Normal ordering in the eyl 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 oneparameter 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.