Editing 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 number]]s. The polynomials generated in a similar way from the [[Lucas numbers]] are called '''Lucas polynomials'''.

==Definition==
These Fibonacci [[polynomial]]s 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#Recurrence_relation|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 [[Generating function#Ordinary generating function|ordinary generating functions]] for the sequences are:<ref>{{MathWorld | urlname=FibonacciPolynomial | title=Fibonacci Polynomial}}</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 sequence]]s 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==
{{main|Lucas sequence}}
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 [https://web.archive.org/web/20170202051159/http://cmimc.org/Documents/Archive/AlgebraSolutions_2016.pdf 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 interpretation==
[[File:pascal_triangle_fibonacci.svg|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 [[domino]]es 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 [[Composition (number theory)|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.

==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]
 }}
<!-- Note: This source idiosyncratically shifts the index by 1, allow for this when checking the formulas. --> 
*{{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}}

==External links==
*{{OEIS el|sequencenumber=A162515|name=Triangle of coefficients of polynomials defined by Binet form|formalname=Triangle of coefficients of polynomials defined by Binet form: P(n,x) = (U^n-L^n)/d, where U=(x+d)/2, L=(x-d)/2, d=(4 + x^2)^(1/2)}}
*{{OEIS el|sequencenumber=A011973|name=Triangle of coefficients of Fibonacci polynomials|formalname=Triangle of numbers {C(n-k,k), n >= 0, 0 <= k <= floor(n/2)}; or, triangle of coefficients of (one version of) Fibonacci polynomials}}

[[Category:Polynomials]]
[[Category:Fibonacci numbers]]