Zero Distribution of Generated Taylor Polynomials

Abstract

Our goal in this paper is to study the zero distribution of a sequence of polynomials whose coefficients satisfy a three-term recurrence. Equivalently, these polynomials are Taylor polynomials of a rational function with a polynomial denominator of degree 2. We will use complex analysis to find the bi-variate generating function for that sequence of Taylor polynomials. With this generating function, we prove that the zeros of these Taylor polynomials lie on one side of the closed disk centered at the origin. The radius of the disk is exactly the reciprocal of the modulus of the smaller zero of the degree two denominator. Finally, we show that the zeros of these Taylor polynomials approach the boundary of the disk above.