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

Abstract

We obtain formulas for sums of powers via Newton's interpolation formula based on backward finite differences of powers. In addition, we note that backward differences are closely related to Eulerian numbers, and Stirling numbers of the second kind. Thus, we express formulas for sums of powers in terms of Eulerian numbers, and Stirling numbers of the second kind.

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/A391210 — Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^j * binomial(k,j) * (3-j)^n. (2026)
• https://oeis.org/A391068 — Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^j * binomial(k,j) * (2-j)^n. (2026)
• https://oeis.org/A389570 — Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^j * binomial(k,j) * (1-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