Integer sequences, functions of slow increase, and the Bell numbers.

In this article we first prove a general theorem on integer sequences An such that the following asymptotic formula holds, An An−1 ∼ Cnf(n) , 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 Bn using well-known formulae. We compare the Bell numbers with a (a > 0) and (n!) (0 0, limx→∞ f(x) = ∞ and with continuous derivative f (x) > 0 . The function f(x) is of slow increase if and only if the following condition holds lim x→∞ f (x) f(x) x = lim x→∞ xf (x) f(x) = 0. (1) Typical functions of slow increase are f(x) = log x, f(x) = log x and f(x) = log log x. Lemma 4. If f(x) is a function of slow increase on the interval [b,∞) then the following asymptotic formula holds