Lucas Polynomial

The Lucas polynomials are the -polynomials obtained by setting and in the Lucas polynomial sequence. It is given explicitly by

(1)

The first few are

			(2)
			(3)
			(4)
			(5)
			(6)

(OEIS A114525).

The Lucas polynomial is implemented in the Wolfram Language as LucasL[n, x].

The Lucas polynomial has generating function

			(7)
			(8)
			(9)

The derivative of is given by

(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

			(11)
			(12)

where the s are Lucas numbers.

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