Abstract
Let Sk(n) denote the sum of the k-th powers of the first n positive integers. In this article, starting from two explicit formulas for Sk(n) involving the Stirling numbers of the second kind, we derive a pair of corresponding recursive formulas for Sk(n) involving the Stirling numbers of the first kind. Our derivation makes use of the orthogonality relations for the Stirling numbers. Furthermore, we obtain yet another recursive formula for Sk(n) in the form of the so-called Newton-Girard identities.
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J L Cereceda has a degree in physics from the University of Zaragoza (Spain). He previously worked on the foundations of quantum mechanics, especially Bell’s theorem without inequalities. More recently, he has been interested in elementary number theory, discrete mathematics, and mathematics education.
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Cereceda, J.L. Sums of Powers of Integers and Stirling Numbers. Reson 27, 769–784 (2022). https://doi.org/10.1007/s12045-022-1371-9
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DOI: https://doi.org/10.1007/s12045-022-1371-9