Fibonacci polynomials

In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials.

Definition

These Fibonacci polynomials are defined by a recurrence relation:
The Lucas polynomials use the same recurrence with different starting values:
They can be defined for negative indices by
The Fibonacci polynomials form a sequence of orthogonal polynomials with and.

Examples

The first few Fibonacci polynomials are:
The first few Lucas polynomials are:

Properties

The degree of F_n is n − 1 and the degree of L_n is n.
The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating F_n at x = 2.
The ordinary generating functions for the sequences are:
:
:
The polynomials can be expressed in terms of Lucas sequences as
:
:
They can also be expressed in terms of Chebyshev polynomials and as
:
:

Identities

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as
Closed form expressions, similar to Binet's formula are:
where
are the solutions of
For Lucas Polynomials n > 0, we have
A relationship between the Fibonacci polynomials and the standard basis polynomials is given by
For example,

Combinatorial interpretation

If F is the coefficient of x^k in F_n, namely
then F is the number of ways an n−1 by 1 rectangle can be tiled with 2 by 1 dominoes and 1 by 1 squares so that exactly k squares are used. Equivalently, F is the number of ways of writing n−1 as an ordered sum involving only 1 and 2, so that 1 is used exactly k times. For example F=4 and 5 can be written in 4 ways, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, as a sum involving only 1 and 2 with 1 used 3 times. By counting the number of times 1 and 2 are both used in such a sum, it is evident that
This gives a way of reading the coefficients from Pascal's triangle as shown on the right.