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:

:<math>F_n(x)= \begin{cases}

0, & \mbox{if } n = 0\\

1, & \mbox{if } n = 1\\

x F_{n - 1}(x) + F_{n - 2}(x),& \mbox{if } n \geq 2

\end{cases}</math>

The Lucas polynomials use the same recurrence with different starting values:

:<math>L_n(x) = \begin{cases}

2, & \mbox{if } n = 0 \\

x, & \mbox{if } n = 1 \\

x L_{n - 1}(x) + L_{n - 2}(x), & \mbox{if } n \geq 2.

\end{cases}</math>

They can be defined for negative indices by

:<math>F_{-n}(x)=(-1)^{n-1}F_{n}(x),</math>

:<math>L_{-n}(x)=(-1)^nL_{n}(x).</math>

The Fibonacci polynomials form a sequence of orthogonal polynomials with <math>A_n=C_n=1</math> and <math>B_n=0</math>.

Examples

The first few Fibonacci polynomials are:

:<math>F_0(x)=0 \,</math>

:<math>F_1(x)=1 \,</math>

:<math>F_2(x)=x \,</math>

:<math>F_3(x)=x^2+1 \,</math>

:<math>F_4(x)=x^3+2x \,</math>

:<math>F_5(x)=x^4+3x^2+1 \,</math>

:<math>F_6(x)=x^5+4x^3+3x \,</math>

The first few Lucas polynomials are:

:<math>L_0(x)=2 \,</math>

:<math>L_1(x)=x \,</math>

:<math>L_2(x)=x^2+2 \,</math>

:<math>L_3(x)=x^3+3x \,</math>

:<math>L_4(x)=x^4+4x^2+2 \,</math>

:<math>L_5(x)=x^5+5x^3+5x \,</math>

:<math>L_6(x)=x^6+6x^4+9x^2 + 2. \,</math>

Properties

  • The degree of F<sub>n</sub> is n&nbsp;&minus;&nbsp;1 and the degree of L<sub>n</sub> is n.
  • The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x&nbsp;=&nbsp;1; Pell numbers are recovered by evaluating F<sub>n</sub> at x&nbsp;=&nbsp;2.
  • The ordinary generating functions for the sequences are:
  • :<math> \sum_{n=0}^\infty F_n(x) t^n = \frac{t}{1-xt-t^2}</math>
  • :<math> \sum_{n=0}^\infty L_n(x) t^n = \frac{2-xt}{1-xt-t^2}.</math>
  • The polynomials can be expressed in terms of Lucas sequences as
  • :<math>F_n(x) = U_n(x,-1),\,</math>
  • :<math>L_n(x) = V_n(x,-1).\,</math>
  • They can also be expressed in terms of Chebyshev polynomials <math>\mathcal{T}_n(x)</math> and <math>\mathcal{U}_n(x)</math> as
  • :<math>F_n(x) = i^{n-1}\cdot\mathcal{U}_{n-1}(\tfrac{-ix}2),\,</math>
  • :<math>L_n(x) = 2\cdot i^n\cdot\mathcal{T}_n(\tfrac{-ix}2),\,</math>

:where <math>i</math> is the imaginary unit.

Identities

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as

:<math>x^n=F_{n+1}(x)+\sum_{k=1}^{\lfloor n/2\rfloor}(-1)^k\left[\binom nk-\binom n{k-1}\right]F_{n+1-2k}(x).</math>

For example,

:<math>x^4 = F_5(x)-3F_3(x)+2F_1(x)\,</math>

:<math>x^5 = F_6(x)-4F_4(x)+5F_2(x)\,</math>

:<math>x^6 = F_7(x)-5F_5(x)+9F_3(x)-5F_1(x)\,</math>

:<math>x^7 = F_8(x)-6F_6(x)+14F_4(x)-14F_2(x)\,</math>

Combinatorial interpretation

thumb|upright=1.25|The coefficients of the Fibonacci polynomials can be read off from a left-justified Pascal's triangle following the diagonals (shown in red). The sums of the coefficients are the Fibonacci numbers.

If F(n,k) is the coefficient of x<sup>k</sup> in F<sub>n</sub>(x), namely

:<math>F_n(x)=\sum_{k=0}^n F(n,k)x^k,\,</math>

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.