A Proof of Symmetry of the Power Sum Polynomials using a Novel Bernoulli Numbers Identity

Abstract: The problem of finding formulas for sums of powers of natural numbers has been of interest to mathematicians for many centuries. Among these is Faulhaber's well-known formula expressing the power sums as polynomials whose coefficients involve Bernoulli numbers. In this paper we give an elementary proof that the sum of $p$-th powers of the first $n$ natural numbers can be expressed as a polynomial in $n$ of degree $p+1$. We also prove a novel identity involving Bernoulli numbers and use it to show symmetry of this polynomial.