# Threefold symmetric Hahn-classical multiple orthogonal polynomials

## TL;DR

This paper characterizes all threefold symmetric Hahn-classical multiple orthogonal polynomials, detailing their properties, asymptotic behavior, and orthogonality measures, extending classical polynomial concepts to multiple orthogonality.

## Contribution

It provides a complete characterization of Hahn-classical threefold symmetric multiple orthogonal polynomials and their orthogonality measures, expanding the theory of classical polynomials.

## Key findings

- Three distinct families of Hahn-classical polynomials identified
- Asymptotic behavior linked to largest zero of the polynomial set
- Orthogonality measures supported on a 3-star in the complex plane

## Abstract

We characterize all the multiple orthogonal threefold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as $2$-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a $3$-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them $2$-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.