  Recently, by the Riordan's 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 Abel's 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.
