On Stirling and Bell numbers of order $1/2$

Abstract

The Stirling numbers of order $1/2$ (of the second kind) introduced by Katugampola are discussed and it is shown that they are given by a scaled subfamily of the generalized Stirling numbers introduced by Hsu and Shiue. This allows to deduce in a straightforward fashion many properties of the Stirling and Bell numbers of order $1/2$. Generalized Stirling numbers of order $1/2$ of the first kind are defined and studied. An analog of the Weyl algebra is introduced and proposed as a natural algebraic setting for normal ordering involving the Stirling numbers of order $1/2$ of both kinds.