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

José L. Cereceda

Published 2023-01-05Version 1

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 modified Lang's formula constitutes a special case of a general relationship discovered by Merca between the power sums, the elementary symmetric functions, and the complete homogeneous symmetric functions.