Discrete Appell-Dunkl sequences and Bernoulli-Dunkl polynomials of the second kind

In a similar way that 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, one can expect that the discrete Appell sequences can be extended to the Dunkl context. In this extension, the role of the ordinary translation is played by the Dunkl translation, that is a much more intricate operator. In this paper, we define discrete Appell-Dunkl sequences of polynomials, and we give some properties and examples. In particular, we show which is the suitable definition for the Bernoulli polynomials of the second kind in the Dunkl context.