Gaussian Generalized Adrien Numbers

Abstract

In this study, we introduce the concept of Gaussian Generalized Adrien numbers, a novel extension within the framework of special number sequences. Our focus centers on two particular instances: the Gaussian Adrien numbers and the Gaussian Adrien-Lucas numbers. We systematically investigate and establish fundamental properties of these sequences, including closed-form identities, recurrence relations, matrix formulations, and Binet-type expressions. Additionally, we derive their generating functions, explore their connections with exponential functions, and present analogues of Simson’s and summation formulas. These results contribute to a deeper algebraic and combinatorial understanding of the Gaussian extensions of Adrien-type numbers and open pathways for further research in number theory and related fields.