Fibonacci Polynomials - Definition

Definition

These Fibonacci polynomials are defined by a recurrence relation:

F_n(x)= \begin{cases}
0, & \mbox{if } n = 0\\
1, & \mbox{if } n = 1\\
x F_{n - 1}(x) + F_{n - 2}(x),& \mbox{if } n \geq 2
\end{cases}

The first few Fibonacci polynomials are:

The Lucas polynomials use the same recurrence with different starting values: L_n(x) = \begin{cases}
2, & \mbox{if } n = 0 \\
x, & \mbox{if } n = 1 \\
x L_{n - 1}(x) + L_{n - 2}(x), & \mbox{if } n \geq 2.
\end{cases}

The first few Lucas polynomials are:

The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating Fn at x = 2. The degrees of Fn is n − 1 and the degree of Ln is n. The ordinary generating function for the sequences are:

The polynomials can be expressed in terms of Lucas sequences as

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