The Relationship between Series $\sum _{i=1}^n{i}^m$ and the Eulerian Numbers

Abstract

In this paper, I study the Eulerian numbers $(A(m,k){)}_{k=1}^m$ and prove the relationship between $\sum _{i=1}^n{i}^m$ and $(A(m,k){)}_{k=1}^m$, to be $\sum _{i=1}^n{i}^m=\sum _{k=1}^{m}A(m,k)(\genfrac{}{}{0ex}{}{m+k}{m+1})$.