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.

Examples

The first few Fibonacci polynomials are:

The first few Lucas polynomials are:

Properties

• The degree of Fn is n1 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 as

• 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 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

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