K-fibonacci sayi dizisi ve polinomunun yeni bir ailesi

In this thesis, a new family of k-Fibonacci polynomial is defined. The Binet formula is defined and proven. The relationship between the Fibonacci polynomial and the k-Fibonacci polynomial is given. Matrix representation is found and some of its terms are tabulated. Also, the Catalan Formula and its generalizations, Cassini, D'ocagne, Honsberger, Vajda formulas were defined and proved. Additionally, these identities are given and proved for the k-Fibonacci number sequence, too. In addition to these, some identities and their connection with the Fibonacci polynomial were defined and proved.