ua nt - p h / 03 03 03 0 v 1 6 M ar 2 00 3 Dobiński - type relations and the Log - normal distribution

We consider sequences of generalized Bell numbers B(n), n = 0, 1,. .. which can be represented by Dobi´nski-type summation formulas, i.e. B(n) = 1 C ∞ k=0 [P (k)] n D(k) , with P (k) a polynomial, D(k) a function of k and C = const. They include the standard Bell numbers (P (k) = k, D(k) = k!, C = e), their generalizations B r,r (n), r = 2, 3,. .. appearing in the normal ordering of powers of boson monomi-als (P (k) = (k+r)! k! , D(k) = k!, C = e), variants of " ordered " Bell numbers B (p) o (n) (P (k) = k, D(k) = (p+1 p) k , C = 1 + p, p=1,2.. .), etc. We demonstrate that for α, β, γ, t positive integers (α, t = 0), B(αn 2 + βn + γ) t is the n-th moment of a positive function on (0, ∞) which is a weighted infinite sum of log-normal distributions.