Unbounded Oscillation of Euler-Gompertz Diophantine Errors from Bell and Gould Numbers

Abstract

We investigate the asymptotic behavior of the Diophantine errors $δB_{n}-A_{n}$, where $δ=-e\textrm{Ei}\left(-1\right)$ denotes the Euler-Gompertz constant and $B_{n}$ and $A_{n}$ are the $n$th Bell and Gould numbers, respectively. These errors have exponential generating function $g_δ\left(z\right)=\exp\left(e^{z}-1\right)\left(δ-{\displaystyle {\textstyle \int_{0}^{z}}\exp\left(1-e^{t}\right)dt}\right)$,$\:z\in\mathbb{C}$, and it is known that $δB_{n}-A_{n}=O(B_{n}\exp(-cn/\left(\ln\left(n\right)\right)^{2}))$ for some $c\in\mathbb{R}_{>0}$, implying $\lim_{n\rightarrow\infty}\left(A_{n}/B_{n}\right)=δ$. In the present work, we prove that $δB_{n}-A_{n}$ oscillates without bound as $n\rightarrow\infty$; that is, both $\limsup_{n\rightarrow\infty}\left(δB_{n}-A_{n}\right)=\infty$ and $\liminf_{n\rightarrow\infty}\left(δB_{n}-A_{n}\right)=-\infty$.