On the bi-periodic k-Jacobsthal and k-Jacobsthal-Lucas numbers
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Abstract
This paper introduces and investigates the bi-periodic Jacobsthal and
Jacobsthal–Lucas sequences, extending the classical Jacobsthal framework by incorporating periodicity and a tunable parameter
. We establish recurrence relations, derive generating functions, and present Binet-type formulas for these generalized sequences. Furthermore, we obtain extensions of well-known identities, including Catalan’s, Cassini’s, and d’Ocagne’s identities. The proposed generalization reveals deeper algebraic structures and periodic patterns, offering potential applications in cryptography, coding theory, and recurrence-based modeling. These findings provide a foundation for future research on combinatorial interpretations and connections with other special sequences.
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References
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