# Laguerre-Angelesco multiple orthogonal polynomials on an $r$-star

## TL;DR

This paper studies Laguerre-Angelesco multiple orthogonal polynomials on an $r$-star, providing explicit formulas, recurrence coefficients, differential equations, and asymptotic zero distributions, and explores their connection to Jacobi-Angelesco polynomials.

## Contribution

It introduces explicit expressions, recurrence relations, and asymptotic analysis for Laguerre-Angelesco polynomials on an $r$-star, linking them to Jacobi-Angelesco polynomials.

## Key findings

- Explicit formulas for polynomials
- Recurrence coefficients derived
- Asymptotic zero distribution obtained

## Abstract

We investigate the type I and type II multiple orthogonal polynomials on an $r$-star with weight function $|x|^{\beta}e^{-x^r}$, with $\beta>-1$. Each measure $\mu_j$, for $1\leq j \leq r$, is supported on the semi-infinite interval $[0,\omega^{j-1}\infty)$ with $\omega=e^{2\pi i/r}$. For both the type I and the type II polynomials we give explicit expressions, the coefficients in the recurrence relation, the differential equation and we obtain the asymptotic zero distribution of the polynomials on the diagonal. Also, we give the connection between the Laguerre-Angelesco polynomials and the Jacobi-Angelesco polynomials on an $r$-star.