On Higher Order Jacobsthal Hyper Complex Numbers

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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