Relationship between alternating sums of powers of integers and sums of powers of integers

Minoru Yamamoto
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 761–767
DOI: 10.7546/nntdm.2025.31.4.761-767
Full paper (PDF, 188 Kb)

Details

Authors and affiliations

Minoru Yamamoto
Department of Mathematics, Faculty of Education, Hirosaki University
1 Bunkyo-cho, Hirosaki, Aomori, 036-8560, Japan

Abstract

In this note, we consider the alternating sums of powers of integers. We write alternating sum of powers of integers as the linear combination of sums of powers of integers. As the coefficients, the special value of the Euler polynomial appears.

Keywords

• Alternating sums of powers of integers
• Sums of powers of integers
• Bernoulli number
• Euler polynomial

2020 Mathematics Subject Classification

• 11B68
• 11B83

References

1. Antonippillai, A. (2024). On sums of powers of integers. Missouri Journal of Mathematical Sciences, 36(2), 130–135.
2. Arakawa, T., Ibukiyama, T., & Kaneko, M. (2014). Bernoulli Numbers and Zeta Functions. Springer, Tokyo.
3. Cereceda, J. L. (2023). Euler polynomials and alternating sums of powers of integers. International Journal of Mathematical Education in Science and Technology, 54, 1132–1145.
4. Cheon, G.-S. (2003). A note on the Bernoulli and Euler polynomials. Applied Mathematics Letters, 16(3), 365–368.
5. Kim, T., Kim, Y.-H., Lee, D.-H., Park, D.-W., & Ro, Y. S. (2005). On the alternating sums of powers of consecutive integers. Proceedings of the Jangjeon Mathematical Society, 8(2), 175–178.
6. Tanaka, Y., & Yamamoto, M. (2016). Representation of triangular numbers using alternation sum of squares and dimensional generalization. Journal of Japan Society of Mathematical Education, 98(3), 3–10, (in Japanese).

Manuscript history

• Received: 19 June 2025
• Revised: 7 October 2025
• Accepted: 28 October 2025
• Online First: 3 November 2025

Copyright information

Ⓒ 2025 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).