In this note, by the umbra calculus method, the Sun and Zagiers congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.
Recently, by the Riordans identity related to tree enumerations, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}(k+1)!(n+1)^{n-k} &=& (n+1)^{n+1}, end{eqnarray*} Sun and Xu derived another analogous one, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}D_{k+1}(n+1
)^{n-k} &=& n^{n+1}, end{eqnarray*} where $D_{k}$ is the number of permutations with no fixed points on ${1,2,dots, k}$. In the paper, we utilize the $lambda$-factorials of $n$, defined by Eriksen, Freij and W$ddot{a}$stlund, to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and another two algebraic proofs. Using the umbral representation of our generalized identity and the Abels binomial formula, we deduce several properties for $lambda$-factorials of $n$ and establish the curious relations between the generating functions of general and exponential types for any sequence of numbers or polynomials.
