Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.4

The Resultant, the Discriminant, and the Derivative of Generalized Fibonacci Polynomials

Rigoberto Flórez
Department of Mathematical Sciences
The Citadel
Charleston, SC 29409

Robinson A. Higuita and Alexander Ramírez
Instituto de Matemáticas
Universidad de Antioquia


A second-order polynomial sequence is of Fibonacci-type (Lucas-type) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these types of sequences are Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, and Chebyshev polynomials.

The resultant of two polynomials is the determinant of the Sylvester matrix and the discriminant of a polynomial p is the resultant of p and its derivative. We study the resultant, the discriminant, and the derivatives of Fibonacci-type polynomials and Lucas-type polynomials as well the resultant of combinations of these two types of polynomials. As a corollary, we give explicit formulas for the resultant, the discriminant, and the derivative for the polynomials mentioned above.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A001629 A001871 A006645 A007701 A045618 A045925 A086804 A093967 A127670 A193678 A317403 A317404 A317405 A317408 A317450 A317451 A318184 A318197.)

Received August 22 2018; revised versions received August 23 2018; March 28 2019; June 15 2019. Published in Journal of Integer Sequences, June 28 2019.

Return to Journal of Integer Sequences home page