Introduces dual-hyperbolic third-order Jacobsthal numbers, revealing new identities and properties.
In this study, we introduce one-parameter generalization of dual-hyperbolic third-order Jacobsthal (or dual-hyperbolic third-order k-Jacobsthal) numbers. We present some identities and properties of them, among others the Binet-type formula, d'Ocagne and Cassini identities. Furthermore, we study the summation formula and generating function for these dual-hyperbolic numbers. The results presented here are a generalizations of the results for the dual-hyperbolic Jacobsthal numbers of order two. New identities for this sequence including its matrix representation are introduced.
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Gamaliel Morales (2026) studied this question.
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