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July 1, 2026Open Access

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

Key Points

The aim is to establish closed formulas for multifold sums of powers of integers using interpolation techniques.
Applying the backward Newton interpolation formula in conjunction with hockey-stick identities for binomial coefficients.
Deriving representations of multifold sums using Stirling and Eulerian numbers.
Providing Mathematica programs for verifying the identities.
Closed formulas for multifold sums of powers were successfully derived using the proposed methods.
New representations in terms of Stirling and Eulerian numbers were obtained.
Verification tools in Wolfram Mathematica were developed for the resulting identities.

Abstract

Abstract In this manuscript, we derive closed formulas for multifold sums of powers of integers by combining the backward Newton interpolation formula with hockey-stick identities for binomial coefficients. We further obtain representations of multifold sums of powers in terms of Stirling numbers of the second kind and Eulerian numbers. Finally, we provide Wolfram Mathematica programs for the efficient verification of the derived identities. Related works 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) OEIS A278075 — Triangle read by rows: T (n, k) = ₉=₀^k (-1) ʲ kj (0-j) ⁿ. (2017) A389570 — Triangle read by rows: T (n, k) = ₉=₀^k (-1) ʲ kj (1-j) ⁿ. (2026) A391068 — Triangle read by rows: T (n, k) = ₉=₀^k (-1) ʲ kj (2-j) ⁿ. (2026) A391210 — Triangle read by rows: T (n, k) = ₉=₀^k (-1) ʲ kj (3-j) ⁿ. (2026) A395604 — Triangle read by rows: T (n, k) = ₉=₀^k (-1) ʲ kj (4-j) ⁿ. (2026) Metadata Initial release date: January 1, 2026. 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, Approximation, Discrete convolution, Combinatorics, Polynomial identities, Central factorial numbers, Stirling numbers, Eulerian numbers, Worpitzky identity, Pascal's triangle, OEIS. License: This work is licensed under a CC BY 4. 0 License. DOI: https: //doi. org/10. 5281/zenodo. 18118011 Web version: https: //kolosovpetro. github. io/sums-of-powers-backward-differences/ Sources: https: //github. com/kolosovpetro/SumsOfPowersViaBackwardFiniteDifferencesAndNewtonFormula ORCID: https: //orcid. org/0000-0002-6544-8880 Email: kolosovp94@gmail. com

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