# Linear combinations of polynomials with three-term recurrence

## TL;DR

This paper investigates the zero distribution of polynomial sums with three-term recurrence relations and extends the analysis to linear combinations of Chebyshev polynomials, identifying conditions for hyperbolicity.

## Contribution

It provides a detailed analysis of zero distributions for polynomials with linear coefficient recurrences and characterizes hyperbolicity conditions for specific Chebyshev polynomial combinations.

## Key findings

- Zero distribution characterized for polynomial sums with three-term recurrence.
- Necessary and sufficient conditions for hyperbolicity of Chebyshev linear combinations.
- Extension of zero distribution analysis to polynomial combinations with linear coefficients.

## Abstract

We study the zero distribution of the sum of the first $n$ polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of $az+b$ for $a,b\in\mathbb{R}$, of Chebyshev polynomials. In particular, we find necessary and sufficient conditions on $a$, $b$ such that this linear combination is hyperbolic.