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

**Download 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

- 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.
- Adegoke, K. (2017). Sums of fourth powers of Fibonacci and Lucas numbers (Preprint), Available online at: https://arxiv.org/pdf/1706.00407.pdf.
- Behera, A., & Panda, G. K. (1999). On the square roots of triangular numbers, Fib. Quart., 37 (2), 98–105.
- 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.
- 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.
- 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.
- Frontczak, R. (2018). Sums of cubes over odd-index Fibonacci numbers, Integers, 18, #A36.
- Melham, R.S. (2000). Alternating sums of fourth powers of Fibonacci and Lucas numbers, Fib. Quart., 38 (3), 254–259.
- 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.
- Rayaguru, S.G., &Panda, G.K.(2018). Some infinite product identities involving balancing and Lucas-balancing numbers, Alabama Journal of Mathematics, 42, 6 pages.
- 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.

## Related papers

## Cite this paper

Rayaguru, S.G., & Panda, G.K. (2019). Sum formulas involving powers of balancing and Lucas-balancing numbers – II. Notes on Number Theory and Discrete Mathematics, 25(3), 102-110, doi: 10.7546/nntdm.2019.25.3.102-110.