The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials

Abstract

A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials. The \emph{resultant} of two polynomials is the determinant of the Sylvester matrix and the \emph{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 combinations of those two types. As a corollary we give explicit formulas for the resultant, the discriminant, and the derivative for the known polynomials mentioned above.