Abstract
We give new representations of various power sums, including the signed/alternating sum of powers of consecutive integers as well as odd numbers. The generalized Eulerian power sums \(E_{n,m}(z):=\sum _{k=1}^n k^mz^k\) are also considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statement
Not applicable.
Notes
Interpreting \(E_m(z)\) as a rational function, the formula does hold for all \(z\in { \mathbb {C}}\setminus \{1\}\).
References
Barman, K., Chakraborty, B., Mortini, R.: Two classical formulas for the sum of powers of consecutive integers via complex analysis. Complex Anal. Syner. 10(5), 1–6 (2024)
Boyadzhiev, K.N.: Close encounters with the Stirling numbers of the second kind. Math. Mag. 85, 252–266 (2012)
Chakraborty, B. and Mortini. R.: Sum of squares of consecutive integers: A proof using complex analysis, to appear in Math. Mag. (2024)
Conway, J.B.: Functions of One Complex Variable I. Springer, New York (1978)
Mortini, R.: Die Potenzreihe \(\sum n^mz^n, m\in {\mathbb{N} }\). Elem. Math. 83, 49–51 (1983)
Mortini, R.: The generalized Eulerian power sums \(\sum _{k=1}^n k^m z^k\). Math. Newsletter, Ramanujan Math. Soc. 34, 29–32 (2023)
Quaintance, J., Gould, H.W.: Combinatorial Identities for Stirling Numbers. The Unpublished Notes of H W Gould, World Scientific, New Jersey (2013)
Funding
The research work of Dr. Bikash Chakraborty was partially supported by the Department of Higher Education, Science and Technology and Biotechnology, Govt. of West Bengal under the sanction order no. 1831 (Sanc.)/STBT-11012 (26)/17/2021-ST SEC dated 11/12/2023.
Author information
Authors and Affiliations
Contributions
Both authors contributed to this manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chakraborty, B., Mortini, R. Representations of Various Power Sums. Results Math 79, 186 (2024). https://doi.org/10.1007/s00025-024-02190-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-024-02190-8