Jacobi-Angelesco multiple orthogonal polynomials on an $r$-star

TL;DR

This paper studies type I multiple orthogonal polynomials on an r-star configuration, providing explicit formulas, recurrence relations, differential equations, and zero distribution asymptotics for these polynomials.

Contribution

It introduces explicit formulas and asymptotic analysis for Jacobi-Angelesco multiple orthogonal polynomials on an r-star, extending understanding of their properties.

Findings

Explicit formulas for the polynomials and recurrence coefficients

Differential equations satisfied by the polynomials

Asymptotic zero distribution results

Abstract

We investigate type I multiple orthogonal polynomials on $r$ intervals which have a common point at the origin and endpoints at the $r$ roots of unity $\omega^j$, $j=0,1,\ldots,r-1$, with $\omega = \exp(2\pi i/r)$. We use the weight function $|x|^\beta (1-x^r)^\alpha$, with $\alpha,\beta >-1$ for the multiple orthogonality relations. We give explicit formulas for the type I multiple orthogonal polynomials, the coefficients in the recurrence relation, the differential equation, and we obtain the asymptotic distribution of the zeros.