# Asymptotic approximations of the continuous Hahn polynomials and their zeros

## TL;DR

This paper develops asymptotic approximations for continuous Hahn polynomials and their zeros using advanced difference equation techniques, providing insights into their behavior as the degree increases infinitely.

## Contribution

It introduces new asymptotic methods for analyzing continuous Hahn polynomials and their zeros, extending existing techniques to this class of orthogonal polynomials.

## Key findings

- Asymptotic formulas for the polynomials as degree grows large
- Approximate locations of zeros for large degrees
- Application of uniform asymptotic expansions in difference equations

## Abstract

Asymptotic approximations for the continuous Hahn polynomials and their zeros as the degree grows to infinity are established via their three-term recurrence relation. The methods are based on the uniform asymptotic expansions for difference equations developed by Wang and Wong (\textit{Numer. Math.}, 2003) and the matching technique in the complex plane developed by Wang (\textit{J. Approx. Theory}, 2014).