# Lucas polynomials

The polynomials (cf. [a1] and [a5]) given by (a1)

They reduce to the Lucas numbers for , and they satisfy several identities, which may be easily proved by induction, e.g.: (a2) (a3) (a4) (a5)
 \(U_{2n}(x)=U_n(x)V_n(x),\,\) (a6)

where denote the Fibonacci polynomials; (a7)

where so that ; and (a8) where denotes the greatest integer in .

The Lucas polynomials are related to the Chebyshev polynomials , , by (a9)

J. Riordan [a9] considered the polynomials and the Lucas-type polynomials (a10) in a derivation of Chebyshev-type pairs of inverse relations. V.E. Hoggatt Jr. and M. Bicknell [a4] found the roots of . These are , . Bicknell [a2] showed that divides if and only if is an odd multiple of . G.E. Bergum and Hoggatt Jr. introduced in [a1] the bivariate Lucas polynomials by the recursion (a11)

generalized (a7) for , and showed that the are irreducible polynomials over the rational numbers if and only if for some positive integer (cf. also Irreducible polynomial). The formula (a12) which may be derived by induction on or by expanding the generating function of , generalizes (a8).

Ch.A. Charalambides [a3] introduced and studied the Lucas and Lucas-type polynomials of order , and . The Lucas-type polynomials of order satisfy the recurrence (a13)

These polynomials have the binomial and multinomial expansions (a14)  where the second summation is taken over all non-negative integers such that , and they are related to the Fibonacci-type polynomials of order (cf. [a6] and [a8] and Fibonacci polynomials), , by (a15)

Furthermore, (a16) where the are the Fibonacci polynomials of order (cf. [a7]). Charalambides [a3] showed that the reliability of a circular consecutive -out-of- : -system, , whose components function independently with probability is given by (a17) How to Cite This Entry:
Lucas polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lucas_polynomials&oldid=19744
This article was adapted from an original article by Andreas N. Philippou (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article