Fibonacci polynomials

In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers.

The first few Fibonacci polynomials are: The first few Lucas polynomials are: As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as[3] Closed form expressions, similar to Binet's formula are:[3] 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[5] For example, 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.

[1] 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.