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 \(\sum_{k=0}^{n}k^ra^kb^{n-k} \binom{n}{k}\), 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 \(\sum_{k=0}^{n} k^r\binom{n}{k} \) 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.