Higher-Order Multiplicative Alqazzaz Sequences over Finite Root Sets and Hypercomplex Algebra

In this paper, we introduce and study higher-order multiplicative Hn-Alqazzazsequences over finite sets of roots of unity. While the second-order case is generated from two preceding terms, we extend the construction to third-, fourth-,and fifth-order recurrences. We then generalize the framework to hypercomplexalgebras, including quaternions, octonions, and sedenions, in order to investigate the effect of non-commutativity and non-associativity on the behavior ofthe resulting sequences. In addition, we explore the geometric consequencesof these sequences through fractal generation in the complex plane. Severalexplicit examples are presented to illustrate the diversity of the algebraic anddynamical structures produced by the proposed model. These results suggestthat Alqazzaz sequences provide a flexible framework for studying multiplicativerecurrences across different algebraic settings