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

We investigate the asymptotic behavior of the Diophantine errors $\delta B_n-A_n$, where $\delta =-e\text{Ei}(-1)$ 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 $g_{\delta }\left(z\right)=\exp (e^z-1)(\delta -{\displaystyle {\int }_0^z\exp (1-e^t)dt})$,$z\in \mathbb{C}$, and it is known that $\delta B_n-A_n=O(B_n\exp (-cn\mathrm{/}{(\ln \left(n\right))}^2))$ for some $c\in {\mathbb{R}}_{>0}$, implying ${\lim }_{n\to \mathrm{\infty }}\left(A_n\mathrm{/}{B}_n\right)=\delta$. In the present work, we prove that $\delta B_n-A_n$ oscillates without bound as $n\to \mathrm{\infty }$; that is, both ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}(\delta B_n-A_n)=\mathrm{\infty }$ and ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}(\delta B_n-A_n)=-\mathrm{\infty }$.