# Sequences of Polynomials

## Fibonacci Polynomials

The Fibonacci polynomials are defined by

F_{0}(x) = 0

F_{1}(x) = 1

F_{n}(x) = xF_{n-1}(x) + F_{n-2}(x)

and the first few are:

F_{0}(x) = 0

F_{1}(x) = 1

F_{2}(x) = x

F_{3}(x) = x^{2} + 1

F_{4}(x) = x^{3} + 2x

F_{5}(x) = x^{4} + 3x^{2} + 1

F_{6}(x) = x^{5} + 4x^{3} + 3x

F_{7}(x) = x^{6} + 5x^{4} + 6x^{2} + 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:

L_{0}(x) = 2

L_{1}(x) = x

L_{n}(x) = xL_{n-1}(x) + L_{n-2}(x)

The first few are:

L_{0}(x) = 2

L_{1}(x) = x

L_{2}(x) = x^{2} + 2

L_{3}(x) = x^{3} + 3x

L_{4}(x) = x^{4} + 4x^{2} + 2

L_{5}(x) = x^{5} + 5x^{3} + 5x

L_{6}(x) = x^{6} + 6x^{4} + 9x^{2} + 2

L_{7}(x) = x^{7} + 7x^{5} + 14x^{3} + 7x

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2018 Aug 27. s.11