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 Dobiński 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.

通过Weyl代数对几何级数的作用，我们建立了Worpitzky恒等式的推广，并为包括古典欧拉多项式在内的多项式族建立了新的递归公式。通过用指数级数代替几何级数，我们获得了对杨比图的车号之和的Dobiński公式的扩展。同样，通过用q-导数运算符替换导数运算符，我们将这些结果扩展到包括q-命中数的q-模拟设置。最后，提供了由一位作者介绍的多项式族的对称性的组合描述和证明。