Fibonacci polynomials

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

Definition

These polynomials are defined by a recurrence relation:

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

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

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

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:

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

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

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

References

• Hoggatt, V.E., jun. (1973). "Roots of Fibonacci polynomials". Fibonacci Quarterly. 11: 271–274. ISSN 0015-0517. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: multiple names: authors list (link)
• Ricci, Paolo Emilio (1995). "Generalized Lucas polynomials and Fibonacci polynomials". Riv. Mat. Univ. Parma, V. Ser. 4: 137–146.

External links

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