 TOPICS # Fibonacci Polynomial The polynomials obtained by setting and in the Lucas polynomial sequence. (The corresponding polynomials are called Lucas polynomials.) They have explicit formula (1)

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

The Fibonacci polynomials are defined by the recurrence relation (2)

with and .

The first few Fibonacci polynomials are   (3)   (4)   (5)   (6)   (7)

(OEIS A049310).

The Fibonacci polynomials have generating function   (8)   (9)   (10)

The Fibonacci polynomials are normalized so that (11)

where the s are Fibonacci numbers. is also given by the explicit sum formula (12)

where is the floor function and is a binomial coefficient.

The derivative of is given by (13)

The Fibonacci polynomials have the divisibility property divides iff divides . For prime , is an irreducible polynomial. The zeros of are for , ..., . For prime , these roots are times the real part of the roots of the th cyclotomic polynomial (Koshy 2001, p. 462).

The identity (14)

for , 3, ... and a Chebyshev polynomial of the second kind gives the identities   (15)   (16)   (17)   (18)

and so on, where gives the sequence 4, 11, 29, ... (OEIS A002878).

The Fibonacci polynomials are related to the Morgan-Voyce polynomials by   (19)   (20)

(Swamy 1968).

Brahmagupta Polynomial, Fibonacci Number, Morgan-Voyce Polynomials

## Related Wolfram sites

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

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## 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