Sums of powers via integration

TL;DR

This paper introduces a recursive algorithm for computing Faulhaber formula coefficients, which express sums of powers as polynomials, and proves its correctness through recurrence relations.

Contribution

It presents a new recursive method for calculating Faulhaber formula coefficients and provides a proof of correctness using recurrence relations.

Findings

Recursive algorithm for Faulhaber coefficients

Proof of correctness via recurrence relations

Efficient computation of power sums as polynomials

Abstract

Sum of powers 1^p+...+n^p, with n and p being natural numbers and n>=1, can be expressed as a polynomial function of n of degree p+1. Such representations are often called Faulhaber formulae. A simple recursive algorithm for computing coefficients of Faulhaber formulae is presented. The correctness of the algorithm is proved by giving a recurrence relation on Faulhaber formulae.