2 research outputs found
Eulerian polynomials via the Weyl algebra action
Through the action of the Weyl algebra on the geometric series, we establish a generalization
of the Worpitzky identity and new recursive formulae for a family of
polynomials including the classical Eulerian polynomials. We obtain an extension
of the Dobi´nski formula for the sum of rook numbers of a Young diagram by replacing
the geometric series with the exponential series. Also, by replacing the derivative
operator with the q-derivative operator, we extend these results to the q-analogue setting
including the q-hit numbers. Finally, a combinatorial description and a proof of
the symmetry of a family of polynomials introduced by one of the authors are provided