Newton's interpolation formula and sums of powers

Abstract In this manuscript, we derive closed formulas for multifold sums of powers of integers by combining the forward 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 https://oeis.org/A131689 — Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows, T(n, k) for 0 <= k <= n. https://oeis.org/A028246 — Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows. https://oeis.org/A038719 — Triangle T(n,k) (0 <= k <= n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion. https://oeis.org/A391552 — Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (3+j)^n. https://oeis.org/A391633 — Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (4+j)^n. https://oeis.org/A391635 — Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (5+j)^n. Metadata Initial release date: December 24, 2025. 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.18040979 Web Version: https://kolosovpetro.github.io/sums-of-powers-newtons-formula/ Sources: https://github.com/kolosovpetro/NewtonsInterpolationFormulaAndSumsOfPowers ORCID: https://orcid.org/0000-0002-6544-8880 Email: [email protected]