Sum formulas involving powers of balancing and Lucas-balancing numbers – II

S. G. Rayaguru and G. K. Panda
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 3, Pages 102-110
DOI: 10.7546/nntdm.2019.25.3.102-110
Full paper (PDF, 146 Kb)

Details

Authors and affiliations

S. G. Rayaguru
Department of Mathematics, National Institute of Technology
Rourkela, India

G. K. Panda
Department of Mathematics, National Institute of Technology
Rourkela, India

Abstract

In this paper, we derive expressions for the sums of first four powers of balancing and Lucas-balancing numbers by using the telescoping summation formula. Further, we use these new results to obtain other closed form expressions studied earlier.

Keywords

• Balancing numbers
• Lucas-balancing numbers
• Telescoping summation formula

2010 Mathematics Subject Classification

• 11B39

References

1. Adegoke, K. (2018). Factored closed-form expressions for the sums of cubes of Fibonacci and Lucas numbers, J. Int. Seq., 21, Article 18.6.8.
2. Adegoke, K. (2017). Sums of fourth powers of Fibonacci and Lucas numbers (Preprint), Available online at: https://arxiv.org/pdf/1706.00407.pdf.
3. Behera, A., & Panda, G. K. (1999). On the square roots of triangular numbers, Fib. Quart., 37 (2), 98–105.
4. Clary, S., & Hemenway, P. D. (1993). On sums of cubes of Fibonacci numbers, in Applications of Fibonacci Numbers, Kluwer Academic Publishers, Dordrecht, The Netherlands, 5, 123–136.
5. Davala, R. K., & Panda, G. K. (2015). On sum and ratio formulas for balancing numbers, Journal of the Ind. Math. Soc., 82 (1–2), 23–32.
6. Frontczak, R. (2018). Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach, Notes Number Theory Discrete Math., 24 (2), 94–103.
7. Frontczak, R. (2018). Sums of cubes over odd-index Fibonacci numbers, Integers, 18, #A36.
8. Melham, R.S. (2000). Alternating sums of fourth powers of Fibonacci and Lucas numbers, Fib. Quart., 38 (3), 254–259.
9. Panda, G. K. (2009). Some fascinating properties of balancing numbers, Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194, 185–189.
10. Rayaguru, S.G., &Panda, G.K.(2018). Some infinite product identities involving balancing and Lucas-balancing numbers, Alabama Journal of Mathematics, 42, 6 pages.
11. Rayaguru, S. G., & Panda, G. K. (2019). Sum formulas involving powers of balancing and Lucas-balancing numbers, Journal of the Ind. Math. Soc., 86 (1–2), 137–160.