Lucas Polynomial
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].](http://mathworld.wolfram.com/images/equations/LucasPolynomial/NumberedEquation1.gif) |
(1)
|
The first few are
(OEIS A114525).
The Lucas polynomial is implemented in the Wolfram Language as LucasL[n, x].
The Lucas polynomial has generating function
The derivative of 
is given by
![(dL_n(x))/(dx)=n/(x^2+4)[xL_n(x)+2L_(n-1)(x)].](http://mathworld.wolfram.com/images/equations/LucasPolynomial/NumberedEquation2.gif) |
(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
where the 
s are Lucas numbers.
REFERENCES:
Koshy, T. Fibonacci and Lucas Numbers with Applications. New York: Wiley, 2001.
Sloane, N. J. A. Sequence A114525 in "The On-Line Encyclopedia of Integer Sequences."
