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 Sk,n≡∑j=1njk=∑r=1(k+1)akrnr. 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.