Abstract
We obtain some new identities for the generalized Fibonacci polynomial by a new approach, namely, the Q(x) matrix.
 These identities including the Cassini type identity and Honsberger type formula can be applied to some polynomial
 sequences such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials and so on, which
 generalize the previous results in references.
Highlights
A second order polynomial sequence Fn(x) is said to be the Fibonacci polynomial if for n ≥ 2 and x ∈ R, Fn(x) = xFn−1(x) + Fn−2(x) with F0(x) = 0 and F1(x) = 1
We obtain some new identities for the generalized Fibonacci polynomial by a new approach, namely, the Q(x) matrix
These identities including the Cassini type identity and Honsberger type formula can be applied to some polynomial sequences such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials and so on, which generalize the previous results in references
Summary
A polynomial sequence Gn(x) in (Florez, Higuita & Mukherjee, 2018; Florez, McAnally & Mukherjee, 2018) is called the generalized Fibonacci polynomial if for n ≥ 2, Gn(x) = c(x)Gn−1(x) + d(x)Gn−2(x) with G0(x) and G1(x), where c(x) and d(x) are fixed non-zero polynomials in Q[x]. If for n ≥ 2, L0(x) = q, L1(x) = b(x) and Ln(x) = c(x)Ln−1(x) + d(x)Ln−2(x), the polynomial sequence Ln(x) is called the Lucas type polynomial, where q ∈ R \ {0} and b(x) is a fixed non-zero polynomial in Q[x] Both Fn(x) and Ln(x) are the generalized Fibonacci polynomials.
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