The Birth of Orbits and Trajectories: A New Era in Fibonacci Studies

This paper introduces a natural generalization of the Fibonacci sequence, encompassing a broader class of recursive sequences defined by six parameters, which form the foundation of its rich and intricate structure. By employing rigorous analysis and leveraging specific differential operators, numerous new properties within this generalized framework are revealed. These findings improve our understanding and offer new insights into this extended class of Fibonacci-like sequences. Furthermore, the research highlights important connections between generalized Fibonacci sequences and other established sequences in number theory. Additionally, the concepts of “trajectories” and “orbits” are introduced. We show that specific trajectories could connect previously isolated sequences and facilitate a unified approach to studying them through trajectory analysis.