In mathematics, the Fibonacci polynomials are a

polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in e ...

which can be considered as a generalization of the

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...

s. The polynomials generated in a similar way from the

Lucas numbers The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci nu ...

are called Lucas polynomials.

Definition

These Fibonacci

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

s are defined by a

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...

:Benjamin & Quinn p. 141 :

F_n(x)= \begin
0, & \mbox n =  0\\
1, & \mbox n =  1\\
x F_(x) + F_(x),& \mbox  n \geq 2
\end

The Lucas polynomials use the same recurrence with different starting values: :

L_n(x) = \begin
2, & \mbox n = 0 \\
x, & \mbox n = 1 \\
x L_(x) + L_(x), & \mbox n \geq 2.
\end

They can be defined for negative indices bySpringer :

F_(x)=(-1)^F_(x),

L_(x)=(-1)^nL_(x).

The Fibonacci polynomials form a sequence of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the cl ...

with

A_n=C_n=1

and

B_n=0

Examples

The first few Fibonacci polynomials are: :

F_0(x)=0 \,

F_1(x)=1 \,

F