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 A_n=C_n=1 and B_n=0.

Examples

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

Properties

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 (in t) 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(n,k) is the coefficient of xk in Fn(x), namely then F(n,k) 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(n,k) 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(6,3)=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.