Monotonicity of zeros of polynomials orthogonal with respect to a discrete measure

TL;DR

This paper proves that zeros of certain orthogonal polynomials move monotonically with respect to a discrete measure's mass point, addressing an open problem in the field.

Contribution

It establishes the monotonicity of zeros of polynomials orthogonal with respect to measures including a discrete mass point, solving a previously open problem.

Findings

Zeros are increasing functions of the mass point a

Addresses an open problem posed by Mourad Ismail

Provides theoretical proof of monotonicity

Abstract

We prove that all zeros of the polynomials orthogonal with respect to a measure $d \mu(x;a) = d \mu(x) + M \delta(x-a)$, where $d\mu$ is a nonatomic positive Borel measure and $M>0$, are increasing functions of the mass point $a$. Thus we solve partially an open problem posed by Mourad Ismail.