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
[edit]These Fibonacci polynomials are defined by a recurrence relation:[1]
The Lucas polynomials use the same recurrence with different starting values:[2]
They can be defined for negative indices by[3]
The Fibonacci polynomials form a sequence of orthogonal polynomials with and .
Examples
[edit]The first few Fibonacci polynomials are:
The first few Lucas polynomials are:
Properties
[edit]- 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:[4]
- The polynomials can be expressed in terms of Lucas sequences as
- They can also be expressed in terms of Chebyshev polynomials and as
- where is the imaginary unit.
Identities
[edit]As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as[3]
Closed form expressions, similar to Binet's formula are:[3]
where
are the solutions (in t) of
For Lucas Polynomials n > 0, we have
A relationship between the Fibonacci polynomials and the standard basis polynomials is given by[5]
For example,
Combinatorial interpretation
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Pascal_triangle_fibonacci.svg/280px-Pascal_triangle_fibonacci.svg.png)
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.[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
This gives a way of reading the coefficients from Pascal's triangle as shown on the right.
References
[edit]- ^ a b Benjamin & Quinn p. 141
- ^ Benjamin & Quinn p. 142
- ^ a b c Springer
- ^ Weisstein, Eric W. "Fibonacci Polynomial". MathWorld.
- ^ A proof starts from page 5 in Algebra Solutions Packet (no author).
- Benjamin, Arthur T.; Quinn, Jennifer J. (2003). "Fibonacci and Lucas Polynomial". Proofs that Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. Vol. 27. Mathematical Association of America. p. 141. ISBN 978-0-88385-333-7.
- Philippou, Andreas N. (2001) [1994], "Fibonacci polynomials", Encyclopedia of Mathematics, EMS Press
- Philippou, Andreas N. (2001) [1994], "Lucas polynomials", Encyclopedia of Mathematics, EMS Press
- Weisstein, Eric W. "Lucas Polynomial". MathWorld.
- 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
[edit]- Hoggatt, V. E.; Bicknell, Marjorie (1973). "Roots of Fibonacci polynomials". Fibonacci Quarterly. 11: 271–274. ISSN 0015-0517. MR 0332645.
- Hoggatt, V. E.; Long, Calvin T. (1974). "Divisibility properties of generalized Fibonacci Polynomials". Fibonacci Quarterly. 12: 113. MR 0352034.
- Ricci, Paolo Emilio (1995). "Generalized Lucas polynomials and Fibonacci polynomials". Rivista di Matematica della Università di Parma. V. Ser. 4: 137–146. MR 1395332.
- Yuan, Yi; Zhang, Wenpeng (2002). "Some identities involving the Fibonacci Polynomials". Fibonacci Quarterly. 40 (4): 314. MR 1920571.
- Cigler, Johann (2003). "q-Fibonacci polynomials". Fibonacci Quarterly (41): 31–40. MR 1962279.
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