Fibonacci polynomials
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:[1]
The first few Fibonacci polynomials are:
The Lucas polynomials use the same recurrence with different starting values:[2]
The first few Lucas polynomials are:
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 degrees of Fn is n − 1 and the degree of Ln is n. The ordinary generating function for the sequences are:[3]
The polynomials can be expressed in terms of Lucas sequences as
Identities
As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities.
First, they can be defined for negative indices by[4]
Other identities include:[4]
Closed form expressions, similar to Binet's formula are:[4]
where
are the solutions (in t) of
Combinatorial interpretation
If F(n,k) is the coefficient of xk in Fn(x), so
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 F(n,k) is equal to the binomial coefficient
when n and k have opposite parity. This gives a way of reading the coefficients from Pascal's triangle as shown on the right.
References
- 1 2 Benjamin & Quinn p. 141
- ↑ Benjamin & Quinn p. 142
- ↑ Weisstein, Eric W., "Fibonacci Polynomial", MathWorld.
- 1 2 3 Springer
- Benjamin, Arthur T.; Quinn, Jennifer J. (2003). "§9.4 Fibonacci and Lucas Polynomial". Proofs that Really Count. MAA. p. 141. ISBN 0-88385-333-7.
- Philippou, Andreas N. (2001), "Fibonacci polynomials", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Philippou, Andreas N. (2001), "Lucas polynomials", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Lucas Polynomial", MathWorld.
Further reading
- 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.
External links
- "Sloane's A162515 : Triangle of coefficients of polynomials defined by Binet form...", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- "Sloane's A011973 : Triangle of coefficients of Fibonacci polynomials.", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.