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
:
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
:
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
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The Fibonacci polynomials form a sequence of orthogonal polynomials with and .
Examples
The first few Fibonacci polynomials are:
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The first few Lucas polynomials are:
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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}}
- :
- :
- The polynomials can be expressed in terms of Lucas sequences as
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- :
- They can also be expressed in terms of Chebyshev polynomials and as
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- :
:where is the imaginary unit.
Identities
{{main|Lucas sequence}}
As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as
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Closed form expressions, similar to Binet's formula are:
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where
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are the solutions (in t) of
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For Lucas Polynomials n > 0, we have
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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)].
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For example,
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Combinatorial interpretation
File:pascal_triangle_fibonacci.svg
If F(n,k) is the coefficient of xk in Fn(x), namely
:
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
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}}
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}}