Touchard polynomials and their applications

The polynomials considered in this paper generalise the Bell polynomials and were introduced by Touchard in order to study some cyclic permutations. For the Touchard polynomials, we give difference-differential equations and recurrence relations, as well as extensions of some well-known properties of the polynomials, their enumerative interpretations, and probabilistic applications, in particular, to analysing a single-server queueing system.