On Ore-Stirling numbers defined by normal ordering in the Ore algebra

Abstract

Normal ordering in the Weyl algebra is related to the Stirling numbers of the second kind, while normal ordering in the shift algebra is related to the unsigned Stirling numbers of the first kind. The Ore algebra -- this name was introduced recently by Patrias and Pylyavskyy -- is an algebra closely related to the Weyl algebra and the shift algebra. We consider a two-parameter family of generalized Ore algebras which comprises all algebras mentioned by specializing the parameters suitably. Analogs of the Stirling numbers -- called Ore-Stirling numbers -- are introduced as normal ordering coefficients in the generalized Ore algebra. In the limit where one parameter vanishes they reduce to the Stirling numbers of the second kind or the unsigned Stirling numbers of the first kind. Choosing the parameters appropriately, a one-parameter family of Ore-Stirling numbers interpolating between Stirling numbers of the second kind and unsigned Stirling numbers of the first kind is found. Several properties of the Ore-Stirling numbers as well as the associated Ore-Bell numbers are discussed.