Fibonacci polynomials

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

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External links

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