On Matrices with Bidimensional Fibonacci Numbers

Eudes Antonio Costa; Paula M.M.C. Catarino

doi:10.2478/amsil-2025-0014

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On Matrices with Bidimensional Fibonacci Numbers

Annales Mathematicae Silesianae

Volume 40 (2026): Issue 1 (March 2026)

By: Eudes Antonio Costa and Paula M.M.C. Catarino

Open Access

|Sep 2025

Abstract

In this paper, the bidimensional extensions of the Fibonacci numbers are explored, along with a detailed examination of their properties, characteristics, and some identities. We introduce and study the matrices with bidimensional Fibonacci numbers, focusing in particular on their recurrence relation, key properties, determinant, and various other identities. It is our purpose to study the matrix version of bidimensional Fibonacci numbers and provide new results and sometimes extensions of some results existing in the literature. We aim to introduce these matrices using the bidimensional Fibonacci numbers and to give the determinant of these matrices.

References

F.R. Alves and P.M. Catarino, Sequência matricial generalizada de Fibonacci e sequência matricial k-Pell: propriedades matriciais, C.Q.D. - Revista Eletrônica Paulista de Matemática 15 (2019), Edição Julho, 39–54.
Search in Google Scholar Back to article
M.K. Azarian, The generating function for the Fibonacci sequence, Missouri J. Math. Sci. 2 (1990), no. 2, 78–79.
Search in Google Scholar Back to article
J.B. Bacani and J.F. Rabago, On generalized Fibonacci numbers, Appl. Math. Sci. (Ruse) 9 (2015), no. 73, 3611–3622.
Search in Google Scholar Back to article
G. Berzsenyi, Gaussian Fibonacci numbers, Fibonacci Quart. 15 (1977), no. 3, 233–236.
Search in Google Scholar Back to article
F. Bezerra, F. Alves, and R. Vieira, Relações recorrentes bidimensionais e tridimensionais de Narayana, C.Q.D. - Revista Eletrônica Paulista de Matemática 18 (2020), Edição Julho, 12–28.
Search in Google Scholar Back to article
A. Borges, P. Catarino, A. Aires, P. Vasco, and H. Campos, Two-by-two matrices involving k-Fibonacci and k-Lucas sequences, Appl. Math. Sci. (Ruse) 8 (2014), no. 34, 1659–1666.
Search in Google Scholar Back to article
P. Catarino, On some identities for k-Fibonacci sequence, Int. J. Contemp. Math. Sci. 9 (2014), no. 1, 37–42.
Search in Google Scholar Back to article
P. Catarino and H. Campos, From Fibonacci sequence to more recent generalisations, in: F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications. IC-MASE 2021, Salamanca, Spain, July 1–2, Springer Proc. Math. Stat., 384, Springer, Cham, 2022, pp. 259–269.
Search in Google Scholar Back to article
P. Catarino, D. Santos, and E. Costa, On t-dimensional Gersenne sequences and their symmetry properties, Symmetry 17 (2025), no. 7, Paper No. 1079, 16 pp.
Search in Google Scholar Back to article
P. Catarino and P. Vasco, Some basic properties and a two-by-two matrix involving the k-Pell numbers, Int. J. Math. Anal. (Ruse) 7 (2013), no. 45, 2209–2215.
Search in Google Scholar Back to article
J. Chimpanzo, P. Catarino, and M. Otero-Espinar, Some identities and generating functions for bidimensional balancing and cobalancing sequences, Univ. J. Math. Appl. 7 (2024), no. 2, 68–75.
Search in Google Scholar Back to article
J. Chimpanzo, P. Catarino, and M. Otero-Espinar, Bidimensional balancing, Lucas-balancing, cobalancing and Lucas-cobalancing numbers via the determinant of a tridiagonal matrix, Indian J. Pure Appl. Math. (2025). https://doi.org/10.1007/s13226-025-00771-z.
Search in Google Scholar Back to article
J. Chimpanzo, P. Catarino, P. Vasco, and A. Borges, Bidimensional extensions of balancing and Lucas-balancing numbers, J. Discrete Math. Sci. Cryptogr. 27 (2024), no. 1, 95–115.
Search in Google Scholar Back to article
J. Chimpanzo, M. Otero-Espinar, A. Borges, P. Vasco, and P. Catarino, Bidimensional extensions of cobalancing and Lucas-cobalancing numbers, Ann. Math. Sil. 38 (2024), no. 2, 241–262.
Search in Google Scholar Back to article
E. Costa, P. Catarino, F. Monteiro, V. Sousa, and D. Santos, Tricomplex Fibonacci numbers: a new family of Fibonacci-type sequences, Mathematics 12 (2024), no. 23, Paper No. 3723, 15 pp.
Search in Google Scholar Back to article
E. Costa, P. Catarino, P. Vasco, and F. Alves, A brief study on the k-dimensional Repunit sequence, Axioms 14 (2025), no. 2, Paper No. 109, 16 pp.
Search in Google Scholar Back to article
