Kuloglu, Bahar; Eser, Engin; Özkan, Engin

In this research, we present a novel family of polynomials that are derived from Fermat numbers. Alongside these newly introduced polynomials, we provide a series of important identities and properties, including the Binet formula and generating functions, which are fundamental to understanding their structure. We also show that Fermat polynomials can be efficiently expressed using matrix representations, enabling an exploration of fundamental properties derived from the invariance of the determinants of these matrices. This approach provides new perspectives on the internal connections within these polynomials. In addition, many different transforms have been considered with the help of polynomials of these new number sequences, and these transforms can play a role in many areas from secure key generation to differential equation solutions. These transformations shed light on the deeper structural properties of the Fermat polynomials and reveal intricate connections between these polynomials and other well-established mathematical constructs, enhancing our understanding of their broader implications in the field of number theory and beyond.

Uysal, Mine; Özkan, Engin

In this paper, we introduce Balancing and Lucas-Balancing hybrid numbers. We examine some identities of Balancing and Lucas-Balancing hybrid numbers. We give some basic definitions and properties related to them. In addition, we find Binet's Formula, Cassini's identity, Catalan's identity, d'Ocagne identity, generating functions, exponential generating functions of the Balancing and Lucas-Balancing hybrid numbers.