INVESTIGATION OF HIGHER ORDER HORADAM NUMBERS, THEIR ASSOCIATED TRANSFORMS AND SELF-SIMILARITY

This paper gives results for higher-order Horadam numbers, which are akin to higher-order Fibonacci sequence, it weaves intricate connections with the classic Horadam, Pell, Jacobsthal, Mersenne, Balancing numbers. Also, all number sequences used in higher-order Horadam numbers, from Fibonacci to Lucas Mersenne to Fermat numbers, exhibit self-similarity at certain points, and so self-similarity has been exploited in these areas to simplify the analysis of complex systems, improve predictive models, and create efficient designs by exploiting the inherent reproducible and scalable nature of self-similar structures. Methods: We introduce the basic properties and results for Horadam numbers, leading to the Catalan transform of higher order Horadam numbers. Results: We give results for Cassini identity, d’Ocagne identity, Vajda identity, Honsberger identity, Melham’s identity. Moreover, this study examines the "Catalan transforms" of these higher-order Horadam numbers, aiming to solidify their comprehension through matrix representations. These powerful tools, along with other equations, act as keys that unlock the heart of these numbers, paving the way for future exploration and deeper insights. © 2025, University of Montenegro Faculty of Natural Sciences and Mathematics. All rights reserved.