Further Applications of Hsu’s Summation Formula

Abstract

In this paper, we first survey the connections between Bell polynomials (numbers) and the derangement polynomials (numbers). Their close relations are mainly based on Hsu’ summation formula. According to this formula, we present some new identities involving harmonic numbers,Bell polynomials (numbers) and the derangement polynomials (numbers).Moreover, we find that the series $\sum _{m\ge 0}(\frac{D_m}{m!}-\frac{1}{e})$ is (absolutely) convergent and their sums are also determined, where $D_m$ is the $mth$ derangement number.