The polynomials obtained by setting and in the Lucas polynomial sequence. (The corresponding polynomials are called Lucas polynomials.) They have explicit formula
(1)
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The Fibonacci polynomial is implemented in the Wolfram Language as Fibonacci[n, x].
The Fibonacci polynomials are defined by the recurrence relation
(2)
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with and .
The first few Fibonacci polynomials are
(3)
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(4)
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(5)
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(6)
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(7)
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(OEIS A049310).
The Fibonacci polynomials have generating function
(8)
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(9)
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(10)
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The Fibonacci polynomials are normalized so that
(11)
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where the s are Fibonacci numbers.
is also given by the explicit sum formula
(12)
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where is the floor function and is a binomial coefficient.
The derivative of is given by
(13)
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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)
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for , 3, ... and a Chebyshev polynomial of the second kind gives the identities
(15)
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(16)
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(17)
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(18)
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and so on, where gives the sequence 4, 11, 29, ... (OEIS A002878).
The Fibonacci polynomials are related to the Morgan-Voyce polynomials by
(19)
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(20)
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(Swamy 1968).