Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach

Robert Frontczak
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 2, Pages 94–103
DOI: 10.7546/nntdm.2018.24.2.94-103
Full paper (PDF, 165 Kb)

Details

Authors and affiliations

Robert Frontczak
Landesbank Baden-Wuerttemberg
Am Hauptbahnhof 2, 70173 Stuttgart, Germany

Abstract

We derive expressions for sums of first, second, third and fourth powers of Fibonacci and Lucas numbers and their alternating versions. On our way of exploration we rediscover some known results and present new. Focusing on third and fourth order power sums, our findings complete those of Clary and Hemenway, Melham and Adegoke.

Keywords

• Fibonacci number
• Lucas number
• Sums of powers

2010 Mathematics Subject Classification

• 11B37
• 11B39

References

1. Adegoke, K., Sums of fourth powers of Fibonacci and Lucas numbers, Preprint, May 2017, Available via arXiv.
2. Adegoke, K., Factored closed-form expressions for the sums of cubes of Fibonacci and Lucas numbers, Preprint, June 2017, Available via arXiv.
3. Clary, S. & Hemenway, P. D. (1993) On sums of cubes of Fibonacci numbers, in Applications of Fibonacci Numbers, Kluwer Academic Publishers, 123–136.
4. Daykin, D. E. & Dresel, L. A. G. (1967) Identities for products of Fibonacci and Lucas numbers, The Fibonacci Quarterly, 5 (4), 367–370.
5. Kilic, E., Omur, N. & Ulutas, Y. (2011) Alternating sums of the powers of Fibonacci and Lucas numbers, Miskolc Mathematical Notes, 12 (1), 87–103.
6. Melham, R.S. (2000) Alternating sums of fourth powers of Fibonacci and Lucas numbers, The Fibonacci Quarterly, 38 (3), 254–259.