# Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients

## TL;DR

This paper analyzes the asymptotic behavior of orthogonal polynomials with slowly oscillating recurrence coefficients, providing explicit formulas for their orthogonality measure and asymptotics in both bounded and unbounded cases.

## Contribution

It introduces a constructive formula for the orthogonality measure density and derives the first order uniform asymptotics for solutions of specific recurrence relations.

## Key findings

- Derived the first order uniform asymptotics for solutions
- Provided a constructive formula for the orthogonality measure density
- Unified treatment of bounded and unbounded cases

## Abstract

We study solutions of three-term recurrence relations whose $N$-step transfer matrices belong to the uniform Stolz class. In particular, we derive the first order of their uniform asymptotics. For orthonormal polynomials we show more. Namely, we find the constructive formula for the density of their orthogonality measure in terms of Tur\'an determinants and we determine their exact asymptotic behavior. We treat both bounded and unbounded cases in a uniform manner.