A pair of generalized Polynomials which generalize the
Lucas Sequence to Polynomials is given by

(1) | |||

(2) |

where

(3) |

(4) |

(5) |

(6) | |||

(7) |

giving

(8) | |||

(9) |

The sequences most commonly considered have , giving

(10) | |||

(11) |

Special cases are given in the following table.

Polynomial 1 | Polynomial 2 | ||

1 | Fibonacci | Lucas | |

1 | Pell | Pell-Lucas | |

1 | Jacobsthal | Jacobsthal | |

Fermat | Fermat-Lucas | ||

Chebyshev Polynomial of the Second Kind | Chebyshev Polynomial of the First Kind |

**References**

Horadam, A. F. ``Extension of a Synthesis for a Class of Polynomial Sequences.'' *Fib. Quart.* **34**,
68-74, 1996.

© 1996-9

1999-05-25