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.
The first few Fibonacci polynomials are:
The Lucas polynomials use the same recurrence with different starting values:
The first few Lucas polynomials are:
The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating Fn at x = 2. The degrees of Fn is n − 1 and the degree of Ln is n. The ordinary generating function for the sequences are:
The polynomials can be expressed in terms of Lucas sequences as
As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities.
First, they can be defined for negative indices by
Other identities include:
Closed form expressions, similar to Binet's formula are:
are the solutions (in t) of
If F(n,k) is the coefficient of xk in Fn(x), so
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 F(n,k) is equal to the binomial coefficient
when n and k have opposite parity. This gives a way of reading the coefficients from Pascal's triangle as shown on the right.
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