On Generalizations of Dickson k-Fibonacci Polynomials

In this study, we define a Dickson k-Fibonacci polynomial inspired by Dickson polynomials and give some terms of these polynomials. Then we present the relations between the terms of Dickson k-Fibonacci polynomials. We find Binet formulas and generating functions for these polynomials. In addition, we give some important identities like Catalan identity, Melham’s identity, and Gelin-Cesaro’s identity. Moreover, Catalan transformation is applied to these polynomials, and their terms are found. Finally, the Hankel transform is applied to the Catalan transform of these polynomials, and the results obtained are associated with known Fibonacci numbers.