An elementary proof of the symmetry of power sum polynomials

Abstract

It is well known that the power sums $1^{k-1}+2^{k-1}+\cdots +n^{k-1}$ are polynomials $p_k(n)$ of degree $k$ in $n$. We provide an elementary proof of the fact that the polynomial $p_k(x)$ is symmetric with respect to the line $x=-1/2$ or the point $(-1/2,0)$, depending on whether $k$ is even or odd, respectively.