Lucas polynomials

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


They reduce to the Lucas numbers for , and they satisfy several identities, which may be easily proved by induction, e.g.:

\(U_{2n}(x)=U_n(x)V_n(x),\,\) (a6)

where denote the Fibonacci polynomials;



so that ; and


where denotes the greatest integer in .

The Lucas polynomials are related to the Chebyshev polynomials , , by


J. Riordan [a9] considered the polynomials and the Lucas-type polynomials


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


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


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


These polynomials have the binomial and multinomial expansions


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




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



[a1] G.E. Bergum, V.E. Hoggatt, Jr., "Irreducibility of Lucas and generalized Lucas polynomials" Fibonacci Quart. , 12 (1974) pp. 95–100
[a2] M. Bicknell, "A primer for the Fibonacci numbers VII" Fibonacci Quart. , 8 (1970) pp. 407–420
[a3] Ch.A. Charalambides, "Lucas numbers and polynomials of order and the length of the longest circular success run" Fibonacci Quart. , 29 (1991) pp. 290–297
[a4] V.E. Hoggatt Jr., M. Bicknell, "Roots of Fibonacci polynomials" Fibonacci Quart. , 11 (1973) pp. 271–274
[a5] E. Lucas, "Theorie de fonctions numeriques simplement periodiques" Amer. J. Math. , 1 (1878) pp. 184–240; 289–321
[a6] A.N. Philippou, "Distributions and Fibonacci polynomials of order , longest runs, and reliability of consecutive--out-of-: systems" A.N. Philippou (ed.) G.E. Bergum (ed.) A.F. Horadam (ed.) , Fibonacci Numbers and Their Applications , Reidel (1986) pp. 203–227
[a7] A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci polynomials of order , multinomial expansions and probability" Internat. J. Math. Math. Sci. , 6 (1983) pp. 545–550
[a8] A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci-type polynomials of order with probability applications" Fibonacci Quart. , 23 (1985) pp. 100–105
[a9] J. Riordan, "Combinatorial Identities" , Wiley (1968)
How to Cite This Entry:
Lucas polynomials. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Andreas N. Philippou (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article