Generalized Binet Formulas, Lucas Polynomials, and Cyclic Constants

Jay Kappraff¹* and Gary W. Adamson²

¹Department of Mathematics, New Jersey Institute of Technology, Newark, NJ 07102, U.S.A.
²P.O. Box 124571, San Diego, CA 92112-4571, U.S.A.
*E-mail address: kappraff@verizon.net

(Received March 11, 2005; Accepted March 15, 2005)

Keywords: Binet's Theorem, Silver Means, Fibonacci Sequence, Pell Sequence, Gauss' Theorem, Pell-Lucas Sequence, Fibonacci Polynomials, Lucas Polynomials

Abstract. Generalizations of Binet's theorem are used to produce generalized Pell sequences from two families of silver means. These Pell sequences are also generated from the family of Fibonacci polynomials. A family of Pell-Lucas sequences are also generated from the family of Lucas polynomials and from another generalization of Binet's formula. A periodic set of cyclic constants are generated from the Lucas polynomials. These cyclic constants are related to the Gauss-Wantzel proof of the constructibility by compass and straightedge of regular polygons.