A Generalization of a Binomial Sum for the Stirling Numbers of the Second Kind

Abstract

A combinatorial sum for the Stirling numbers of the second kind is generalized. This generalization provides a new explicit formula for the binomial sum $\sum _{k=0}^n{k}^r{a}^k{b}^{n-k}(\genfrac{}{}{0ex}{}{n}{k})$, where $a,b\in \mathbb{R}–\{0\}$ and $n,r\in \mathbb{N}$. As relevant special cases, simple explicit expressions for both the binomial sum $\sum _{k=0}^n{k}^r(\genfrac{}{}{0ex}{}{n}{k})$ and the raw moment of order $r$ of the binomial distribution $B(n,p)$ are given. All these sums are expressed in terms of generalized $r$-permutations.