A Note on Dual Generalized Adrien Numbers

Abstract

In this study, we introduce and develop the concept of Dual Adrien numbers, with particular emphasis on two fundamental cases: the Dual Adrien sequence and the Dual Adrien–Lucas sequence. We conduct a systematic investigation of their structural and analytical properties, encompassing algebraic identities, matrix representations, recurrence relations, Binet-type formulas, generating functions, exponential expressions, Simson-type identities, and summation formulas. By establishing these results, we aim to construct a coherent and mathematically rigorous framework for the study of Dual Adrien numbers. Furthermore, we highlight their intrinsic connections with classical recurrence families, thereby situating them within the broader landscape of hypercomplex sequence analysis. This work not only extends the theory of generalized number sequences into the dual-number algebraic setting but also provides new tools and perspectives that may inspire further research in recurrence relations, combinatorial identities, and hypercomplex algebraic structures.