
Chapter 4
Generalizations of Stirling Numbers
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 ...