(0,1) probability distribution). Investigating the

through recurrence relations, we find that at least one of the pair {

} and at least two of each of the sets {

}, {

}, and {

} are transcendental, implying analogous results for the sequence

. We then show, via a theorem of Shidlovskii, that the

are algebraically independent, and therefore transcendental, for all

, implying that at least one of each pair, {

} and {

}, and at least two of the triple {

}, are transcendental for all

. Further analysis of the

and

reveals that the values

are transcendental infinitely often, with the density of the set of transcendental terms having asymptotic lower bound

. Finally, we provide parallel results for the sequences

and

satisfying the "non-alternating analogue" equation