Lucas Polynomial

The Lucas polynomials are the -polynomials obtained by setting and in the Lucas polynomial sequence. For a nonnegative integer , they are given explicitly by

(1)

Thus . The first few with 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)

For , the derivative of satisfies the polynomial identity

(10)

For positive integers and , divides iff is an odd number times . For an odd prime number , is an irreducible polynomial. For , the roots of are for , ..., . For an odd prime number , except for the root 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.