Sums of powers via backward finite differences and Newton's formula

Abstract

We obtain formulas for sums of powers via Newton’s interpolation formula based on backward finite differences of powers.
In addition, we note that backward differences are closely related to Eulerian numbers, and Stirling numbers of the second kind.
Thus, we express formulas for sums of powers in terms of Eulerian numbers, and Stirling numbers of the second kind.

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Metadata

DOI: https://doi.org/10.5281/zenodo.18118011

MSC2010: 05A19, 05A10, 11B83, 03C40.

Keywords: Sums of powers, Newton's interpolation formula, Finite differences, Binomial coefficients, Faulhaber's formula,
Bernoulli numbers, Bernoulli polynomials, Interpolation, Combinatorics, Central factorial numbers, OEIS, Stirling numbers,
Eulerian numbers, Worpitzky identity.