IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

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.

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