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

Abstract In this manuscript, we derive closed formulas for multifold sums of powers of integers by combining the central Newton interpolation formula with hockey-stick identities for binomial coefficients. We further 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) OEIS https://oeis.org/A387597 — Triangle read by rows: T(n,k) = Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (0+k-j)^(2n). https://oeis.org/A392337 — Triangle read by rows: T(n,k) = Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (1+k-j)^(2n). https://oeis.org/A390029 — Triangle read by rows: T(n,k) = Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (2+k-j)^(2n). Metadata Initial Release Date: January 3, 2026. MSC2010: 05A19, 05A10, 41A15, 11B83. 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.18096789 Web Version: https://kolosovpetro.github.io/sums-of-powers-central-differences/ Sources: https://github.com/kolosovpetro/SumsOfPowersViaCentralFiniteDifferencesAndNewtonFormula ORCID: https://orcid.org/0000-0002-6544-8880 Email: kolosovp94@gmail.com