Asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions

TL;DR

This paper investigates the asymptotic behavior of zeros of certain multiple orthogonal polynomials linked to Macdonald functions, revealing their convergence to a specific equilibrium measure described by an algebraic equation.

Contribution

It provides a detailed analysis of the zero distribution for these polynomials and explicitly characterizes the limiting measure through a vector equilibrium problem.

Findings

Zeros converge to an explicit equilibrium measure

The equilibrium measure is described by an algebraic equation

Results extend understanding of orthogonal polynomials related to special functions

Abstract

We study the asymptotic zero distribution of type II multiple orthogonal polynomials associated with two Macdonald functions (modified Bessel functions of the second kind). Based on the four-term recurrence relation, it is shown that, after proper scaling, the sequence of normalized zero counting measures converges weakly to the first component of a vector of two measures which satisfies a vector equilibrium problem with two external fields. We also give the explicit formula for the equilibrium vector in terms of solutions of an algebraic equation.