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

2018/08/03 by Flórez, Rigoberto, Higuita, Robinson, Ramírez, Alexander · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1808.01264

Abstract

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 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 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 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.

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