The Generalized Stirling and Bell Numbers Revisited

Toufik Mansour Department of Mathematics University of Haifa 31905 Haifa Israel

                toufik@math.haifa.ac.il

Matthias Schork Camillo-Sitte-Weg 25 60488 Frankfurt Germany

                mschork@member.ams.org

Mark Shattuck Department of Mathematics University of Tennessee Knoxville, TN 37996 USA

                 shattuck@math.utk.edu

Abstract

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 𝔅_{s;h|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 polynomials of negative order are introduced and studied. Finally, a q-analogue 𝔖_{s;h}(n, k|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|q) are special cases of the type-II p, q-analogue of generalized Stirling numbers introduced by Remmel 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.