Lucas Polynomial
The Lucas polynomials are the 👁 w
-polynomials obtained by setting
👁 p(x)=x
and 👁 q(x)=1
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].](https://mathworld.wolfram.com/images/equations/LucasPolynomial/NumberedEquation1.svg){kind=link}
The first few are
| 👁 L_1(x) | 👁 = | 👁 x |
(2)
|
| 👁 L_2(x) | 👁 = | 👁 x^2+2 |
(3)
|
| 👁 L_3(x) | 👁 = | 👁 x^3+3x |
(4)
|
| 👁 L_4(x) | 👁 = | 👁 x^4+4x^2+2 |
(5)
|
| 👁 L_5(x) | 👁 = | 👁 x^5+5x^3+5x |
(6)
|
(OEIS A114525).
The Lucas polynomial is implemented in the Wolfram Language as [n, x].
The Lucas polynomial has generating function
| 👁 G(x,t) | 👁 = | 👁 (1+t^2)/(1-t^2-tx) |
(7)
|
| 👁 Image | 👁 = | 👁 sum_(n=0)^(infty)L_n(x)t^n |
(8)
|
| 👁 Image | 👁 = | 👁 1+xt+(x^2+2)t^2+(x^3+3x)t^3+.... |
(9)
|
The derivative of 👁 L_n(x)
is given by
![👁 (dL_n(x))/(dx)=n/(x^2+4)[xL_n(x)+2L_(n-1)(x)].](https://mathworld.wolfram.com/images/equations/LucasPolynomial/NumberedEquation2.svg){kind=link}
The Lucas polynomials have the divisibility property that 👁 L_n(x)
divides 👁 L_m(x)
iff 👁 m
is an odd multiple of 👁 n
. For prime 👁 p
, 👁 L_p(x)/x
is an irreducible
polynomial. The zeros of 👁 L_n(x)
are 👁 2isin(kpi/n)
for 👁 k=1
, ..., 👁 n-1
. For prime 👁 p
, except for the root of 0, these roots are 👁 2i
times the imaginary part of the roots of the 👁 p
th cyclotomic polynomial
(Koshy 2001, p. 464).
The corresponding 👁 W
polynomials are called Fibonacci
polynomials. The Lucas polynomials satisfy
where the 👁 L_n
s
are Lucas numbers.
See also
Fibonacci Polynomial, Lucas Number, Lucas Polynomial SequenceExplore with Wolfram|Alpha
More things to try:
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."Referenced on Wolfram|Alpha
Lucas PolynomialCite this as:
Weisstein, Eric W. "Lucas Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LucasPolynomial.html