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 us with a new explicit formula for the binomial sum ∑k=0 n=krak(k n where a, b ∈ ℝ- {0}, n, r ∈ ℕ. As relevant special cases, simple explicit expressions for both the binomial sum ∑k=0 n kr(k n) 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.