Explicit bounds for Bell numbers and their ratios

TL;DR

This paper provides explicit bounds and asymptotic analysis of Bell numbers, including ratios and convergence rates, using Lambert W function, with applications to elementary bounds and growth behavior.

Contribution

It introduces new explicit bounds and asymptotic forms for Bell numbers, unifying previous dispersed results and improving understanding of their growth and ratios.

Findings

Derived explicit lower and upper bounds for Bell numbers

Established asymptotic forms involving Lambert W function

Calculated convergence rate of ratios of consecutive Bell numbers

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 the consecutive Bell numbers is derived. The main challenge was to obtain satisfactory constants, as the Bell numbers grow rapidly, while the convergence rates are rather slow.