Linear combinations of polynomials with three-term recurrence

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.