On generalized Fibonacci and Lucas polynomials

Let hðxÞ be a polynomial with real coefficients. We introduce hðxÞ-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these hðxÞ-Fibonacci polynomials. We also introduce hðxÞ-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix QhðxÞ that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.