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.) They have explicit formula

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

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

The first few Fibonacci polynomials are

(OEIS A049310).

The Fibonacci polynomials have generating function

The Fibonacci polynomials are normalized so that

where the F_ns are Fibonacci numbers.

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

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

The derivative of F_n(x) is given by

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

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

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

(Swamy 1968).

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

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

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