Stirling-Dunkl numbers

Abstract

The Appell sequences of polynomials can be extended to the Dunkl context, where the ordinary derivative is replaced by Dunkl operator on the real line, and the exponential function is replaced by the so-called Dunkl kernel. In a similar way, the discrete Appell sequences can be extended to the Dunkl context, where the role of the ordinary translation is played by the Dunkl translation, that is a much more intricate operator. In particular, this allows to define the falling factorial polynomials in the Dunkl context. Some numbers closely related to falling factorial are the so called Stirling numbers of the first kind and of the second kind, as well as the Bell numbers and the Bell polynomials. In this paper, we define these numbers and polynomials in the Dunkl context, and prove some of their properties.