Explicit upper bounds for Touchard polynomials and Bell numbers.

We obtain explicit upper bounds for the Touchard polynomials T n (x) , for x > 0 . When applied to the Bell numbers B n = T n (1) , such bounds are asymptotically sharp. A simple probabilistic approach based on estimates of moments of nonnegative random variables is used. Applications giving upper bounds for the moments of a certain subset of Jakimovski-Leviatan operators are also provided. [ABSTRACT FROM AUTHOR]