Fibonacci polynomials

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

<math>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\ge3 \end{matrix}\right.<math>

The first few Fibonacci polynomials are:

<math>F_1(x)=1 \,<math>

<math>F_2(x)=x \,<math>

<math>F_3(x)=x^2+1 \,<math>

<math>F_4(x)=x^3+2x \,<math>

<math>F_5(x)=x^4+3x^2+1 \,<math>

<math>F_6(x)=x^5+4x^3+3x \,<math>

The Fibonacci numbers are recovered by evaluating the polynomials at x = 1.

See also

• Polynomial sequences