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

## TL;DR

This paper provides necessary and sufficient conditions for all zeros of polynomial sequences generated by a specific three-term recurrence relation to be real, extending understanding of zero distribution in such sequences.

## Contribution

It establishes the exact criteria for the reality of zeros in polynomial sequences defined by a general three-term recurrence relation with polynomial coefficients.

## Key findings

- Derived necessary and sufficient conditions for real zeros
- Generalized previous results to arbitrary polynomial coefficients
- Enhanced understanding of zero distribution in polynomial sequences

## 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.