Stirling polynomial

The Stirling polynomials $S_k(t)$ are defined by $$S_k(t)=k! \displaystyle\sum_{j=0}^k (-1)^{k-j}\displaystyle\sum_{m=j}^k {{x+m} \choose m}{m \choose j}L_{k+m}^{(-k-j)}(-j),$$ where $L_{k+m}^{(-k-j)}$ denotes an associated Laguerre polynomial.