A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers

Year 2026, Volume: 9 Issue: 1 , 23 - 30 , 30.03.2026

Gamaliel Morales

https://doi.org/10.33401/fujma.1783934

https://izlik.org/JA22CL64CC

Abstract

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.

Keywords

Binet’s formula , Cassini’s identity , Dual-hyperbolic number , Recurrence relation , Third-order Jacobsthal number

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