Fibonacci polynomials Information & Fibonacci polynomials Links at HealthHaven.com

In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalisation of the Fibonacci numbers.

[edit] Definition

These polynomials are defined by a recurrence relation:

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

[edit] Properties

The first few Fibonacci polynomials are:

$F_1(x)=1 \,$

$F_2(x)=x \,$

$F_3(x)=x^2+1 \,$

$F_4(x)=x^3+2x \,$

$F_5(x)=x^4+3x^2+1 \,$

$F_6(x)=x^5+4x^3+3x \,$

The Fibonacci numbers are recovered by evaluating the polynomials at x = 1. The degree of F_n is n − 1. The ordinary generating function for the sequence is

$\sum_{n=0}^\infty F_n(x) t^n = \frac{t}{1-t(x+t)}.$

[edit] Lucas polynomials

The associated Lucas polynomials L_n(x) have a similar relationship to the Lucas numbers. They satisfy the same recurrence relationship, with different starting values:

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

The first few Lucas polynomials are:

$L_1(x)=x \,$

$L_2(x)=x^2+2 \,$

$L_3(x)=x^3+3x \,$

$L_4(x)=x^4+4x^2+2 \,$

$L_5(x)=x^5+5x^3+5x \,$

$L_6(x)=x^6+6x^4+9x^2 + 2 \,$

The Lucas numbers are recovered by evaluating the polynomials at x = 1. The degree of L_n is n. The ordinary generating function for the sequence is

$\sum_{n=0}^\infty L_n(x) t^n = \frac{2-xt}{1-t(x+t)} .$

[edit] References

Hoggatt, V.E., jun.; Bicknell, Marjorie (1973). "Roots of Fibonacci polynomials.". Fibonacci Quarterly 11: 271–274. ISSN 0015-0517.
Ricci, Paolo Emilio (1995). "Generalized Lucas polynomials and Fibonacci polynomials". Riv. Mat. Univ. Parma, V. Ser. 4: 137–146.

[edit] External links

Weisstein, Eric W., "Fibonacci Polynomial" from MathWorld.
Weisstein, Eric W., "Lucas Polynomial" from MathWorld.

Contents

[edit] Definition

[edit] Properties

[edit] Lucas polynomials

[edit] References

[edit] External links