Journal of Advances in Mathematics and Computer Science

In this study, we introduce and rigorously define a new class of number sequences known as Hyperbolic Adrien numbers, with particular emphasis on two distinct cases: the Hyperbolic Adrien numbers and the Hyperbolic Adrien-Lucas numbers. These sequences are constructed through hyperbolic analogues of classical Adrien and Adrien-Lucas formulations, offering novel perspectives within the framework of hypercomplex analysis. Following their definition, we conduct a comprehensive investigation into their structural and algebraic properties. Specifically, we derive and analyze a range of identities, explore their matrix representations, establish recurrence relations, and formulate explicit expressions via Binet-type formulas. Furthermore, we develop their generating functions and exponential representations, and examine their behavior through Simson-type identities and summation formulas. These results not only enrich the theoretical landscape of hyperbolic number sequences but also provide foundational tools for potential applications in discrete mathematics and mathematical physics.