Polynomial formula for sums of powers of integers

Abstract

In this article, it is shown that for any positive integer k ≥ 1, there exist unique real numbers a _kr , r= 1, 2,…, (k+1), such that for any integer n ≥ 1

$${S_{k,n}} \equiv \sum\limits_{j = 1}^n {{j^k}} = \sum\limits_{r = 1}^{\left( {k + 1} \right)} {{a_{kr}}{n^r}} .$$

The numbers a _kr are computed explicitly for r = k + 1, k, k - 1,…, (k - 10). This fully determines the polynomials for k = 1, 2,…, 12. The cases k = 1, 2, 3 are well known and available in high school algebra books.