Fibonacci polynomials

In mathematics, Fibonacci polynomials are a generalization of Fibonacci numbers. These polynomials are defined by:

$F_{n}(x)=\left\{{\begin{matrix}1,\qquad \qquad \qquad \qquad &{\mbox{if }}n=1\\x,\qquad \qquad \qquad \qquad &{\mbox{if }}n=2\\xF_{n-1}(x)+F_{n-2}(x),&{\mbox{if }}n\geq 3\end{matrix}}\right.$

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.

See also

• Polynomial sequences