Boole-Dunkl polynomials and generalizations

Appell sequences of polynomials can be extended to the Dunkl context replacing the ordinary derivative by the Dunkl operator on the real line, and the exponential function by the Dunkl kernel. In a similar way, discrete Appell sequences can be extended to the Dunkl context; here, the role of the ordinary translation is played by the Dunkl translation, which is a much more intricate operator. Some sequences as the falling factorials or the Bernoulli polynomials of the second kind have already been extended and investigated in the mathematical literature. In this paper, we study the discrete Appell version of the Euler polynomials, usually known as Euler polynomials of the second kind or Boole polynomials. We show how to define the Dunkl extension of these polynomials (and some of their generalizations), and prove some relevant properties and relations with other polynomials and with Stirling-Dunkl numbers.