Discatenated and lacunary recurrences

Hakan Akkuş, Ömür Deveci, Engin Özkan and Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 1, Pages 8–19
DOI: 10.7546/nntdm.2024.30.1.8-19
Full paper (PDF, 944 Kb)

Details

Authors and affiliations

Hakan Akkuş
Department of Mathematics, Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University
Yalnızbağ Campus, 24100, Erzincan, Türkiye

Ömür Deveci
Department of Mathematics, Faculty of Sciences and Letters, Kafkas University
36100 Kars, Türkiye

Engin Özkan
Department of Mathematics, Faculty of Sciences and Letters, Erzincan Binali Yıldırım University
Yalnızbağ Campus, 24100, Erzincan, Türkiye

Anthony G. Shannon
Warrane College, The University of New South Wales
Kensington, NSW 2033, Australia

Abstract

Recursive sequences with gaps have been studied previously. This paper considers some elementary properties of such sequences where the gaps have been created on a regular basis from sequence to sequence – ‘discatenated’ (systematic gaps) and ‘lacunary’ (general gaps). In particular, their generating functions are developed in order to open up their general terms and relations with other properties.

Keywords

• Fibonacci numbers
• Lucas sequences
• Lacunary
• Primordial sequences
• Recurrence relations

2020 Mathematics Subject Classification

• 11B39
• 11B0F

References

1. Aistleitner, C., Berkes, I., & Tichy, R. (2023). Lacunary sequences in analysis, probability and number theory. arXiv preprint arXiv:2301.05561.
2. Barakat, R. (1964). The matrix operator ex and the Lucas polynomials. Journal of Mathematics and Physics, 43(1–4), 332–335.
3. Borwein, J. M., & Corless, R. M. (1996). The Encyclopedia of Integer Sequences (NJA Sloane and Simon Plouffe). SIAM Review, 38(2), 333–337.
4. Craveiro, I. M., Spreafico, E. V. P., & Rachidi, M. (2023). New approaches of (q, k)-Fibonacci–Pell sequences via linear difference equations. Applications. Notes on Number Theory and Discrete Mathematics, 29(4), 647–669.
5. Howard, F. T. (1998). Lacunary recurrences for sums of powers of integers. The Fibonacci Quarterly, 36(5), 435–442.
6. Lehmer, D. H. (1935). Lacunary recurrence formulas for the numbers of Bernoulli and Euler. Annals of Mathematics, 36(3), 637–649.
7. Luca, F., & Shorey, T. N. (2005). Diophantine equations with products of consecutive terms in Lucas sequences. Journal of Number Theory, 114(2), 298–311.
8. Lucas, E. (1891). Théorie des Nombre. Gauthier-Villars.
9. Malik, S. A., Kumar, A., & Zargar, B. A. (2022). Zero-free regions for lacunary type polynomials. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 57(3), 172–182.
10. Özkan, E., & Altun, İ. (2019). Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials. Communications in Algebra, 47(10), 4020–4030.
11. Özkan, E., Aydın, H., & Dikici, R. (2003). 3-step Fibonacci series modulo m. Applied Mathematics and Computation, 143(1), 165–172.
12. Shannon, A. G. (1972). Iterative formulas associated with generalized third-order
    recurrence relations. SIAM Journal on Applied Mathematics, 23(3), 364–368.
13. Shannon, A. G. (1980). Some lacunary recurrence relations. The Fibonacci Quarterly, 18(1), 73–79.
14. Shannon, A. G. (2019). Applications of Mollie Horadam’s generalized integers to Fermatian and Fibonacci numbers. Notes on Number Theory and Discrete Mathematics, 25(2), 113–126.
15. Sharma, S. K. (2022). New lacunary sequence spaces defined by fractional difference operator. International Journal of Nonlinear Analysis and Applications, 13(2), 2413–2424.
16. Taştan, M., Özkan, E., & Shannon, A. G. (2021). The generalized k-Fibonacci polynomials and generalized k-Lucas polynomials. Notes on Number Theory and Discrete Mathematics, 27(2), 148–158.
17. Whitney, R. E. (1970). On a class of difference equations. The Fibonacci Quarterly, 8, 470–475.
18. Yılmaz, N. Ş., Włoch, A., Özkan, E., & Strzałka, D. (2023). On doubled and quadrupled Fibonacci type sequences. Annales Mathematicae Silesianae, 38, 1–15.
19. Young, P. T. (2003). On lacunary recurrences. The Fibonacci Quarterly, 41(1), 41–47.

Manuscript history

• Received: 8 January 2023
• Revised: 14 February 2023
• Accepted: 17 February 2024
• Online First: 23 February 2024

Copyright information

Ⓒ 2024 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).