Infinite sums associated with certain Lucas sequences

S. G. Rayaguru and G. K. Panda
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 2, Pages 152–158
DOI: 10.7546/nntdm.2020.26.2.152-158
Full paper (PDF, 180 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

We obtain some closed form expressions for certain infinite sums related to Lucas and associated Lucas sequences. Some of our results generalize existing results concerning infinite sums involving Lucas sequences and others are variants of some existing results.

Keywords

• Lucas sequences
• Convergence of infinite series

2010 Mathematics Subject Classification

• 11B39

References

1. André-Jeannin, R. (1990). Lambert series and the summations of reciprocals in certain Fibonacci–Lucas-type sequences, Fib. Quart., 28 (3), 223–226.
2. Bruckman, P. S., & Good, I. J. (1976). A generalization of a series of de Morgan, with applications of Fibonacci type, Fib. Quart., 14 (3), 193–196.
3. 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.
4. Farhi, B. (2019). Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., 22 (1), Article No. 19.1.6.
5. Frontczak, R. (2018). A Note on Hybrid Convolutions Involving Balancing and Lucas-balancing Numbers, Appl. Math. Sci., 12 (25), 2001–2008.
6. Frontczak, R. (2018). Sums of Balancing and Lucas-balancing Numbers with Binomial Coefficients, Int. J. Math. Anal., 12 (12), 585–594.
7. Hu, H., Sun, Z. W., & Liu, J. X. (2001). Reciprocal sums of second order recurrent
   sequences, Fib. Quart., 39 (3), 214–220.
8. Komatsu, T., & Panda, G.K. (2018). On several kinds of sums of balancing numbers, Preprint, arXiv: 1608.05918v3 [math.NT].
9. Rayaguru, S. G., & Panda, G. K. (2018). Some Infinite Product Identities Involving Balancing and Lucas-balancing Numbers, Alabama Journal of Mathematics, 42.
10. 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.
11. Rayaguru, S. G., & Panda, G. K. (2019). Sum Formulas Involving Powers of Balancing and Lucas-balancing Numbers-II, Notes on Number Theory and Discrete Math., 25 (3), 102–110.