Mehler-Heine asymptotics for multiple orthogonal polynomials

TL;DR

This paper extends Mehler-Heine asymptotics to multiple orthogonal polynomials, revealing their behavior near interval endpoints involves generalized Bessel functions, broadening understanding beyond classical cases.

Contribution

It introduces generalized Bessel function asymptotics for multiple orthogonal polynomials, including Jacobi-Angelesco, Jacobi-Pi ilde{neiro}, and others, near their support edges.

Findings

Asymptotic behavior involves generalized Bessel functions.

Results apply to various classes of multiple orthogonal polynomials.

Extends classical Mehler-Heine asymptotics to new polynomial families.

Abstract

Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard edge involves Bessel functions $J_\alpha$. We show that the asymptotic behavior near the endpoint of the interval of (one of) the measures for multiple orthogonal polynomials involves a generalization of the Bessel function. The multiple orthogonal polynomials considered are Jacobi-Angelesco polynomials, Jacobi-Pi\~neiro polynomials, multiple Laguerre polynomials, multiple orthogonal polynomials associated with modified Bessel functions (of the first and second kind), and multiple orthogonal polynomials associated with Meijer $G$-functions.