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arXiv · 2301.02141

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

José L. Cereceda
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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.

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José L. Cereceda. 2023-05-08. A refinement of Lang's formula for the sum of powers of integers. https://doi.org/10.12988/imf.2023.912382

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