Explicit bounds for Bell numbers and their ratios

Abstract

In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main results correspond to two asymptotic forms expressed by means of the Lambert W function. As an application, some straightforward elementary bounds are derived. Additionally, an absolute convergence rate of the ratio of consecutive Bell numbers is derived. One of the main challenges was to obtain satisfactory constants, as the Bell numbers grow rapidly, while the convergence rates are rather slow.