ABSTRACT
In this chapter we recall the definition and basic properties of the classical Stirling and Bell numbers. Many of these properties were already discussed in preceding chapters. For instance, the Stirling numbers of the second kind, S(n, k), were introduced as the numbers counting set partitions of the set [n] having k blocks, and later considerations showed that they appear as connection coefficients when writing xn in terms of (x)k; see (1.3). We recall that the numbers S(n, k) also appear as normal ordering coefficients for (XD)n and as rook numbers of particular Ferrers boards. For the Bell numbers, which are closely related to S(n, k), several properties were also considered in the preceding chapters. In the present chapter, the definition and basic properties of the Stirling numbers (of both kinds) and the Bell numbers are collected in one place (Section 3.1) for the convenience of the reader and for easy reference in later chapters. All results presented in this section are well-known and can be found, for instance, in [230, 280]. Historical remarks concerning the classical Stirling and Bell numbers can be found in Chapter 1. In Section 3.2 several properties of Stirling and Bell numbers are collected which are found less often in textbooks. We discuss, for example, the classical Dobin´ski formula (found in many books) as well as Spivey’s Bell number relation discovered in 2008. Following Milne, a q-deformation of Stirling and Bell numbers is introduced in Section 3.3 and several of its properties are discussed. The first qdeformation of Stirling numbers was introduced by Carlitz in 1933 and later studied by him again in 1948, and Gould presented a systematic account in 1961. A further generalization was introduced in 1991 by Wachs and White, namely (p, q)-deformed Stirling numbers. For p = 1, they reduce to q-deformed Stirling numbers. In Section 3.4 we define (p, q)-deformed Stirling numbers and state some of their properties.
