An Operational Approach to the Generalized Rencontres Polynomials

In this paper, we study the umbral operators J, M and N associated with the generalized rencontres polynomials D(m) n (x). We obtain their representations in terms of the differential operator Dx and the shift operator E. Then, by using these representations, we obtain some combinatorial and differential identities for the generalized rencontres polynomials. Finally, we extend these results to some related polynomials and, in particular, to the generalized permutation polynomials P(m)n (x) and the generalized arrangement polynomials A(m)n (x).