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

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 $P_0(z)=1,P_{-1}(z)=\dots =P_{-k+1}(z)=0,$ 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.

10 pages, 2 figures