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

TL;DR

This paper refines Lang's explicit formula for the sum of powers of integers, connecting it to symmetric functions and providing new applications of these relationships.

Contribution

It offers a slight refinement of Lang's formula and links it to known symmetric function identities, expanding its theoretical understanding.

Findings

Refined Lang's formula as a special case of symmetric function relationships

Established connections between power sums and elementary/homogeneous symmetric functions

Presented applications of the symmetric function framework to sum of powers

Abstract

In 2011, W. Lang derived a novel, explicit formula for the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$ involving simultaneously 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 refined 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. In addition, we provide several applications of this general relationship.