Non-real zeros of polynomials in a polynomial sequence satisfying a three-term recurrence relation

TL;DR

This paper investigates the zeros of polynomial sequences generated by a three-term recurrence relation, demonstrating that such sequences always contain polynomials with non-real zeros.

Contribution

It establishes the existence of polynomials with non-real zeros within sequences generated by specific three-term recurrence relations.

Findings

Polynomials with non-real zeros always exist in the sequence.

The recurrence relation involves coprime real polynomials A(z) and B(z).

The result applies for k > 2 with standard initial conditions.

Abstract

This paper discusses the location of zeros of polynomials in a polynomial sequence $\{P_n(z)\}$ generated by a three-term recurrence relation of the form $P_n(z)+ B(z)P_{n-1}(z) +A(z) P_{n-k}(z)=0$ with $k>2$ and the standard initial conditions $P_{0}(z)=1, P_{-1}(z)=\ldots=P_{-k+1}(z)=0,$ where $A(z)$ and $B(z)$ are arbitrary coprime real polynomials. We show that there always exist polynomials in $\{P_n(z)\}$ with non-real zeros.