A Note on Hybrid Numbers with Generalized Hybrid k-Pell Numbers as Coefficients

Year 2025, Volume: 8 Issue: 3, 125 - 135, 23.09.2025

Elen Viviani Pereira Spreafico , Eudes Antonio Costa , Paula Maria Machado Cruz Catarino

https://doi.org/10.33434/cams.1655752

Abstract

In this study, we define a new generalization of the hybrid $k$-Pell sequence consisting of hybrid numbers with generalized hybrid $k$-Pell numbers as coefficients. We establish some algebraic properties and also the Binet formula, generating function, and exponential generating function related to this new sequence. In addition, some identities are provided as sum identities, and Catalan, Cassini, and d'Ocagne's identities. The particular cases are studied, namely, the hybrid numbers with hybrid $k$-Pell numbers as coefficients, the hybrid numbers with hybrid $k$-Pell--Lucas numbers as coefficients, and the hybrid numbers with hybrid Modified $k$-Pell numbers as coefficients.

Keywords

Binet , Generating function , Hybrid numbers , Hybrid $k$-Pell sequence , Identities

Thanks

The first author expresses their sincere thanks to the Federal University of Mato Grosso do Sul – UFMS/MEC – Brazil for their valuable support. The second author was partially supported by PROPESQ-UFT. The last author is member of the Research Centre CMAT-UTAD (Polo of Research Centre CMAT - Centre of Mathematics of University of Minho) and thanks the Portuguese Funds through FCT -- Fundação para a Ciência e a Tecnologia, within the Project UID/00013: Mathematics Centre of the University of Minho (CMAT/UM).

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