Split quaternionic representations of horadam sequences and their binet, generating function, and cassini-type identities

Abstract

This study establishes a novel algebraic connection between Horadam numbers and the split quaternion algebra. To this end, two fundamental constructs are introduced: the Fibonacci Sq,r-split quaternions and the Horadam sq,r-split quaternions, which generalize Horadam numbers within the framework of split quaternions. Several key results are derived, including the Binet-like formula, the generating function, and the Cassini-type identity, each revealing new structural relationships between these algebraic systems. The findings not only extend the scope of Horadam sequences into the realm of hypercomplex numbers but also open potential applications in number theory and algebraic analysis. & Idot;skender & Ouml;zt & uuml;rk and Hasan & Ccedil;ak & imath;r contributed equally to this study.