M. Edson and O. Yayenie, A new generalization of Fibonacci sequence and extended Binet's formula, Integers 9 (2009), no. 6, 639–654.
Search in Google Scholar Back to article
S. Falcón, On the generating matrices of the k-Fibonacci numbers, Proyecciones 32 (2013), no. 4, 347–357.
Search in Google Scholar Back to article
S. Falcón, On the extended (k, t)-Fibonacci numbers, J. Adv. Math. Comput. Sci. 39 (2024), no. 7, 81–89.
Search in Google Scholar Back to article
S. Falcón and Á. Plaza, On the Fibonacci k-numbers, Chaos Solitons Fractals 32 (2007), no. 5, 1615–1624.
Search in Google Scholar Back to article
H.W. Gould, A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonacci Quart. 19 (1981), no. 3, 250–257.
Search in Google Scholar Back to article
C.J. Harman, Complex Fibonacci numbers, Fibonacci Quart. 19 (1981), no. 1, 82–86.
Search in Google Scholar Back to article
A.F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly 68 (1961), no. 5, 455–459.
Search in Google Scholar Back to article
A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), no. 3, 289–291.
Search in Google Scholar Back to article
A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (1965), no. 3, 161–176.
Search in Google Scholar Back to article
H. Hosoya, Fibonacci triangle, Fibonacci Quart. 14 (1976), no. 2, 173–179.
Search in Google Scholar Back to article
J.H. Jordan, Gaussian Fibonacci and Lucas numbers, Fibonacci Quart. 3 (1965), no. 4, 315–318.
Search in Google Scholar Back to article
D. Kalman and R. Mena, The Fibonacci numbers–exposed, Math. Mag. 76 (2003), no. 3, 167–181.
Search in Google Scholar Back to article
C.H. King, Some Properties of the Fibonacci Numbers, Master's thesis, San Jose State College, 1960.
Search in Google Scholar Back to article
C. Kızılateş, P. Catarino, and N. Tuğlu, On the bicomplex generalized Tribonacci quaternions, Mathematics 7 (2019), no. 1, Paper No. 80, 8 pp.
Search in Google Scholar Back to article
T. Koshy, Fibonacci and Lucas Numbers with Applications, Volume 1, John Wiley & Sons, Hoboken, NJ, 2017.
Search in Google Scholar Back to article
B. Kuloğlu and E. Özkan, On the (p, q)-Narayana n-dimensional recurrences, J. Sci. Arts 23 (2023), no. 3, 707–714.
Search in Google Scholar Back to article
R. Oliveira and F. Alves, Os números Gaussianos de Fibonacci e relações recorrentes bidimensionais, Rev. Thema 16 (2019), no. 4, 745–754.
Search in Google Scholar Back to article
R. Oliveira, F. Alves, and R. Paiva, Identidades bi e tridimensionais para os números de Fibonacci na forma complexa, C.Q.D. - Revista Eletrônica Paulista de Matemática 11 (2017), Edição Dezembro, 91–106.
Search in Google Scholar Back to article
S. Pethe and A.F. Horadam, Generalised Gaussian Fibonacci numbers, Bull. Aust. Math. Soc. 33 (1986), no. 1, 37–48.
Search in Google Scholar Back to article
B. Singh, O. Sikhwal, and Y. Gupta, Generalized Fibonacci–Lucas sequence, Turk. J. Anal. Number Theory 2 (2014), no. 6, 193–197.
Search in Google Scholar Back to article
N.J. Sloane et al., The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation Inc., https://oeis.org.
Search in Google Scholar Back to article
S. Uygun, Complex Jacobsthal numbers in two dimension, Sarajevo J. Math. 20(33) (2024), no. 2, 219–229.
Search in Google Scholar Back to article
S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Courier Corporation, Chelmsford, MA, 2008.
Search in Google Scholar Back to article
R. Vieira, F. Alves, and P. Catarino, Relações bidimensionais e identidades da sequência de Leonardo, Rev. Sergipana Mat. Educ. Mat. 4 (2019), no. 2, 156–173.
Search in Google Scholar Back to article
A. Wani, V. Badshah, G. Rathore, and P. Catarino, Generalized Fibonacci and k-Pell matrix sequences, Punjab Univ. J. Math. (Lahore) 51 (2020), no. 1, 17–28.
Search in Google Scholar Back to article
O. Yayenie, A note on generalized Fibonacci sequences, Appl. Math. Comput. 217 (2011), no. 12, 5603–5611.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/amsil-2025-0014 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107

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Language: English

Page range: 59 - 76

Submitted on: Apr 8, 2025

Accepted on: Sep 4, 2025

Published on: Sep 22, 2025

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

11B39,

11B37,

11C20,

05A05

Related subjects:

Mathematics,

General mathematics

© 2025 Eudes Antonio Costa, Paula M.M.C. Catarino, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 40 (2026): Issue 1 (March 2026)