Sums of powers of integers: A complete framework for closed formulas

Abstract. This manuscript originates from the historical development of the sums of powers problem, briefly tracing its evolution from Archimedes (287-212 BC) to Donald Knuth's work Johann Faulhaber and sums of powers (1993), highlighting the transition from purely arithmetic methods to modern enumerative and algebraic-combinatorial approaches. The main results establish an algorithm for deriving closed formulas for multifold sums of powers of integers by combining variations of Newton's interpolation formula with hockey-stick family identities for binomial coefficients. Related projects Newton's interpolation formula and sums of powers (2025) Sums of powers via central finite differences and Newton's formula (2025) Sums of powers via backward finite differences and Newton's formula (2026) Sums of powers of integers: A complete framework for closed formulas (2026) Metadata Initial release date: 04-Jun-2026. MSC2010: 05A10, 11B68, 11B73, 11B83. Keywords: Sums of powers, Newton's interpolation formula, Finite differences, Binomial coefficients, Faulhaber's formula, Bernoulli numbers, Bernoulli polynomials, Interpolation, Central factorial numbers, Stirling numbers, Eulerian numbers, OEIS. License: This work is licensed under a CC BY 4.0 License DOI: https://doi.org/10.5281/zenodo.20548019 Web Version: https://kolosovpetro.github.io/math/sums-of-powers-framework-for-closed-forms GitHub: https://github.com/kolosovpetro/SumsOfPowersACompleteFrameworkForClosedForms ORCID: https://orcid.org/0000-0002-6544-8880 Email: [email protected]