On Higher Order Jacobsthal Hyper Complex Numbers

Year 2024, Volume: 16 Issue: 1, 35 - 44, 30.06.2024

Hayrullah Özimamoğlu

https://doi.org/10.47000/tjmcs.1195463

Abstract

In this work, we define a new class of hyper complex numbers whose components are higher order Jacobsthal numbers, and call such numbers as the higher order Jacobsthal $ 2^{s} $-ions. We obtain some algebraic properties of the higher order Jacobsthal $ 2^{s} $-ions such as recurrence relation, Binet-like formula, generating function, exponential generating function, Vajda's identity, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. Morever we derive the matrix representation of the higher order Jacobsthal $ 2^{s} $-ions, and so prove Cassini's identity as a further type.

Keywords

Hyper complex numbers, higher order Jacobsthal numbers, higher order Jacobsthal $ 2^{s} $-ions, Binet-like formula, recurrence relation

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