# 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) |

#### References

[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: http://encyclopediaofmath.org/index.php?title=Lucas_polynomials&oldid=19744