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Fibonacci polynomials - Wikiwand

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.

These Fibonacci polynomials are defined by a recurrence relation:[1]

{\displaystyle F_{n}(x)={\begin{cases}0,&{\mbox{if }}n=0\\1,&{\mbox{if }}n=1\\xF_{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:[2]

{\displaystyle L_{n}(x)={\begin{cases}2,&{\mbox{if }}n=0\\x,&{\mbox{if }}n=1\\xL_{n-1}(x)+L_{n-2}(x),&{\mbox{if }}n\geq 2.\end{cases}}}

They can be defined for negative indices by[3]

{\displaystyle F_{-n}(x)=(-1)^{n-1}F_{n}(x),}
{\displaystyle L_{-n}(x)=(-1)^{n}L_{n}(x).}

The Fibonacci polynomials form a sequence of orthogonal polynomials with {\displaystyle A_{n}=C_{n}=1} and {\displaystyle B_{n}=0}.

The first few Fibonacci polynomials are:

{\displaystyle F_{0}(x)=0\,}
{\displaystyle F_{1}(x)=1\,}
{\displaystyle F_{2}(x)=x\,}
{\displaystyle F_{3}(x)=x^{2}+1\,}
{\displaystyle F_{4}(x)=x^{3}+2x\,}
{\displaystyle F_{5}(x)=x^{4}+3x^{2}+1\,}
{\displaystyle F_{6}(x)=x^{5}+4x^{3}+3x\,}

The first few Lucas polynomials are:

{\displaystyle L_{0}(x)=2\,}
{\displaystyle L_{1}(x)=x\,}
{\displaystyle L_{2}(x)=x^{2}+2\,}
{\displaystyle L_{3}(x)=x^{3}+3x\,}
{\displaystyle L_{4}(x)=x^{4}+4x^{2}+2\,}
{\displaystyle L_{5}(x)=x^{5}+5x^{3}+5x\,}
{\displaystyle L_{6}(x)=x^{6}+6x^{4}+9x^{2}+2.\,}
  • 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.
where {\displaystyle i} is the imaginary unit.

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as[3]

{\displaystyle F_{m+n}(x)=F_{m+1}(x)F_{n}(x)+F_{m}(x)F_{n-1}(x)\,}
{\displaystyle L_{m+n}(x)=L_{m}(x)L_{n}(x)-(-1)^{n}L_{m-n}(x)\,}
{\displaystyle F_{n+1}(x)F_{n-1}(x)-F_{n}(x)^{2}=(-1)^{n}\,}
{\displaystyle F_{2n}(x)=F_{n}(x)L_{n}(x).\,}

Closed form expressions, similar to Binet's formula are:[3]

{\displaystyle F_{n}(x)={\frac {\alpha (x)^{n}-\beta (x)^{n}}{\alpha (x)-\beta (x)}},\,L_{n}(x)=\alpha (x)^{n}+\beta (x)^{n},}

where

{\displaystyle \alpha (x)={\frac {x+{\sqrt {x^{2}+4}}}{2}},\,\beta (x)={\frac {x-{\sqrt {x^{2}+4}}}{2}}}

are the solutions (in t) of

{\displaystyle t^{2}-xt-1=0.\,}

For Lucas Polynomials n > 0, we have

{\displaystyle 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[5]

{\displaystyle x^{n}=F_{n+1}(x)+\sum _{k=1}^{\lfloor n/2\rfloor }(-1)^{k}\left[{\binom {n}{k}}-{\binom {n}{k-1}}\right]F_{n+1-2k}(x).}

For example,

{\displaystyle x^{4}=F_{5}(x)-3F_{3}(x)+2F_{1}(x)\,}
{\displaystyle x^{5}=F_{6}(x)-4F_{4}(x)+5F_{2}(x)\,}
{\displaystyle x^{6}=F_{7}(x)-5F_{5}(x)+9F_{3}(x)-5F_{1}(x)\,}
{\displaystyle x^{7}=F_{8}(x)-6F_{6}(x)+14F_{4}(x)-14F_{2}(x)\,}
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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 xk in Fn(x), namely

{\displaystyle 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.[1] 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 {\displaystyle F(n,k)={\begin{cases}\displaystyle {\binom {{\frac {1}{2}}(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.