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:

$F_{n}(x)={\begin{cases}0,&{\mbox{if }}n=0\\1,&{\mbox{if }}n=1\\xF_{n-1}(x)+F_{n-2}(x),&{\mbox{if }}n\geq 2\end{cases}}$

The Lucas polynomials use the same recurrence with different starting values:

$L_{n}(x)={\begin{cases}2,&{\mbox{if }}n=0\\x,&{\mbox{if }}n=1\\xL_{n-1}(x)+L_{n-2}(x),&{\mbox{if }}n\geq 2.\end{cases}}$

They can be defined for negative indices by

$F_{-n}(x)=(-1)^{n-1}F_{n}(x),$

$L_{-n}(x)=(-1)^{n}L_{n}(x).$

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 F_n is n − 1 and the degree of L_n is n.

• The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating F_n at x = 2.

• The ordinary generating function for the sequences are:

  $\sum _{n=0}^{\infty }F_{n}(x)t^{n}={\frac {t}{1-xt-t^{2}}}$

  $\sum _{n=0}^{\infty }L_{n}(x)t^{n}={\frac {2-xt}{1-xt-t^{2}}}.$

• 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 article: Lucas sequence

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as

$F_{m+n}(x)=F_{m+1}(x)F_{n}(x)+F_{m}(x)F_{n-1}(x)\,$

$L_{m+n}(x)=L_{m}(x)L_{n}(x)-(-1)^{n}L_{m-n}(x)\,$

$F_{n+1}(x)F_{n-1}(x)-F_{n}(x)^{2}=(-1)^{n}\,$

$F_{2n}(x)=F_{n}(x)L_{n}(x).\,$

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 by

$x^{n}=F_{n+1}(x)+\sum _{k=1}^{\lfloor n/2\rfloor }(-1)^{k}\left[{\binom {n}{k}}-{\binom {n}{k-1}}\right]F_{n+1-2k}(x).$

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

The coefficients of the Fibonacci polynomials can be read off from Pascal's triangle following the "shallow" diagonals (shown in red). The sums of the coefficients are the Fibonacci numbers.

If F(n,k) is the coefficient of x^k in F_n(x), so

$F_{n}(x)=\sum _{k=0}^{n}F(n,k)x^{k},\,$

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) is equal to the binomial coefficient

$F(n,k)={\binom {\tfrac {n+k-1}{2}}{k}}$

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

• Benjamin, Arthur T.; Quinn, Jennifer J. (2003). "§9.4 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. Adv Differ Equ 2018, 126 (2018). https://doi.org/10.1186/s13662-018-1527-9

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

• OEIS sequence A162515 (Triangle of coefficients of polynomials defined by Binet form)
• OEIS sequence A011973 (Triangle of coefficients of Fibonacci polynomials)