ON A RECURRENCE RELATION FOR THE SUMS OF POWERS OF INTEGERS
Keywords:
sums of powers of integers, recursive formula, odd powers, even powers, symmetry of the power sum polynomials.DOI:
https://doi.org/10.17654/0973563123011Abstract
Recently, Thomas and Namboothiri [1] derived a recurrence identity expressing an exponential power sum with negative powers in terms of another exponential power sum with positive powers. From this result, the authors obtained a corresponding recurrence relation for the ordinary power sums In this short note, we provide an alternative simple proof of the latter recurrence. Our proof is based on the following two ingredients: (i) an expression for and (ii) the symmetry property of the power sum polynomials
Received: May 9, 2023
Revised: May 22, 2023
References
N. E. Thomas and K. V. Namboothiri, On an exponential power sum, preprint (2023), available at https://arxiv.org/abs/2303.1085v1
D. R. Snow, Some identities for sums of powers of integers, Proceedings of Utah Academy of Sciences, Arts, and Letters, 52 Part 2, (1975), pp. 29-31.
N. J. Newsome, M. S. Nogin and A. H. Sabuwala, A proof of symmetry of the power sum polynomials using a novel Bernoulli number identity, Journal of Integer Sequences 20 (2017), Article 17.6.6.
M. El-Mikkawy and F. Atlan, Notes on the power sum Ann. Pure Appl. Math. 9 (2015), 215-232.
D. Treeby, Optimal block stacking and combinatorial identities via Archimedes’ method, PhD Thesis, Monash University, 2018.
F. Martínez de la Rosa, Proof without words: sums of cubes in puzzle form, Far East Journal of Mathematical Education 23 (2022), 7.
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