Incubator escapee wiki

Fibonacci polynomials

Article Discussion

Template:Short description 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.

DefinitionEdit

These Fibonacci polynomials are defined by a recurrence relation:<ref name=BQ141>Benjamin & Quinn p. 141</ref>

<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:<ref>Benjamin & Quinn p. 142</ref>

<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<ref name=Springer Invalid citation.>Springer</ref>

<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>.

ExamplesEdit

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>

PropertiesEdit

  • 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.

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:FibonacciPolynomial%7CFibonacciPolynomial.html}} |title = Fibonacci Polynomial |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>

  • <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.

IdentitiesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as<ref name=Springer/>

<math>F_{m+n}(x)=F_{m+1}(x)F_n(x)+F_m(x)F_{n-1}(x)\,</math>
<math>L_{m+n}(x)=L_m(x)L_n(x)-(-1)^nL_{m-n}(x)\,</math>
<math>F_{n+1}(x)F_{n-1}(x)- F_n(x)^2=(-1)^n\,</math>
<math>F_{2n}(x)=F_n(x)L_n(x).\,</math>

Closed form expressions, similar to Binet's formula are:<ref name=Springer/>

<math>F_n(x)=\frac{\alpha(x)^n-\beta(x)^n}{\alpha(x)-\beta(x)},\,L_n(x)=\alpha(x)^n+\beta(x)^n,</math>

where

<math>\alpha(x)=\frac{x+\sqrt{x^2+4}}{2},\,\beta(x)=\frac{x-\sqrt{x^2+4}}{2}</math>

are the solutions (in t) of

<math>t^2-xt-1=0.\,</math>

For Lucas Polynomials n > 0, we have

<math>L_n(x)=\sum_{k=0}^{\lfloor n/2\rfloor} \frac{n}{n-k} \binom{n-k}{k} x^{n-2k}.</math>

A relationship between the Fibonacci polynomials and the standard basis polynomials is given by<ref>A proof starts from page 5 in Algebra Solutions Packet (no author).</ref>

<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 interpretationEdit

File:Pascal triangle fibonacci.svg
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

<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.<ref name=BQ141/> 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 <math>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}</math>

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

ReferencesEdit

Template:Reflist

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:LucasPolynomial%7CLucasPolynomial.html}} |title = Lucas Polynomial |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

Further readingEdit

External linksEdit

Retrieved from "https://zalansite.site/mediawiki/index.php?title=Fibonacci_polynomials&oldid=298390"