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).
