Fibonacci Polynomial

The $W$ Polynomials obtained by setting $p(x)=x$ and $q(x)=1$ in the Lucas Polynomial Sequence. (The corresponding $w$ Polynomials are called Lucas Polynomials.) The Fibonacci polynomials are defined by the Recurrence Relation

$$F_{n+1}(x)=xF_n(x)+F_{n-1}(x),$$ (1)

with $F_1(x)=1$ and $F_2(x)=x$. They are also given by the explicit sum formula

$$F_n(x)=\sum_{j=0}^{\left\lfloor{(n-1)/2}\right\rfloor }{n-j-1\choose j}x^{n-2j-1},$$ (2)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function and ${n\choose m}$ is a Binomial Coefficient. The first few Fibonacci polynomials are

$$F_1(x) = 1$$

$$F_2(x) = x$$

$$F_3(x) = x^2+1$$

$$F_4(x) = x^3+2x$$

$$F_5(x) = x^4+3x^2+1.$$

The Fibonacci polynomials are normalized so that

$$F_n(1)=F_n,$$ (3)

where the $F_n$s are Fibonacci Numbers.

The Fibonacci polynomials are related to the Morgan-Voyce Polynomials by

$$F_{2n+1}(x) = b_n(x^2)$$ (4)

$$F_{2n+n2}(x) = x B_n(x^2)$$ (5)

(Swamy 1968).

See also Brahmagupta Polynomial, Fibonacci Number, Morgan-Voyce Polynomial

References

Swamy, M. N. S. ``Further Properties of Morgan-Voyce Polynomials.'' Fib. Quart. 6, 167-175, 1968.