We provide tiling proofs for some relations between Fibonacci and Lucas numbers, as requested by Benjamin and Quinn in their text, Proofs that Really Count. Extending our arguments yields Gibonacci generalizations of these identities.

In this paper, we define new families of Generalized Fibonacci polynomials and Lucas develop some elegant properties these families. We also find the relationships between family generalized k-Fibonacci known polynomials. Furthermore, generalizations in matrix representation. Then establish Cassini’s Identities for their Finally, suggest avenues further research.

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric have been studied in the literature with help generating functions their functional equations. In this paper, using (p,q)–Fibonacci (p,q)–Lucas Changhee numbers, we define (p,q)–Fibonacci–Changhee polynomials (p,q)–Lucas–Changhee respectively. We obtain some important identities relations these new...

In this study, we obtain a new identities for Fibonacci numbers F2n, F−2n, F2n+1, F−2n+1 and Lucas numbers L2n, L−2n, L2n+1, L−2n+1 when n ≥ 1.

In this paper, the matrices related to Fibonacci, Lucas, Pell, and Pell-Lucas numbers have been examined. By using these new identities integer sequences investigated.