A Study On Hyperbolic Generalized Edouard Numbers

In this study, we introduce the generalized hyperbolic Edouard numbers, a novel class of sequences governed by fourth-order recurrence relations, which extend traditional frameworks through enriched algebraic and combinatorial structures. We analyze several noteworthy special cases, including the hyperbolic Edouard numbers and hyperbolic Edouard–Lucas numbers, both of which exhibit complex recurrence dynamics and reveal intriguing mathematical identities. A comprehensive set of structural properties is presented, including closed-form (Binet-type) expressions, ordinary and exponential generating functions, Simson-type identities, summation formulas, and matrix representations. These formulations provide deep analytical insight into the sequences’ intrinsic patterns and behavior. Moreover, the matrix-based approach offers an elegant avenue for further theoretical developments and practical applications. The sequences proposed here hold promise for interdisciplinary applications across various domains. In quantum physics, they may inform models of coherence, entanglement, and periodic evolution in chaotic systems. In biology, recurrence-based frameworks contribute to understanding gene regulation, population dynamics, and memory encoding. In statistics, they support time-series analysis and complex dependence modeling. Collectively, our findings lay the groundwork for future exploration into higher-order recurrence systems, symbolic regression, and machine learning architectures, bridging discrete mathematics with continuous systems.