Fibonacci polynomials

{{Short description|Sequence of polynomials defined recursively}}

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:Benjamin & Quinn p. 141

: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:Benjamin & Quinn p. 142

: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 byInvalid citation.>Springer

: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 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.
  • The ordinary generating functions for the sequences are:{{MathWorld | urlname=FibonacciPolynomial | title=Fibonacci Polynomial}}
  • : \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|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 byA 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)].

: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

File:pascal_triangle_fibonacci.svg

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

{{reflist}}

  • {{cite book

| title = Proofs that Really Count: The Art of Combinatorial Proof

| first1 = Arthur T.

| last1 = Benjamin

| author1-link = Arthur T. Benjamin

| first2 = Jennifer J.

| last2 = Quinn

| author2-link = Jennifer Quinn

| publisher = Mathematical Association of America

| series = Dolciani Mathematical Expositions

| volume = 27

| year = 2003

| isbn = 978-0-88385-333-7

| chapter = Fibonacci and Lucas Polynomial

| page = [https://archive.org/details/proofsthatreally0000benj/page/141 141]

}}

  • {{SpringerEOM|title=Fibonacci polynomials

|id=Fibonacci_polynomials&oldid=14185|last=Philippou|first=Andreas N.}}

  • {{SpringerEOM|title=Lucas polynomials

|id=Lucas_polynomials&oldid=17297|last=Philippou|first=Andreas N.}}

  • {{MathWorld | urlname=LucasPolynomial| title=Lucas Polynomial}}
  • 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

  • {{cite journal | first1=V. E.|last1=Hoggatt | authorlink=Verner Emil Hoggatt, Jr. | first2=Marjorie | last2=Bicknell | title=Roots of Fibonacci polynomials. | journal=Fibonacci Quarterly | volume=11 | pages=271–274 | year=1973 | issn=0015-0517| mr=0332645 }}
  • {{cite journal | first1=V. E.|last1=Hoggatt | first2=Calvin T. | last2=Long | title=Divisibility properties of generalized Fibonacci Polynomials | journal=Fibonacci Quarterly | volume=12 | page=113 | year=1974 | mr=0352034 }}
  • {{cite journal | last1=Ricci |first1=Paolo Emilio | title=Generalized Lucas polynomials and Fibonacci polynomials | journal=Rivista di Matematica della Università di Parma|series= V. Ser. | volume=4 | year=1995 | pages=137–146 |mr=1395332 }}
  • {{cite journal|last1=Yuan |first1=Yi |last2=Zhang|first2=Wenpeng |journal=Fibonacci Quarterly| year=2002 |title =Some identities involving the Fibonacci Polynomials |page=314|mr=1920571|volume=40|issue=4}}
  • {{cite journal|first1=Johann|last1=Cigler |journal=Fibonacci Quarterly |year=2003 |mr=1962279 | pages=31–40|number=41|title=q-Fibonacci polynomials}}