Bell Numbers, Log-concavity, and Log-convexity

Let fb k (n)g 1 n=0 be the Bell numbers of order k. It is proved that the sequence fb k (n)=n!g 1 n=0 is log-concave and the sequence fb k (n)g 1 n=0 is log-convex, or equivalently, the following inequalities hold for all n 0, 1 b k (n + 2)b k (n) b k (n + 1) 2 n + 2 n + 1 : Let f(n)g 1 n=0 be a sequence of positive numbers with (0) = 1. We show that if f(n)g 1 n=0 is log-convex, then (n)(m) (n + m); 8n;m 0: On the other hand, if f(n)=n!g 1 n=0 is log-concave, then (n + m) ? n + m n (n)(m); 8n;m 0: In particular, we have the following inequalities for the Bell numbers b k (n)b k (m) b k (n + m) ? n + m n b k (n)b k (m); 8n;m 0: Then we apply these results to white noise distribution theory. 1. The main theorems For an integer k 2, let exp k (x) denote the k-times iterated exponential function exp k (x) = exp ? exp ? exp(x) | {z } k?times : Let fB k (n)g 1 n=0 be the sequence of numbers given in the power series of exp k (x) exp k (x) = 1 X n=0 B k (n) n! x n : (1) The Bell numbers fb k (n)g 1 n=0 of order k are deened by b k (n) = B k (n) exp k (0) ; n 0: The numbers b 2 (n); n 0; with k = 2 are usually known as the Bell numbers. The rst few terms of these numbers are 1; 1; 2; 5; 15; 52; 203. Note that exp 2 (0) = e and so we have e e x ?1 = 1 X n=0 b 2 (n) n! x n : (2)