Fibonacci polynomials

Definition

These Fibonacci polynomials are defined by a recurrence relation:

: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}

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

: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}

They can be defined for negative indices by :F_{-n}(x)=(-1)^{n-1}F_{n}(x), :L_{-n}(x)=(-1)^nL_{n}(x).

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

Examples

The first few Fibonacci polynomials are: :F_0(x)=0 , :F_1(x)=1 , :F_2(x)=x , :F_3(x)=x^2+1 , :F_4(x)=x^3+2x , :F_5(x)=x^4+3x^2+1 , :F_6(x)=x^5+4x^3+3x ,

The first few Lucas polynomials are: :L_0(x)=2 , :L_1(x)=x , :L_2(x)=x^2+2 , :L_3(x)=x^3+3x , :L_4(x)=x^4+4x^2+2 , :L_5(x)=x^5+5x^3+5x , :L_6(x)=x^6+6x^4+9x^2 + 2. ,

Properties

• The degree of F**n is n − 1 and the degree of L**n is n.

• The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating F**n at x = 2.

• The ordinary generating functions for the sequences are:

• : \sum_{n=0}^\infty F_n(x) t^n = \frac{t}{1-xt-t^2}

• : \sum_{n=0}^\infty L_n(x) t^n = \frac{2-xt}{1-xt-t^2}.

• The polynomials can be expressed in terms of Lucas sequences as

• :F_n(x) = U_n(x,-1),,

• :L_n(x) = V_n(x,-1).,

• They can also be expressed in terms of Chebyshev polynomials \mathcal{T}_n(x) and \mathcal{U}_n(x) as

• :F_n(x) = i^{n-1}\cdot\mathcal{U}_{n-1}(\tfrac{-ix}2),,

• :L_n(x) = 2\cdot i^n\cdot\mathcal{T}_n(\tfrac{-ix}2),, :where i is the imaginary unit.

Identities

Main article: Lucas sequence

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as :F_{m+n}(x)=F_{m+1}(x)F_n(x)+F_m(x)F_{n-1}(x), :L_{m+n}(x)=L_m(x)L_n(x)-(-1)^nL_{m-n}(x), :F_{n+1}(x)F_{n-1}(x)- F_n(x)^2=(-1)^n, :F_{2n}(x)=F_n(x)L_n(x)., Closed form expressions, similar to Binet's formula are: :F_n(x)=\frac{\alpha(x)^n-\beta(x)^n}{\alpha(x)-\beta(x)},,L_n(x)=\alpha(x)^n+\beta(x)^n, where :\alpha(x)=\frac{x+\sqrt{x^2+4}}{2},,\beta(x)=\frac{x-\sqrt{x^2+4}}{2} are the solutions (in t) of :t^2-xt-1=0., For Lucas Polynomials n 0, we have :L_n(x)=\sum_{k=0}^{\lfloor n/2\rfloor} \frac{n}{n-k} \binom{n-k}{k} x^{n-2k}.

A relationship between the Fibonacci polynomials and the standard basis polynomials is given by :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). For example, :x^4 = F_5(x)-3F_3(x)+2F_1(x), :x^5 = F_6(x)-4F_4(x)+5F_2(x), :x^6 = F_7(x)-5F_5(x)+9F_3(x)-5F_1(x), :x^7 = F_8(x)-6F_6(x)+14F_4(x)-14F_2(x),

Combinatorial interpretation

If F(n,k) is the coefficient of xk in Fn(x), namely :F_n(x)=\sum_{k=0}^n F(n,k)x^k,, 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)=\begin{cases}\displaystyle\binom{\frac12(n+k-1)}{k} &\text{if }n \not\equiv k \pmod 2,\[12pt] 0 &\text{else}. \end{cases}

This gives a way of reading the coefficients from Pascal's triangle as shown on the right.

References

• {{cite book | author1-link = Arthur T. Benjamin | author2-link = Jennifer Quinn
• {{SpringerEOM|title=Fibonacci polynomials
• {{SpringerEOM|title=Lucas polynomials
• 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

References

1. Benjamin & Quinn p. 141
2. Benjamin & Quinn p. 142
3. "Fibonacci Polynomial".
4. A proof starts from page 5 in [https://web.archive.org/web/20170202051159/http://cmimc.org/Documents/Archive/AlgebraSolutions_2016.pdf Algebra Solutions Packet (no author)].