## The strong divisibility property and the resultant of generalized Fibonacci polynomials

- Oct. 12, 2018
- 2:30 p.m.
- LeConte 310

## 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. Those are known as *generalized Fibonacci polynomials* GFP.
Some known examples are: Fibobacci Polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials,
Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials, Vieta and Vieta-Lucas polynomials.

It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the *strong divisibility property*. However, this property does not hold for every second order recursive sequence. We give a characterization of GFPs that satisfy the strong divisibility property. We also give formulas to evaluate the gcd of GFPs that do not satisfy the strong divisibility property.

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. In this talk we discuss closed formulas for
the resultant, the discriminant, and the derivatives of Fibonacci-type polynomials and Lucas-type polynomials. As a corollary we
give explicit formulas for the resultant, the discriminant, and the derivative some known examples of GFPs.
Joint work with R. Higuita, N. McAnally, A. Mukherjee and R. Ramirez.