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
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2016 May 21.
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