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Lucas Polynomial


LucasPolynomial

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].
(1)

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 LucasL[n, x].

The Lucas polynomial has generating function

G(x,t)=(1+t^2)/(1-t^2-tx)
(7)
=sum_(n=0)^(infty)L_n(x)t^n
(8)
=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)].
(10)

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 pth cyclotomic polynomial (Koshy 2001, p. 464).

The corresponding W polynomials are called Fibonacci polynomials. The Lucas polynomials satisfy

L_n(0)=1+(-1)^n
(11)
L_n(1)=L_n,
(12)

where the L_ns are Lucas numbers.


See also

Fibonacci Polynomial, Lucas Number, Lucas Polynomial Sequence

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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 Polynomial

Cite this as:

Weisstein, Eric W. "Lucas Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LucasPolynomial.html

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