The $r$-Bell numbers

The notion of generalized Bell numbers was appeared in several works but there is no a systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic properties of these numbers which in any cases generalize the similar properties of the Bell numbers. Most of these properties seems to be new. It turns out that in a paper of Whitehead these numbers appeared in a very dierent context. In addition, we introduce the so-called