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).
