Common zeros of polynomials satisfying a recurrence of order two

TL;DR

This paper characterizes the common zeros of polynomial sequences defined by second-order recurrences, analyzing their root distribution and focusing on real zeros for polynomials with real coefficients.

Contribution

It provides a new characterization of common zeros in second-order polynomial recurrences, advancing understanding of their root distribution.

Findings

Characterization of common zeros for second-order polynomial recurrences

Analysis of real common zeros for polynomials with real coefficients

Contribution to the understanding of root distribution in recursive polynomial sequences

Abstract

We give a characterization of common zeros of a sequence of univariate polynomials $W_n(z)$ defined by a recurrence of order two with polynomial coefficients, and with $W_0(z)=1$. Real common zeros for such polynomials with real coefficients are studied further. This paper contributes to the study of root distribution of recursive polynomial sequences.