TOPICS
Search

Fibonacci Polynomial


FibonacciPolynomials

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.) They have explicit formula

 F_n(x)=((x+sqrt(x^2+4))^n-(x-sqrt(x^2+4))^n)/(2^nsqrt(x^2+4)).
(1)

The Fibonacci polynomial F_n(x) is implemented in the Wolfram Language as Fibonacci[n, x].

The Fibonacci polynomials are defined by the recurrence relation

 F_(n+1)(x)=xF_n(x)+F_(n-1)(x),
(2)

with F_1(x)=1 and F_2(x)=x.

The first few Fibonacci polynomials are

F_1(x)=1
(3)
F_2(x)=x
(4)
F_3(x)=x^2+1
(5)
F_4(x)=x^3+2x
(6)
F_5(x)=x^4+3x^2+1
(7)

(OEIS A049310).

The Fibonacci polynomials have generating function

G(x,t)=t/(1-t^2-tx)
(8)
=sum_(n=0)^(infty)F_n(x)t^n
(9)
=t+xt^2+(x^2+1)t^3+(x^3+2x)t^4+....
(10)

The Fibonacci polynomials are normalized so that

 F_n(1)=F_n,
(11)

where the F_ns are Fibonacci numbers.

F_n(x) is also given by the explicit sum formula

 F_n(x)=sum_(j=0)^(|_(n-1)/2_|)(n-j-1; j)x^(n-2j-1),
(12)

where |_x_| is the floor function and (n; m) is a binomial coefficient.

The derivative of F_n(x) is given by

 (dF_n(x))/(dx)=(2nF_(n-1)(x)+(n-1)xF_n(x))/(x^2+4).
(13)

The Fibonacci polynomials have the divisibility property F_n(x) divides F_m(x) iff n divides m. For prime p, F_p(x) is an irreducible polynomial. The zeros of F_n(x) are 2icos(kpi/n) for k=1, ..., n-1. For prime p, these roots are 2i times the real part of the roots of the pth cyclotomic polynomial (Koshy 2001, p. 462).

The identity

 F_n(U_(p-1)(1/2sqrt(5)))=(F_(np))/(F_p),
(14)

for p=1, 3, ... and U_n(x) a Chebyshev polynomial of the second kind gives the identities

F_n(4)=(F_(3n))/(F_3)
(15)
F_n(11)=(F_(5n))/(F_5)
(16)
F_n(29)=(F_(7n))/(F_7)
(17)
F_n(76)=(F_(9n))/(F_9)
(18)

and so on, where U_(p-1)(1/2sqrt(5)) gives the sequence 4, 11, 29, ... (OEIS A002878).

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

F_(2n+1)(x)=b_n(x^2)
(19)
F_(2n+2)(x)=xB_n(x^2)
(20)

(Swamy 1968).


See also

Brahmagupta Polynomial, Fibonacci Number, Morgan-Voyce Polynomials

Related Wolfram sites

http://functions.wolfram.com/Polynomials/Fibonacci2/, http://functions.wolfram.com/HypergeometricFunctions/Fibonacci2General/

Explore with Wolfram|Alpha

References

Koshy, T. Fibonacci and Lucas Numbers with Applications. New York: Wiley, 2001.Sloane, N. J. A. Sequence A002878/M3420 in "The On-Line Encyclopedia of Integer Sequences."Swamy, M. N. S. "Further Properties of Morgan-Voyce Polynomials." Fib. Quart. 6, 167-175, 1968.

Referenced on Wolfram|Alpha

Fibonacci Polynomial

Cite this as:

Weisstein, Eric W. "Fibonacci Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FibonacciPolynomial.html

Subject classifications