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

Abstract

In 2011, W. Lang derived a novel, explicit formula for the sum of powers of integerssimultaneously the Stirling numbers of the first and second kind. In this note, we first recall and then slightly refine Lang's formula for S k (n). As it turns out, the modified Lang's formula constitutes a special case of a well-known relationship between the power sums, the elementary symmetric functions, and the complete homogeneous symmetric functions. Furthermore, we provide several applications of this general relationship.