Some binomial-sum identities for the generalized bi-periodic Fibonacci sequences

Ho-Hon Leung
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 199-208
DOI: 10.7546/nntdm.2020.26.1.199-208
Download full paper: PDF, 169 Kb


Authors and affiliations

Ho-Hon Leung
Department of Mathematical Sciences, United Arab Emirates University
Al-Ain, United Arab Emirates


A bi-periodic sequence is a sequence which satisfies different recurrence relations
depending on whether the n-th term considered is odd or even. In this paper, we investigate the properties of the generalized bi-periodic Fibonacci sequences. It is a generalization of the biperiodic Fibonacci sequences defined by Edson and Yayenie. We derive binomial-sum identities or the generalized bi-periodic Fibonacci sequences by matrix method. Our identities generalize binomial-sum identities derived by Edson and Yayenie for the case of bi-periodic Fibonacci sequences.


  • Fibonacci sequence
  • Matrix method
  • Bionmial sum

2010 Mathematics Subject Classification

  • 11B39
  • 05A15


  1. Bacon, M., Cook, C. & Graves, R. (2016). Using matrices to derive identities for recursive sequences, Fibonacci Quart., 54(3), 204–216.
  2. Deveci, O. (2016). The Pell-Circulant Sequences and Their Applications, Maejo Int. J. Sci. Technol., 10, 284–293.
  3. Edson, M. & Yayenie, O. (2009). A new generalizations of Fibonacci sequences and extended Binet’s Formula, Integers, 9, 639–654.
  4. Gould, H.W. (1981) A history of the Fibonacci Q-matrix and a higher dimensional problem, Fibonacci Quart., 19, 250–257.
  5. Hoggatt, V. E. & Bicknell, M. (1980). A matrix generation of Fibonacci identities for F2nk, A collection of manuscripts related to the Fibonacci sequence, Fibonacci Assoc., 114–124.
  6. Horadam, A. F. (1965). Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quart., 3, 161–176.
  7. Khomovsky, D. (2018). A method for obtaining Fibonacci identities, Integers, 18 , Paper No. A42, 9 pages.
  8. Ruggles, I. D. (1963.) Some Fibonacci results using Fibonacci-type sequences, Fibonacci Quart., 1, 75–80.
  9. Tan, E. & Ekin, A. B. (2017). Some Identities On Conditional Sequences By Using Matrix Method, Miskolc Math. Notes, 18, 469–477.
  10. Tan, E. (2017). On bi-periodic Fibonacci and Lucas numbers by matrix method, Ars Combin., 133, 107–113.
  11. Tan, E. & Leung, H.-H. (2020). Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences, Adv. Difference Equ., Paper No. 26, 11 pages.
  12. Waddill, M. E. (1974). Matrices and Generalized Fibonacci Sequences, Fibonacci Quart., 55, 381–386.
  13. Yayenie, O. (2011). A note on generalized Fibonacci sequence, Appl. Math. Comput., 217, 5603–5611.

Related papers

Cite this paper

Leung, H.-H. (2020). Some binomial-sum identities for the generalized bi-periodic Fibonacci sequences. Notes on Number Theory and Discrete Mathematics, 26(1), 199-208, doi: 10.7546/nntdm.2020.26.1.199-208.

Comments are closed.