The Lucas polynomials are the 
-polynomials obtained by setting
and 
in the Lucas
polynomial sequence. It is given explicitly by
![L_n(x)=2^(-n)[(x-sqrt(x^2+4))^n+(x+sqrt(x^2+4))^n].](/images/equations/LucasPolynomial/NumberedEquation1.svg) |
(1)
|
The first few are
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(2)
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(3)
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(4)
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(5)
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(6)
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(OEIS A114525).
The Lucas polynomial is implemented in the Wolfram Language as LucasL[n, x].
The Lucas polynomial has generating function
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(7)
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(8)
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(9)
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The derivative of 
is given by
![(dL_n(x))/(dx)=n/(x^2+4)[xL_n(x)+2L_(n-1)(x)].](/images/equations/LucasPolynomial/NumberedEquation2.svg) |
(10)
|
The Lucas polynomials have the divisibility property that 
divides 
iff 
is an odd multiple of 
. For prime 
, 
is an irreducible
polynomial. The zeros of 
are 
for 
, ..., 
. For prime 
, except for the root of 0, these roots are 
times the imaginary part of the roots of the 
th cyclotomic polynomial
(Koshy 2001, p. 464).
The corresponding 
polynomials are called Fibonacci
polynomials. The Lucas polynomials satisfy
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(11)
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(12)
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where the 
s
are Lucas numbers.
