Fibonacci polynomials

Fibonacci polynomials are the polynomials defined by:

$F_{1}(x)=1$

$F_{2}(x)=x$

$F_{n}(x)=xF_{n-1}(x)+F_{n-2}(x)\quad {\hbox{for all }}n\geq 3$

The first few Fibonacci polynomials are:

$F_{1}(x)=1$

$F_{2}(x)=x$

$F_{3}(x)=x^{2}+1$

$F_{4}(x)=x^{3}+2x$

$F_{5}(x)=x^{4}+3x^{2}+1$

$F_{6}(x)=x^{5}+4x^{3}+3x$

Putting $x=1$ one recovers the Fibonacci numbers.

See also

• Polynomial sequences