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

Abstract

We investigate the asymptotic behavior of the Diophantine errors $\delta B_n-A_n$, where $\delta =-e\text{Ei}\left(-1\right)$ 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 $g_{\delta }\left(z\right)=\exp \left(e^z-1\right)\left(\delta -{\int }_0^z\exp \left(1-e^t\right)dt\right)$,$z\in \mathbb{C}$, and it is known that $\delta B_n-A_n=O(B_n\exp (-cn/{\left(\ln \left(n\right)\right)}^2))$ for some $c\in {\mathbb{R}}_{>0}$, implying ${\lim }_{n\to \infty }\left(A_n/B_n\right)=\delta$. In the present work, we prove that $\delta B_n-A_n$ oscillates without bound as $n\to \infty$; that is, both ${\mathrm{limsup}}_{n\to \infty }\left(\delta B_n-A_n\right)=\infty$ and ${\mathrm{liminf}}_{n\to \infty }\left(\delta B_n-A_n\right)=-\infty$.