Sums of Powers of Integers and Stirling Numbers

Abstract

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integerswe derive a couple of infinite families of explicit formulas for S k (n). One of the families involves the r-Stirling numbers of the second kind k j r , j = 0, 1, . . . , k, while the other involves their duals k j −r , with both families of formulas being indexed by the non-negative integer r. As a by-product, we obtain three additional formulas for S k (n) in terms of the numbers k j n+m , k j n−m (where m is any given non-negative integer), and k j k−j , respectively. Moreover, we provide a pair of formulas for the Bernoulli polynomials B k (x) involving the numbers k j x .

A refinement of Lang's formula for the sum of powers of integers

“…( 8) can be readily obtained from the Newton-Girard identities (cf. Exercise 2 of [3]). Let {x 1 , x 2 , .…”