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:Benjamin & Quinn p. 141
The Lucas polynomials use the same recurrence with different starting values:Benjamin & Quinn p. 142
They can be defined for negative indices bySpringer
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 Fn is n − 1 and the degree of Ln is n.
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 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
where is the imaginary unit.
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 byA proof starts from page 5 in Algebra Solutions Packet (no author).
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.
References
Jin, Z. On the Lucas polynomials and some of their new identities. Advances in Differential Equations 2018, 126 (2018). https://doi.org/10.1186/s13662-018-1527-9
Further reading
External links