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, for example, recurrence relations, generating functions, Dobi?ski formula, and Spivey formula. The even Bell polynomials of order 1/2 are shown to be closely related to generalized Laguerre polynomials 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 where the Stirling numbers of order 1/2 of both kinds appear as ordering coefficients. This algebra contains the Weyl algebra as a subalgebra.
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