Integer Sequences, Functions of Slow Increase and the Bell Numbers

Rafael Jakimczuk División Matemática Universidad Nacional de Luján Buenos Aires Argentina

             jakimczu@mail.unlu.edu.ar

In memory of my sister Fedra Marina Jakimczuk (1970–2010)

Abstract

In this article we first prove a general theorem on integer sequences A_{n} such that the following asymptotic formula holds,

\frac{A_{n}}{A_{n−1}} ∼ Cn^{α}f(n)^{β},

where f(x) is a function of slow increase, C > 0, α > 0 and β is a real number.

We also obtain some results on the Bell numbers B_{n} using well-known formulae. We compare the Bell numbers with a^{n} (a > 0) and (n!)^{h} (0 < h ≤ 1).

Finally, applying the general statements proved in the article we obtain the formula

B_{n+1} ∼ e (B_{n})^{1+ \frac{1}{n}} .