Criterion of reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Abstract

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence $\{P_i{\}}_{i=1}^{\infty }$ generated by a three-term recurrence relation $P_i(x)+Q_1(x)P_{i-1}(x)+Q_2(x)P_{i-2}(x)=0$ with the standard initial conditions $P_0(x)=1,P_{-1}(x)=0,$ where $Q_1(x)$ and $Q_2(x)$ are arbitrary real polynomials.

10 pages, 6 figures, To appear in Czechoslovak mathematical journal, typos corrected, references added