Sequences of Polynomials  

Fibonacci Polynomials

The Fibonacci polynomials are defined by

F0(x) = 0
F1(x) = 1
Fn(x) = xFn-1(x) + Fn-2(x)

and the first few are:

F0(x) = 0
F1(x) = 1
F2(x) = x
F3(x) = x2 + 1
F4(x) = x3 + 2x
F5(x) = x4 + 3x2 + 1
F6(x) = x5 + 4x3 + 3x
F7(x) = x6 + 5x4 + 6x2 + 1

These are discussed furtehr in my article on 2nd-Order Linear Recurrence Sequences.

Lucas Polynomials

The Lucas polynomials are defined the same way except starting with 2 and x:

L0(x) = 2
L1(x) = x
Ln(x) = xLn-1(x) + Ln-2(x)

The first few are:

L0(x) = 2
L1(x) = x
L2(x) = x2 + 2
L3(x) = x3 + 3x
L4(x) = x4 + 4x2 + 2
L5(x) = x5 + 5x3 + 5x
L6(x) = x6 + 6x4 + 9x2 + 2
L7(x) = x7 + 7x5 + 14x3 + 7x


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