A note on the generalized bi-periodic Lucas-balancing numbers

Authors

DOI:

https://doi.org/10.14393/BEJOM-v6-2025-75273

Keywords:

Lucas-balancing sequence, bi-periodic sequence, identities, analytic representations

Abstract

In this study, we introduce a new class of integers called the sequence of generalized bi-periodic Lucas-balancing numbers, which extends the well-known sequence of Lucas-balancing numbers. We present several fundamental properties, including the deduction of the corresponding generating function, as well as homogeneous and non-homogeneous recurrence relations associated with this new sequence. We also formulate generalized versions of Binet's formulas for these numbers. In addition, we investigated the validity of several classical identities within this new context, such as the Tagiuri-Vajda, d'Ocagne, Catalan, and Cassini identities, considering the two different cases of the discriminant value of the equation polynomial associated with the recurrence relation. These extensions contribute to a structural and algebraic deepening of the properties of Lucas-balancing numbers.

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Author Biographies

  • Eudes Antonio Costa, Universidade Federal do Tocantins Campus Arraias, Colegiado de Matemática

    Adjunct Professor at the Federal University of Tocantins, Arraias Campus, Mathematics Course. He has a post-doctorate in Mathematics from the Federal University of Ceará (2019) and a PhD in Mathematics from the University of Brasília (2013). He has experience with Teacher Training (PROFMAT, Degree Courses and Improvement Courses) and Mathematical Olympiads (OBM and OBMEP).

  • Elen Viviani Pereira Spreafico, Universidade Federal de Mato Grosso do Sul, Campo Grande-MS, Brasil

    She holds a degree in Mathematics from the São Paulo State University Júlio de Mesquita Filho (2007), a Master's degree (2010), a PhD (2014) in Applied Mathematics from the State University of Campinas and a Post-Doctorate (2023) in Mathematics from the University of Trás-os-Montes and Alto Douro-UTAD. He is currently a professor at the Federal University of Mato Grosso do Sul, Campo Grande Campus - MS. He has works in the area of ​​Mathematics, with an emphasis on Applied Mathematics, working mainly on the theme of Additive Number Theory, Difference Equations, and Numerical Sequences.

  • Paula Maria Machado Cruz Catarino, Departamento de Matemática, Universidade de Trás-os-Montes e Douro, Vila Real, Portugal

    PhD in Mathematics at Essex University, UK, area of specialization in semigroups - algebra. She is a Full Professor in the Department of Mathematics, School of Sciences and Technology, University of Trás-os-Montes and Alto Douro (UTAD), Vila Real, Portugal. She is a Collaborator Member of the Laboratory of Didactics of Science and Technology of UTAD / CIDTFF of University of Aveiro and an Integrated Member of pole of Mathematics Research Center CMAT-UTAD of Centre of Mathematics CMAT of University of Minho, Braga, Portugal. The interests of the research done and published are mainly included in the areas of linear algebra, semigroups, number theory, ethnomatematics, mathematical education and history of mathematics.

References

M. Edson and O. Yayenie. “A New Generalization of Fibonacci Sequence & Extended Binet’s Formula”. In: Integers 9.6 (2009), pp. 639–654. DOI: 10 . 1515 / INTEG . 2009.051.

G. Bilgici. “Two generalizations of Lucas sequence”. In: Applied Mathematics and Computation 245 (2014), pp. 526–538.

S. Uygun and H. Karatas. “A new generalization of Pell-Lucas numbers (bi-periodic Pell-Lucas sequence)”. In: Communications in Mathematics and Applications 10.3 (2019), pp. 1–12.

S. Uygun and E. Owusu. “A new generalization of Jacobsthal numbers (bi-periodic Jacobsthal sequences)”. In: Journal of Mathematical Analysis 7.5 (2016), pp. 28–39.

P. M. M. C. Catarino and E. V. Spreafico. “A Note on Bi-Periodic Leonardo Sequence”. In: Armenian Journal of Mathematics 16.5 (2023), pp. 1–17.

A. Behera and G. K. Panda. “On the square roots of triangular numbers”. In: Fibonacci Quarterly 37 (1999), pp. 98–105.

P. K. Ray. “Balancing and Cobalancing Numbers”. Ph.D. thesis. Rourkela, India: Department of Mathematics, 2009.

A. Behera and P. Ray. “On the Properties of Balancing and Lucas-Balancing p-Numbers”. In: Iranian Journal of Mathematical Sciences and Informatics 17.2 (2022), pp. 147–163.

Y. Choo. “Relations between generalized bi-periodic Fibonacci and Lucas sequences”. In: Mathematics 8.9 (2020), p. 1527.

K. Liptai. “Lucas-balancing Numbers”. In: Acta Mathematica Universitatis Ostraviensis 14 (2006), pp. 43–47.

K. Liptai et al. “Generalized balancing numbers”. In: Indagationes Mathematicae 20.1 (2009), pp. 87–100.

P. Olajos. “Properties of Balancing, Cobalancing and Generalized Balancing Numbers”. In: Annales Mathematicae et Informaticae 37 (2010), pp. 125–138.

G. K. Panda and P. K. Ray. “Some links of balancing and cobalancing numbers with Pell and associated Pell numbers”. In: Bulletin of the Institute of Mathematics, Academia Sinica (New Series) 6.1 (2011), pp. 41–72.

S. Sriram and P. Veeramallan. “Generalized bi-periodic balancing numbers”. In: Advances and Applications in Mathematical Sciences 21.8 (2022), pp. 4515–4522.

D. Tasci and E. Sevgi. “Bi-periodic balancing numbers”. In: Journal of Science and Arts 50 (2020), pp. 75–84.

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Published

2025-06-16

Issue

Vol. 6 (2025): Brazilian Electronic Journal of Mathematics

Section

Pure Mathematics

License

Copyright (c) 2025 Eudes Antonio Costa, Elen Viviani Pereira Spreafico, Paula Maria Machado Cruz Catarino

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

  1. Articles published from 2025 onwards are licensed under the CC BY 4.0 license. By submitting material for publication, authors automatically agree to the journal’s editorial guidelines and affirm that the text has been properly reviewed. Simultaneous submission of articles to other journals is prohibited, as is the translation of articles published in this journal into another language without proper authorization.
  2. Articles published prior to 2025 are licensed under the CC BY-NC 4.0 license.

How to Cite

COSTA, Eudes Antonio; SPREAFICO, Elen Viviani Pereira; CATARINO, Paula Maria Machado Cruz. A note on the generalized bi-periodic Lucas-balancing numbers. BRAZILIAN ELECTRONIC JOURNAL OF MATHEMATICS, Uberlândia, Minas Gerais, v. 6, p. 1–15, 2025. DOI: 10.14393/BEJOM-v6-2025-75273. Disponível em: https://seer.ufu.br/index.php/BEJOM/article/view/e75273. Acesso em: 8 jul. 2025.

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