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

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) = \sum_{j=0}^{k} (-1)^j \binom{k}{j} (0-j)^n. (2017)
• A389570 — Triangle read by rows: T(n,k) = \sum_{j=0}^{k} (-1)^j \binom{k}{j} (1-j)^n. (2026)
• A391068 — Triangle read by rows: T(n,k) = \sum_{j=0}^{k} (-1)^j \binom{k}{j} (2-j)^n. (2026)
• A391210 — Triangle read by rows: T(n,k) = \sum_{j=0}^{k} (-1)^j \binom{k}{j} (3-j)^n. (2026)
• A395604 — Triangle read by rows: T(n,k) = \sum_{j=0}^{k} (-1)^j \binom{k}{j} (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