The generalized stirling and Bell numbers revisited

The generalized Stirling numbers S_s;h(n, k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n, k; α,β, r) considered by Hsu and Shiue. From this relation, several properties of S_s;h(n, k) and the associated Bell numbers B_s;h(n) and Bell polynomials Bs_;h{pipe}n(x) are derived. The particular case s = 2 and h = -1 corresponding to the meromorphic Weyl algebra is treated explicitly and its connection to Bessel numbers and Bessel polynomials is shown. The dual case s = -1 and h = 1 is connected to Hermite polynomials. For the general case, a close connection to the Touchard polynomials of higher order recently introduced by Dattoli et al. is established, and Touchard polyno- mials of negative order are introduced and studied. Finally, a q-analogue Ss;h(n, k{pipe}q) is introduced and first properties are established, e.g., the recursion relation and an explicit expression. It is shown that the q-deformed numbers S_s;h(n, k{pipe}q) are special cases of the type-II p, q-analogue of generalized Stirling numbers introduced by Rem- mel and Wachs, providing the analogue to the undeformed case (q = 1). Furthermore, several special cases are discussed explicitly, in particular, the case s = 2 and h = -1 corresponding to the q-meromorphic Weyl algebra considered by Diaz and Pariguan.