On Fibonacci and Lucas Quasi-quaternions: Theoretical insights and graphical representations

Open Journal of Mathematical Sciences, no.10, pp.625-644, 2026 (Scopus)

In this paper, we introduce Fibonacci and Lucas quasi-quaternions by combining classical number

sequences with the structure of quasi-quaternion algebra. We investigate their fundamental algebraic

properties, including real and imaginary parts, conjugates, norms, and recurrence relations. We establish

Binet-type formulas, generating functions, and sum formulas for these sequences in the quasi-quaternionic

setting. In addition, we derive several classical identities, such as the Cassini, Catalan, d’Ocagne, Vajda, and

Honsberger identities, adapted to Fibonacci and Lucas quasi-quaternions. Furthermore, we present matrix

representations of these structures and obtain explicit expressions for the powers of the associated matrices.

We also consider De Moivre-type formulas in the quasi-quaternion framework and analyze the behavior of

these sequences under repeated operations. The graphical representations complement the theoretical results

by illustrating the structural and asymptotic behavior of these quasi-quaternionic sequences.