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

TL;DR

This paper provides an elementary proof that power sum polynomials are symmetric and introduces a new Bernoulli number identity, advancing understanding of classical formulas for sums of powers.

Contribution

It presents a novel Bernoulli number identity and uses it to prove the symmetry of power sum polynomials, offering new insights into Faulhaber's formula.

Findings

Established the polynomial degree as p+1 for power sums

Proved symmetry of power sum polynomials using the new identity

Provided an elementary proof of the polynomial expression

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.