On (k,t)-Fibonacci–François numbers

Élis G. C. Mesquita, Francisco R. V. Alves, Eudes A. Costa
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 354–373
DOI: 10.7546/nntdm.2026.32.2.354-373
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Authors and affiliations

Élis G. C. Mesquita

Department of Mathematics, Federal University of Tocantins Campus Palmas
77020-021, Tocantins, Brazil

Francisco R. V. Alves

Department of Mathematics, Federal Institute of Education, Science and Technology of State of Ceará
Fortaleza 60040-531, Brazil

Eudes A. Costa

Department of Mathematics, Federal University of Tocantins Campus of Arraias
77330-000 Tocantins, Brazil

Abstract

In this article, we present a study of a new member of the family of k-Fibonacci numbers, which we call the (k,t)-Fibonacci–François sequence \{w_{n}^{(k,t)}\}_{ {n\ge0}}. This sequence is defined by the same k-Fibonacci recurrence relation with the initial terms w_{0}^{(k,t)} = k-2 and w_{1}^{(k,t)} = k^2 - k + 2 + t. We describe the structure of this family of sequences by providing explicit formulas and establishing several related algebraic identities. In addition, we derive a Binet-type formula, and extend several classical identities, including those of Tagiuri–Vajda and d’Ocagne, as well as some expressions for the negative indices. Furthermore, we investigate fundamental properties of this family, obtaining limit identities for the ratios of successive terms, as well as summation formulas for the partial sums of the (k, t)-Fibonacci–François sequence.

Keywords

  • François sequence
  • Generalized François sequence
  • Generalized k-Fibonacci sequence
  • Generating function
  • Tagiuri–Vajda identity

2020 Mathematics Subject Classification

  • 11B37
  • 11B39

References

  1. Alp, Y. (2024). On the generalized Francois numbers. Turkish Journal of Mathematics and Computer Science, 16(2), 346–353.
  2. Catarino, P. M. M. C. (2014). On some identities for k-Fibonacci sequence. International Journal of Contemporary Mathematical Sciences, 9(1), 37–42.
  3. Cennet, B., & Hasan, K. (2010). On the properties of k-Fibonacci numbers. International Journal of Contemporary Mathematical Sciences, 5(22), 1097–1105.
  4. Dışkaya, O., Menken, H., & Catarino, P. M. M. C. (2023). On the hyperbolic Leonardo and hyperbolic Francois quaternions. Journal of New Theory, 42, 74–85.
  5. Edouard, L. (1878). Théorie des fonctions numériques simplement périodiques. American Journal of Mathematics, 1(4), 289–321.
  6. Falcón, S. (2011). On the k-Lucas numbers. International Journal of Contemporary Mathematical Sciences, 6(21), 1039–1050.
  7. Falcón, S. (2016). The k–Fibonacci difference sequences. Chaos, Solitons & Fractals, 87(C), 153–157.
  8. Falcón, S., & Plaza, Á. (2007). On the Fibonacci k-numbers. Chaos, Solitons & Fractals, 32(5), 1615–1624.
  9. Falcón, S., & Plaza, Á. (2007). The k-Fibonacci sequence and the Pascal 2-triangle. Chaos, Solitons and Fractals, 33(1), 38–49.
  10. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161–176.
  11. Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications (2nd edition). Wiley.
  12. Panwar, Y. (2021). A note on the generalized k-Fibonacci sequence. NATURENGS, 2(2), 29–39.
  13. Shannon, A. G., & Horadam, A. F. (1972). Some properties of third-order recurrence relations. The Fibonacci Quarterly, 10(2), 135–145.
  14. Sloane, N. J. A. (2026). The On-line Encyclopedia of Integer Sequences. Available online at: http://oeis.org.
  15. Vajda, S. (1989). Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications. Ellis Horwood Ltd., Chichester.

Manuscript history

  • Received: 18 November 2025
  • Revised: 16 May 2026
  • Accepted: 2 June 2026
  • Online First: 5 June 2026

Copyright information

Ⓒ 2026 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).

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Cite this paper

Mesquita, É. G. C., Alves, F. R. V., & Costa, E. A. (2026). On (k,t)-Fibonacci–François numbers. Notes on Number Theory and Discrete Mathematics, 32(2), 354-373, DOI: 10.7546/nntdm.2026.32.2.354-373.

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