Published June 27, 2026
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Newton's interpolation formula and sums of powers
Authors/Creators
Description
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: kolosovp94@gmail.com
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Additional details
Software
- Repository URL
- https://github.com/kolosovpetro/NewtonsInterpolationFormulaAndSumsOfPowers
- Development Status
- Active
References
- Knuth, D. E. (1993). Johann Faulhaber and sums of powers. Mathematics of Computation, 61(203), 277–294. https://arxiv.org/abs/math/9207222
- Newton, I., & Chittenden, N. W. (1850). Newton's Principia: The mathematical principles of natural philosophy. New-York: D. Adee. https://archive.org/details/bub_gb_KaAIAAAAIAAJ/page/466/mode/2up
- Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete mathematics: A foundation for computer science (2nd ed.). Addison-Wesley Publishing Company, Inc. https://archive.org/details/concrete-mathematics
- Pfaff, T. J. (2007). Deriving a formula for sums of powers of integers. Pi Mu Epsilon Journal, 12(7), 425–430. https://www.jstor.org/stable/24340705
- Sloane, N. J. A., et al. (2003). The on-line encyclopedia of integer sequences. https://oeis.org/
- Cereceda, J. L. (2022). Sums of powers of integers and generalized Stirling numbers of the second kind. arXiv preprint arXiv:2211.11648. https://arxiv.org/abs/2211.11648
- Worpitzky, J. (1883). Studien über die bernoullischen und eulerschen zahlen. Journal für die reine und angewandte Mathematik, 94, 203–232. http://eudml.org/doc/148532
- Steffensen, J. F. (1927). Interpolation. Williams & Wilkins. https://www.amazon.com/-/de/Interpolation-Second-Dover-Books-Mathematics-ebook/dp/B00GHQVON8
- Carlitz, L., & Riordan, J. (1963). The divided central differences of zero. Canadian Journal of Mathematics, 15, 94–100. https://doi.org/10.4153/CJM-1963-010-8
- Riordan, J. (1968). Combinatorial identities (Vol. 217). Wiley New York. https://www.amazon.com/-/de/Combinatorial-Identities-Probability-Mathematical-Statistics/dp/0471722758
- Scheuer, M. (2020). Reference request: identity in central factorial numbers. https://math.stackexchange.com/a/3665722/463487
- Kolosov, P. (2025). Mathematica programs for finite differences, Stirling numbers, and sums of powers. GitHub repository. https://github.com/kolosovpetro/NewtonsInterpolationFormulaAndSumsOfPowers/tree/main/mathematica
- Kolosov, P. (2025). Newton's interpolation formula and sums of powers. Zenodo. https://doi.org/10.5281/zenodo.18040979
- Kolosov, P. (2025). Sums of powers via central finite differences and Newton's formula. Zenodo. https://doi.org/10.5281/zenodo.18096789
- Kolosov, P. (2026). Sums of powers via backward finite differences and Newton's formula. Zenodo. https://doi.org/10.5281/zenodo.18118011