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تسجيل مستخدم جديد$lambda$-factorials of $n$
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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.
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