ABSTRACT
In this chapter several generalizations of Stirling and Bell numbers are considered. We concentrate on those generalizations which will be important later on.
The first starting point for generalization is the operational interpretation of Stirling numbers. Considering instead of (XD)n powers of other operators gives rise to variants of generalizations of Stirling numbers. Beginning with Scherk in 1823, different generalizations have been considered by Carlitz, Comtet, McCoy, and Toscano, to name just the most prominent authors. We discuss them in Section 4.1 as well as more complex operational relations and their connection with different families of polynomials. For instance, Bell polynomials and Hermite polynomials. We also describe combinatorial aspects of some of these generalized Stirling numbers and consider q-analogs for most of them.
