Application of the generalized Rodrigue formula in combinatorial analysis

The article considers the generalized Rodrigues formula, which allows to define some important polynomial families applied in combinatorial analysis. This formula is used to obtain recurrence correlations and generating functions. In particular, from this point of view it is possible to study generalized Eulerian polynomials and consider their properties. In order to combinatorially interpret the coefficients of these polynomials the authors use generalized permutations of Gessel - Stanley and root marked r-angle cactuses. The article also considers finite-difference and q-analogues of the generalized Rodrigues' formula, by which, in particular, the authors study the q-analogs of exponential polynomials and Eulerian polynomials and their properties.