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

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Published

2025-06-16

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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